Research ArticleFixed Point Results for 119866-120572-Contractive Maps with Applicationto Boundary Value Problems

Nawab Hussain1 Vahid Parvaneh2 and Jamal Rezaei Roshan3

1 Department of Mathematics King Abdulaziz University PO Box 80203 Jeddah 21589 Saudi Arabia2Department of Mathematics Gilan-E-Gharb Branch Islamic Azad University Gilan-E-Gharb Iran3Department of Mathematics Qaemshahr Branch Islamic Azad University Qaemshahr Iran

Correspondence should be addressed to Jamal Rezaei Roshan jmlroshangmailcom

Received 9 March 2014 Accepted 29 March 2014 Published 7 May 2014

Academic Editor M Mursaleen

Copyright copy 2014 Nawab Hussain et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We unify the concepts of G-metric metric-like and b-metric to define new notion of generalized b-metric-like space and discussits topological and structural properties In addition certain fixed point theorems for two classes of G-120572-admissible contractivemappings in such spaces are obtained and some new fixed point results are derived in corresponding partially ordered spaceMoreover some examples and an application to the existence of a solution for the first-order periodic boundary value problem areprovided here to illustrate the usability of the obtained results

1 Introduction and MathematicalPreliminaries

The concept of a 119887-metric space was introduced by Czerwik[1] After that several interesting results about the existenceof fixed point for single-valued and multivalued operators in(ordered) 119887-metric spaces have been obtained (see eg [2ndash11])

Definition 1 (see [1]) Let 119883 be a (nonempty) set and 119904 ge 1 agiven real number A function 119889 119883times119883 rarr R+ is a 119887-metricon119883 if for all 119909 119910 119911 isin 119883 the following conditions hold

(b1) 119889(119909 119910) = 0 if and only if 119909 = 119910

(b2) 119889(119909 119910) = 119889(119910 119909)

(b3) 119889(119909 119911) le 119904[119889(119909 119910) + 119889(119910 119911)]

In this case the pair (119883 119889) is called a 119887-metric space

The concept of a generalized metric space or a 119866-metricspace was introduced by Mustafa and Sims [12]

Definition 2 (see [12]) Let 119883 be a nonempty set and 119866 119883 times

119883 times 119883 rarr R+ a function satisfying the following properties(1198661) 119866(119909 119910 119911) = 0 if and only if 119909 = 119910 = 119911

(1198662) 0 lt 119866(119909 119909 119910) for all 119909 119910 isin 119883 with 119909 = 119910

(1198663) 119866(119909 119909 119910) le 119866(119909 119910 119911) for all x 119910 119911 isin 119883 with 119910 = 119911

(1198664) 119866(119909 119910 119911) = 119866(119901119909 119910 119911) where119901 is any permutationof 119909 119910 119911 (symmetry in all three variables)

(1198665) 119866(119909 119910 119911) le 119866(119909 119886 119886)+119866(119886 119910 119911) for all119909 119910 119911 119886 isin 119883

(rectangle inequality)

Then the function 119866 is called a 119866-metric on 119883 and thepair (119883 119866) is called a 119866-metric space

Definition 3 (see [13]) A metric-like on a nonempty set 119883 isa mapping 120590 119883 times119883 rarr R+ such that for all 119909 119910 119911 isin 119883 thefollowing hold

(1205901) 120590(119909 119910) = 0 implies 119909 = 119910

(1205902) 120590(119909 119910) = 120590(119910 119909)

(1205903) 120590(119909 119910) le 120590(119909 119911) + 120590(119911 119910)

The pair (119883 120590) is called a metric-like space

Below we give some examples of metric-like spaces

Example 4 (see [14]) Let 119883 = [0 1] Then the mapping 1205901

119883times119883 rarr R+ defined by 1205901(119909 119910) = 119909+119910minus119909119910 is a metric-like

on119883

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 585964 14 pageshttpdxdoiorg1011552014585964

2 The Scientific World Journal

Example 5 (see [14]) Let119883 = R then the mappings 120590119894 119883 times

119883 rarr R+(119894 isin 2 3 4) defined by

1205902(119909 119910) = |119909| +

10038161003816100381610038161199101003816100381610038161003816+ 119886

1205903(119909 119910) = |119909 minus 119887| +

1003816100381610038161003816119910 minus 119887

1003816100381610038161003816

1205904(119909 119910) = 119909

2

+ 1199102

(1)

are metric-likes on119883 where 119886 ge 0 and 119887 isin R

Definition 6 (see [15]) Let 119883 be a nonempty set and 119904 ge 1

a given real number A function 120590119887 119883 times 119883 rarr R+ is a 119887-

metric-like if for all 119909 119910 119911 isin 119883 the following conditions aresatisfied

(1205901198871) 120590119887(119909 119910) = 0 implies 119909 = 119910

(1205901198872) 120590119887(119909 119910) = 120590

119887(119910 119909)

(1205901198873) 120590119887(119909 119910) le 119904[120590

119887(119909 119911) + 120590

119887(119911 119910)]

A 119887-metric-like space is a pair (119883 120590119887) such that 119883 is a

nonempty set and 120590119887is a 119887-metric-like on119883 The number 119904 is

called the coefficient of (119883 120590119887)

In a 119887-metric-like space (119883 120590119887) if 119909 119910 isin 119883 and 120590

119887(119909 119910) =

0 then 119909 = 119910 but the converse may not be true and 120590119887(119909 119909)

may be positive for all 119909 isin 119883 It is clear that every 119887-metricspace is a 119887-metric-like space with the same coefficient 119904 butnot conversely in general

Example 7 (see [8]) Let119883 = R+ let 119901 gt 1 be a constant andlet 120590119887 119883 times 119883 rarr R+ be defined by

120590119887(119909 119910) = (119909 + 119910)

119901

forall119909 119910 isin 119883 (2)

Then (119883 120590119887) is a 119887-metric-like space with coefficient 119904 =

2119901minus1

The following propositions help us to construct somemore examples of 119887-metric-like spaces

Proposition 8 (see [8]) Let (119883 120590) be a metric-like space and120590119887(119909 119910) = [120590(119909 119910)]

119901 where 119901 gt 1 is a real number Then 120590119887

is a 119887-metric-like with coefficient 119904 = 2119901minus1

From the above proposition and Examples 4 and 5 wehave the following examples of 119887-metric-like spaces

Example 9 (see [8]) Let 119883 = [0 1] Then the mapping 1205901198871

119883 times 119883 rarr R+ defined by 1205901198871(119909 119910) = (119909 + 119910 minus 119909119910)

119901 where119901 gt 1 is a real number is a 119887-metric-like on119883with coefficient119904 = 2119901minus1

Example 10 (see [8]) Let 119883 = R Then the mappings 120590119887119894

119883 times 119883 rarr R+(119894 isin 2 3 4) defined by

1205901198872

(119909 119910) = (|119909| +10038161003816100381610038161199101003816100381610038161003816+ 119886)119901

1205901198873

(119909 119910) = (|119909 minus 119887| +1003816100381610038161003816119910 minus 119887

1003816100381610038161003816)119901

1205901198874

(119909 119910) = (1199092

+ 1199102

)

119901

(3)

are 119887-metric-like on119883 where 119901 gt 1 119886 ge 0 and 119887 isin R

Each 119887-metric-like 120590119887on119883 generates a topology 120591

120590119887on119883

whose base is the family of all open 120590119887-balls 119861

120590119887(119909 120576) 119909 isin

119883 120576 gt 0 where 119861120590119887(119909 120576) = 119910 isin 119883 |120590

119887(119909 119910)minus120590

119887(119909 119909)| lt 120576

for all 119909 isin 119883 and 120576 gt 0Now we introduce the concept of generalized 119887-metric-

like space or 119866120590119887-metric space as a proper generalization of

both of the concepts of 119887-metric-like spaces and 119866-metricspaces

Definition 11 Let 119883 be a nonempty set Suppose that amapping 119866

120590119887 119883 times 119883 times 119883 rarr R+ satisfies the following

(1198661205901198871) 119866120590119887(119909 119910 119911) = 0 implies 119909 = 119910 = 119911

(1198661205901198872) 119866120590119887(119909 119910 119911) = 119866

120590119887(119901119909 119910 119911) where 119901 is any permu-

tation of 119909 119910 119911 (symmetry in all three variables)

(1198661205901198873) 119866120590119887(119909 119910 119911) le 119904[119866

120590119887(119909 119886 119886) + 119866

120590119887(119886 119910 119911)] for all

119909 119910 119911 119886 isin 119883 (rectangle inequality)

Then 119866120590119887

is called a 119866120590119887-metric and (119883 119866

120590119887) is called a

generalized 119887-metric-like spaceThe following proposition will be useful in constructing

examples of a generalized 119887-metric-like space

Proposition 12 Let (119883 120590119887) be a 119887-metric-like space with

coefficient 119904 Then

119866119898

120590119887

(119909 119910 119911) = max 120590119887(119909 119910) 120590

119887(119910 119911) 120590

119887(119911 119909)

119866119904

120590119887

(119909 119910 119911) = 120590119887(119909 119910) + 120590

119887(119910 119911) + 120590

119887(119911 119909)

(4)

are two generalized 119887-metric-like functions on119883

Proof It is clear that 119866119898120590119887

and 119866119904

120590119887

satisfy conditions 1198661205901198871 and

1198661205901198872 of Definition 11 So we only show that 119866

1205901198873 is satisfied

by 119866119898

120590119887

and 119866119904

120590119887

Let 119909 119910 119911 119886 isin 119883 Then using the triangularinequality in 119887-metric-like spaces we have

119866119898

120590119887

(119909 119910 119911) = max 120590119887(119909 119910) 120590

119887(119910 119911) 120590

119887(119911 119909)

le max 119904 (120590119887(119909 119886) + 120590

119887(119886 119910))

120590119887(119910 119911) 119904 (120590

119887(119911 119886) + 120590

119887(119886 119909))

le max 119904 (120590119887(119909 119886) + 120590

119887(119886 119910))

119904 (120590119887(119910 119911) + 120590

119887(119886 119886))

119904 (120590119887(119911 119886) + 120590

119887(119886 119909))

= 119904max 120590119887(119909 119886) 120590

119887(119886 119886) 120590

119887(119886 119909)

+ 119904max 120590119887(119886 119910) 120590

119887(119910 119911) 120590

119887(119911 119886)

= 119904 (119866119898

120590119887(119909 119886 119886) + 119866

119898

120590119887

(119886 119910 119911))

(5)

The Scientific World Journal 3

Also

119866119904

120590119887

(119909 119910 119911) = 120590119887(119909 119910) + 120590

119887(119910 119911) + 120590

119887(119911 119909)

le 119904 (120590119887(119909 119886) + 120590

119887(119886 119910))

+ 120590119887(119910 119911) + 119904 (120590

119887(119911 119886) + 120590

119887(119886 119909))

le 119904 (120590119887(119909 119886) + 120590

119887(119886 119910))

+ 119904 (120590119887(119910 119911) + 120590

119887(119886 119886))

+ 119904 (120590119887(119911 119886) + 120590

119887(119886 119909))

= 119904 (120590119887(119909 119886) + 120590

119887(119886 119886) + 120590

119887(119886 119909))

+ 119904 (120590119887(119886 119910) 120590

119887(119910 119911) 120590

119887(119911 119886))

= 119904 (119866119904

120590119887(119909 119886 119886) + 119866

119904

120590119887

(119886 119910 119911))

(6)

According to the above proposition we provide someexamples of generalized 119887-metric-like spaces

Example 13 Let 119883 = R+ let 119901 gt 1 be a constant and let119866119898

120590119887

119866119904

120590119887

119883 times 119883 times 119883 rarr R+ be defined by

119866119898

120590119887

(119909 119910 119911) = max (119909 + 119910)119901

(119910 + 119911)119901

(119911 + 119909)119901

119866119904

120590119887

(119909 y 119911) = (119909 + 119910)119901

+ (119910 + 119911)119901

+ (119911 + 119909)119901

(7)

for all 119909 119910 119911 isin 119883Then (119883 119866119898

120590119887

) and (119883 119866119904

120590119887

) are generalized119887-metric-like spaces with coefficient 119904 = 2

119901minus1 Note that for119909 = 119910 = 119911 gt 0119866119898

120590119887

(119909 119909 119909) = (2119909)119901

gt 0 and 119866119904

120590119887

(119909 119909 119909) =

3(2119909)119901

gt 0

Example 14 Let 119883 = [0 1] Then the mappings 1198661198981205901198871

119866119904

1205901198871

119883 times 119883 times 119883 rarr R+ defined by

119866119898

1205901198871

(119909 119910 119911)

= max (119909 + 119910 minus 119909119910)119901

(119910 + 119911 minus 119910119911)119901

(119911 + 119909 minus 119911119909)119901

119866119904

1205901198871

(119909 119910 119911)

= (119909 + 119910 minus 119909119910)119901

+ (119910 + 119911 minus 119910119911)119901

+ (119911 + 119909 minus 119911119909)119901

(8)

where 119901 gt 1 is a real number are generalized 119887-metric-likespaces with coefficient 119904 = 2

119901minus1

By some straight forward calculations we can establishthe following

Proposition 15 Let 119883 be a 119866120590119887-metric space Then for each

119909 119910 119911 119886 isin 119883 it follows that

(1) 119866120590119887(119909 119910 119910) gt 0 for 119909 = 119910

(2) 119866120590119887(119909 119910 119911) le 119904(119866

120590119887(119909 119909 119910) + 119866

120590119887(119909 119909 119911))

(3) 119866120590119887(119909 119910 119910) le 2119904119866

120590119887(119910 119909 119909)

(4) 119866120590119887(119909 119910 119911) le 119904119866

120590119887(119909 119886 119886) + 119904

2119866120590119887(119910 119886 119886) +

1199042119866120590119887(119911 119886 119886)

Definition 16 Let (119883 119866120590119887) be a 119866

120590119887-metric space Then for

any 119909 isin 119883 and 119903 gt 0 the 119866120590119887-ball with center 119909 and radius 119903

is

119861119866120590119887

(119909 119903) = 119910 isin 119883 |

10038161003816100381610038161003816119866120590119887

(119909 119909 119910) minus 119866120590119887

(119909 119909 119909)

10038161003816100381610038161003816lt +119903

(9)

The family of all 119866120590119887-balls

ϝ = 119861119866120590119887

(119909 119903) | 119909 isin 119883 119903 gt 0 (10)

is a base of a topology 120591(119866120590119887) on119883 whichwe call it119866

120590119887-metric

topology

Definition 17 Let (119883 119866120590119887) be a 119866

120590119887-metric space Let 119909

119899 be

a sequence in119883 Consider the following

(1) A point119909 isin 119883 is said to be a limit of the sequence 119909119899

denoted by 119909119899

rarr 119909 if lim119899119898rarrinfin

119866120590119887(119909 119909119899 119909119898) =

119866120590119887(119909 119909 119909)

(2) 119909119899 is said to be a 119866

120590119887-Cauchy sequence if

lim119899119898rarrinfin

119866120590119887(119909119899 119909119898 119909119898) exists (and is finite)

(3) (119883 119866120590119887) is said to be 119866

120590119887-complete if every 119866

120590119887-

Cauchy sequence in 119883 is 119866120590119887-convergent to an 119909 isin 119883

such that

lim119899119898rarrinfin

119866120590119887

(119909119899 119909119898 119909119898) = lim119899rarrinfin

119866120590119887

(119909119899 119909 119909) = 119866

120590119887(119909 119909 119909)

(11)

Using the above definitions one can easily prove thefollowing proposition

Proposition 18 Let (119883 119866120590119887) be a 119866

120590119887-metric space Then for

any sequence 119909119899 in X and a point 119909 isin 119883 the following are

equivalent

(1) 119909119899 is 119866120590119887-convergent to 119909

(2) 119866120590119887(119909119899 119909119899 119909) rarr 119866

120590119887(119909 119909 119909) as 119899 rarr infin

(3) 119866120590119887(119909119899 119909 119909) rarr 119866

120590119887(119909 119909 119909) as 119899 rarr infin

Definition 19 Let (119883 119866120590119887) and (119883

1015840 1198661015840

120590119887

) be two generalized119887-metric like spaces and let 119891 (119883 119866

120590119887) rarr (119883

1015840 1198661015840

120590119887

) bea mapping Then 119891 is said to be 119866

120590119887-continuous at a point

119886 isin 119883 if for a given 120576 gt 0 there exists 120575 gt 0 suchthat 119909 isin 119883 and |119866

120590119887(119886 119886 119909) minus 119866

120590119887(119886 119886 119886)| lt 120575 imply

that |1198661015840120590119887

(119891(119886) 119891(119886) 119891(119909)) minus 1198661015840

120590119887

(119891(119886) 119891(119886) 119891(119886))| lt 120576 Themapping 119891 is 119866

120590119887-continuous on 119883 if it is 119866

120590119887-continuous at

all 119886 isin 119883 For simplicity we say that 119891 is continuous

Proposition 20 Let (119883 119866120590119887) and (119883

1015840 1198661015840

120590119887

) be two generalized119887-metric like spaces Then a mapping 119891 119883 rarr 119883

1015840 is 119866120590119887-

continuous at a point 119909 isin 119883 if and only if it is 119866120590119887-sequentially

continuous at 119909 that is whenever 119909119899 is 119866120590119887-convergent to 119909

119891(119909119899) is 119866

120590119887-convergent to 119891(119909)

4 The Scientific World Journal

We need the following simple lemma about the 119866120590119887-

convergent sequences in the proof of our main results

Lemma 21 Let (119883 119866120590119887) be a 119866

120590119887-metric space and suppose

that 119909119899 119910119899 and 119911

119899 are 119866

120590119887-convergent to 119909 119910 and 119911

respectively Then we have1

1199043119866120590119887

(119909 119910 119911) minus

1

1199042119866120590119887

(119909 119909 119909)

minus

1

119904

119866120590119887

(119910 119910 119910) minus 119866120590119887

(119911 119911 119911)

le lim inf119899rarrinfin

119866120590119887

(119909119899 119910119899 119911119899)

le lim sup119899rarrinfin

119866120590119887

(119909119899 119910119899 119911119899)

le 1199043

119866120590119887

(119909 119910 119911) + 119904119866120590b

(119909 119909 119909)

+ 1199042

119866120590119887

(119910 119910 119910) + 1199043

119866120590119887

(119911 119911 119911)

(12)

In particular if 119910119899 = 119911

119899 = 119886 are constant then

1

119904

119866120590119887

(119909 119886 119886) minus 119866120590119887

(119909 119909 119909)

le lim inf119899rarrinfin

119866120590119887

(119909119899 119886 119886) le lim sup

119899rarrinfin

119866120590119887

(119909119899 119886 119886)

le 119904119866120590119887

(119909 119886 119886) + 119904119866120590119887

(119909 119909 119909)

(13)

Proof Using the rectangle inequality we obtain

119866120590119887

(119909 119910 119911) le 119904119866120590119887

(119909 119909119899 119909119899) + 1199042

119866120590119887

(119910 119910119899 119910119899)

+ 1199043

119866120590119887

(119911 119911119899 119911119899) + 1199043

119866120590119887

(119909119899 119910119899 119911119899)

119866120590119887

(119909119899 119910119899 119911119899) le 119904119866

120590119887(119909119899 119909 119909) + 119904

2

119866120590119887

(119910119899 119910 119910)

+ 1199043

119866120590119887

(119911119899 119911 119911) + 119904

3

119866120590119887

(119909 119910 119911)

(14)

Taking the lower limit as 119899 rarr infin in the first inequality andthe upper limit as 119899 rarr infin in the second inequality we obtainthe desired result

If 119910119899 = 119911

119899 = 119886 then

119866120590119887

(119909 119886 119886) le 119904119866120590119887

(119909 119909119899 119909119899) + 119904119866

120590119887(119909119899 119886 119886)

119866120590119887

(119909119899 119886 119886) le 119904119866

120590119887(119909119899 119909 119909) + 119904119866

120590119887(119909 119886 119886)

(15)

Again taking the lower limit as 119899 rarr infin in the first inequalityand the upper limit as 119899 rarr infin in the second inequality weobtain the desired result

2 Main Results

Samet et al [16] defined the notion of 120572-admissible mappingsand proved the following result

Definition 22 Let119879 be a self-mapping on119883 and 120572 119883times119883 rarr

[0infin) a function We say that 119879 is an 120572-admissible mappingif

119909 119910 isin 119883 120572 (119909 119910) ge 1 997904rArr 120572 (119879119909 119879119910) ge 1 (16)

Denote withΨ1015840 the family of all nondecreasing functions

120595 [0infin) rarr [0infin) such that suminfin119899=1

120595119899(119905) lt infin for all 119905 gt 0

where 120595119899 is the 119899th iterate of 120595

Theorem 23 Let (119883 119889) be a complete metric space and 119879 an120572-admissible mapping Assume that

120572 (119909 119910) 119889 (119879119909 119879119910) le 120595 (119889 (119909 119910)) (17)

where 120595 isin Ψ1015840 Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 120572(119909

0 1198791199090) ge 1

(ii) either 119879 is continuous or for any sequence 119909119899 in 119883

with 120572(119909119899 119909119899+1

) ge 1 for all 119899 isin Ncup0 such that 119909119899rarr

119909 as 119899 rarr infin we have 120572(119909119899 119909) ge 1 for all 119899 isin N cup 0

Then 119879 has a fixed point

For more details on 120572-admissible mappings we refer thereader to [17ndash20]

Definition 24 (see [21]) Let (119883 119866) be a 119866-metric space let119891 be a self-mapping on 119883 and let 120572 119883

3rarr [0infin) be a

function We say that 119891 is a 119866-120572-admissible mapping if

119909 119910 119911 isin 119883 120572 (119909 119910 119911) ge 1 997904rArr 120572 (119891119909 119891119910 119891119911) ge 1

(18)

Motivated by [22] letF denote the class of all functions120573 [0infin) rarr [0 1119904) satisfying the following condition

lim sup119899rarrinfin

120573 (119905119899) =

1

119904

997904rArr lim sup119899rarrinfin

119905119899= 0 (19)

Definition 25 Let 119891 119883 rarr 119883 and 120572 119883 times 119883 times 119883 rarr RWe say that 119891 is a rectangular 119866-120572-admissible mapping if

(T1) 120572(119909 119910 119911) ge 1 implies 120572(119891119909 119891119910 119891119911) ge 1 119909 119910 119911 isin 119883

(T2) 120572(119909119910119910)ge1120572(119910119911119911)ge1

implies 120572(119909 119911 119911) ge 1 119909 119910 119911 isin 119883

From now on let 120572 1198833rarr [0infin) be a function and

119872119904(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119891119909 119891119909)119866120590119887

(119910 119891119910 119891119910)119866120590119887

(119911 119891119911 119891119911)

1 + 11990421198662

120590119887

(119891119909 119891119910 119891119911)

(20)

Theorem 26 Let (119883 119866120590119887) be a 119866

120590119887-complete generalized 119887-

metric-like space and let 119891 119883 rarr 119883 be a rectangular 119866-120572-admissible mapping Suppose that

119904120572 (119909 119910 119911) 119866120590119887

(119891119909 119891119910 119891119911)

le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911)

(21)

for all 119909 119910 119911 isin 119883

The Scientific World Journal 5

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 120572(119909

0 1198911199090 1198911199090) ge 1

(ii) 119891 is continuous and for any sequence 119909119899 in 119883 with

120572(119909119899 119909119899+1

119909119899+1

) ge 1 for all 119899 isin Ncup0 such that 119909119899rarr

119909 as 119899 rarr infin we have 120572(119909 119909 119909) ge 1 for all 119899 isin Ncup0

Then 119891 has a fixed point

Proof Let 1199090isin 119883 be such that 120572(119909

0 1198911199090 1198911199090) ge 1 Define

a sequence 119909119899 by 119909

119899= 1198911198991199090for all 119899 isin N Since 119891 is a 119866-

120572-admissible mapping and 120572(1199090 1199091 1199091) = 120572(119909

0 1198911199090 1198911199090) ge

1 we deduce that 120572(1199091 1199092 1199092) = 120572(119891119909

0 1198911199091 1198911199091) ge 1

Continuing this process we get 120572(119909119899 119909119899+1

119909119899+1

) ge 1 for all119899 isin N cup 0

Step I We will show that lim119899rarrinfin

119866120590119887(119909119899 119909119899+1

119909119899+1

) = 0 If119909119899= 119909119899+1

for some 119899 isin N then 119909119899= 119891119909119899 Thus 119909

119899is a fixed

point of 119891 Therefore we assume that 119909119899

= 119909119899+1

for all 119899 isin NSince 120572(119909

119899 119909119899+1

119909119899+1

) ge 1 for each 119899 isin N then we canapply (21) which yields

119904119866120590119887

(119909119899 119909119899+1

119909119899+1

)

= 119904119866120590119887

(119891119909119899minus1

119891119909119899 119891119909119899)

le 120573 (119866120590119887

(119909119899minus1

119909119899 119909119899))119872119904(119909119899minus1

119909119899 119909119899)

lt

1

119904

119866120590119887

(119909119899minus1

119909119899 119909119899)

le 119866120590119887

(119909119899minus1

119909119899 119909119899)

(22)

119872119904(119909119899minus1

119909119899 119909119899)

= max 119866120590119887

(119909119899minus1

119909119899 119909119899)

(119866120590119887

(119909119899minus1

119891119909119899minus1

119891119909119899minus1

)

times119866120590119887

(119909119899 119891119909119899 119891119909119899) 119866120590119887

(119909119899 119891119909119899 119891119909119899))

times(1 + 1199042

1198662

120590119887

(119891119909119899minus1

119891119909119899 119891119909119899))

minus1

= max 119866120590119887

(119909119899minus1

119909119899 119909119899)

(119866120590119887

(119909119899minus1

119909119899 119909119899)

times119866120590119887

(119909119899 119909119899+1

119909119899+1

) 119866120590119887

(119909119899 119909119899+1

119909119899+1

))

times(1 + 1199042

1198662

120590119887

(119909119899 119909119899+1

x119899+1

))

minus1

= 119866120590119887

(119909119899minus1

119909119899 119909119899)

(23)

Therefore 119866120590119887(119909119899 119909119899+1

119909119899+1

) is a decreasing and boundedsequence of nonnegative real numbers Then there exists 119903 ge

0 such that lim119899rarrinfin

119866120590119887(119909119899 119909119899+1

119909119899+1

) = 119903 Letting 119899 rarr infin in(22) we have

119904119903 le 119903 (24)

Since 119904 gt 1 we deduce that 119903 = 0 that is

lim119899rarrinfin

119866120590119887

(119909119899 119909119899+1

119909119899+1

) = 0 (25)

By Proposition 15(2) we conclude that

lim119899rarrinfin

119866120590119887

(119909119899 119909119899 119909119899+1

) = 0 (26)

Step II Now we prove that the sequence 119909119899 is a119866

120590119887-Cauchy

sequence For this purpose we will show that

lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898) = 0 (27)

Using the rectangular inequality with (21) (as 120572(119909119899 119909119898 119909119898) ge

1 since 119891 is a rectangular119866-120572-admissible mapping) we have

119866120590119887

(119909119899 119909119898 119909119898)

le 119904119866120590119887

(119909119899 119909119899+1

119909119899+1

)

+ 1199042

119866120590119887

(119909119899+1

119909119898+1

119909119898+1

) + 1199042

119866120590119887

(119909119898+1

119909119898 119909119898)

le 119904119866120590119887

(119909119899 119909119899+1

119909119899+1

)

+ 119904120573 (119866120590119887

(119909119899 119909119898 119909119898))119872119904(119909119899 119909119898 119909119898)

+ 1199042

119866120590119887

(119909119898+1

119909119898 119909119898)

(28)

Taking limit as 119898 119899 rarr infin in the above inequality andapplying (25) and (26) we have

lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

le 119904 lim sup119899119898rarrinfin

120573 (119866120590119887

(119909119899 119909119898 119909119898)) lim sup119899119898rarrinfin

119872119904(119909119899 119909119898 119909119898)

(29)

Here

119866120590119887

(119909119899 119909119898 119909119898)

le 119872119904(119909119899 119909119898 119909119898)

= max

119866120590119887

(119909119899 119909119898 119909119898)

119866120590119887

(119909119899 119891119909119899 119891119909119899) [119866120590119887

(119909119898 119891119909119898 119891119909119898)]

2

1 + 11990421198662

120590119887

(119891119909119899 119891119909119898 119891119909119898)

= max

119866120590119887

(119909119899 119909119898 119909119898)

119866120590119887

(119909119899 119909119899+1

119909119899+1

) [119866120590119887

(119909119898 119909119898+1

119909119898+1

)]

2

1 + 11990421198662

120590119887

(119909119899+1

119909119898+1

119909119898+1

)

(30)

6 The Scientific World Journal

Letting119898 119899 rarr infin in the above inequality we get

lim sup119898119899rarrinfin

119872119904(119909119899 119909119898 119909119898) = lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898) (31)

Hence from (29) and (31) we obtain

lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

le 119904lim sup119898119899rarrinfin

120573 (119866120590119887

(119909119899 119909119898 119909119898)) lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

(32)

If lim sup119898119899rarrinfin

119866120590119887(119909119899 119909119898 119909119898) = 0 then we get

1

119904

le lim sup119898119899rarrinfin

120573 (119866120590119887

(119909119899 119909119898 119909119898)) (33)

Since 120573 isin F we deduce that

lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898) = 0 (34)

which is a contradiction Consequently 119909119899 is a 119866

120590119887-Cauchy

sequence in119883 Since (119883 119866120590119887) is119866120590119887-complete there exists 119906 isin

119883 such that 119909119899

rarr 119906 as 119899 rarr infin Now from (34) and 119866120590119887-

completeness of119883

lim119899rarrinfin

119866120590119887

(119906 119909119899 119909119899) = lim119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

= 119866120590119887

(119906 119906 119906) = 0

(35)

Step III Now we show that 119906 is a fixed point of 119891Using the rectangle inequality we get

119866120590119887

(119906 119891119906 119891119906) le 119904119866120590119887

(119906 119891119909119899 119891119909119899) + 119904119866

120590119887(119891119909119899 119891119906 119891119906)

(36)

Letting 119899 rarr infin and using the continuity of 119891 and (35) weobtain

119866120590119887

(119906 119891119906 119891119906) le 119904 lim119899rarrinfin

G120590119887

(119906 119891119909119899 119891119909119899)

+ 119904 lim119899rarrinfin

119866120590119887

(119891119909119899 119891119906 119891119906)

= 119904119866120590119887

(119891119906 119891119906 119891119906)

(37)

Note that from (21) as 120572(119906 119906 119906) ge 1 we have

119904119866120590119887

(119891119906 119891119906 119891119906) le 120573 (119866120590119887

(119906 119906 119906))119872119904(119906 119906 119906) (38)

where by (37)

119872119904(119906 119906 119906)

= max119866120590119887

(119906 119906 119906)

119866120590119887

(119906 119891119906 119891119906)119866120590119887

(119906 119891119906 119891119906)119866120590119887

(119906 119891119906 119891119906)

1 + 11990421198662

120590119887

(119891119906 119891119906 119891119906)

lt 119866120590119887

(119906 119891119906 119891119906)

(39)

Hence as 120573(119905) le 1 for all 119905 isin [0infin) we have 119904119866120590119887(119891119906

119891119906 119891119906) le 119866120590119887(119906 119891119906 119891119906) Thus by (37) we obtain that

119904119866120590119887(119891119906 119891119906 119891119906) = 119866

120590119887(119906 119891119906 119891119906) But then using (38) we

get that

119866120590119887

(119906 119891119906 119891119906) = 119904119866120590119887

(119891119906 119891119906 119891119906)

le 120573 (119866120590119887

(119906 119906 119906))119872119904(119906 119906 119906)

lt

1

119904

119866120590119887

(119906 119891119906 119891119906)

(40)

which is a contradiction Hence we have 119891119906 = 119906 Thus 119906 is afixed point of 119891

We replace condition (ii) in Theorem 26 by regularity ofthe space119883

Theorem 27 Under the same hypotheses of Theorem 26instead of condition (119894119894) assume that whenever 119909

119899 in 119883 is a

sequence such that 120572(119909119899 119909119899+1

119909119899+1

) ge 1 for all 119899 isin Ncup0 and119909119899rarr 119909 as 119899 rarr infin one has 120572(119909

119899 119909 119909) ge 1 for all 119899 isin Ncup0

Then 119891 has a fixed point

Proof Repeating the proof of Theorem 26 we can constructa sequence 119909

119899 in 119883 such that 120572(119909

119899 119909119899+1

119909119899+1

) ge 1 for all119899 isin N cup 0 and 119909

119899rarr 119906 isin 119883 for some 119906 isin 119883 Using the

assumption on 119883 we have 120572(119909119899 119906 119906) ge 1 for all 119899 isin N cup 0

Now we show that 119906 = 119891119906 By Lemma 21 and (35)

119904 [

1

119904

119866120590119887

(119906 119891119906 119891119906) minus 119866120590119887

(119906 119906 119906)]

le 119904lim sup119899rarrinfin

119866120590119887

(119909119899+1

119891119906 119891119906)

le lim sup119899rarrinfin

(120573 (119866120590119887

(119909119899 119906 119906))119872

119904(119909119899 119906 119906))

le

1

119904

lim sup119899rarrinfin

119872119904(119909119899 119906 119906)

(41)

wherelim119899rarrinfin

119872119904(119909119899 119906 119906)

= lim119899rarrinfin

max

119866120590119887

(119909119899 119906 119906)

[119866120590119887

(119909119899 119891119909119899 119891119909119899)] [119866120590119887

(119906 119891119906 119891119906)]

2

1 + 11990421198662

120590119887

(119891119909119899 119891119906 119891119906)

= lim119899rarrinfin

max

119866120590119887

(119909119899 119906 119906)

[119866120590119887

(119909119899 119909119899+1

119909119899+1

)] [119866120590119887

(119906 119891119906 119891119906)]

2

1 + 11990421198662

120590119887

(119909119899+1

119891119906 119891119906)

= max 119866120590119887

(119906 119906 119906) 0 = 0 (see (25) and (35))

(42)

Therefore we deduce that119866120590119887(119906 119891119906 119891119906) le 0 Hence we have

119906 = 119891119906

The Scientific World Journal 7

A mapping 120593 [0infin) rarr [0infin) is called a comparisonfunction if it is increasing and 120593

119899(119905) rarr 0 as 119899 rarr infin for any

119905 isin [0infin) (see eg [23 24] for more details and examples)

Definition 28 A function 120593 [0infin) rarr [0infin) is said to bea (119888)-comparison function if

(1198881) 120593 is increasing

(1198882) there exists 119896

0isin N 119886 isin (0 1) and a convergent series

of nonnegative terms suminfin

119896=1V119896such that 120593

119896+1(119905) le

119886120593119896(119905) + V

119896for 119896 ge 119896

0and any 119905 isin [0infin)

Later Berinde [5] introduced the notion of (119887)-comparison function as a generalization of (119888)-comparisonfunction

Definition 29 (see [5]) Let 119904 ge 1 be a real number Amapping120593 [0infin) rarr [0infin) is called a (119887)-comparison function ifthe following conditions are fulfilled

(1) 120593 is increasing

(2) there exist 1198960isin N 119886 isin (0 1) and a convergent series

of nonnegative terms suminfin119896=1

V119896such that 119904119896+1120593119896+1(119905) le

119886119904119896120593119896(119905) + V

119896for 119896 ge 119896

0and any 119905 isin [0infin)

Let Ψ119887be the class of all (119887)-comparison functions 120593

[0infin) rarr [0infin) It is clear that the notion of (119887)-comparison function coincideswith (119888)-comparison functionfor 119904 = 1

Lemma 30 (see [25]) If 120593 [0infin) rarr [0infin) is a (119887)-comparison function then we have the following

(1) the series suminfin119896=0

119904119896120593119896(119905) converges for any 119905 isin R

+

(2) the function 119887119904 [0infin) rarr [0infin) defined by 119887

119904(119905) =

suminfin

119896=0119904119896120593119896(119905) 119905 isin [0infin) is increasing and continuous

at 0

Remark 31 It is easy to see that if 120593 isin Ψ119887 then we have 120593(0) =

0 and 120593(119905) lt 119905 for each 119905 gt 0 and 120593 is continuous at 0

In the next example we present a class of (119887)-comparisonfunctions

Example 32 Any function of the form120595(119905) = ln((119886119904)119905+1) forall 119905 isin [0infin) where 0 lt 119886 lt 1 is a (119887)-comparison function

Proof From the part (1) of Lemma 30 the necessary condi-tion is that the series suminfin

119896=0119904119896120593119896(119905) converges for any 119905 isin R

+

But for each 119905 gt 0 and 119896 ge 1 we have

119904119896

120593119896

(119905) = 119904119896

120593 (120593119896minus1

(119905))

= 119904119896 ln(

119886

119904

120593119896minus1

(119905) + 1) le 119904119896 119886

119904

120593119896minus1

(119905)

le sdot sdot sdot le 119904119896

(

119886

119904

)

119896

119905 = 119886119896

119905

(43)

So according to the comparison test of the series we shouldhave 119886 lt 1 On the other hand we have

119904119896+1

120593119896+1

(119905) = 119904119896+1

120593 (120593119896

(119905))

= 119904119896+1 ln(

119886

119904

120593119896

(119905) + 1)

le 119904119896+1 119886

119904

120593119896

(119905) = 119904119896

119886120593119896

(119905)

(44)

Therefore for any convergent series of nonnegative termssuminfin

119896=1V119896and each 119896 ge 119896

0= 1 we have

119904119896+1

120593119896+1

(119905) le 119886119904119896

120593119896

(119905) le 119886119904119896

120593119896

(119905) + V119896 (45)

For example for 119904 = 2 and 119886 = 12 the function 120595(119905) =

ln(1199054 + 1) is a (119887)-comparison function

Theorem 33 Let (119883 119866120590119887) be a 119866

120590119887-complete generalized 119887-

metric-like space and let 119891 119883 rarr 119883 be a 119866-120572-admissiblemapping Suppose that

119904120572 (119909 119910 119911) 119866120590119887

(119891119909 119891119910 119891119911) le 120595 (119872119904(119909 119910 119911)) (46)

for all 119909 119910 119911 isin 119883 where 120595 isin Ψ119887and

119872119904(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119909 119891119909)119866120590119887

(119910 119910 119891119910)119866120590119887

(119911 119911 119891119911)

1 + 211990421198662

120590119887

(119891119909 119891119910 119891119911)

(47)

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 120572(119909

0 1198911199090 11989121199090) ge 1

(ii)

(a) 119891 is continuous and for any sequence 119909119899 in119883

with 120572(119909119899 119909119899+1

119909119899+2

) ge 1 for all 119899 isin Ncup0 suchthat 119909

119899rarr 119909 as 119899 rarr infin one has 120572(119909 119909 119909) ge 1

for all 119899 isin N cup 0(b) assume that whenever 119909

119899 in 119883 is a sequence

such that 120572(119909119899 119909119899+1

119909119899+2

) ge 1 for all 119899 isin Ncup0

and 119909119899rarr 119909 as 119899 rarr infin one has 120572(119909

119899 119909 119909) ge 1

for all 119899 isin N cup 0

Then 119891 has a fixed point

Proof Let 1199090isin 119883 be such that 120572(119909

0 1198911199090 11989121199090) ge 1 Define

a sequence 119909119899 by 119909

119899= 1198911198991199090for all 119899 isin N Since 119891 is a 119866-120572-

admissible mapping and 120572(1199090 1199091 1199092) = 120572(119909

0 1198911199090 11989121199090) ge

1 we deduce that 120572(1199091 1199092 1199093) = 120572(119891119909

0 1198911199091 1198911199092) ge 1

Continuing this process we get 120572(119909119899 119909119899+1

119909119899+2

) ge 1 for all119899 isin N cup 0

8 The Scientific World Journal

If there exists 1198990isin N such that 119909

1198990= 1199091198990+1

then 1199091198990

=

1198911199091198990and so we have nothing to prove Hence for all 119899 isin N

we assume that 119909119899

= 119909119899+1

Step I (Cauchyness of 119909119899) As 120572(119909

119899 119909119899+1

119909119899+2

) ge 1 for all119899 ge 0 using condition (46) we obtain

119904119866120590119887

(119909119899 119909119899+1

119909119899+2

)

le 119904120572 (119909119899minus1

119909119899 119909119899+1

) 119866120590119887

(119909119899 119909119899+1

119909119899+2

)

= 119904120572 (119909119899minus1

119909119899 119909119899+1

) 119866120590119887

(119891119909119899minus1

119891119909119899 119891119909119899+1

)

le 120595 (119872119904(119909119899minus1

119909119899 119909119899+1

))

(48)

Using Proposition 15(2) as 119909119899

= 119909119899+1

we get

119872119904(119909119899minus1

119909119899 119909119899+1

)

= max 119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

(119866120590119887

(119909119899minus1

119909119899minus1

119891119909119899minus1

)

times 119866120590119887

(119909119899 119909119899 119891119909119899)

times119866120590119887

(119909119899+1

119909119899+1

119891119909119899+1

))

times(1 + 21199042

1198662

120590119887

(119891119909119899minus1

119891119909119899 119891119909119899+1

))

minus1

= max 119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

(119866120590119887

(119909119899minus1

119909119899minus1

119909119899)

times 119866120590119887

(119909119899 119909119899 119909119899+1

)

times119866120590119887

(119909119899+1

119909119899+1

119909119899+2

))

times(1 + 21199042

1198662

120590119887

(119909119899 119909119899+1

119909119899+2

))

minus1

le max119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

2119904119866120590119887

(119909119899minus1

119909119899 119909119899+1

) 1198662

120590119887

(119909119899 119909119899+1

119909119899+2

)

1 + 2s21198662120590119887

(119909119899 119909119899+1

119909119899+2

)

= 119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

(49)

Hence119866120590119887

(119909119899 119909119899+1

119909119899+2

)

le 119904119866120590119887

(119909119899 119909119899+1

119909119899+2

) le 120595 (119866120590119887

(119909119899minus1

119909119899 119909119899+1

))

(50)

By induction since 119909119899

= 119909119899+1

we get that

119866120590119887

(119909119899 119909119899+1

119909119899+2

)

le 120595 (119866120590119887

(119909119899minus1

119909119899 119909119899+1

))

le 1205952

(119866120590119887

(119909119899minus2

119909119899minus1

119909119899))

le sdot sdot sdot le 120595119899

(119866120590119887

(1199090 1199091 1199092))

(51)

Let 120576 gt 0 be arbitrary Then there exists a natural number119873such that

infin

sum

119899=119873

119904119899

120595119899

(119866120590119887

(1199090 1199091 1199092)) lt

120576

2119904

(52)

Let 119898 gt 119899 ge 119873 Then by the rectangular inequality andProposition 15(2) as 119909

119899= 119909119899+1

we get

119866120590119887

(119909119899 119909119898 119909119898)

le 119904119866120590119887

(119909119899 119909119899+1

119909119899+1

) + 1199042

119866120590119887

(119909119899+1

119909119899+2

119909119899+2

)

+ sdot sdot sdot + 119904119898minus119899minus1

119866120590119887

(119909119898minus1

119909119898 119909119898)

le 2119904 (119904119866120590119887

(119909119899 119909119899 119909119899+1

) + 1199042

119866120590119887

(119909119899+1

119909119899+1

119909119899+2

)

+ sdot sdot sdot + 119904119898minus119899minus1

119866120590119887

(119909119898minus1

119909119898minus1

119909119898))

le 2119904 (119904119866120590119887

(119909119899 119909119899+1

119909119899+2

) + 1199042

119866120590119887

(119909119899+1

119909119899+2

119909119899+3

)

+ sdot sdot sdot + 119904119898minus119899minus1

119866120590119887

(119909119898minus1

119909119898 119909119898+1

))

le 2119904

119898minus2

sum

119896=119899

119904119896minus119899+1

120595119896

(119866120590119887

(1199090 1199091 1199092))

le 2119904

infin

sum

119896=119899

119904119896

120595119896

(119866120590119887

(1199090 1199091 1199092)) lt 120576

(53)

Consequently 119909119899 is a 119866

120590119887-Cauchy sequence in 119883 Since

(119883 119866120590119887) is 119866120590119887-complete so there exists 119906 isin 119883 such that

lim119899rarrinfin

119866120590119887

(119906 119909119899 119909119899) = lim119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

= 119866120590119887

(119906 119906 119906) = 0

(54)

Step II Now we show that 119906 is a fixed point of 119891 Suppose tothe contrary that is 119891119906 = 119906 then we have 119866

120590119887(119906 119906 119891119906) gt 0

Let the part (a) of (ii) holdsUsing the rectangle inequality we get

119866120590119887

(119906 119906 119891119906) le 119904119866120590119887

(119891119906 119891119909119899 119891119909119899) + 119904119866

120590119887(119891119909119899 119906 119906)

(55)

Letting 119899 rarr infin and using the continuity of 119891 we get

119866120590119887

(119906 119906 119891119906) le 119904119866120590119887

(119891119906 119891119906 119891119906) (56)

From (46) and part (a) of condition (ii) we have

119904119866120590119887

(119891119906 119891119906 119891119906) le 120595 (119872119904(119906 119906 119906)) (57)

where by using (56) we have

119872119904(119906 119906 119906)

= max119866120590119887

(119906 119906 119906)

119866120590119887

(119906 119906 119891119906)119866120590119887

(119906 119906 119891119906)119866120590119887

(119906 119906 119891119906)

1 + 211990421198662

120590119887

(119891119906 119891119906 119891119906)

le 119866120590119887

(119906 119906 119891119906)

(58)

Hence from properties of 120595 119904119866120590119887(119891119906 119891119906 119891119906) le

119866120590119887(119906 119906 119891119906) Thus by (56) we obtain that

119866120590119887

(119906 119906 119891119906) = 119904119866120590119887

(119891119906 119891119906 119891119906) (59)

The Scientific World Journal 9

Moreover (57) yields that 119866120590119887(119906 119906 119891119906) le 120595(119872

119904(119906 119906 119906)) le

120595(119866120590119887(119906 119906 119891119906))This is impossible according to our assump-

tions on 120595 Hence we have 119891119906 = 119906 Thus 119906 is a fixed point of119891

Now let part (b) of (ii) holdsAs 119909119899 is a sequence such that 120572(119909

119899 119909119899+1

119909119899+2

) ge 1 for all119899 isin N cup 0 and 119909

119899rarr 119909 as 119899 rarr infin we have 120572(119909

119899 119909 119909) ge 1

for all 119899 isin N cup 0Now we show that 119906 = 119891119906 By (46) we have

119904119866120590119887

(119891119906 119891119906 119909119899) le 119904120572 (119906 119906 119909

119899minus1) 119866120590119887

(119891119906 119891119906 119891119909119899minus1

)

le 120595 (119872119904(119906 119906 119909

119899minus1))

(60)

where

119872119904(119906 119906 119909

119899minus1)

= max

119866120590119887

(119906 119906 119909119899minus1

)

[119866120590119887

(119906 119906 119891119906)]

2

119866120590119887

(119909119899minus1

119909119899minus1

119891119909119899minus1

)

1 + [119904119866120590119887

(119891119906 119891119906 119891119909119899minus1

)]

2

(61)

Letting 119899 rarr infin in the above inequality and using (54) andLemma 21 we get

lim119899rarrinfin

119872119904(119906 119906 119909

119899minus1) = 0 (62)

Again taking the upper limit as 119899 rarr infin in (60) and using(62) and Lemma 21 we obtain

119904 [

1

119904

119866120590119887

(119906 119891119906 119891119906) minus 119866120590119887

(119906 119906 119906)]

le 119904 lim sup119899rarrinfin

119866120590119887

(119909119899 119891119906 119891119906)

le lim supnrarrinfin

120595 (119872119904(119906 119906 119909

119899minus1))

= 120595(lim sup119899rarrinfin

119872119904(119906 119906 119909

119899minus1))

= 120595 (0) = 0

(63)

So we get 119866120590119887(119906 119891119906 119891119906) = 0 That is 119891119906 = 119906

Let (119883 119866120590119887 ⪯) be a partially ordered 119866

120590119887-metric-like

space We say that 119879 is an increasing mapping on 119883 ifforall119909 119910 isin 119883 119909 ⪯ 119910 rArr 119879(119909) ⪯ 119879(119910) [26] Fixed pointtheorems for monotone operators in ordered metric spacesarewidely investigated andhave found various applications indifferential and integral equations (see [27ndash30] and referencestherein) From the results proved above we derive thefollowing new results in partially ordered 119866

120590119887-metric-like

space

Theorem 34 Let (119883 119866120590119887 ⪯) be a partially ordered 119866

120590119887-

complete generalized 119887-metric-like space and let 119891 119883 rarr 119883

be an increasing mapping Suppose that

119904119866120590119887

(119891119909 119891119910 119891119911) le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911) (64)

for all 119909 119910 119911 isin 119883 with 119909 ⪯ 119910 ⪯ 119911 where

119872119904(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119891119909 119891119909)119866120590119887

(119910 119891119910 119891119910)119866120590119887

(119911 119891119911 119891119911)

1 + 11990421198662

120590119887

(119891119909 119891119910 119891119911)

(65)

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) 119891 is continuous or assume that whenever 119909119899 in 119883 is

an increasing sequence such that 119909119899rarr 119909 as 119899 rarr infin

one has 119909119899⪯ 119909 for all 119899 isin N cup 0

Then 119891 has a fixed point

Proof Define 120572 119883 times 119883 times 119883 rarr [0 +infin) by

120572 (119909 119910 119911) =

1 if119909 ⪯ 119910 ⪯ 119911

0 otherwise(66)

First we prove that 119891 is a triangular 120572-admissible mappingHence we assume that 120572(119909 119910 119911) ge 1 Therefore we have 119909 ⪯

119910 ⪯ 119911 Since 119891 is increasing we get 119891119909 ⪯ 119891119910 ⪯ 119891119911 that is120572(119891119909 119891119910 119891119911) ge 1 Also let 120572(119909 119910 119910) ge 1 and 120572(119910 119911 119911) ge 1then 119909 ⪯ 119910 and 119910 ⪯ 119911 Consequently we deduce that 119909 ⪯ 119911that is 120572(119909 119911 119911) ge 1 Thus 119891 is a triangular 120572-admissiblemapping Since 119891 satisfies (64) so by the definition of 120572 wehave

119904120572 (119909 119910 119911) 119866120590119887

(119891119909 119891119910 119891119911)

le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911)

(67)

for all 119909 119910 119911 isin 119883 Therefore 119891 satisfies the contractivecondition (21) From (i) there exists 119909

0isin 119883 such that

1199090⪯ 1198911199090 that is 120572(119909

0 1198911199090 1198911199090) ge 1 According to (ii) we

conclude that all the conditions of Theorems 26 and 27 aresatisfied and so 119891 has a fixed point

Similarly using Theorem 33 we can prove followingresult

Theorem 35 Let (119883 119866120590119887 ⪯) be an ordered 119866

120590119887-complete

generalized 119887-metric-like space and let 119891 119883 rarr 119883 be anincreasing mapping Suppose that

119904119866120590119887

(119891119909 119891119910 119891119911) le 120595 (119872119904(119909 119910 119911)) (68)

for all 119909 119910 119911 isin 119883 with 119909 ⪯ 119910 ⪯ 119911 where 120595 isin Ψ119887and

119872119904(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119909 119891119909)119866120590119887

(119910 119910 119891119910)119866120590119887

(119911 119911 119891119911)

1 + 211990421198662

120590119887

(119891119909 119891119910 119891119911)

(69)

10 The Scientific World Journal

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii)

(a) 119891 is continuous(b) assume that whenever x

119899 in 119883 is an increasing

sequence such that 119909119899rarr 119909 as 119899 rarr infin one has

119909119899⪯ 119909 for all 119899 isin N cup 0

Then 119891 has a fixed point

We conclude this section by presenting some examplesthat illustrate our results

Example 36 Let 119883 = [0 (minus1 + radic5)2] be endowed withthe usual ordering onR Define the generalized 119887-metric-likefunction 119866

120590119887given by

119866120590119887

(119909 119910 119911) = max (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

(70)

with 119904 = 2 Consider the mapping 119891 119883 rarr 119883 defined by119891(119909) = (14)119909

radic119890minus(119909+((minus1+radic5)2))

2 and the function 120573 isin Fgiven by 120573(119905) = (12)119890

minus119905 119905 gt 0 and 120573(0) isin [0 12) Itis easy to see that 119891 is an increasing function on 119883 Weshow that 119891 is 119866

120590119887-continuous on 119883 By Proposition 20 it is

sufficient to show that119891 is119866120590119887-sequentially continuous on119883

Let 119909119899 be a sequence in119883 such that lim

119899rarrinfin119866120590119887(119909119899 119909 119909) =

119866120590119887(119909 119909 119909) so we have max(lim

119899rarrinfin119909119899

+ 119909)2

41199092 =

41199092 and equivalently lim

119899rarrinfin119909119899

= 120572 le 119909 On the otherhand we have

lim119899rarrinfin

119866120590119887

(119891119909119899 119891119909 119891119909)

= lim119899rarrinfin

max (119891119909119899+ 119891119909)

2

41198912

119909

= lim119899rarrinfin

max 1

16

(119909119899

radic119890minus(119909119899+(minus1+

radic5)2)2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= max 1

16

( lim119899rarrinfin

119909119899

radic119890minus(lim119899rarrinfin119909119899+(minus1+radic5)2)

2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= max 1

16

(120572radic119890minus(120572+(minus1+radic5)2)

2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

=

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= 119866120590119887

(119891119909 119891119909 119891119909)

(71)

So 119891 is 119866120590119887-sequentially continuous on119883

For all elements 119909 119910 119911 isin 119883 and the fact that 119892(119909) =

1199092119890minus1199092

is an increasing function on119883 we have

119904119866120590119887

(119891119909 119891119910 119891119911)

= 2max (119891119909 + 119891119910)2

(119891119910 + 119891119911)2

(119891119911 + 119891119909)2

= 2max(

1

4

119909radic119890minus(119909+(minus1+radic5)2)

2

+

1

4

119910radic119890minus(119910+(minus1+radic5)2)

2

)

2

(

1

4

119910radic119890minus(119910+(minus1+radic5)2)

2

+

1

4

119911radic119890minus(119911+(minus1+radic5)2)

2

)

2

(

1

4

119911radic119890minus(119911+(minus1+radic5)2)

2

+

1

4

119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

le 2max(

1

4

119909radic119890minus(119909+119910)

2

+

1

4

119910radic119890minus(119910+119909)

2

)

2

(

1

4

119910radic119890minus(119910+119911)

2

+

1

4

119911radic119890minus(119911+119910)

2

)

2

(

1

4

119911radic119890minus(119911+119909)

2

+

1

4

119909radic119890minus(119909+119911)

2

)

2

=

1

8

max (119909 + 119910)2

119890minus(119909+119910)

2

(119910 + 119911)2

119890minus(119910+119911)

2

(119911 + 119909)2

119890minus(119911+119909)

2

=

1

8

119890minusmax(119909+119910)2 (119910+119911)2 (119911+119909)2

timesmax (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

le

1

2

eminusmax(119909+119910)2 (119910+119911)2(119911+119909)2

timesmax (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

= 120573 (119866120590119887

(119909 119910 119911)) 119866120590119887

(119909 119910 119911)

le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911)

(72)

Hence119891 satisfies all the assumptions ofTheorem 34 and thusit has a fixed point (which is 119906 = 0)

Example 37 Let 119883 = [0 1] with the usual ordering on RDefine the generalized 119887-metric-like function 119866

120590119887given by

119866120590119887(119909 119910 119911) = max(119909 + 119910)

2

(119910 + 119911)2

(119911 + 119909)2

with 119904 = 2Consider the mapping 119891 119883 rarr 119883 defined by 119891(119909) =

The Scientific World Journal 11

ln(1 + 119909119890minus119909

4) and the function 120595 isin Ψ119887given by 120595(119905) =

(18)119905 119905 ge 0 It is easy to see that 119891 is increasing functionNow we show that 119891 is a 119866

120590119887-continuous function on119883

Let 119909119899 be a sequence in 119883 such that lim

119899rarrinfin119866120590119887(119909119899

119909 119909) = 119866120590119887(119909 119909 119909) so we havemax(lim

119899rarrinfin119909119899+119909)2

41199092 =

41199092 and equally lim

119899rarrinfin119909119899= 120572 le 119909 On the other hand we

have

lim119899rarrinfin

119866120590119887

(119891119909119899 119891119909 119891119909)

= lim119899rarrinfin

max (119891119909119899+ 119891119909)

2

41198912

119909

= max(ln(1 +

lim119899rarrinfin

119909119899119890minuslim119899rarrinfin119909119899

4

)

+ ln(1 +

119909119890minus119909

4

))

2

4(ln(1 +

119909119890minusx

4

))

2

= max(ln(1 +

120572119890minus120572

4

) + ln(1 +

119909119890minus119909

4

))

2

4(ln(1 +

119909119890minus119909

4

))

2

= 4(ln(1 +

119909119890minus119909

4

))

2

= 119866120590119887

(119891119909 119891119909 119891119909)

(73)

So 119891 is 119866120590119887-sequentially continuous on119883

For all elements 119909 119910 119911 isin 119883 we have

119904119866120590119887

(119891119909 119891119910 119891119911)

= 2max(ln(1 +

119909119890minus119909

4

) + ln(1 +

119910119890minus119910

4

))

2

(ln(1 +

119910119890minus119910

4

) + ln(1 +

119911119890minus119911

4

))

2

(ln(1 +

119911119890minus119911

4

) + ln(1 +

119909119890minus119909

4

))

2

le 2max(

119909119890minus119909

4

+

119910119890minus119910

4

)

2

(

119910119890minus119910

4

+

119911119890minus119911

4

)

2

(

119911119890minus119911

4

+

119909119890minus119909

4

)

2

le

1

8

max (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

= 120595 (119866120590119887

(119909 119910 119911))

le 120595 (119872119904(119909 119910 119911))

(74)

Hence119891 satisfies all the assumptions ofTheorem 35 and thusit has a fixed point (which is 119906 = 0)

3 Application

In this section we present an application of our results toestablish the existence of a solution to a periodic boundaryvalue problem (see [30 31])

Let 119883 = 119862([0 119879]) be the set of all real continuousfunctions on [0T] We first endow119883 with the 119887-metric-like

120590119887(119906 V) = sup

119905isin[0119879]

(|119906 (119905)| + |V (119905)|)119901 (75)

for all 119906 V isin 119883 where 119901 gt 1 and then we endow it with thegeneralized 119887-metric-like 119866

120590119887defined by

119866120590119887

(119906 V 119908)

= max sup119905isin[0119879]

(|119906 (119905)| + |V (119905)|)119901

sup119905isin[0119879]

(|V (119905)| + |119908 (119905)|)119901

sup119905isin[0119879]

(|119908 (119905)| + |119906 (119905)|)119901

(76)

Clearly (119883 119866120590119887) is a complete generalized 119887-metric-like space

with parameter 119870 = 2119901minus1 We equip 119862([0 119879]) with a partial

order given by119909 ⪯ 119910 iff119909 (119905) le 119910 (119905) forall119905 isin [0 119879] (77)

Moreover as in [30] it is proved that (119862([0 119879]) ⪯) is regularthat is whenever 119909

119899 in119883 is an increasing sequence such that

119909119899rarr 119909 as 119899 rarr infin we have 119909

119899⪯ 119909 for all 119899 isin N cup 0

Consider the first-order periodic boundary value prob-lem

1199091015840

(119905) = 119891 (119905 119909 (119905)) 119909 (0) = 119909 (119879) (78)

where 119905 isin 119868 = [0 119879] with 119879 gt 0 and 119891 119868 times R rarr R is acontinuous function

A lower solution for (78) is a function 120572 isin 1198621[0 119879] such

that1205721015840

(119905) le 119891 (119905 120572 (119905))

120572 (0) le 120572 (119879)

(79)

where 119905 isin 119868 = [0 119879]Assume that there exists 120582 gt 0 such that for all 119909 119910 isin

119862[0 119879] we have1003816100381610038161003816119891 (119905 119909 (119905)) + 120582119909 (119905)

1003816100381610038161003816+1003816100381610038161003816119891 (119905 119910 (119905)) + 120582119910 (119905)

1003816100381610038161003816

le

120582

2119901minus1

119901radicln(

119886

2119901minus1

(|119909 (119905)| +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)119901

+ 1)

(80)

where 0 le 119886 lt 1 Then the existence of a lower solution for(78) provides the existence of a solution of (78)

Problem (78) can be rewritten as

1199091015840

(119905) + 120582119909 (119905) = 119891 (119905 119909 (119905)) + 120582119909 (119905)

119909 (0) = 119909 (119879)

(81)

Consider1199091015840

(119905) + 120582119909 (119905) = 120575119909(119905) = 119865 (119905 119909 (119905))

119909 (0) = 119909 (119879)

(82)

where 119905 isin 119868

12 The Scientific World Journal

Using variation of parameters formula we get

119909 (119905) = 119909 (0) 119890minus120582119905

+ int

119905

0

119890minus120582(119905minus119904)

120575119909(119904) 119889119904 (83)

which yields

119909 (119879) = 119909 (0) 119890minus120582119879

+ int

119879

0

119890minus120582(119879minus119904)

120575119909(119904) 119889119904 (84)

Since 119909(0) = 119909(119879) we get

119909 (0) [1 minus 119890minus120582119879

] = 119890minus120582119879

int

119879

0

119890120582(119904)

120575119909(119904) 119889119904 (85)

or

119909 (0) =

1

119890120582119879

minus 1

int

119879

0

119890120582119904

120575119909(119904) 119889119904 (86)

Substituting the value of 119909(0) in (83) we arrive at

119909 (119905) = int

119879

0

119866 (119905 119904) 120575119909(119904) 119889119904 (87)

where

119866 (119905 119904) =

119890120582(119879+119904minus119905)

119890120582119879

minus 1

0 le 119904 le 119905 le 119879

119890120582(119904minus119905)

119890120582119879

minus 1

0 le 119905 le 119904 le 119879

(88)

Now define the operator 119878 119862[0 119879] rarr 119862[0 119879] as

119878119909 (119905) = int

119879

0

119866 (119905 119904) 119865 (119904 119909 (119904)) 119889119904 (89)

The mapping 119878 is increasing [31] Note that if 119906 isin 119862[0 119879] is afixed point of 119878 then 119906 isin 119862

1[0 119879] is a solution of (78)

Let 119909 119910 119911 isin 119862[0 119879] Then we have

2119901minus1

[|119878119909 (119905)| +1003816100381610038161003816119878119910 (119905)

1003816100381610038161003816]

= 2119901minus1

[

100381610038161003816100381610038161003816100381610038161003816

int

119879

0

119866 (119905 119904) 119865 (119904 119909 (119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

119879

0

119866 (119905 119904) 119865 (119904 119910 (119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

]

le 2119901minus1

int

119879

0

|119866 (119905 119904)|

times [|119865 (119904 119909 (119904))| +1003816100381610038161003816119865 (119904 119910 (119904))

1003816100381610038161003816] 119889119904

le 2119901minus1

int

119879

0

|119866 (119905 119904)|

120582

2119901minus1

times119901radicln(

119886

2119901minus1

(|119909 (119905)| +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)119901

+ 1)119889119904

le 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [int

119905

0

119890120582(119879+119904minus119905)

119890120582119879

minus 1

119889119904 + int

119879

119905

119890120582(119904minus119905)

119890120582119879

minus 1

119889119904]

= 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [

1

120582 (119890120582119879

minus 1)

(119890120582(119879+119904minus119905)10038161003816

100381610038161003816

119905

0

+ 119890120582(119904minus119905)10038161003816

100381610038161003816

119879

119905

)]

= 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [

1

120582 (119890120582119879

minus 1)

(119890120582119879

minus 119890120582(119879minus119905)

+ 119890120582(119879minus119905)

minus 1)]

=119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

le119901radicln(

119886

2119901minus1

119866120590119887

(119909 119910 119911) + 1)

le119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(90)

Similarly

2119901minus1 1003816

100381610038161003816119878119910 (119905)

1003816100381610038161003816+ |119878119911 (119905)| le

119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

|119878119909 (119905)| + |119878119911 (119905)| le119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(91)

where

119872(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119878119909 119878119909) 119866120590119887

(119910 119878119910 119878119910)119866120590119887

(119911 119878119911 119878119911)

1 + 22119901minus1

1198662

120590119887

(119878119909 119878119910 119878119911)

(92)

Equivalently

(2119901minus1

|119878119909 (119905)| +1003816100381610038161003816119878119910 (119905)

1003816100381610038161003816)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(2119901minus1 1003816

100381610038161003816119878119910 (119905)

1003816100381610038161003816+ |119878119911 (119905)|)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(2119901minus1

|119878119909 (119905)| + |119878119911 (119905)|)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(93)

which yields that

2119901minus1

120590119887(119878119909 119878119910) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

120590119887(119878119910 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

120590119887(119878119909 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(94)

The Scientific World Journal 13

Finally it is easy to obtain that

2119901minus1G120590119887

(119878119909 119878119910 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1) (95)

Finally since120572 is a lower solution for (78) so it is easy to showthat 120572 ⪯ 119878(120572) [31]

Hence the hypotheses of Theorem 35 are satisfied with120595(119905) = ln((1198862119901minus1)119905 + 1) where 0 le 119886 lt 1 Hence there existsa fixed point 119909 isin 119862[0 119879] such that 119878119909 = 119909 which is a solutionto periodic boundary value problem (78)

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) KingAbdulaziz University JeddahTherefore the firstauthor acknowledges with thanks DSR KAU for financialsupport

References

[1] S Czerwik ldquoContraction mappings in b-metric spacesrdquo ActaMathematica et Informatica Universitatis Ostraviensis vol 1 pp5ndash11 1993

[2] V Parvaneh J R Roshan and S Radenovic ldquoExistence of tri-pled coincidence points in ordered b-metric spaces and an app-lication to a system of integral equationsrdquo Fixed Point Theoryand Applications vol 2013 article 130 2013

[3] J R Roshan V Parvaneh S Sedghi N Shobkolaei and WShatanawi ldquoCommon fixed points of almost generalized con-tractive mappings in ordered b-metric spacesrdquo Fixed PointTheory and Applications vol 2013 article 159 2013

[4] H Aydi M F Bota E Karapınar and S Moradi ldquoA commonfixed point for weak 120601-contractions on b-metric spacesrdquo FixedPoint Theory vol 13 no 2 pp 337ndash346 2012

[5] V Berinde ldquoSequences of operators and fixed points in quasi-metric spacesrdquo Studia Universitatis ldquoBabes-Bolyairdquo Mathemat-ica vol 16 no 4 pp 23ndash27 1996

[6] N Hussain D ETHoric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 article 126 2012

[7] N Hussain and M H Shah ldquoKKMmappings in cone b-metricspacesrdquo Computers and Mathematics with Applications vol 62no 4 pp 1677ndash1684 2011

[8] N Hussain J R Roshan V Parvaneh and Z Kadelburg ldquoFixedpoints of contractive mappings in b-metric-like spacesrdquo TheScientific World Journal vol 2014 Article ID 471827 15 pages2014

[9] M A Khamsi and N Hussain ldquoKKMmappings in metric typespacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 73 no 9 pp 3123ndash3129 2010

[10] M A Khamsi ldquoRemarks on cone metric spaces and fixed pointtheorems of contractive mappingsrdquo Fixed Point Theory andApplications vol 2010 Article ID 315398 7 pages 2010

[11] Z Mustafa J R Roshan V Parvaneh and Z Kadelburg ldquoSomecommon fixed point results in ordered partial b-metric spacesrdquoJournal of Inequalities and Applications vol 2013 article 5622013

[12] Z Mustafa and B Sims ldquoA new approach to generalized metricspacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash297 2006

[13] A Amini-Harandi ldquoMetric-like spaces partial metric spacesand fixed pointsrdquo Fixed PointTheory and Applications vol 2012article 204 2012

[14] N Shobkolaei S Sedghi J R Roshan and N Hussain ldquoSuzukitype fixed point results in metric-like spacesrdquo Journal of Func-tion Spaces and Applications vol 2013 Article ID 143686 9pages 2013

[15] M A Alghamdi N Hussain and P Salimi ldquoFixed point andcoupled fixed point theorems on b-metric-like spacesrdquo Journalof Inequalities and Applications vol 2013 article 402 2013

[16] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572 minus 120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

[17] P Salimi and E Karapınar ldquoSuzuki-Edelstein type contractionsvia auxiliary functionsrdquoMathematical Problems in Engineeringvol 2013 Article ID 648528 8 pages 2013

[18] P Salimi C Vetro and P Vetro ldquoFixed point theorems fortwisted (120572 120573)-120595-contractive type mappings and applicationsrdquoFilomat vol 27 no 4 pp 605ndash615 2013

[19] P Salimi C Vetro and P Vetro ldquoSome new fixed point resultsin non-Archimedean fuzzy metric spacesrdquo Nonlinear AnalysisModelling and Control vol 18 no 3 pp 344ndash358 2013

[20] N Hussain P Salimi and A Latif ldquoFixed point results forsingle and set-valued 120572-120578-120595-contractive mappingsrdquo Fixed PointTheory and Applications vol 2013 article 212 2013

[21] M A Alghamdi and E Karapınar ldquoG-120573-120595 contractive-typemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 article 70 2013

[22] M Geraghty ldquoOn contractive mappingsrdquo Proceedings of theAmerican Mathematical Society vol 40 pp 604ndash608 1973

[23] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001

[24] N Hussain Z Kadelburg S Radenovic and F R Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012

[25] V Berinde ldquoGeneralized contractions in quasimetric spacesrdquo inSeminar on Fixed Point Theory Preprint no 3 pp 3ndash9 1993

[26] L Ciric M Abbas R Saadati and N Hussain ldquoCommon fixedpoints of almost generalized contractive mappings in orderedmetric spacesrdquo Applied Mathematics and Computation vol 217no 12 pp 5784ndash5789 2011

[27] R P Agarwal N Hussain and M A Taoudi ldquoFixed point the-orems in ordered Banach spaces and applications to nonlinearintegral equationsrdquo Abstract and Applied Analysis vol 2012Article ID 245872 15 pages 2012

[28] N Hussain A R Khan and R P Agarwal ldquoKrasnoselrsquoskii andKy Fan type fixed point theorems in ordered Banach spacesrdquoJournal of Nonlinear and Convex Analysis vol 11 no 3 pp 475ndash489 2010

[29] N Hussain and M A Taoudi ldquoKrasnoselrsquoskii-type fixed pointtheorems with applications to Volterra integral equationsrdquoFixed Point Theory and Applications vol 2013 article 196 2013

14 The Scientific World Journal

[30] J J Nieto and R Rodrıguez-Lopez ldquoContractivemapping theo-rems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[31] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysisTheoryMethods andApplications vol 71 no 7-8 pp 3403ndash3410 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 3

Also

119866119904

120590119887

(119909 119910 119911) = 120590119887(119909 119910) + 120590

119887(119910 119911) + 120590

119887(119911 119909)

le 119904 (120590119887(119909 119886) + 120590

119887(119886 119910))

+ 120590119887(119910 119911) + 119904 (120590

119887(119911 119886) + 120590

119887(119886 119909))

le 119904 (120590119887(119909 119886) + 120590

119887(119886 119910))

+ 119904 (120590119887(119910 119911) + 120590

119887(119886 119886))

+ 119904 (120590119887(119911 119886) + 120590

119887(119886 119909))

= 119904 (120590119887(119909 119886) + 120590

119887(119886 119886) + 120590

119887(119886 119909))

+ 119904 (120590119887(119886 119910) 120590

119887(119910 119911) 120590

119887(119911 119886))

= 119904 (119866119904

120590119887(119909 119886 119886) + 119866

119904

120590119887

(119886 119910 119911))

(6)

According to the above proposition we provide someexamples of generalized 119887-metric-like spaces

Example 13 Let 119883 = R+ let 119901 gt 1 be a constant and let119866119898

120590119887

119866119904

120590119887

119883 times 119883 times 119883 rarr R+ be defined by

119866119898

120590119887

(119909 119910 119911) = max (119909 + 119910)119901

(119910 + 119911)119901

(119911 + 119909)119901

119866119904

120590119887

(119909 y 119911) = (119909 + 119910)119901

+ (119910 + 119911)119901

+ (119911 + 119909)119901

(7)

for all 119909 119910 119911 isin 119883Then (119883 119866119898

120590119887

) and (119883 119866119904

120590119887

) are generalized119887-metric-like spaces with coefficient 119904 = 2

119901minus1 Note that for119909 = 119910 = 119911 gt 0119866119898

120590119887

(119909 119909 119909) = (2119909)119901

gt 0 and 119866119904

120590119887

(119909 119909 119909) =

3(2119909)119901

gt 0

Example 14 Let 119883 = [0 1] Then the mappings 1198661198981205901198871

119866119904

1205901198871

119883 times 119883 times 119883 rarr R+ defined by

119866119898

1205901198871

(119909 119910 119911)

= max (119909 + 119910 minus 119909119910)119901

(119910 + 119911 minus 119910119911)119901

(119911 + 119909 minus 119911119909)119901

119866119904

1205901198871

(119909 119910 119911)

= (119909 + 119910 minus 119909119910)119901

+ (119910 + 119911 minus 119910119911)119901

+ (119911 + 119909 minus 119911119909)119901

(8)

where 119901 gt 1 is a real number are generalized 119887-metric-likespaces with coefficient 119904 = 2

119901minus1

By some straight forward calculations we can establishthe following

Proposition 15 Let 119883 be a 119866120590119887-metric space Then for each

119909 119910 119911 119886 isin 119883 it follows that

(1) 119866120590119887(119909 119910 119910) gt 0 for 119909 = 119910

(2) 119866120590119887(119909 119910 119911) le 119904(119866

120590119887(119909 119909 119910) + 119866

120590119887(119909 119909 119911))

(3) 119866120590119887(119909 119910 119910) le 2119904119866

120590119887(119910 119909 119909)

(4) 119866120590119887(119909 119910 119911) le 119904119866

120590119887(119909 119886 119886) + 119904

2119866120590119887(119910 119886 119886) +

1199042119866120590119887(119911 119886 119886)

Definition 16 Let (119883 119866120590119887) be a 119866

120590119887-metric space Then for

any 119909 isin 119883 and 119903 gt 0 the 119866120590119887-ball with center 119909 and radius 119903

is

119861119866120590119887

(119909 119903) = 119910 isin 119883 |

10038161003816100381610038161003816119866120590119887

(119909 119909 119910) minus 119866120590119887

(119909 119909 119909)

10038161003816100381610038161003816lt +119903

(9)

The family of all 119866120590119887-balls

ϝ = 119861119866120590119887

(119909 119903) | 119909 isin 119883 119903 gt 0 (10)

is a base of a topology 120591(119866120590119887) on119883 whichwe call it119866

120590119887-metric

topology

Definition 17 Let (119883 119866120590119887) be a 119866

120590119887-metric space Let 119909

119899 be

a sequence in119883 Consider the following

(1) A point119909 isin 119883 is said to be a limit of the sequence 119909119899

denoted by 119909119899

rarr 119909 if lim119899119898rarrinfin

119866120590119887(119909 119909119899 119909119898) =

119866120590119887(119909 119909 119909)

(2) 119909119899 is said to be a 119866

120590119887-Cauchy sequence if

lim119899119898rarrinfin

119866120590119887(119909119899 119909119898 119909119898) exists (and is finite)

(3) (119883 119866120590119887) is said to be 119866

120590119887-complete if every 119866

120590119887-

Cauchy sequence in 119883 is 119866120590119887-convergent to an 119909 isin 119883

such that

lim119899119898rarrinfin

119866120590119887

(119909119899 119909119898 119909119898) = lim119899rarrinfin

119866120590119887

(119909119899 119909 119909) = 119866

120590119887(119909 119909 119909)

(11)

Using the above definitions one can easily prove thefollowing proposition

Proposition 18 Let (119883 119866120590119887) be a 119866

120590119887-metric space Then for

any sequence 119909119899 in X and a point 119909 isin 119883 the following are

equivalent

(1) 119909119899 is 119866120590119887-convergent to 119909

(2) 119866120590119887(119909119899 119909119899 119909) rarr 119866

120590119887(119909 119909 119909) as 119899 rarr infin

(3) 119866120590119887(119909119899 119909 119909) rarr 119866

120590119887(119909 119909 119909) as 119899 rarr infin

Definition 19 Let (119883 119866120590119887) and (119883

1015840 1198661015840

120590119887

) be two generalized119887-metric like spaces and let 119891 (119883 119866

120590119887) rarr (119883

1015840 1198661015840

120590119887

) bea mapping Then 119891 is said to be 119866

120590119887-continuous at a point

119886 isin 119883 if for a given 120576 gt 0 there exists 120575 gt 0 suchthat 119909 isin 119883 and |119866

120590119887(119886 119886 119909) minus 119866

120590119887(119886 119886 119886)| lt 120575 imply

that |1198661015840120590119887

(119891(119886) 119891(119886) 119891(119909)) minus 1198661015840

120590119887

(119891(119886) 119891(119886) 119891(119886))| lt 120576 Themapping 119891 is 119866

120590119887-continuous on 119883 if it is 119866

120590119887-continuous at

all 119886 isin 119883 For simplicity we say that 119891 is continuous

Proposition 20 Let (119883 119866120590119887) and (119883

1015840 1198661015840

120590119887

) be two generalized119887-metric like spaces Then a mapping 119891 119883 rarr 119883

1015840 is 119866120590119887-

continuous at a point 119909 isin 119883 if and only if it is 119866120590119887-sequentially

continuous at 119909 that is whenever 119909119899 is 119866120590119887-convergent to 119909

119891(119909119899) is 119866

120590119887-convergent to 119891(119909)

4 The Scientific World Journal

We need the following simple lemma about the 119866120590119887-

convergent sequences in the proof of our main results

Lemma 21 Let (119883 119866120590119887) be a 119866

120590119887-metric space and suppose

that 119909119899 119910119899 and 119911

119899 are 119866

120590119887-convergent to 119909 119910 and 119911

respectively Then we have1

1199043119866120590119887

(119909 119910 119911) minus

1

1199042119866120590119887

(119909 119909 119909)

minus

1

119904

119866120590119887

(119910 119910 119910) minus 119866120590119887

(119911 119911 119911)

le lim inf119899rarrinfin

119866120590119887

(119909119899 119910119899 119911119899)

le lim sup119899rarrinfin

119866120590119887

(119909119899 119910119899 119911119899)

le 1199043

119866120590119887

(119909 119910 119911) + 119904119866120590b

(119909 119909 119909)

+ 1199042

119866120590119887

(119910 119910 119910) + 1199043

119866120590119887

(119911 119911 119911)

(12)

In particular if 119910119899 = 119911

119899 = 119886 are constant then

1

119904

119866120590119887

(119909 119886 119886) minus 119866120590119887

(119909 119909 119909)

le lim inf119899rarrinfin

119866120590119887

(119909119899 119886 119886) le lim sup

119899rarrinfin

119866120590119887

(119909119899 119886 119886)

le 119904119866120590119887

(119909 119886 119886) + 119904119866120590119887

(119909 119909 119909)

(13)

Proof Using the rectangle inequality we obtain

119866120590119887

(119909 119910 119911) le 119904119866120590119887

(119909 119909119899 119909119899) + 1199042

119866120590119887

(119910 119910119899 119910119899)

+ 1199043

119866120590119887

(119911 119911119899 119911119899) + 1199043

119866120590119887

(119909119899 119910119899 119911119899)

119866120590119887

(119909119899 119910119899 119911119899) le 119904119866

120590119887(119909119899 119909 119909) + 119904

2

119866120590119887

(119910119899 119910 119910)

+ 1199043

119866120590119887

(119911119899 119911 119911) + 119904

3

119866120590119887

(119909 119910 119911)

(14)

Taking the lower limit as 119899 rarr infin in the first inequality andthe upper limit as 119899 rarr infin in the second inequality we obtainthe desired result

If 119910119899 = 119911

119899 = 119886 then

119866120590119887

(119909 119886 119886) le 119904119866120590119887

(119909 119909119899 119909119899) + 119904119866

120590119887(119909119899 119886 119886)

119866120590119887

(119909119899 119886 119886) le 119904119866

120590119887(119909119899 119909 119909) + 119904119866

120590119887(119909 119886 119886)

(15)

Again taking the lower limit as 119899 rarr infin in the first inequalityand the upper limit as 119899 rarr infin in the second inequality weobtain the desired result

2 Main Results

Samet et al [16] defined the notion of 120572-admissible mappingsand proved the following result

Definition 22 Let119879 be a self-mapping on119883 and 120572 119883times119883 rarr

[0infin) a function We say that 119879 is an 120572-admissible mappingif

119909 119910 isin 119883 120572 (119909 119910) ge 1 997904rArr 120572 (119879119909 119879119910) ge 1 (16)

Denote withΨ1015840 the family of all nondecreasing functions

120595 [0infin) rarr [0infin) such that suminfin119899=1

120595119899(119905) lt infin for all 119905 gt 0

where 120595119899 is the 119899th iterate of 120595

Theorem 23 Let (119883 119889) be a complete metric space and 119879 an120572-admissible mapping Assume that

120572 (119909 119910) 119889 (119879119909 119879119910) le 120595 (119889 (119909 119910)) (17)

where 120595 isin Ψ1015840 Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 120572(119909

0 1198791199090) ge 1

(ii) either 119879 is continuous or for any sequence 119909119899 in 119883

with 120572(119909119899 119909119899+1

) ge 1 for all 119899 isin Ncup0 such that 119909119899rarr

119909 as 119899 rarr infin we have 120572(119909119899 119909) ge 1 for all 119899 isin N cup 0

Then 119879 has a fixed point

For more details on 120572-admissible mappings we refer thereader to [17ndash20]

Definition 24 (see [21]) Let (119883 119866) be a 119866-metric space let119891 be a self-mapping on 119883 and let 120572 119883

3rarr [0infin) be a

function We say that 119891 is a 119866-120572-admissible mapping if

119909 119910 119911 isin 119883 120572 (119909 119910 119911) ge 1 997904rArr 120572 (119891119909 119891119910 119891119911) ge 1

(18)

Motivated by [22] letF denote the class of all functions120573 [0infin) rarr [0 1119904) satisfying the following condition

lim sup119899rarrinfin

120573 (119905119899) =

1

119904

997904rArr lim sup119899rarrinfin

119905119899= 0 (19)

Definition 25 Let 119891 119883 rarr 119883 and 120572 119883 times 119883 times 119883 rarr RWe say that 119891 is a rectangular 119866-120572-admissible mapping if

(T1) 120572(119909 119910 119911) ge 1 implies 120572(119891119909 119891119910 119891119911) ge 1 119909 119910 119911 isin 119883

(T2) 120572(119909119910119910)ge1120572(119910119911119911)ge1

implies 120572(119909 119911 119911) ge 1 119909 119910 119911 isin 119883

From now on let 120572 1198833rarr [0infin) be a function and

119872119904(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119891119909 119891119909)119866120590119887

(119910 119891119910 119891119910)119866120590119887

(119911 119891119911 119891119911)

1 + 11990421198662

120590119887

(119891119909 119891119910 119891119911)

(20)

Theorem 26 Let (119883 119866120590119887) be a 119866

120590119887-complete generalized 119887-

metric-like space and let 119891 119883 rarr 119883 be a rectangular 119866-120572-admissible mapping Suppose that

119904120572 (119909 119910 119911) 119866120590119887

(119891119909 119891119910 119891119911)

le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911)

(21)

for all 119909 119910 119911 isin 119883

The Scientific World Journal 5

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 120572(119909

0 1198911199090 1198911199090) ge 1

(ii) 119891 is continuous and for any sequence 119909119899 in 119883 with

120572(119909119899 119909119899+1

119909119899+1

) ge 1 for all 119899 isin Ncup0 such that 119909119899rarr

119909 as 119899 rarr infin we have 120572(119909 119909 119909) ge 1 for all 119899 isin Ncup0

Then 119891 has a fixed point

Proof Let 1199090isin 119883 be such that 120572(119909

0 1198911199090 1198911199090) ge 1 Define

a sequence 119909119899 by 119909

119899= 1198911198991199090for all 119899 isin N Since 119891 is a 119866-

120572-admissible mapping and 120572(1199090 1199091 1199091) = 120572(119909

0 1198911199090 1198911199090) ge

1 we deduce that 120572(1199091 1199092 1199092) = 120572(119891119909

0 1198911199091 1198911199091) ge 1

Continuing this process we get 120572(119909119899 119909119899+1

119909119899+1

) ge 1 for all119899 isin N cup 0

Step I We will show that lim119899rarrinfin

119866120590119887(119909119899 119909119899+1

119909119899+1

) = 0 If119909119899= 119909119899+1

for some 119899 isin N then 119909119899= 119891119909119899 Thus 119909

119899is a fixed

point of 119891 Therefore we assume that 119909119899

= 119909119899+1

for all 119899 isin NSince 120572(119909

119899 119909119899+1

119909119899+1

) ge 1 for each 119899 isin N then we canapply (21) which yields

119904119866120590119887

(119909119899 119909119899+1

119909119899+1

)

= 119904119866120590119887

(119891119909119899minus1

119891119909119899 119891119909119899)

le 120573 (119866120590119887

(119909119899minus1

119909119899 119909119899))119872119904(119909119899minus1

119909119899 119909119899)

lt

1

119904

119866120590119887

(119909119899minus1

119909119899 119909119899)

le 119866120590119887

(119909119899minus1

119909119899 119909119899)

(22)

119872119904(119909119899minus1

119909119899 119909119899)

= max 119866120590119887

(119909119899minus1

119909119899 119909119899)

(119866120590119887

(119909119899minus1

119891119909119899minus1

119891119909119899minus1

)

times119866120590119887

(119909119899 119891119909119899 119891119909119899) 119866120590119887

(119909119899 119891119909119899 119891119909119899))

times(1 + 1199042

1198662

120590119887

(119891119909119899minus1

119891119909119899 119891119909119899))

minus1

= max 119866120590119887

(119909119899minus1

119909119899 119909119899)

(119866120590119887

(119909119899minus1

119909119899 119909119899)

times119866120590119887

(119909119899 119909119899+1

119909119899+1

) 119866120590119887

(119909119899 119909119899+1

119909119899+1

))

times(1 + 1199042

1198662

120590119887

(119909119899 119909119899+1

x119899+1

))

minus1

= 119866120590119887

(119909119899minus1

119909119899 119909119899)

(23)

Therefore 119866120590119887(119909119899 119909119899+1

119909119899+1

) is a decreasing and boundedsequence of nonnegative real numbers Then there exists 119903 ge

0 such that lim119899rarrinfin

119866120590119887(119909119899 119909119899+1

119909119899+1

) = 119903 Letting 119899 rarr infin in(22) we have

119904119903 le 119903 (24)

Since 119904 gt 1 we deduce that 119903 = 0 that is

lim119899rarrinfin

119866120590119887

(119909119899 119909119899+1

119909119899+1

) = 0 (25)

By Proposition 15(2) we conclude that

lim119899rarrinfin

119866120590119887

(119909119899 119909119899 119909119899+1

) = 0 (26)

Step II Now we prove that the sequence 119909119899 is a119866

120590119887-Cauchy

sequence For this purpose we will show that

lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898) = 0 (27)

Using the rectangular inequality with (21) (as 120572(119909119899 119909119898 119909119898) ge

1 since 119891 is a rectangular119866-120572-admissible mapping) we have

119866120590119887

(119909119899 119909119898 119909119898)

le 119904119866120590119887

(119909119899 119909119899+1

119909119899+1

)

+ 1199042

119866120590119887

(119909119899+1

119909119898+1

119909119898+1

) + 1199042

119866120590119887

(119909119898+1

119909119898 119909119898)

le 119904119866120590119887

(119909119899 119909119899+1

119909119899+1

)

+ 119904120573 (119866120590119887

(119909119899 119909119898 119909119898))119872119904(119909119899 119909119898 119909119898)

+ 1199042

119866120590119887

(119909119898+1

119909119898 119909119898)

(28)

Taking limit as 119898 119899 rarr infin in the above inequality andapplying (25) and (26) we have

lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

le 119904 lim sup119899119898rarrinfin

120573 (119866120590119887

(119909119899 119909119898 119909119898)) lim sup119899119898rarrinfin

119872119904(119909119899 119909119898 119909119898)

(29)

Here

119866120590119887

(119909119899 119909119898 119909119898)

le 119872119904(119909119899 119909119898 119909119898)

= max

119866120590119887

(119909119899 119909119898 119909119898)

119866120590119887

(119909119899 119891119909119899 119891119909119899) [119866120590119887

(119909119898 119891119909119898 119891119909119898)]

2

1 + 11990421198662

120590119887

(119891119909119899 119891119909119898 119891119909119898)

= max

119866120590119887

(119909119899 119909119898 119909119898)

119866120590119887

(119909119899 119909119899+1

119909119899+1

) [119866120590119887

(119909119898 119909119898+1

119909119898+1

)]

2

1 + 11990421198662

120590119887

(119909119899+1

119909119898+1

119909119898+1

)

(30)

6 The Scientific World Journal

Letting119898 119899 rarr infin in the above inequality we get

lim sup119898119899rarrinfin

119872119904(119909119899 119909119898 119909119898) = lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898) (31)

Hence from (29) and (31) we obtain

lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

le 119904lim sup119898119899rarrinfin

120573 (119866120590119887

(119909119899 119909119898 119909119898)) lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

(32)

If lim sup119898119899rarrinfin

119866120590119887(119909119899 119909119898 119909119898) = 0 then we get

1

119904

le lim sup119898119899rarrinfin

120573 (119866120590119887

(119909119899 119909119898 119909119898)) (33)

Since 120573 isin F we deduce that

lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898) = 0 (34)

which is a contradiction Consequently 119909119899 is a 119866

120590119887-Cauchy

sequence in119883 Since (119883 119866120590119887) is119866120590119887-complete there exists 119906 isin

119883 such that 119909119899

rarr 119906 as 119899 rarr infin Now from (34) and 119866120590119887-

completeness of119883

lim119899rarrinfin

119866120590119887

(119906 119909119899 119909119899) = lim119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

= 119866120590119887

(119906 119906 119906) = 0

(35)

Step III Now we show that 119906 is a fixed point of 119891Using the rectangle inequality we get

119866120590119887

(119906 119891119906 119891119906) le 119904119866120590119887

(119906 119891119909119899 119891119909119899) + 119904119866

120590119887(119891119909119899 119891119906 119891119906)

(36)

Letting 119899 rarr infin and using the continuity of 119891 and (35) weobtain

119866120590119887

(119906 119891119906 119891119906) le 119904 lim119899rarrinfin

G120590119887

(119906 119891119909119899 119891119909119899)

+ 119904 lim119899rarrinfin

119866120590119887

(119891119909119899 119891119906 119891119906)

= 119904119866120590119887

(119891119906 119891119906 119891119906)

(37)

Note that from (21) as 120572(119906 119906 119906) ge 1 we have

119904119866120590119887

(119891119906 119891119906 119891119906) le 120573 (119866120590119887

(119906 119906 119906))119872119904(119906 119906 119906) (38)

where by (37)

119872119904(119906 119906 119906)

= max119866120590119887

(119906 119906 119906)

119866120590119887

(119906 119891119906 119891119906)119866120590119887

(119906 119891119906 119891119906)119866120590119887

(119906 119891119906 119891119906)

1 + 11990421198662

120590119887

(119891119906 119891119906 119891119906)

lt 119866120590119887

(119906 119891119906 119891119906)

(39)

Hence as 120573(119905) le 1 for all 119905 isin [0infin) we have 119904119866120590119887(119891119906

119891119906 119891119906) le 119866120590119887(119906 119891119906 119891119906) Thus by (37) we obtain that

119904119866120590119887(119891119906 119891119906 119891119906) = 119866

120590119887(119906 119891119906 119891119906) But then using (38) we

get that

119866120590119887

(119906 119891119906 119891119906) = 119904119866120590119887

(119891119906 119891119906 119891119906)

le 120573 (119866120590119887

(119906 119906 119906))119872119904(119906 119906 119906)

lt

1

119904

119866120590119887

(119906 119891119906 119891119906)

(40)

which is a contradiction Hence we have 119891119906 = 119906 Thus 119906 is afixed point of 119891

We replace condition (ii) in Theorem 26 by regularity ofthe space119883

Theorem 27 Under the same hypotheses of Theorem 26instead of condition (119894119894) assume that whenever 119909

119899 in 119883 is a

sequence such that 120572(119909119899 119909119899+1

119909119899+1

) ge 1 for all 119899 isin Ncup0 and119909119899rarr 119909 as 119899 rarr infin one has 120572(119909

119899 119909 119909) ge 1 for all 119899 isin Ncup0

Then 119891 has a fixed point

Proof Repeating the proof of Theorem 26 we can constructa sequence 119909

119899 in 119883 such that 120572(119909

119899 119909119899+1

119909119899+1

) ge 1 for all119899 isin N cup 0 and 119909

119899rarr 119906 isin 119883 for some 119906 isin 119883 Using the

assumption on 119883 we have 120572(119909119899 119906 119906) ge 1 for all 119899 isin N cup 0

Now we show that 119906 = 119891119906 By Lemma 21 and (35)

119904 [

1

119904

119866120590119887

(119906 119891119906 119891119906) minus 119866120590119887

(119906 119906 119906)]

le 119904lim sup119899rarrinfin

119866120590119887

(119909119899+1

119891119906 119891119906)

le lim sup119899rarrinfin

(120573 (119866120590119887

(119909119899 119906 119906))119872

119904(119909119899 119906 119906))

le

1

119904

lim sup119899rarrinfin

119872119904(119909119899 119906 119906)

(41)

wherelim119899rarrinfin

119872119904(119909119899 119906 119906)

= lim119899rarrinfin

max

119866120590119887

(119909119899 119906 119906)

[119866120590119887

(119909119899 119891119909119899 119891119909119899)] [119866120590119887

(119906 119891119906 119891119906)]

2

1 + 11990421198662

120590119887

(119891119909119899 119891119906 119891119906)

= lim119899rarrinfin

max

119866120590119887

(119909119899 119906 119906)

[119866120590119887

(119909119899 119909119899+1

119909119899+1

)] [119866120590119887

(119906 119891119906 119891119906)]

2

1 + 11990421198662

120590119887

(119909119899+1

119891119906 119891119906)

= max 119866120590119887

(119906 119906 119906) 0 = 0 (see (25) and (35))

(42)

Therefore we deduce that119866120590119887(119906 119891119906 119891119906) le 0 Hence we have

119906 = 119891119906

The Scientific World Journal 7

A mapping 120593 [0infin) rarr [0infin) is called a comparisonfunction if it is increasing and 120593

119899(119905) rarr 0 as 119899 rarr infin for any

119905 isin [0infin) (see eg [23 24] for more details and examples)

Definition 28 A function 120593 [0infin) rarr [0infin) is said to bea (119888)-comparison function if

(1198881) 120593 is increasing

(1198882) there exists 119896

0isin N 119886 isin (0 1) and a convergent series

of nonnegative terms suminfin

119896=1V119896such that 120593

119896+1(119905) le

119886120593119896(119905) + V

119896for 119896 ge 119896

0and any 119905 isin [0infin)

Later Berinde [5] introduced the notion of (119887)-comparison function as a generalization of (119888)-comparisonfunction

Definition 29 (see [5]) Let 119904 ge 1 be a real number Amapping120593 [0infin) rarr [0infin) is called a (119887)-comparison function ifthe following conditions are fulfilled

(1) 120593 is increasing

(2) there exist 1198960isin N 119886 isin (0 1) and a convergent series

of nonnegative terms suminfin119896=1

V119896such that 119904119896+1120593119896+1(119905) le

119886119904119896120593119896(119905) + V

119896for 119896 ge 119896

0and any 119905 isin [0infin)

Let Ψ119887be the class of all (119887)-comparison functions 120593

[0infin) rarr [0infin) It is clear that the notion of (119887)-comparison function coincideswith (119888)-comparison functionfor 119904 = 1

Lemma 30 (see [25]) If 120593 [0infin) rarr [0infin) is a (119887)-comparison function then we have the following

(1) the series suminfin119896=0

119904119896120593119896(119905) converges for any 119905 isin R

+

(2) the function 119887119904 [0infin) rarr [0infin) defined by 119887

119904(119905) =

suminfin

119896=0119904119896120593119896(119905) 119905 isin [0infin) is increasing and continuous

at 0

Remark 31 It is easy to see that if 120593 isin Ψ119887 then we have 120593(0) =

0 and 120593(119905) lt 119905 for each 119905 gt 0 and 120593 is continuous at 0

In the next example we present a class of (119887)-comparisonfunctions

Example 32 Any function of the form120595(119905) = ln((119886119904)119905+1) forall 119905 isin [0infin) where 0 lt 119886 lt 1 is a (119887)-comparison function

Proof From the part (1) of Lemma 30 the necessary condi-tion is that the series suminfin

119896=0119904119896120593119896(119905) converges for any 119905 isin R

+

But for each 119905 gt 0 and 119896 ge 1 we have

119904119896

120593119896

(119905) = 119904119896

120593 (120593119896minus1

(119905))

= 119904119896 ln(

119886

119904

120593119896minus1

(119905) + 1) le 119904119896 119886

119904

120593119896minus1

(119905)

le sdot sdot sdot le 119904119896

(

119886

119904

)

119896

119905 = 119886119896

119905

(43)

So according to the comparison test of the series we shouldhave 119886 lt 1 On the other hand we have

119904119896+1

120593119896+1

(119905) = 119904119896+1

120593 (120593119896

(119905))

= 119904119896+1 ln(

119886

119904

120593119896

(119905) + 1)

le 119904119896+1 119886

119904

120593119896

(119905) = 119904119896

119886120593119896

(119905)

(44)

Therefore for any convergent series of nonnegative termssuminfin

119896=1V119896and each 119896 ge 119896

0= 1 we have

119904119896+1

120593119896+1

(119905) le 119886119904119896

120593119896

(119905) le 119886119904119896

120593119896

(119905) + V119896 (45)

For example for 119904 = 2 and 119886 = 12 the function 120595(119905) =

ln(1199054 + 1) is a (119887)-comparison function

Theorem 33 Let (119883 119866120590119887) be a 119866

120590119887-complete generalized 119887-

metric-like space and let 119891 119883 rarr 119883 be a 119866-120572-admissiblemapping Suppose that

119904120572 (119909 119910 119911) 119866120590119887

(119891119909 119891119910 119891119911) le 120595 (119872119904(119909 119910 119911)) (46)

for all 119909 119910 119911 isin 119883 where 120595 isin Ψ119887and

119872119904(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119909 119891119909)119866120590119887

(119910 119910 119891119910)119866120590119887

(119911 119911 119891119911)

1 + 211990421198662

120590119887

(119891119909 119891119910 119891119911)

(47)

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 120572(119909

0 1198911199090 11989121199090) ge 1

(ii)

(a) 119891 is continuous and for any sequence 119909119899 in119883

with 120572(119909119899 119909119899+1

119909119899+2

) ge 1 for all 119899 isin Ncup0 suchthat 119909

119899rarr 119909 as 119899 rarr infin one has 120572(119909 119909 119909) ge 1

for all 119899 isin N cup 0(b) assume that whenever 119909

119899 in 119883 is a sequence

such that 120572(119909119899 119909119899+1

119909119899+2

) ge 1 for all 119899 isin Ncup0

and 119909119899rarr 119909 as 119899 rarr infin one has 120572(119909

119899 119909 119909) ge 1

for all 119899 isin N cup 0

Then 119891 has a fixed point

Proof Let 1199090isin 119883 be such that 120572(119909

0 1198911199090 11989121199090) ge 1 Define

a sequence 119909119899 by 119909

119899= 1198911198991199090for all 119899 isin N Since 119891 is a 119866-120572-

admissible mapping and 120572(1199090 1199091 1199092) = 120572(119909

0 1198911199090 11989121199090) ge

1 we deduce that 120572(1199091 1199092 1199093) = 120572(119891119909

0 1198911199091 1198911199092) ge 1

Continuing this process we get 120572(119909119899 119909119899+1

119909119899+2

) ge 1 for all119899 isin N cup 0

8 The Scientific World Journal

If there exists 1198990isin N such that 119909

1198990= 1199091198990+1

then 1199091198990

=

1198911199091198990and so we have nothing to prove Hence for all 119899 isin N

we assume that 119909119899

= 119909119899+1

Step I (Cauchyness of 119909119899) As 120572(119909

119899 119909119899+1

119909119899+2

) ge 1 for all119899 ge 0 using condition (46) we obtain

119904119866120590119887

(119909119899 119909119899+1

119909119899+2

)

le 119904120572 (119909119899minus1

119909119899 119909119899+1

) 119866120590119887

(119909119899 119909119899+1

119909119899+2

)

= 119904120572 (119909119899minus1

119909119899 119909119899+1

) 119866120590119887

(119891119909119899minus1

119891119909119899 119891119909119899+1

)

le 120595 (119872119904(119909119899minus1

119909119899 119909119899+1

))

(48)

Using Proposition 15(2) as 119909119899

= 119909119899+1

we get

119872119904(119909119899minus1

119909119899 119909119899+1

)

= max 119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

(119866120590119887

(119909119899minus1

119909119899minus1

119891119909119899minus1

)

times 119866120590119887

(119909119899 119909119899 119891119909119899)

times119866120590119887

(119909119899+1

119909119899+1

119891119909119899+1

))

times(1 + 21199042

1198662

120590119887

(119891119909119899minus1

119891119909119899 119891119909119899+1

))

minus1

= max 119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

(119866120590119887

(119909119899minus1

119909119899minus1

119909119899)

times 119866120590119887

(119909119899 119909119899 119909119899+1

)

times119866120590119887

(119909119899+1

119909119899+1

119909119899+2

))

times(1 + 21199042

1198662

120590119887

(119909119899 119909119899+1

119909119899+2

))

minus1

le max119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

2119904119866120590119887

(119909119899minus1

119909119899 119909119899+1

) 1198662

120590119887

(119909119899 119909119899+1

119909119899+2

)

1 + 2s21198662120590119887

(119909119899 119909119899+1

119909119899+2

)

= 119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

(49)

Hence119866120590119887

(119909119899 119909119899+1

119909119899+2

)

le 119904119866120590119887

(119909119899 119909119899+1

119909119899+2

) le 120595 (119866120590119887

(119909119899minus1

119909119899 119909119899+1

))

(50)

By induction since 119909119899

= 119909119899+1

we get that

119866120590119887

(119909119899 119909119899+1

119909119899+2

)

le 120595 (119866120590119887

(119909119899minus1

119909119899 119909119899+1

))

le 1205952

(119866120590119887

(119909119899minus2

119909119899minus1

119909119899))

le sdot sdot sdot le 120595119899

(119866120590119887

(1199090 1199091 1199092))

(51)

Let 120576 gt 0 be arbitrary Then there exists a natural number119873such that

infin

sum

119899=119873

119904119899

120595119899

(119866120590119887

(1199090 1199091 1199092)) lt

120576

2119904

(52)

Let 119898 gt 119899 ge 119873 Then by the rectangular inequality andProposition 15(2) as 119909

119899= 119909119899+1

we get

119866120590119887

(119909119899 119909119898 119909119898)

le 119904119866120590119887

(119909119899 119909119899+1

119909119899+1

) + 1199042

119866120590119887

(119909119899+1

119909119899+2

119909119899+2

)

+ sdot sdot sdot + 119904119898minus119899minus1

119866120590119887

(119909119898minus1

119909119898 119909119898)

le 2119904 (119904119866120590119887

(119909119899 119909119899 119909119899+1

) + 1199042

119866120590119887

(119909119899+1

119909119899+1

119909119899+2

)

+ sdot sdot sdot + 119904119898minus119899minus1

119866120590119887

(119909119898minus1

119909119898minus1

119909119898))

le 2119904 (119904119866120590119887

(119909119899 119909119899+1

119909119899+2

) + 1199042

119866120590119887

(119909119899+1

119909119899+2

119909119899+3

)

+ sdot sdot sdot + 119904119898minus119899minus1

119866120590119887

(119909119898minus1

119909119898 119909119898+1

))

le 2119904

119898minus2

sum

119896=119899

119904119896minus119899+1

120595119896

(119866120590119887

(1199090 1199091 1199092))

le 2119904

infin

sum

119896=119899

119904119896

120595119896

(119866120590119887

(1199090 1199091 1199092)) lt 120576

(53)

Consequently 119909119899 is a 119866

120590119887-Cauchy sequence in 119883 Since

(119883 119866120590119887) is 119866120590119887-complete so there exists 119906 isin 119883 such that

lim119899rarrinfin

119866120590119887

(119906 119909119899 119909119899) = lim119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

= 119866120590119887

(119906 119906 119906) = 0

(54)

Step II Now we show that 119906 is a fixed point of 119891 Suppose tothe contrary that is 119891119906 = 119906 then we have 119866

120590119887(119906 119906 119891119906) gt 0

Let the part (a) of (ii) holdsUsing the rectangle inequality we get

119866120590119887

(119906 119906 119891119906) le 119904119866120590119887

(119891119906 119891119909119899 119891119909119899) + 119904119866

120590119887(119891119909119899 119906 119906)

(55)

Letting 119899 rarr infin and using the continuity of 119891 we get

119866120590119887

(119906 119906 119891119906) le 119904119866120590119887

(119891119906 119891119906 119891119906) (56)

From (46) and part (a) of condition (ii) we have

119904119866120590119887

(119891119906 119891119906 119891119906) le 120595 (119872119904(119906 119906 119906)) (57)

where by using (56) we have

119872119904(119906 119906 119906)

= max119866120590119887

(119906 119906 119906)

119866120590119887

(119906 119906 119891119906)119866120590119887

(119906 119906 119891119906)119866120590119887

(119906 119906 119891119906)

1 + 211990421198662

120590119887

(119891119906 119891119906 119891119906)

le 119866120590119887

(119906 119906 119891119906)

(58)

Hence from properties of 120595 119904119866120590119887(119891119906 119891119906 119891119906) le

119866120590119887(119906 119906 119891119906) Thus by (56) we obtain that

119866120590119887

(119906 119906 119891119906) = 119904119866120590119887

(119891119906 119891119906 119891119906) (59)

The Scientific World Journal 9

Moreover (57) yields that 119866120590119887(119906 119906 119891119906) le 120595(119872

119904(119906 119906 119906)) le

120595(119866120590119887(119906 119906 119891119906))This is impossible according to our assump-

tions on 120595 Hence we have 119891119906 = 119906 Thus 119906 is a fixed point of119891

Now let part (b) of (ii) holdsAs 119909119899 is a sequence such that 120572(119909

119899 119909119899+1

119909119899+2

) ge 1 for all119899 isin N cup 0 and 119909

119899rarr 119909 as 119899 rarr infin we have 120572(119909

119899 119909 119909) ge 1

for all 119899 isin N cup 0Now we show that 119906 = 119891119906 By (46) we have

119904119866120590119887

(119891119906 119891119906 119909119899) le 119904120572 (119906 119906 119909

119899minus1) 119866120590119887

(119891119906 119891119906 119891119909119899minus1

)

le 120595 (119872119904(119906 119906 119909

119899minus1))

(60)

where

119872119904(119906 119906 119909

119899minus1)

= max

119866120590119887

(119906 119906 119909119899minus1

)

[119866120590119887

(119906 119906 119891119906)]

2

119866120590119887

(119909119899minus1

119909119899minus1

119891119909119899minus1

)

1 + [119904119866120590119887

(119891119906 119891119906 119891119909119899minus1

)]

2

(61)

Letting 119899 rarr infin in the above inequality and using (54) andLemma 21 we get

lim119899rarrinfin

119872119904(119906 119906 119909

119899minus1) = 0 (62)

Again taking the upper limit as 119899 rarr infin in (60) and using(62) and Lemma 21 we obtain

119904 [

1

119904

119866120590119887

(119906 119891119906 119891119906) minus 119866120590119887

(119906 119906 119906)]

le 119904 lim sup119899rarrinfin

119866120590119887

(119909119899 119891119906 119891119906)

le lim supnrarrinfin

120595 (119872119904(119906 119906 119909

119899minus1))

= 120595(lim sup119899rarrinfin

119872119904(119906 119906 119909

119899minus1))

= 120595 (0) = 0

(63)

So we get 119866120590119887(119906 119891119906 119891119906) = 0 That is 119891119906 = 119906

Let (119883 119866120590119887 ⪯) be a partially ordered 119866

120590119887-metric-like

space We say that 119879 is an increasing mapping on 119883 ifforall119909 119910 isin 119883 119909 ⪯ 119910 rArr 119879(119909) ⪯ 119879(119910) [26] Fixed pointtheorems for monotone operators in ordered metric spacesarewidely investigated andhave found various applications indifferential and integral equations (see [27ndash30] and referencestherein) From the results proved above we derive thefollowing new results in partially ordered 119866

120590119887-metric-like

space

Theorem 34 Let (119883 119866120590119887 ⪯) be a partially ordered 119866

120590119887-

complete generalized 119887-metric-like space and let 119891 119883 rarr 119883

be an increasing mapping Suppose that

119904119866120590119887

(119891119909 119891119910 119891119911) le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911) (64)

for all 119909 119910 119911 isin 119883 with 119909 ⪯ 119910 ⪯ 119911 where

119872119904(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119891119909 119891119909)119866120590119887

(119910 119891119910 119891119910)119866120590119887

(119911 119891119911 119891119911)

1 + 11990421198662

120590119887

(119891119909 119891119910 119891119911)

(65)

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) 119891 is continuous or assume that whenever 119909119899 in 119883 is

an increasing sequence such that 119909119899rarr 119909 as 119899 rarr infin

one has 119909119899⪯ 119909 for all 119899 isin N cup 0

Then 119891 has a fixed point

Proof Define 120572 119883 times 119883 times 119883 rarr [0 +infin) by

120572 (119909 119910 119911) =

1 if119909 ⪯ 119910 ⪯ 119911

0 otherwise(66)

First we prove that 119891 is a triangular 120572-admissible mappingHence we assume that 120572(119909 119910 119911) ge 1 Therefore we have 119909 ⪯

119910 ⪯ 119911 Since 119891 is increasing we get 119891119909 ⪯ 119891119910 ⪯ 119891119911 that is120572(119891119909 119891119910 119891119911) ge 1 Also let 120572(119909 119910 119910) ge 1 and 120572(119910 119911 119911) ge 1then 119909 ⪯ 119910 and 119910 ⪯ 119911 Consequently we deduce that 119909 ⪯ 119911that is 120572(119909 119911 119911) ge 1 Thus 119891 is a triangular 120572-admissiblemapping Since 119891 satisfies (64) so by the definition of 120572 wehave

119904120572 (119909 119910 119911) 119866120590119887

(119891119909 119891119910 119891119911)

le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911)

(67)

for all 119909 119910 119911 isin 119883 Therefore 119891 satisfies the contractivecondition (21) From (i) there exists 119909

0isin 119883 such that

1199090⪯ 1198911199090 that is 120572(119909

0 1198911199090 1198911199090) ge 1 According to (ii) we

conclude that all the conditions of Theorems 26 and 27 aresatisfied and so 119891 has a fixed point

Similarly using Theorem 33 we can prove followingresult

Theorem 35 Let (119883 119866120590119887 ⪯) be an ordered 119866

120590119887-complete

generalized 119887-metric-like space and let 119891 119883 rarr 119883 be anincreasing mapping Suppose that

119904119866120590119887

(119891119909 119891119910 119891119911) le 120595 (119872119904(119909 119910 119911)) (68)

for all 119909 119910 119911 isin 119883 with 119909 ⪯ 119910 ⪯ 119911 where 120595 isin Ψ119887and

119872119904(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119909 119891119909)119866120590119887

(119910 119910 119891119910)119866120590119887

(119911 119911 119891119911)

1 + 211990421198662

120590119887

(119891119909 119891119910 119891119911)

(69)

10 The Scientific World Journal

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii)

(a) 119891 is continuous(b) assume that whenever x

119899 in 119883 is an increasing

sequence such that 119909119899rarr 119909 as 119899 rarr infin one has

119909119899⪯ 119909 for all 119899 isin N cup 0

Then 119891 has a fixed point

We conclude this section by presenting some examplesthat illustrate our results

Example 36 Let 119883 = [0 (minus1 + radic5)2] be endowed withthe usual ordering onR Define the generalized 119887-metric-likefunction 119866

120590119887given by

119866120590119887

(119909 119910 119911) = max (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

(70)

with 119904 = 2 Consider the mapping 119891 119883 rarr 119883 defined by119891(119909) = (14)119909

radic119890minus(119909+((minus1+radic5)2))

2 and the function 120573 isin Fgiven by 120573(119905) = (12)119890

minus119905 119905 gt 0 and 120573(0) isin [0 12) Itis easy to see that 119891 is an increasing function on 119883 Weshow that 119891 is 119866

120590119887-continuous on 119883 By Proposition 20 it is

sufficient to show that119891 is119866120590119887-sequentially continuous on119883

Let 119909119899 be a sequence in119883 such that lim

119899rarrinfin119866120590119887(119909119899 119909 119909) =

119866120590119887(119909 119909 119909) so we have max(lim

119899rarrinfin119909119899

+ 119909)2

41199092 =

41199092 and equivalently lim

119899rarrinfin119909119899

= 120572 le 119909 On the otherhand we have

lim119899rarrinfin

119866120590119887

(119891119909119899 119891119909 119891119909)

= lim119899rarrinfin

max (119891119909119899+ 119891119909)

2

41198912

119909

= lim119899rarrinfin

max 1

16

(119909119899

radic119890minus(119909119899+(minus1+

radic5)2)2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= max 1

16

( lim119899rarrinfin

119909119899

radic119890minus(lim119899rarrinfin119909119899+(minus1+radic5)2)

2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= max 1

16

(120572radic119890minus(120572+(minus1+radic5)2)

2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

=

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= 119866120590119887

(119891119909 119891119909 119891119909)

(71)

So 119891 is 119866120590119887-sequentially continuous on119883

For all elements 119909 119910 119911 isin 119883 and the fact that 119892(119909) =

1199092119890minus1199092

is an increasing function on119883 we have

119904119866120590119887

(119891119909 119891119910 119891119911)

= 2max (119891119909 + 119891119910)2

(119891119910 + 119891119911)2

(119891119911 + 119891119909)2

= 2max(

1

4

119909radic119890minus(119909+(minus1+radic5)2)

2

+

1

4

119910radic119890minus(119910+(minus1+radic5)2)

2

)

2

(

1

4

119910radic119890minus(119910+(minus1+radic5)2)

2

+

1

4

119911radic119890minus(119911+(minus1+radic5)2)

2

)

2

(

1

4

119911radic119890minus(119911+(minus1+radic5)2)

2

+

1

4

119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

le 2max(

1

4

119909radic119890minus(119909+119910)

2

+

1

4

119910radic119890minus(119910+119909)

2

)

2

(

1

4

119910radic119890minus(119910+119911)

2

+

1

4

119911radic119890minus(119911+119910)

2

)

2

(

1

4

119911radic119890minus(119911+119909)

2

+

1

4

119909radic119890minus(119909+119911)

2

)

2

=

1

8

max (119909 + 119910)2

119890minus(119909+119910)

2

(119910 + 119911)2

119890minus(119910+119911)

2

(119911 + 119909)2

119890minus(119911+119909)

2

=

1

8

119890minusmax(119909+119910)2 (119910+119911)2 (119911+119909)2

timesmax (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

le

1

2

eminusmax(119909+119910)2 (119910+119911)2(119911+119909)2

timesmax (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

= 120573 (119866120590119887

(119909 119910 119911)) 119866120590119887

(119909 119910 119911)

le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911)

(72)

Hence119891 satisfies all the assumptions ofTheorem 34 and thusit has a fixed point (which is 119906 = 0)

Example 37 Let 119883 = [0 1] with the usual ordering on RDefine the generalized 119887-metric-like function 119866

120590119887given by

119866120590119887(119909 119910 119911) = max(119909 + 119910)

2

(119910 + 119911)2

(119911 + 119909)2

with 119904 = 2Consider the mapping 119891 119883 rarr 119883 defined by 119891(119909) =

The Scientific World Journal 11

ln(1 + 119909119890minus119909

4) and the function 120595 isin Ψ119887given by 120595(119905) =

(18)119905 119905 ge 0 It is easy to see that 119891 is increasing functionNow we show that 119891 is a 119866

120590119887-continuous function on119883

Let 119909119899 be a sequence in 119883 such that lim

119899rarrinfin119866120590119887(119909119899

119909 119909) = 119866120590119887(119909 119909 119909) so we havemax(lim

119899rarrinfin119909119899+119909)2

41199092 =

41199092 and equally lim

119899rarrinfin119909119899= 120572 le 119909 On the other hand we

have

lim119899rarrinfin

119866120590119887

(119891119909119899 119891119909 119891119909)

= lim119899rarrinfin

max (119891119909119899+ 119891119909)

2

41198912

119909

= max(ln(1 +

lim119899rarrinfin

119909119899119890minuslim119899rarrinfin119909119899

4

)

+ ln(1 +

119909119890minus119909

4

))

2

4(ln(1 +

119909119890minusx

4

))

2

= max(ln(1 +

120572119890minus120572

4

) + ln(1 +

119909119890minus119909

4

))

2

4(ln(1 +

119909119890minus119909

4

))

2

= 4(ln(1 +

119909119890minus119909

4

))

2

= 119866120590119887

(119891119909 119891119909 119891119909)

(73)

So 119891 is 119866120590119887-sequentially continuous on119883

For all elements 119909 119910 119911 isin 119883 we have

119904119866120590119887

(119891119909 119891119910 119891119911)

= 2max(ln(1 +

119909119890minus119909

4

) + ln(1 +

119910119890minus119910

4

))

2

(ln(1 +

119910119890minus119910

4

) + ln(1 +

119911119890minus119911

4

))

2

(ln(1 +

119911119890minus119911

4

) + ln(1 +

119909119890minus119909

4

))

2

le 2max(

119909119890minus119909

4

+

119910119890minus119910

4

)

2

(

119910119890minus119910

4

+

119911119890minus119911

4

)

2

(

119911119890minus119911

4

+

119909119890minus119909

4

)

2

le

1

8

max (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

= 120595 (119866120590119887

(119909 119910 119911))

le 120595 (119872119904(119909 119910 119911))

(74)

Hence119891 satisfies all the assumptions ofTheorem 35 and thusit has a fixed point (which is 119906 = 0)

3 Application

In this section we present an application of our results toestablish the existence of a solution to a periodic boundaryvalue problem (see [30 31])

Let 119883 = 119862([0 119879]) be the set of all real continuousfunctions on [0T] We first endow119883 with the 119887-metric-like

120590119887(119906 V) = sup

119905isin[0119879]

(|119906 (119905)| + |V (119905)|)119901 (75)

for all 119906 V isin 119883 where 119901 gt 1 and then we endow it with thegeneralized 119887-metric-like 119866

120590119887defined by

119866120590119887

(119906 V 119908)

= max sup119905isin[0119879]

(|119906 (119905)| + |V (119905)|)119901

sup119905isin[0119879]

(|V (119905)| + |119908 (119905)|)119901

sup119905isin[0119879]

(|119908 (119905)| + |119906 (119905)|)119901

(76)

Clearly (119883 119866120590119887) is a complete generalized 119887-metric-like space

with parameter 119870 = 2119901minus1 We equip 119862([0 119879]) with a partial

order given by119909 ⪯ 119910 iff119909 (119905) le 119910 (119905) forall119905 isin [0 119879] (77)

Moreover as in [30] it is proved that (119862([0 119879]) ⪯) is regularthat is whenever 119909

119899 in119883 is an increasing sequence such that

119909119899rarr 119909 as 119899 rarr infin we have 119909

119899⪯ 119909 for all 119899 isin N cup 0

Consider the first-order periodic boundary value prob-lem

1199091015840

(119905) = 119891 (119905 119909 (119905)) 119909 (0) = 119909 (119879) (78)

where 119905 isin 119868 = [0 119879] with 119879 gt 0 and 119891 119868 times R rarr R is acontinuous function

A lower solution for (78) is a function 120572 isin 1198621[0 119879] such

that1205721015840

(119905) le 119891 (119905 120572 (119905))

120572 (0) le 120572 (119879)

(79)

where 119905 isin 119868 = [0 119879]Assume that there exists 120582 gt 0 such that for all 119909 119910 isin

119862[0 119879] we have1003816100381610038161003816119891 (119905 119909 (119905)) + 120582119909 (119905)

1003816100381610038161003816+1003816100381610038161003816119891 (119905 119910 (119905)) + 120582119910 (119905)

1003816100381610038161003816

le

120582

2119901minus1

119901radicln(

119886

2119901minus1

(|119909 (119905)| +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)119901

+ 1)

(80)

where 0 le 119886 lt 1 Then the existence of a lower solution for(78) provides the existence of a solution of (78)

Problem (78) can be rewritten as

1199091015840

(119905) + 120582119909 (119905) = 119891 (119905 119909 (119905)) + 120582119909 (119905)

119909 (0) = 119909 (119879)

(81)

Consider1199091015840

(119905) + 120582119909 (119905) = 120575119909(119905) = 119865 (119905 119909 (119905))

119909 (0) = 119909 (119879)

(82)

where 119905 isin 119868

12 The Scientific World Journal

Using variation of parameters formula we get

119909 (119905) = 119909 (0) 119890minus120582119905

+ int

119905

0

119890minus120582(119905minus119904)

120575119909(119904) 119889119904 (83)

which yields

119909 (119879) = 119909 (0) 119890minus120582119879

+ int

119879

0

119890minus120582(119879minus119904)

120575119909(119904) 119889119904 (84)

Since 119909(0) = 119909(119879) we get

119909 (0) [1 minus 119890minus120582119879

] = 119890minus120582119879

int

119879

0

119890120582(119904)

120575119909(119904) 119889119904 (85)

or

119909 (0) =

1

119890120582119879

minus 1

int

119879

0

119890120582119904

120575119909(119904) 119889119904 (86)

Substituting the value of 119909(0) in (83) we arrive at

119909 (119905) = int

119879

0

119866 (119905 119904) 120575119909(119904) 119889119904 (87)

where

119866 (119905 119904) =

119890120582(119879+119904minus119905)

119890120582119879

minus 1

0 le 119904 le 119905 le 119879

119890120582(119904minus119905)

119890120582119879

minus 1

0 le 119905 le 119904 le 119879

(88)

Now define the operator 119878 119862[0 119879] rarr 119862[0 119879] as

119878119909 (119905) = int

119879

0

119866 (119905 119904) 119865 (119904 119909 (119904)) 119889119904 (89)

The mapping 119878 is increasing [31] Note that if 119906 isin 119862[0 119879] is afixed point of 119878 then 119906 isin 119862

1[0 119879] is a solution of (78)

Let 119909 119910 119911 isin 119862[0 119879] Then we have

2119901minus1

[|119878119909 (119905)| +1003816100381610038161003816119878119910 (119905)

1003816100381610038161003816]

= 2119901minus1

[

100381610038161003816100381610038161003816100381610038161003816

int

119879

0

119866 (119905 119904) 119865 (119904 119909 (119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

119879

0

119866 (119905 119904) 119865 (119904 119910 (119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

]

le 2119901minus1

int

119879

0

|119866 (119905 119904)|

times [|119865 (119904 119909 (119904))| +1003816100381610038161003816119865 (119904 119910 (119904))

1003816100381610038161003816] 119889119904

le 2119901minus1

int

119879

0

|119866 (119905 119904)|

120582

2119901minus1

times119901radicln(

119886

2119901minus1

(|119909 (119905)| +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)119901

+ 1)119889119904

le 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [int

119905

0

119890120582(119879+119904minus119905)

119890120582119879

minus 1

119889119904 + int

119879

119905

119890120582(119904minus119905)

119890120582119879

minus 1

119889119904]

= 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [

1

120582 (119890120582119879

minus 1)

(119890120582(119879+119904minus119905)10038161003816

100381610038161003816

119905

0

+ 119890120582(119904minus119905)10038161003816

100381610038161003816

119879

119905

)]

= 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [

1

120582 (119890120582119879

minus 1)

(119890120582119879

minus 119890120582(119879minus119905)

+ 119890120582(119879minus119905)

minus 1)]

=119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

le119901radicln(

119886

2119901minus1

119866120590119887

(119909 119910 119911) + 1)

le119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(90)

Similarly

2119901minus1 1003816

100381610038161003816119878119910 (119905)

1003816100381610038161003816+ |119878119911 (119905)| le

119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

|119878119909 (119905)| + |119878119911 (119905)| le119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(91)

where

119872(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119878119909 119878119909) 119866120590119887

(119910 119878119910 119878119910)119866120590119887

(119911 119878119911 119878119911)

1 + 22119901minus1

1198662

120590119887

(119878119909 119878119910 119878119911)

(92)

Equivalently

(2119901minus1

|119878119909 (119905)| +1003816100381610038161003816119878119910 (119905)

1003816100381610038161003816)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(2119901minus1 1003816

100381610038161003816119878119910 (119905)

1003816100381610038161003816+ |119878119911 (119905)|)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(2119901minus1

|119878119909 (119905)| + |119878119911 (119905)|)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(93)

which yields that

2119901minus1

120590119887(119878119909 119878119910) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

120590119887(119878119910 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

120590119887(119878119909 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(94)

The Scientific World Journal 13

Finally it is easy to obtain that

2119901minus1G120590119887

(119878119909 119878119910 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1) (95)

Finally since120572 is a lower solution for (78) so it is easy to showthat 120572 ⪯ 119878(120572) [31]

Hence the hypotheses of Theorem 35 are satisfied with120595(119905) = ln((1198862119901minus1)119905 + 1) where 0 le 119886 lt 1 Hence there existsa fixed point 119909 isin 119862[0 119879] such that 119878119909 = 119909 which is a solutionto periodic boundary value problem (78)

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) KingAbdulaziz University JeddahTherefore the firstauthor acknowledges with thanks DSR KAU for financialsupport

References

[1] S Czerwik ldquoContraction mappings in b-metric spacesrdquo ActaMathematica et Informatica Universitatis Ostraviensis vol 1 pp5ndash11 1993

[2] V Parvaneh J R Roshan and S Radenovic ldquoExistence of tri-pled coincidence points in ordered b-metric spaces and an app-lication to a system of integral equationsrdquo Fixed Point Theoryand Applications vol 2013 article 130 2013

[3] J R Roshan V Parvaneh S Sedghi N Shobkolaei and WShatanawi ldquoCommon fixed points of almost generalized con-tractive mappings in ordered b-metric spacesrdquo Fixed PointTheory and Applications vol 2013 article 159 2013

[4] H Aydi M F Bota E Karapınar and S Moradi ldquoA commonfixed point for weak 120601-contractions on b-metric spacesrdquo FixedPoint Theory vol 13 no 2 pp 337ndash346 2012

[5] V Berinde ldquoSequences of operators and fixed points in quasi-metric spacesrdquo Studia Universitatis ldquoBabes-Bolyairdquo Mathemat-ica vol 16 no 4 pp 23ndash27 1996

[6] N Hussain D ETHoric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 article 126 2012

[7] N Hussain and M H Shah ldquoKKMmappings in cone b-metricspacesrdquo Computers and Mathematics with Applications vol 62no 4 pp 1677ndash1684 2011

[8] N Hussain J R Roshan V Parvaneh and Z Kadelburg ldquoFixedpoints of contractive mappings in b-metric-like spacesrdquo TheScientific World Journal vol 2014 Article ID 471827 15 pages2014

[9] M A Khamsi and N Hussain ldquoKKMmappings in metric typespacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 73 no 9 pp 3123ndash3129 2010

[10] M A Khamsi ldquoRemarks on cone metric spaces and fixed pointtheorems of contractive mappingsrdquo Fixed Point Theory andApplications vol 2010 Article ID 315398 7 pages 2010

[11] Z Mustafa J R Roshan V Parvaneh and Z Kadelburg ldquoSomecommon fixed point results in ordered partial b-metric spacesrdquoJournal of Inequalities and Applications vol 2013 article 5622013

[12] Z Mustafa and B Sims ldquoA new approach to generalized metricspacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash297 2006

[13] A Amini-Harandi ldquoMetric-like spaces partial metric spacesand fixed pointsrdquo Fixed PointTheory and Applications vol 2012article 204 2012

[14] N Shobkolaei S Sedghi J R Roshan and N Hussain ldquoSuzukitype fixed point results in metric-like spacesrdquo Journal of Func-tion Spaces and Applications vol 2013 Article ID 143686 9pages 2013

[15] M A Alghamdi N Hussain and P Salimi ldquoFixed point andcoupled fixed point theorems on b-metric-like spacesrdquo Journalof Inequalities and Applications vol 2013 article 402 2013

[16] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572 minus 120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

[17] P Salimi and E Karapınar ldquoSuzuki-Edelstein type contractionsvia auxiliary functionsrdquoMathematical Problems in Engineeringvol 2013 Article ID 648528 8 pages 2013

[18] P Salimi C Vetro and P Vetro ldquoFixed point theorems fortwisted (120572 120573)-120595-contractive type mappings and applicationsrdquoFilomat vol 27 no 4 pp 605ndash615 2013

[19] P Salimi C Vetro and P Vetro ldquoSome new fixed point resultsin non-Archimedean fuzzy metric spacesrdquo Nonlinear AnalysisModelling and Control vol 18 no 3 pp 344ndash358 2013

[20] N Hussain P Salimi and A Latif ldquoFixed point results forsingle and set-valued 120572-120578-120595-contractive mappingsrdquo Fixed PointTheory and Applications vol 2013 article 212 2013

[21] M A Alghamdi and E Karapınar ldquoG-120573-120595 contractive-typemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 article 70 2013

[22] M Geraghty ldquoOn contractive mappingsrdquo Proceedings of theAmerican Mathematical Society vol 40 pp 604ndash608 1973

[23] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001

[24] N Hussain Z Kadelburg S Radenovic and F R Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012

[25] V Berinde ldquoGeneralized contractions in quasimetric spacesrdquo inSeminar on Fixed Point Theory Preprint no 3 pp 3ndash9 1993

[26] L Ciric M Abbas R Saadati and N Hussain ldquoCommon fixedpoints of almost generalized contractive mappings in orderedmetric spacesrdquo Applied Mathematics and Computation vol 217no 12 pp 5784ndash5789 2011

[27] R P Agarwal N Hussain and M A Taoudi ldquoFixed point the-orems in ordered Banach spaces and applications to nonlinearintegral equationsrdquo Abstract and Applied Analysis vol 2012Article ID 245872 15 pages 2012

[28] N Hussain A R Khan and R P Agarwal ldquoKrasnoselrsquoskii andKy Fan type fixed point theorems in ordered Banach spacesrdquoJournal of Nonlinear and Convex Analysis vol 11 no 3 pp 475ndash489 2010

[29] N Hussain and M A Taoudi ldquoKrasnoselrsquoskii-type fixed pointtheorems with applications to Volterra integral equationsrdquoFixed Point Theory and Applications vol 2013 article 196 2013

14 The Scientific World Journal

[30] J J Nieto and R Rodrıguez-Lopez ldquoContractivemapping theo-rems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[31] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysisTheoryMethods andApplications vol 71 no 7-8 pp 3403ndash3410 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 The Scientific World Journal

We need the following simple lemma about the 119866120590119887-

convergent sequences in the proof of our main results

Lemma 21 Let (119883 119866120590119887) be a 119866

120590119887-metric space and suppose

that 119909119899 119910119899 and 119911

119899 are 119866

120590119887-convergent to 119909 119910 and 119911

respectively Then we have1

1199043119866120590119887

(119909 119910 119911) minus

1

1199042119866120590119887

(119909 119909 119909)

minus

1

119904

119866120590119887

(119910 119910 119910) minus 119866120590119887

(119911 119911 119911)

le lim inf119899rarrinfin

119866120590119887

(119909119899 119910119899 119911119899)

le lim sup119899rarrinfin

119866120590119887

(119909119899 119910119899 119911119899)

le 1199043

119866120590119887

(119909 119910 119911) + 119904119866120590b

(119909 119909 119909)

+ 1199042

119866120590119887

(119910 119910 119910) + 1199043

119866120590119887

(119911 119911 119911)

(12)

In particular if 119910119899 = 119911

119899 = 119886 are constant then

1

119904

119866120590119887

(119909 119886 119886) minus 119866120590119887

(119909 119909 119909)

le lim inf119899rarrinfin

119866120590119887

(119909119899 119886 119886) le lim sup

119899rarrinfin

119866120590119887

(119909119899 119886 119886)

le 119904119866120590119887

(119909 119886 119886) + 119904119866120590119887

(119909 119909 119909)

(13)

Proof Using the rectangle inequality we obtain

119866120590119887

(119909 119910 119911) le 119904119866120590119887

(119909 119909119899 119909119899) + 1199042

119866120590119887

(119910 119910119899 119910119899)

+ 1199043

119866120590119887

(119911 119911119899 119911119899) + 1199043

119866120590119887

(119909119899 119910119899 119911119899)

119866120590119887

(119909119899 119910119899 119911119899) le 119904119866

120590119887(119909119899 119909 119909) + 119904

2

119866120590119887

(119910119899 119910 119910)

+ 1199043

119866120590119887

(119911119899 119911 119911) + 119904

3

119866120590119887

(119909 119910 119911)

(14)

Taking the lower limit as 119899 rarr infin in the first inequality andthe upper limit as 119899 rarr infin in the second inequality we obtainthe desired result

If 119910119899 = 119911

119899 = 119886 then

119866120590119887

(119909 119886 119886) le 119904119866120590119887

(119909 119909119899 119909119899) + 119904119866

120590119887(119909119899 119886 119886)

119866120590119887

(119909119899 119886 119886) le 119904119866

120590119887(119909119899 119909 119909) + 119904119866

120590119887(119909 119886 119886)

(15)

Again taking the lower limit as 119899 rarr infin in the first inequalityand the upper limit as 119899 rarr infin in the second inequality weobtain the desired result

2 Main Results

Samet et al [16] defined the notion of 120572-admissible mappingsand proved the following result

Definition 22 Let119879 be a self-mapping on119883 and 120572 119883times119883 rarr

[0infin) a function We say that 119879 is an 120572-admissible mappingif

119909 119910 isin 119883 120572 (119909 119910) ge 1 997904rArr 120572 (119879119909 119879119910) ge 1 (16)

Denote withΨ1015840 the family of all nondecreasing functions

120595 [0infin) rarr [0infin) such that suminfin119899=1

120595119899(119905) lt infin for all 119905 gt 0

where 120595119899 is the 119899th iterate of 120595

Theorem 23 Let (119883 119889) be a complete metric space and 119879 an120572-admissible mapping Assume that

120572 (119909 119910) 119889 (119879119909 119879119910) le 120595 (119889 (119909 119910)) (17)

where 120595 isin Ψ1015840 Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 120572(119909

0 1198791199090) ge 1

(ii) either 119879 is continuous or for any sequence 119909119899 in 119883

with 120572(119909119899 119909119899+1

) ge 1 for all 119899 isin Ncup0 such that 119909119899rarr

119909 as 119899 rarr infin we have 120572(119909119899 119909) ge 1 for all 119899 isin N cup 0

Then 119879 has a fixed point

For more details on 120572-admissible mappings we refer thereader to [17ndash20]

Definition 24 (see [21]) Let (119883 119866) be a 119866-metric space let119891 be a self-mapping on 119883 and let 120572 119883

3rarr [0infin) be a

function We say that 119891 is a 119866-120572-admissible mapping if

119909 119910 119911 isin 119883 120572 (119909 119910 119911) ge 1 997904rArr 120572 (119891119909 119891119910 119891119911) ge 1

(18)

Motivated by [22] letF denote the class of all functions120573 [0infin) rarr [0 1119904) satisfying the following condition

lim sup119899rarrinfin

120573 (119905119899) =

1

119904

997904rArr lim sup119899rarrinfin

119905119899= 0 (19)

Definition 25 Let 119891 119883 rarr 119883 and 120572 119883 times 119883 times 119883 rarr RWe say that 119891 is a rectangular 119866-120572-admissible mapping if

(T1) 120572(119909 119910 119911) ge 1 implies 120572(119891119909 119891119910 119891119911) ge 1 119909 119910 119911 isin 119883

(T2) 120572(119909119910119910)ge1120572(119910119911119911)ge1

implies 120572(119909 119911 119911) ge 1 119909 119910 119911 isin 119883

From now on let 120572 1198833rarr [0infin) be a function and

119872119904(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119891119909 119891119909)119866120590119887

(119910 119891119910 119891119910)119866120590119887

(119911 119891119911 119891119911)

1 + 11990421198662

120590119887

(119891119909 119891119910 119891119911)

(20)

Theorem 26 Let (119883 119866120590119887) be a 119866

120590119887-complete generalized 119887-

metric-like space and let 119891 119883 rarr 119883 be a rectangular 119866-120572-admissible mapping Suppose that

119904120572 (119909 119910 119911) 119866120590119887

(119891119909 119891119910 119891119911)

le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911)

(21)

for all 119909 119910 119911 isin 119883

The Scientific World Journal 5

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 120572(119909

0 1198911199090 1198911199090) ge 1

(ii) 119891 is continuous and for any sequence 119909119899 in 119883 with

120572(119909119899 119909119899+1

119909119899+1

) ge 1 for all 119899 isin Ncup0 such that 119909119899rarr

119909 as 119899 rarr infin we have 120572(119909 119909 119909) ge 1 for all 119899 isin Ncup0

Then 119891 has a fixed point

Proof Let 1199090isin 119883 be such that 120572(119909

0 1198911199090 1198911199090) ge 1 Define

a sequence 119909119899 by 119909

119899= 1198911198991199090for all 119899 isin N Since 119891 is a 119866-

120572-admissible mapping and 120572(1199090 1199091 1199091) = 120572(119909

0 1198911199090 1198911199090) ge

1 we deduce that 120572(1199091 1199092 1199092) = 120572(119891119909

0 1198911199091 1198911199091) ge 1

Continuing this process we get 120572(119909119899 119909119899+1

119909119899+1

) ge 1 for all119899 isin N cup 0

Step I We will show that lim119899rarrinfin

119866120590119887(119909119899 119909119899+1

119909119899+1

) = 0 If119909119899= 119909119899+1

for some 119899 isin N then 119909119899= 119891119909119899 Thus 119909

119899is a fixed

point of 119891 Therefore we assume that 119909119899

= 119909119899+1

for all 119899 isin NSince 120572(119909

119899 119909119899+1

119909119899+1

) ge 1 for each 119899 isin N then we canapply (21) which yields

119904119866120590119887

(119909119899 119909119899+1

119909119899+1

)

= 119904119866120590119887

(119891119909119899minus1

119891119909119899 119891119909119899)

le 120573 (119866120590119887

(119909119899minus1

119909119899 119909119899))119872119904(119909119899minus1

119909119899 119909119899)

lt

1

119904

119866120590119887

(119909119899minus1

119909119899 119909119899)

le 119866120590119887

(119909119899minus1

119909119899 119909119899)

(22)

119872119904(119909119899minus1

119909119899 119909119899)

= max 119866120590119887

(119909119899minus1

119909119899 119909119899)

(119866120590119887

(119909119899minus1

119891119909119899minus1

119891119909119899minus1

)

times119866120590119887

(119909119899 119891119909119899 119891119909119899) 119866120590119887

(119909119899 119891119909119899 119891119909119899))

times(1 + 1199042

1198662

120590119887

(119891119909119899minus1

119891119909119899 119891119909119899))

minus1

= max 119866120590119887

(119909119899minus1

119909119899 119909119899)

(119866120590119887

(119909119899minus1

119909119899 119909119899)

times119866120590119887

(119909119899 119909119899+1

119909119899+1

) 119866120590119887

(119909119899 119909119899+1

119909119899+1

))

times(1 + 1199042

1198662

120590119887

(119909119899 119909119899+1

x119899+1

))

minus1

= 119866120590119887

(119909119899minus1

119909119899 119909119899)

(23)

Therefore 119866120590119887(119909119899 119909119899+1

119909119899+1

) is a decreasing and boundedsequence of nonnegative real numbers Then there exists 119903 ge

0 such that lim119899rarrinfin

119866120590119887(119909119899 119909119899+1

119909119899+1

) = 119903 Letting 119899 rarr infin in(22) we have

119904119903 le 119903 (24)

Since 119904 gt 1 we deduce that 119903 = 0 that is

lim119899rarrinfin

119866120590119887

(119909119899 119909119899+1

119909119899+1

) = 0 (25)

By Proposition 15(2) we conclude that

lim119899rarrinfin

119866120590119887

(119909119899 119909119899 119909119899+1

) = 0 (26)

Step II Now we prove that the sequence 119909119899 is a119866

120590119887-Cauchy

sequence For this purpose we will show that

lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898) = 0 (27)

Using the rectangular inequality with (21) (as 120572(119909119899 119909119898 119909119898) ge

1 since 119891 is a rectangular119866-120572-admissible mapping) we have

119866120590119887

(119909119899 119909119898 119909119898)

le 119904119866120590119887

(119909119899 119909119899+1

119909119899+1

)

+ 1199042

119866120590119887

(119909119899+1

119909119898+1

119909119898+1

) + 1199042

119866120590119887

(119909119898+1

119909119898 119909119898)

le 119904119866120590119887

(119909119899 119909119899+1

119909119899+1

)

+ 119904120573 (119866120590119887

(119909119899 119909119898 119909119898))119872119904(119909119899 119909119898 119909119898)

+ 1199042

119866120590119887

(119909119898+1

119909119898 119909119898)

(28)

Taking limit as 119898 119899 rarr infin in the above inequality andapplying (25) and (26) we have

lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

le 119904 lim sup119899119898rarrinfin

120573 (119866120590119887

(119909119899 119909119898 119909119898)) lim sup119899119898rarrinfin

119872119904(119909119899 119909119898 119909119898)

(29)

Here

119866120590119887

(119909119899 119909119898 119909119898)

le 119872119904(119909119899 119909119898 119909119898)

= max

119866120590119887

(119909119899 119909119898 119909119898)

119866120590119887

(119909119899 119891119909119899 119891119909119899) [119866120590119887

(119909119898 119891119909119898 119891119909119898)]

2

1 + 11990421198662

120590119887

(119891119909119899 119891119909119898 119891119909119898)

= max

119866120590119887

(119909119899 119909119898 119909119898)

119866120590119887

(119909119899 119909119899+1

119909119899+1

) [119866120590119887

(119909119898 119909119898+1

119909119898+1

)]

2

1 + 11990421198662

120590119887

(119909119899+1

119909119898+1

119909119898+1

)

(30)

6 The Scientific World Journal

Letting119898 119899 rarr infin in the above inequality we get

lim sup119898119899rarrinfin

119872119904(119909119899 119909119898 119909119898) = lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898) (31)

Hence from (29) and (31) we obtain

lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

le 119904lim sup119898119899rarrinfin

120573 (119866120590119887

(119909119899 119909119898 119909119898)) lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

(32)

If lim sup119898119899rarrinfin

119866120590119887(119909119899 119909119898 119909119898) = 0 then we get

1

119904

le lim sup119898119899rarrinfin

120573 (119866120590119887

(119909119899 119909119898 119909119898)) (33)

Since 120573 isin F we deduce that

lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898) = 0 (34)

which is a contradiction Consequently 119909119899 is a 119866

120590119887-Cauchy

sequence in119883 Since (119883 119866120590119887) is119866120590119887-complete there exists 119906 isin

119883 such that 119909119899

rarr 119906 as 119899 rarr infin Now from (34) and 119866120590119887-

completeness of119883

lim119899rarrinfin

119866120590119887

(119906 119909119899 119909119899) = lim119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

= 119866120590119887

(119906 119906 119906) = 0

(35)

Step III Now we show that 119906 is a fixed point of 119891Using the rectangle inequality we get

119866120590119887

(119906 119891119906 119891119906) le 119904119866120590119887

(119906 119891119909119899 119891119909119899) + 119904119866

120590119887(119891119909119899 119891119906 119891119906)

(36)

Letting 119899 rarr infin and using the continuity of 119891 and (35) weobtain

119866120590119887

(119906 119891119906 119891119906) le 119904 lim119899rarrinfin

G120590119887

(119906 119891119909119899 119891119909119899)

+ 119904 lim119899rarrinfin

119866120590119887

(119891119909119899 119891119906 119891119906)

= 119904119866120590119887

(119891119906 119891119906 119891119906)

(37)

Note that from (21) as 120572(119906 119906 119906) ge 1 we have

119904119866120590119887

(119891119906 119891119906 119891119906) le 120573 (119866120590119887

(119906 119906 119906))119872119904(119906 119906 119906) (38)

where by (37)

119872119904(119906 119906 119906)

= max119866120590119887

(119906 119906 119906)

119866120590119887

(119906 119891119906 119891119906)119866120590119887

(119906 119891119906 119891119906)119866120590119887

(119906 119891119906 119891119906)

1 + 11990421198662

120590119887

(119891119906 119891119906 119891119906)

lt 119866120590119887

(119906 119891119906 119891119906)

(39)

Hence as 120573(119905) le 1 for all 119905 isin [0infin) we have 119904119866120590119887(119891119906

119891119906 119891119906) le 119866120590119887(119906 119891119906 119891119906) Thus by (37) we obtain that

119904119866120590119887(119891119906 119891119906 119891119906) = 119866

120590119887(119906 119891119906 119891119906) But then using (38) we

get that

119866120590119887

(119906 119891119906 119891119906) = 119904119866120590119887

(119891119906 119891119906 119891119906)

le 120573 (119866120590119887

(119906 119906 119906))119872119904(119906 119906 119906)

lt

1

119904

119866120590119887

(119906 119891119906 119891119906)

(40)

which is a contradiction Hence we have 119891119906 = 119906 Thus 119906 is afixed point of 119891

We replace condition (ii) in Theorem 26 by regularity ofthe space119883

Theorem 27 Under the same hypotheses of Theorem 26instead of condition (119894119894) assume that whenever 119909

119899 in 119883 is a

sequence such that 120572(119909119899 119909119899+1

119909119899+1

) ge 1 for all 119899 isin Ncup0 and119909119899rarr 119909 as 119899 rarr infin one has 120572(119909

119899 119909 119909) ge 1 for all 119899 isin Ncup0

Then 119891 has a fixed point

Proof Repeating the proof of Theorem 26 we can constructa sequence 119909

119899 in 119883 such that 120572(119909

119899 119909119899+1

119909119899+1

) ge 1 for all119899 isin N cup 0 and 119909

119899rarr 119906 isin 119883 for some 119906 isin 119883 Using the

assumption on 119883 we have 120572(119909119899 119906 119906) ge 1 for all 119899 isin N cup 0

Now we show that 119906 = 119891119906 By Lemma 21 and (35)

119904 [

1

119904

119866120590119887

(119906 119891119906 119891119906) minus 119866120590119887

(119906 119906 119906)]

le 119904lim sup119899rarrinfin

119866120590119887

(119909119899+1

119891119906 119891119906)

le lim sup119899rarrinfin

(120573 (119866120590119887

(119909119899 119906 119906))119872

119904(119909119899 119906 119906))

le

1

119904

lim sup119899rarrinfin

119872119904(119909119899 119906 119906)

(41)

wherelim119899rarrinfin

119872119904(119909119899 119906 119906)

= lim119899rarrinfin

max

119866120590119887

(119909119899 119906 119906)

[119866120590119887

(119909119899 119891119909119899 119891119909119899)] [119866120590119887

(119906 119891119906 119891119906)]

2

1 + 11990421198662

120590119887

(119891119909119899 119891119906 119891119906)

= lim119899rarrinfin

max

119866120590119887

(119909119899 119906 119906)

[119866120590119887

(119909119899 119909119899+1

119909119899+1

)] [119866120590119887

(119906 119891119906 119891119906)]

2

1 + 11990421198662

120590119887

(119909119899+1

119891119906 119891119906)

= max 119866120590119887

(119906 119906 119906) 0 = 0 (see (25) and (35))

(42)

Therefore we deduce that119866120590119887(119906 119891119906 119891119906) le 0 Hence we have

119906 = 119891119906

The Scientific World Journal 7

A mapping 120593 [0infin) rarr [0infin) is called a comparisonfunction if it is increasing and 120593

119899(119905) rarr 0 as 119899 rarr infin for any

119905 isin [0infin) (see eg [23 24] for more details and examples)

Definition 28 A function 120593 [0infin) rarr [0infin) is said to bea (119888)-comparison function if

(1198881) 120593 is increasing

(1198882) there exists 119896

0isin N 119886 isin (0 1) and a convergent series

of nonnegative terms suminfin

119896=1V119896such that 120593

119896+1(119905) le

119886120593119896(119905) + V

119896for 119896 ge 119896

0and any 119905 isin [0infin)

Later Berinde [5] introduced the notion of (119887)-comparison function as a generalization of (119888)-comparisonfunction

Definition 29 (see [5]) Let 119904 ge 1 be a real number Amapping120593 [0infin) rarr [0infin) is called a (119887)-comparison function ifthe following conditions are fulfilled

(1) 120593 is increasing

(2) there exist 1198960isin N 119886 isin (0 1) and a convergent series

of nonnegative terms suminfin119896=1

V119896such that 119904119896+1120593119896+1(119905) le

119886119904119896120593119896(119905) + V

119896for 119896 ge 119896

0and any 119905 isin [0infin)

Let Ψ119887be the class of all (119887)-comparison functions 120593

[0infin) rarr [0infin) It is clear that the notion of (119887)-comparison function coincideswith (119888)-comparison functionfor 119904 = 1

Lemma 30 (see [25]) If 120593 [0infin) rarr [0infin) is a (119887)-comparison function then we have the following

(1) the series suminfin119896=0

119904119896120593119896(119905) converges for any 119905 isin R

+

(2) the function 119887119904 [0infin) rarr [0infin) defined by 119887

119904(119905) =

suminfin

119896=0119904119896120593119896(119905) 119905 isin [0infin) is increasing and continuous

at 0

Remark 31 It is easy to see that if 120593 isin Ψ119887 then we have 120593(0) =

0 and 120593(119905) lt 119905 for each 119905 gt 0 and 120593 is continuous at 0

In the next example we present a class of (119887)-comparisonfunctions

Example 32 Any function of the form120595(119905) = ln((119886119904)119905+1) forall 119905 isin [0infin) where 0 lt 119886 lt 1 is a (119887)-comparison function

Proof From the part (1) of Lemma 30 the necessary condi-tion is that the series suminfin

119896=0119904119896120593119896(119905) converges for any 119905 isin R

+

But for each 119905 gt 0 and 119896 ge 1 we have

119904119896

120593119896

(119905) = 119904119896

120593 (120593119896minus1

(119905))

= 119904119896 ln(

119886

119904

120593119896minus1

(119905) + 1) le 119904119896 119886

119904

120593119896minus1

(119905)

le sdot sdot sdot le 119904119896

(

119886

119904

)

119896

119905 = 119886119896

119905

(43)

So according to the comparison test of the series we shouldhave 119886 lt 1 On the other hand we have

119904119896+1

120593119896+1

(119905) = 119904119896+1

120593 (120593119896

(119905))

= 119904119896+1 ln(

119886

119904

120593119896

(119905) + 1)

le 119904119896+1 119886

119904

120593119896

(119905) = 119904119896

119886120593119896

(119905)

(44)

Therefore for any convergent series of nonnegative termssuminfin

119896=1V119896and each 119896 ge 119896

0= 1 we have

119904119896+1

120593119896+1

(119905) le 119886119904119896

120593119896

(119905) le 119886119904119896

120593119896

(119905) + V119896 (45)

For example for 119904 = 2 and 119886 = 12 the function 120595(119905) =

ln(1199054 + 1) is a (119887)-comparison function

Theorem 33 Let (119883 119866120590119887) be a 119866

120590119887-complete generalized 119887-

metric-like space and let 119891 119883 rarr 119883 be a 119866-120572-admissiblemapping Suppose that

119904120572 (119909 119910 119911) 119866120590119887

(119891119909 119891119910 119891119911) le 120595 (119872119904(119909 119910 119911)) (46)

for all 119909 119910 119911 isin 119883 where 120595 isin Ψ119887and

119872119904(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119909 119891119909)119866120590119887

(119910 119910 119891119910)119866120590119887

(119911 119911 119891119911)

1 + 211990421198662

120590119887

(119891119909 119891119910 119891119911)

(47)

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 120572(119909

0 1198911199090 11989121199090) ge 1

(ii)

(a) 119891 is continuous and for any sequence 119909119899 in119883

with 120572(119909119899 119909119899+1

119909119899+2

) ge 1 for all 119899 isin Ncup0 suchthat 119909

119899rarr 119909 as 119899 rarr infin one has 120572(119909 119909 119909) ge 1

for all 119899 isin N cup 0(b) assume that whenever 119909

119899 in 119883 is a sequence

such that 120572(119909119899 119909119899+1

119909119899+2

) ge 1 for all 119899 isin Ncup0

and 119909119899rarr 119909 as 119899 rarr infin one has 120572(119909

119899 119909 119909) ge 1

for all 119899 isin N cup 0

Then 119891 has a fixed point

Proof Let 1199090isin 119883 be such that 120572(119909

0 1198911199090 11989121199090) ge 1 Define

a sequence 119909119899 by 119909

119899= 1198911198991199090for all 119899 isin N Since 119891 is a 119866-120572-

admissible mapping and 120572(1199090 1199091 1199092) = 120572(119909

0 1198911199090 11989121199090) ge

1 we deduce that 120572(1199091 1199092 1199093) = 120572(119891119909

0 1198911199091 1198911199092) ge 1

Continuing this process we get 120572(119909119899 119909119899+1

119909119899+2

) ge 1 for all119899 isin N cup 0

8 The Scientific World Journal

If there exists 1198990isin N such that 119909

1198990= 1199091198990+1

then 1199091198990

=

1198911199091198990and so we have nothing to prove Hence for all 119899 isin N

we assume that 119909119899

= 119909119899+1

Step I (Cauchyness of 119909119899) As 120572(119909

119899 119909119899+1

119909119899+2

) ge 1 for all119899 ge 0 using condition (46) we obtain

119904119866120590119887

(119909119899 119909119899+1

119909119899+2

)

le 119904120572 (119909119899minus1

119909119899 119909119899+1

) 119866120590119887

(119909119899 119909119899+1

119909119899+2

)

= 119904120572 (119909119899minus1

119909119899 119909119899+1

) 119866120590119887

(119891119909119899minus1

119891119909119899 119891119909119899+1

)

le 120595 (119872119904(119909119899minus1

119909119899 119909119899+1

))

(48)

Using Proposition 15(2) as 119909119899

= 119909119899+1

we get

119872119904(119909119899minus1

119909119899 119909119899+1

)

= max 119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

(119866120590119887

(119909119899minus1

119909119899minus1

119891119909119899minus1

)

times 119866120590119887

(119909119899 119909119899 119891119909119899)

times119866120590119887

(119909119899+1

119909119899+1

119891119909119899+1

))

times(1 + 21199042

1198662

120590119887

(119891119909119899minus1

119891119909119899 119891119909119899+1

))

minus1

= max 119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

(119866120590119887

(119909119899minus1

119909119899minus1

119909119899)

times 119866120590119887

(119909119899 119909119899 119909119899+1

)

times119866120590119887

(119909119899+1

119909119899+1

119909119899+2

))

times(1 + 21199042

1198662

120590119887

(119909119899 119909119899+1

119909119899+2

))

minus1

le max119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

2119904119866120590119887

(119909119899minus1

119909119899 119909119899+1

) 1198662

120590119887

(119909119899 119909119899+1

119909119899+2

)

1 + 2s21198662120590119887

(119909119899 119909119899+1

119909119899+2

)

= 119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

(49)

Hence119866120590119887

(119909119899 119909119899+1

119909119899+2

)

le 119904119866120590119887

(119909119899 119909119899+1

119909119899+2

) le 120595 (119866120590119887

(119909119899minus1

119909119899 119909119899+1

))

(50)

By induction since 119909119899

= 119909119899+1

we get that

119866120590119887

(119909119899 119909119899+1

119909119899+2

)

le 120595 (119866120590119887

(119909119899minus1

119909119899 119909119899+1

))

le 1205952

(119866120590119887

(119909119899minus2

119909119899minus1

119909119899))

le sdot sdot sdot le 120595119899

(119866120590119887

(1199090 1199091 1199092))

(51)

Let 120576 gt 0 be arbitrary Then there exists a natural number119873such that

infin

sum

119899=119873

119904119899

120595119899

(119866120590119887

(1199090 1199091 1199092)) lt

120576

2119904

(52)

Let 119898 gt 119899 ge 119873 Then by the rectangular inequality andProposition 15(2) as 119909

119899= 119909119899+1

we get

119866120590119887

(119909119899 119909119898 119909119898)

le 119904119866120590119887

(119909119899 119909119899+1

119909119899+1

) + 1199042

119866120590119887

(119909119899+1

119909119899+2

119909119899+2

)

+ sdot sdot sdot + 119904119898minus119899minus1

119866120590119887

(119909119898minus1

119909119898 119909119898)

le 2119904 (119904119866120590119887

(119909119899 119909119899 119909119899+1

) + 1199042

119866120590119887

(119909119899+1

119909119899+1

119909119899+2

)

+ sdot sdot sdot + 119904119898minus119899minus1

119866120590119887

(119909119898minus1

119909119898minus1

119909119898))

le 2119904 (119904119866120590119887

(119909119899 119909119899+1

119909119899+2

) + 1199042

119866120590119887

(119909119899+1

119909119899+2

119909119899+3

)

+ sdot sdot sdot + 119904119898minus119899minus1

119866120590119887

(119909119898minus1

119909119898 119909119898+1

))

le 2119904

119898minus2

sum

119896=119899

119904119896minus119899+1

120595119896

(119866120590119887

(1199090 1199091 1199092))

le 2119904

infin

sum

119896=119899

119904119896

120595119896

(119866120590119887

(1199090 1199091 1199092)) lt 120576

(53)

Consequently 119909119899 is a 119866

120590119887-Cauchy sequence in 119883 Since

(119883 119866120590119887) is 119866120590119887-complete so there exists 119906 isin 119883 such that

lim119899rarrinfin

119866120590119887

(119906 119909119899 119909119899) = lim119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

= 119866120590119887

(119906 119906 119906) = 0

(54)

Step II Now we show that 119906 is a fixed point of 119891 Suppose tothe contrary that is 119891119906 = 119906 then we have 119866

120590119887(119906 119906 119891119906) gt 0

Let the part (a) of (ii) holdsUsing the rectangle inequality we get

119866120590119887

(119906 119906 119891119906) le 119904119866120590119887

(119891119906 119891119909119899 119891119909119899) + 119904119866

120590119887(119891119909119899 119906 119906)

(55)

Letting 119899 rarr infin and using the continuity of 119891 we get

119866120590119887

(119906 119906 119891119906) le 119904119866120590119887

(119891119906 119891119906 119891119906) (56)

From (46) and part (a) of condition (ii) we have

119904119866120590119887

(119891119906 119891119906 119891119906) le 120595 (119872119904(119906 119906 119906)) (57)

where by using (56) we have

119872119904(119906 119906 119906)

= max119866120590119887

(119906 119906 119906)

119866120590119887

(119906 119906 119891119906)119866120590119887

(119906 119906 119891119906)119866120590119887

(119906 119906 119891119906)

1 + 211990421198662

120590119887

(119891119906 119891119906 119891119906)

le 119866120590119887

(119906 119906 119891119906)

(58)

Hence from properties of 120595 119904119866120590119887(119891119906 119891119906 119891119906) le

119866120590119887(119906 119906 119891119906) Thus by (56) we obtain that

119866120590119887

(119906 119906 119891119906) = 119904119866120590119887

(119891119906 119891119906 119891119906) (59)

The Scientific World Journal 9

Moreover (57) yields that 119866120590119887(119906 119906 119891119906) le 120595(119872

119904(119906 119906 119906)) le

120595(119866120590119887(119906 119906 119891119906))This is impossible according to our assump-

tions on 120595 Hence we have 119891119906 = 119906 Thus 119906 is a fixed point of119891

Now let part (b) of (ii) holdsAs 119909119899 is a sequence such that 120572(119909

119899 119909119899+1

119909119899+2

) ge 1 for all119899 isin N cup 0 and 119909

119899rarr 119909 as 119899 rarr infin we have 120572(119909

119899 119909 119909) ge 1

for all 119899 isin N cup 0Now we show that 119906 = 119891119906 By (46) we have

119904119866120590119887

(119891119906 119891119906 119909119899) le 119904120572 (119906 119906 119909

119899minus1) 119866120590119887

(119891119906 119891119906 119891119909119899minus1

)

le 120595 (119872119904(119906 119906 119909

119899minus1))

(60)

where

119872119904(119906 119906 119909

119899minus1)

= max

119866120590119887

(119906 119906 119909119899minus1

)

[119866120590119887

(119906 119906 119891119906)]

2

119866120590119887

(119909119899minus1

119909119899minus1

119891119909119899minus1

)

1 + [119904119866120590119887

(119891119906 119891119906 119891119909119899minus1

)]

2

(61)

Letting 119899 rarr infin in the above inequality and using (54) andLemma 21 we get

lim119899rarrinfin

119872119904(119906 119906 119909

119899minus1) = 0 (62)

Again taking the upper limit as 119899 rarr infin in (60) and using(62) and Lemma 21 we obtain

119904 [

1

119904

119866120590119887

(119906 119891119906 119891119906) minus 119866120590119887

(119906 119906 119906)]

le 119904 lim sup119899rarrinfin

119866120590119887

(119909119899 119891119906 119891119906)

le lim supnrarrinfin

120595 (119872119904(119906 119906 119909

119899minus1))

= 120595(lim sup119899rarrinfin

119872119904(119906 119906 119909

119899minus1))

= 120595 (0) = 0

(63)

So we get 119866120590119887(119906 119891119906 119891119906) = 0 That is 119891119906 = 119906

Let (119883 119866120590119887 ⪯) be a partially ordered 119866

120590119887-metric-like

space We say that 119879 is an increasing mapping on 119883 ifforall119909 119910 isin 119883 119909 ⪯ 119910 rArr 119879(119909) ⪯ 119879(119910) [26] Fixed pointtheorems for monotone operators in ordered metric spacesarewidely investigated andhave found various applications indifferential and integral equations (see [27ndash30] and referencestherein) From the results proved above we derive thefollowing new results in partially ordered 119866

120590119887-metric-like

space

Theorem 34 Let (119883 119866120590119887 ⪯) be a partially ordered 119866

120590119887-

complete generalized 119887-metric-like space and let 119891 119883 rarr 119883

be an increasing mapping Suppose that

119904119866120590119887

(119891119909 119891119910 119891119911) le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911) (64)

for all 119909 119910 119911 isin 119883 with 119909 ⪯ 119910 ⪯ 119911 where

119872119904(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119891119909 119891119909)119866120590119887

(119910 119891119910 119891119910)119866120590119887

(119911 119891119911 119891119911)

1 + 11990421198662

120590119887

(119891119909 119891119910 119891119911)

(65)

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) 119891 is continuous or assume that whenever 119909119899 in 119883 is

an increasing sequence such that 119909119899rarr 119909 as 119899 rarr infin

one has 119909119899⪯ 119909 for all 119899 isin N cup 0

Then 119891 has a fixed point

Proof Define 120572 119883 times 119883 times 119883 rarr [0 +infin) by

120572 (119909 119910 119911) =

1 if119909 ⪯ 119910 ⪯ 119911

0 otherwise(66)

First we prove that 119891 is a triangular 120572-admissible mappingHence we assume that 120572(119909 119910 119911) ge 1 Therefore we have 119909 ⪯

119910 ⪯ 119911 Since 119891 is increasing we get 119891119909 ⪯ 119891119910 ⪯ 119891119911 that is120572(119891119909 119891119910 119891119911) ge 1 Also let 120572(119909 119910 119910) ge 1 and 120572(119910 119911 119911) ge 1then 119909 ⪯ 119910 and 119910 ⪯ 119911 Consequently we deduce that 119909 ⪯ 119911that is 120572(119909 119911 119911) ge 1 Thus 119891 is a triangular 120572-admissiblemapping Since 119891 satisfies (64) so by the definition of 120572 wehave

119904120572 (119909 119910 119911) 119866120590119887

(119891119909 119891119910 119891119911)

le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911)

(67)

for all 119909 119910 119911 isin 119883 Therefore 119891 satisfies the contractivecondition (21) From (i) there exists 119909

0isin 119883 such that

1199090⪯ 1198911199090 that is 120572(119909

0 1198911199090 1198911199090) ge 1 According to (ii) we

conclude that all the conditions of Theorems 26 and 27 aresatisfied and so 119891 has a fixed point

Similarly using Theorem 33 we can prove followingresult

Theorem 35 Let (119883 119866120590119887 ⪯) be an ordered 119866

120590119887-complete

generalized 119887-metric-like space and let 119891 119883 rarr 119883 be anincreasing mapping Suppose that

119904119866120590119887

(119891119909 119891119910 119891119911) le 120595 (119872119904(119909 119910 119911)) (68)

for all 119909 119910 119911 isin 119883 with 119909 ⪯ 119910 ⪯ 119911 where 120595 isin Ψ119887and

119872119904(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119909 119891119909)119866120590119887

(119910 119910 119891119910)119866120590119887

(119911 119911 119891119911)

1 + 211990421198662

120590119887

(119891119909 119891119910 119891119911)

(69)

10 The Scientific World Journal

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii)

(a) 119891 is continuous(b) assume that whenever x

119899 in 119883 is an increasing

sequence such that 119909119899rarr 119909 as 119899 rarr infin one has

119909119899⪯ 119909 for all 119899 isin N cup 0

Then 119891 has a fixed point

We conclude this section by presenting some examplesthat illustrate our results

Example 36 Let 119883 = [0 (minus1 + radic5)2] be endowed withthe usual ordering onR Define the generalized 119887-metric-likefunction 119866

120590119887given by

119866120590119887

(119909 119910 119911) = max (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

(70)

with 119904 = 2 Consider the mapping 119891 119883 rarr 119883 defined by119891(119909) = (14)119909

radic119890minus(119909+((minus1+radic5)2))

2 and the function 120573 isin Fgiven by 120573(119905) = (12)119890

minus119905 119905 gt 0 and 120573(0) isin [0 12) Itis easy to see that 119891 is an increasing function on 119883 Weshow that 119891 is 119866

120590119887-continuous on 119883 By Proposition 20 it is

sufficient to show that119891 is119866120590119887-sequentially continuous on119883

Let 119909119899 be a sequence in119883 such that lim

119899rarrinfin119866120590119887(119909119899 119909 119909) =

119866120590119887(119909 119909 119909) so we have max(lim

119899rarrinfin119909119899

+ 119909)2

41199092 =

41199092 and equivalently lim

119899rarrinfin119909119899

= 120572 le 119909 On the otherhand we have

lim119899rarrinfin

119866120590119887

(119891119909119899 119891119909 119891119909)

= lim119899rarrinfin

max (119891119909119899+ 119891119909)

2

41198912

119909

= lim119899rarrinfin

max 1

16

(119909119899

radic119890minus(119909119899+(minus1+

radic5)2)2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= max 1

16

( lim119899rarrinfin

119909119899

radic119890minus(lim119899rarrinfin119909119899+(minus1+radic5)2)

2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= max 1

16

(120572radic119890minus(120572+(minus1+radic5)2)

2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

=

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= 119866120590119887

(119891119909 119891119909 119891119909)

(71)

So 119891 is 119866120590119887-sequentially continuous on119883

For all elements 119909 119910 119911 isin 119883 and the fact that 119892(119909) =

1199092119890minus1199092

is an increasing function on119883 we have

119904119866120590119887

(119891119909 119891119910 119891119911)

= 2max (119891119909 + 119891119910)2

(119891119910 + 119891119911)2

(119891119911 + 119891119909)2

= 2max(

1

4

119909radic119890minus(119909+(minus1+radic5)2)

2

+

1

4

119910radic119890minus(119910+(minus1+radic5)2)

2

)

2

(

1

4

119910radic119890minus(119910+(minus1+radic5)2)

2

+

1

4

119911radic119890minus(119911+(minus1+radic5)2)

2

)

2

(

1

4

119911radic119890minus(119911+(minus1+radic5)2)

2

+

1

4

119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

le 2max(

1

4

119909radic119890minus(119909+119910)

2

+

1

4

119910radic119890minus(119910+119909)

2

)

2

(

1

4

119910radic119890minus(119910+119911)

2

+

1

4

119911radic119890minus(119911+119910)

2

)

2

(

1

4

119911radic119890minus(119911+119909)

2

+

1

4

119909radic119890minus(119909+119911)

2

)

2

=

1

8

max (119909 + 119910)2

119890minus(119909+119910)

2

(119910 + 119911)2

119890minus(119910+119911)

2

(119911 + 119909)2

119890minus(119911+119909)

2

=

1

8

119890minusmax(119909+119910)2 (119910+119911)2 (119911+119909)2

timesmax (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

le

1

2

eminusmax(119909+119910)2 (119910+119911)2(119911+119909)2

timesmax (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

= 120573 (119866120590119887

(119909 119910 119911)) 119866120590119887

(119909 119910 119911)

le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911)

(72)

Hence119891 satisfies all the assumptions ofTheorem 34 and thusit has a fixed point (which is 119906 = 0)

Example 37 Let 119883 = [0 1] with the usual ordering on RDefine the generalized 119887-metric-like function 119866

120590119887given by

119866120590119887(119909 119910 119911) = max(119909 + 119910)

2

(119910 + 119911)2

(119911 + 119909)2

with 119904 = 2Consider the mapping 119891 119883 rarr 119883 defined by 119891(119909) =

The Scientific World Journal 11

ln(1 + 119909119890minus119909

4) and the function 120595 isin Ψ119887given by 120595(119905) =

(18)119905 119905 ge 0 It is easy to see that 119891 is increasing functionNow we show that 119891 is a 119866

120590119887-continuous function on119883

Let 119909119899 be a sequence in 119883 such that lim

119899rarrinfin119866120590119887(119909119899

119909 119909) = 119866120590119887(119909 119909 119909) so we havemax(lim

119899rarrinfin119909119899+119909)2

41199092 =

41199092 and equally lim

119899rarrinfin119909119899= 120572 le 119909 On the other hand we

have

lim119899rarrinfin

119866120590119887

(119891119909119899 119891119909 119891119909)

= lim119899rarrinfin

max (119891119909119899+ 119891119909)

2

41198912

119909

= max(ln(1 +

lim119899rarrinfin

119909119899119890minuslim119899rarrinfin119909119899

4

)

+ ln(1 +

119909119890minus119909

4

))

2

4(ln(1 +

119909119890minusx

4

))

2

= max(ln(1 +

120572119890minus120572

4

) + ln(1 +

119909119890minus119909

4

))

2

4(ln(1 +

119909119890minus119909

4

))

2

= 4(ln(1 +

119909119890minus119909

4

))

2

= 119866120590119887

(119891119909 119891119909 119891119909)

(73)

So 119891 is 119866120590119887-sequentially continuous on119883

For all elements 119909 119910 119911 isin 119883 we have

119904119866120590119887

(119891119909 119891119910 119891119911)

= 2max(ln(1 +

119909119890minus119909

4

) + ln(1 +

119910119890minus119910

4

))

2

(ln(1 +

119910119890minus119910

4

) + ln(1 +

119911119890minus119911

4

))

2

(ln(1 +

119911119890minus119911

4

) + ln(1 +

119909119890minus119909

4

))

2

le 2max(

119909119890minus119909

4

+

119910119890minus119910

4

)

2

(

119910119890minus119910

4

+

119911119890minus119911

4

)

2

(

119911119890minus119911

4

+

119909119890minus119909

4

)

2

le

1

8

max (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

= 120595 (119866120590119887

(119909 119910 119911))

le 120595 (119872119904(119909 119910 119911))

(74)

Hence119891 satisfies all the assumptions ofTheorem 35 and thusit has a fixed point (which is 119906 = 0)

3 Application

In this section we present an application of our results toestablish the existence of a solution to a periodic boundaryvalue problem (see [30 31])

Let 119883 = 119862([0 119879]) be the set of all real continuousfunctions on [0T] We first endow119883 with the 119887-metric-like

120590119887(119906 V) = sup

119905isin[0119879]

(|119906 (119905)| + |V (119905)|)119901 (75)

for all 119906 V isin 119883 where 119901 gt 1 and then we endow it with thegeneralized 119887-metric-like 119866

120590119887defined by

119866120590119887

(119906 V 119908)

= max sup119905isin[0119879]

(|119906 (119905)| + |V (119905)|)119901

sup119905isin[0119879]

(|V (119905)| + |119908 (119905)|)119901

sup119905isin[0119879]

(|119908 (119905)| + |119906 (119905)|)119901

(76)

Clearly (119883 119866120590119887) is a complete generalized 119887-metric-like space

with parameter 119870 = 2119901minus1 We equip 119862([0 119879]) with a partial

order given by119909 ⪯ 119910 iff119909 (119905) le 119910 (119905) forall119905 isin [0 119879] (77)

Moreover as in [30] it is proved that (119862([0 119879]) ⪯) is regularthat is whenever 119909

119899 in119883 is an increasing sequence such that

119909119899rarr 119909 as 119899 rarr infin we have 119909

119899⪯ 119909 for all 119899 isin N cup 0

Consider the first-order periodic boundary value prob-lem

1199091015840

(119905) = 119891 (119905 119909 (119905)) 119909 (0) = 119909 (119879) (78)

where 119905 isin 119868 = [0 119879] with 119879 gt 0 and 119891 119868 times R rarr R is acontinuous function

A lower solution for (78) is a function 120572 isin 1198621[0 119879] such

that1205721015840

(119905) le 119891 (119905 120572 (119905))

120572 (0) le 120572 (119879)

(79)

where 119905 isin 119868 = [0 119879]Assume that there exists 120582 gt 0 such that for all 119909 119910 isin

119862[0 119879] we have1003816100381610038161003816119891 (119905 119909 (119905)) + 120582119909 (119905)

1003816100381610038161003816+1003816100381610038161003816119891 (119905 119910 (119905)) + 120582119910 (119905)

1003816100381610038161003816

le

120582

2119901minus1

119901radicln(

119886

2119901minus1

(|119909 (119905)| +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)119901

+ 1)

(80)

where 0 le 119886 lt 1 Then the existence of a lower solution for(78) provides the existence of a solution of (78)

Problem (78) can be rewritten as

1199091015840

(119905) + 120582119909 (119905) = 119891 (119905 119909 (119905)) + 120582119909 (119905)

119909 (0) = 119909 (119879)

(81)

Consider1199091015840

(119905) + 120582119909 (119905) = 120575119909(119905) = 119865 (119905 119909 (119905))

119909 (0) = 119909 (119879)

(82)

where 119905 isin 119868

12 The Scientific World Journal

Using variation of parameters formula we get

119909 (119905) = 119909 (0) 119890minus120582119905

+ int

119905

0

119890minus120582(119905minus119904)

120575119909(119904) 119889119904 (83)

which yields

119909 (119879) = 119909 (0) 119890minus120582119879

+ int

119879

0

119890minus120582(119879minus119904)

120575119909(119904) 119889119904 (84)

Since 119909(0) = 119909(119879) we get

119909 (0) [1 minus 119890minus120582119879

] = 119890minus120582119879

int

119879

0

119890120582(119904)

120575119909(119904) 119889119904 (85)

or

119909 (0) =

1

119890120582119879

minus 1

int

119879

0

119890120582119904

120575119909(119904) 119889119904 (86)

Substituting the value of 119909(0) in (83) we arrive at

119909 (119905) = int

119879

0

119866 (119905 119904) 120575119909(119904) 119889119904 (87)

where

119866 (119905 119904) =

119890120582(119879+119904minus119905)

119890120582119879

minus 1

0 le 119904 le 119905 le 119879

119890120582(119904minus119905)

119890120582119879

minus 1

0 le 119905 le 119904 le 119879

(88)

Now define the operator 119878 119862[0 119879] rarr 119862[0 119879] as

119878119909 (119905) = int

119879

0

119866 (119905 119904) 119865 (119904 119909 (119904)) 119889119904 (89)

The mapping 119878 is increasing [31] Note that if 119906 isin 119862[0 119879] is afixed point of 119878 then 119906 isin 119862

1[0 119879] is a solution of (78)

Let 119909 119910 119911 isin 119862[0 119879] Then we have

2119901minus1

[|119878119909 (119905)| +1003816100381610038161003816119878119910 (119905)

1003816100381610038161003816]

= 2119901minus1

[

100381610038161003816100381610038161003816100381610038161003816

int

119879

0

119866 (119905 119904) 119865 (119904 119909 (119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

119879

0

119866 (119905 119904) 119865 (119904 119910 (119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

]

le 2119901minus1

int

119879

0

|119866 (119905 119904)|

times [|119865 (119904 119909 (119904))| +1003816100381610038161003816119865 (119904 119910 (119904))

1003816100381610038161003816] 119889119904

le 2119901minus1

int

119879

0

|119866 (119905 119904)|

120582

2119901minus1

times119901radicln(

119886

2119901minus1

(|119909 (119905)| +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)119901

+ 1)119889119904

le 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [int

119905

0

119890120582(119879+119904minus119905)

119890120582119879

minus 1

119889119904 + int

119879

119905

119890120582(119904minus119905)

119890120582119879

minus 1

119889119904]

= 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [

1

120582 (119890120582119879

minus 1)

(119890120582(119879+119904minus119905)10038161003816

100381610038161003816

119905

0

+ 119890120582(119904minus119905)10038161003816

100381610038161003816

119879

119905

)]

= 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [

1

120582 (119890120582119879

minus 1)

(119890120582119879

minus 119890120582(119879minus119905)

+ 119890120582(119879minus119905)

minus 1)]

=119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

le119901radicln(

119886

2119901minus1

119866120590119887

(119909 119910 119911) + 1)

le119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(90)

Similarly

2119901minus1 1003816

100381610038161003816119878119910 (119905)

1003816100381610038161003816+ |119878119911 (119905)| le

119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

|119878119909 (119905)| + |119878119911 (119905)| le119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(91)

where

119872(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119878119909 119878119909) 119866120590119887

(119910 119878119910 119878119910)119866120590119887

(119911 119878119911 119878119911)

1 + 22119901minus1

1198662

120590119887

(119878119909 119878119910 119878119911)

(92)

Equivalently

(2119901minus1

|119878119909 (119905)| +1003816100381610038161003816119878119910 (119905)

1003816100381610038161003816)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(2119901minus1 1003816

100381610038161003816119878119910 (119905)

1003816100381610038161003816+ |119878119911 (119905)|)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(2119901minus1

|119878119909 (119905)| + |119878119911 (119905)|)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(93)

which yields that

2119901minus1

120590119887(119878119909 119878119910) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

120590119887(119878119910 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

120590119887(119878119909 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(94)

The Scientific World Journal 13

Finally it is easy to obtain that

2119901minus1G120590119887

(119878119909 119878119910 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1) (95)

Finally since120572 is a lower solution for (78) so it is easy to showthat 120572 ⪯ 119878(120572) [31]

Hence the hypotheses of Theorem 35 are satisfied with120595(119905) = ln((1198862119901minus1)119905 + 1) where 0 le 119886 lt 1 Hence there existsa fixed point 119909 isin 119862[0 119879] such that 119878119909 = 119909 which is a solutionto periodic boundary value problem (78)

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) KingAbdulaziz University JeddahTherefore the firstauthor acknowledges with thanks DSR KAU for financialsupport

References

[1] S Czerwik ldquoContraction mappings in b-metric spacesrdquo ActaMathematica et Informatica Universitatis Ostraviensis vol 1 pp5ndash11 1993

[2] V Parvaneh J R Roshan and S Radenovic ldquoExistence of tri-pled coincidence points in ordered b-metric spaces and an app-lication to a system of integral equationsrdquo Fixed Point Theoryand Applications vol 2013 article 130 2013

[3] J R Roshan V Parvaneh S Sedghi N Shobkolaei and WShatanawi ldquoCommon fixed points of almost generalized con-tractive mappings in ordered b-metric spacesrdquo Fixed PointTheory and Applications vol 2013 article 159 2013

[4] H Aydi M F Bota E Karapınar and S Moradi ldquoA commonfixed point for weak 120601-contractions on b-metric spacesrdquo FixedPoint Theory vol 13 no 2 pp 337ndash346 2012

[5] V Berinde ldquoSequences of operators and fixed points in quasi-metric spacesrdquo Studia Universitatis ldquoBabes-Bolyairdquo Mathemat-ica vol 16 no 4 pp 23ndash27 1996

[6] N Hussain D ETHoric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 article 126 2012

[7] N Hussain and M H Shah ldquoKKMmappings in cone b-metricspacesrdquo Computers and Mathematics with Applications vol 62no 4 pp 1677ndash1684 2011

[8] N Hussain J R Roshan V Parvaneh and Z Kadelburg ldquoFixedpoints of contractive mappings in b-metric-like spacesrdquo TheScientific World Journal vol 2014 Article ID 471827 15 pages2014

[9] M A Khamsi and N Hussain ldquoKKMmappings in metric typespacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 73 no 9 pp 3123ndash3129 2010

[10] M A Khamsi ldquoRemarks on cone metric spaces and fixed pointtheorems of contractive mappingsrdquo Fixed Point Theory andApplications vol 2010 Article ID 315398 7 pages 2010

[11] Z Mustafa J R Roshan V Parvaneh and Z Kadelburg ldquoSomecommon fixed point results in ordered partial b-metric spacesrdquoJournal of Inequalities and Applications vol 2013 article 5622013

[12] Z Mustafa and B Sims ldquoA new approach to generalized metricspacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash297 2006

[13] A Amini-Harandi ldquoMetric-like spaces partial metric spacesand fixed pointsrdquo Fixed PointTheory and Applications vol 2012article 204 2012

[14] N Shobkolaei S Sedghi J R Roshan and N Hussain ldquoSuzukitype fixed point results in metric-like spacesrdquo Journal of Func-tion Spaces and Applications vol 2013 Article ID 143686 9pages 2013

[15] M A Alghamdi N Hussain and P Salimi ldquoFixed point andcoupled fixed point theorems on b-metric-like spacesrdquo Journalof Inequalities and Applications vol 2013 article 402 2013

[16] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572 minus 120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

[17] P Salimi and E Karapınar ldquoSuzuki-Edelstein type contractionsvia auxiliary functionsrdquoMathematical Problems in Engineeringvol 2013 Article ID 648528 8 pages 2013

[18] P Salimi C Vetro and P Vetro ldquoFixed point theorems fortwisted (120572 120573)-120595-contractive type mappings and applicationsrdquoFilomat vol 27 no 4 pp 605ndash615 2013

[19] P Salimi C Vetro and P Vetro ldquoSome new fixed point resultsin non-Archimedean fuzzy metric spacesrdquo Nonlinear AnalysisModelling and Control vol 18 no 3 pp 344ndash358 2013

[20] N Hussain P Salimi and A Latif ldquoFixed point results forsingle and set-valued 120572-120578-120595-contractive mappingsrdquo Fixed PointTheory and Applications vol 2013 article 212 2013

[21] M A Alghamdi and E Karapınar ldquoG-120573-120595 contractive-typemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 article 70 2013

[22] M Geraghty ldquoOn contractive mappingsrdquo Proceedings of theAmerican Mathematical Society vol 40 pp 604ndash608 1973

[23] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001

[24] N Hussain Z Kadelburg S Radenovic and F R Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012

[25] V Berinde ldquoGeneralized contractions in quasimetric spacesrdquo inSeminar on Fixed Point Theory Preprint no 3 pp 3ndash9 1993

[26] L Ciric M Abbas R Saadati and N Hussain ldquoCommon fixedpoints of almost generalized contractive mappings in orderedmetric spacesrdquo Applied Mathematics and Computation vol 217no 12 pp 5784ndash5789 2011

[27] R P Agarwal N Hussain and M A Taoudi ldquoFixed point the-orems in ordered Banach spaces and applications to nonlinearintegral equationsrdquo Abstract and Applied Analysis vol 2012Article ID 245872 15 pages 2012

[28] N Hussain A R Khan and R P Agarwal ldquoKrasnoselrsquoskii andKy Fan type fixed point theorems in ordered Banach spacesrdquoJournal of Nonlinear and Convex Analysis vol 11 no 3 pp 475ndash489 2010

[29] N Hussain and M A Taoudi ldquoKrasnoselrsquoskii-type fixed pointtheorems with applications to Volterra integral equationsrdquoFixed Point Theory and Applications vol 2013 article 196 2013

14 The Scientific World Journal

[30] J J Nieto and R Rodrıguez-Lopez ldquoContractivemapping theo-rems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[31] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysisTheoryMethods andApplications vol 71 no 7-8 pp 3403ndash3410 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 5

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 120572(119909

0 1198911199090 1198911199090) ge 1

(ii) 119891 is continuous and for any sequence 119909119899 in 119883 with

120572(119909119899 119909119899+1

119909119899+1

) ge 1 for all 119899 isin Ncup0 such that 119909119899rarr

119909 as 119899 rarr infin we have 120572(119909 119909 119909) ge 1 for all 119899 isin Ncup0

Then 119891 has a fixed point

Proof Let 1199090isin 119883 be such that 120572(119909

0 1198911199090 1198911199090) ge 1 Define

a sequence 119909119899 by 119909

119899= 1198911198991199090for all 119899 isin N Since 119891 is a 119866-

120572-admissible mapping and 120572(1199090 1199091 1199091) = 120572(119909

0 1198911199090 1198911199090) ge

1 we deduce that 120572(1199091 1199092 1199092) = 120572(119891119909

0 1198911199091 1198911199091) ge 1

Continuing this process we get 120572(119909119899 119909119899+1

119909119899+1

) ge 1 for all119899 isin N cup 0

Step I We will show that lim119899rarrinfin

119866120590119887(119909119899 119909119899+1

119909119899+1

) = 0 If119909119899= 119909119899+1

for some 119899 isin N then 119909119899= 119891119909119899 Thus 119909

119899is a fixed

point of 119891 Therefore we assume that 119909119899

= 119909119899+1

for all 119899 isin NSince 120572(119909

119899 119909119899+1

119909119899+1

) ge 1 for each 119899 isin N then we canapply (21) which yields

119904119866120590119887

(119909119899 119909119899+1

119909119899+1

)

= 119904119866120590119887

(119891119909119899minus1

119891119909119899 119891119909119899)

le 120573 (119866120590119887

(119909119899minus1

119909119899 119909119899))119872119904(119909119899minus1

119909119899 119909119899)

lt

1

119904

119866120590119887

(119909119899minus1

119909119899 119909119899)

le 119866120590119887

(119909119899minus1

119909119899 119909119899)

(22)

119872119904(119909119899minus1

119909119899 119909119899)

= max 119866120590119887

(119909119899minus1

119909119899 119909119899)

(119866120590119887

(119909119899minus1

119891119909119899minus1

119891119909119899minus1

)

times119866120590119887

(119909119899 119891119909119899 119891119909119899) 119866120590119887

(119909119899 119891119909119899 119891119909119899))

times(1 + 1199042

1198662

120590119887

(119891119909119899minus1

119891119909119899 119891119909119899))

minus1

= max 119866120590119887

(119909119899minus1

119909119899 119909119899)

(119866120590119887

(119909119899minus1

119909119899 119909119899)

times119866120590119887

(119909119899 119909119899+1

119909119899+1

) 119866120590119887

(119909119899 119909119899+1

119909119899+1

))

times(1 + 1199042

1198662

120590119887

(119909119899 119909119899+1

x119899+1

))

minus1

= 119866120590119887

(119909119899minus1

119909119899 119909119899)

(23)

Therefore 119866120590119887(119909119899 119909119899+1

119909119899+1

) is a decreasing and boundedsequence of nonnegative real numbers Then there exists 119903 ge

0 such that lim119899rarrinfin

119866120590119887(119909119899 119909119899+1

119909119899+1

) = 119903 Letting 119899 rarr infin in(22) we have

119904119903 le 119903 (24)

Since 119904 gt 1 we deduce that 119903 = 0 that is

lim119899rarrinfin

119866120590119887

(119909119899 119909119899+1

119909119899+1

) = 0 (25)

By Proposition 15(2) we conclude that

lim119899rarrinfin

119866120590119887

(119909119899 119909119899 119909119899+1

) = 0 (26)

Step II Now we prove that the sequence 119909119899 is a119866

120590119887-Cauchy

sequence For this purpose we will show that

lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898) = 0 (27)

Using the rectangular inequality with (21) (as 120572(119909119899 119909119898 119909119898) ge

1 since 119891 is a rectangular119866-120572-admissible mapping) we have

119866120590119887

(119909119899 119909119898 119909119898)

le 119904119866120590119887

(119909119899 119909119899+1

119909119899+1

)

+ 1199042

119866120590119887

(119909119899+1

119909119898+1

119909119898+1

) + 1199042

119866120590119887

(119909119898+1

119909119898 119909119898)

le 119904119866120590119887

(119909119899 119909119899+1

119909119899+1

)

+ 119904120573 (119866120590119887

(119909119899 119909119898 119909119898))119872119904(119909119899 119909119898 119909119898)

+ 1199042

119866120590119887

(119909119898+1

119909119898 119909119898)

(28)

Taking limit as 119898 119899 rarr infin in the above inequality andapplying (25) and (26) we have

lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

le 119904 lim sup119899119898rarrinfin

120573 (119866120590119887

(119909119899 119909119898 119909119898)) lim sup119899119898rarrinfin

119872119904(119909119899 119909119898 119909119898)

(29)

Here

119866120590119887

(119909119899 119909119898 119909119898)

le 119872119904(119909119899 119909119898 119909119898)

= max

119866120590119887

(119909119899 119909119898 119909119898)

119866120590119887

(119909119899 119891119909119899 119891119909119899) [119866120590119887

(119909119898 119891119909119898 119891119909119898)]

2

1 + 11990421198662

120590119887

(119891119909119899 119891119909119898 119891119909119898)

= max

119866120590119887

(119909119899 119909119898 119909119898)

119866120590119887

(119909119899 119909119899+1

119909119899+1

) [119866120590119887

(119909119898 119909119898+1

119909119898+1

)]

2

1 + 11990421198662

120590119887

(119909119899+1

119909119898+1

119909119898+1

)

(30)

6 The Scientific World Journal

Letting119898 119899 rarr infin in the above inequality we get

lim sup119898119899rarrinfin

119872119904(119909119899 119909119898 119909119898) = lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898) (31)

Hence from (29) and (31) we obtain

lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

le 119904lim sup119898119899rarrinfin

120573 (119866120590119887

(119909119899 119909119898 119909119898)) lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

(32)

If lim sup119898119899rarrinfin

119866120590119887(119909119899 119909119898 119909119898) = 0 then we get

1

119904

le lim sup119898119899rarrinfin

120573 (119866120590119887

(119909119899 119909119898 119909119898)) (33)

Since 120573 isin F we deduce that

lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898) = 0 (34)

which is a contradiction Consequently 119909119899 is a 119866

120590119887-Cauchy

sequence in119883 Since (119883 119866120590119887) is119866120590119887-complete there exists 119906 isin

119883 such that 119909119899

rarr 119906 as 119899 rarr infin Now from (34) and 119866120590119887-

completeness of119883

lim119899rarrinfin

119866120590119887

(119906 119909119899 119909119899) = lim119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

= 119866120590119887

(119906 119906 119906) = 0

(35)

Step III Now we show that 119906 is a fixed point of 119891Using the rectangle inequality we get

119866120590119887

(119906 119891119906 119891119906) le 119904119866120590119887

(119906 119891119909119899 119891119909119899) + 119904119866

120590119887(119891119909119899 119891119906 119891119906)

(36)

Letting 119899 rarr infin and using the continuity of 119891 and (35) weobtain

119866120590119887

(119906 119891119906 119891119906) le 119904 lim119899rarrinfin

G120590119887

(119906 119891119909119899 119891119909119899)

+ 119904 lim119899rarrinfin

119866120590119887

(119891119909119899 119891119906 119891119906)

= 119904119866120590119887

(119891119906 119891119906 119891119906)

(37)

Note that from (21) as 120572(119906 119906 119906) ge 1 we have

119904119866120590119887

(119891119906 119891119906 119891119906) le 120573 (119866120590119887

(119906 119906 119906))119872119904(119906 119906 119906) (38)

where by (37)

119872119904(119906 119906 119906)

= max119866120590119887

(119906 119906 119906)

119866120590119887

(119906 119891119906 119891119906)119866120590119887

(119906 119891119906 119891119906)119866120590119887

(119906 119891119906 119891119906)

1 + 11990421198662

120590119887

(119891119906 119891119906 119891119906)

lt 119866120590119887

(119906 119891119906 119891119906)

(39)

Hence as 120573(119905) le 1 for all 119905 isin [0infin) we have 119904119866120590119887(119891119906

119891119906 119891119906) le 119866120590119887(119906 119891119906 119891119906) Thus by (37) we obtain that

119904119866120590119887(119891119906 119891119906 119891119906) = 119866

120590119887(119906 119891119906 119891119906) But then using (38) we

get that

119866120590119887

(119906 119891119906 119891119906) = 119904119866120590119887

(119891119906 119891119906 119891119906)

le 120573 (119866120590119887

(119906 119906 119906))119872119904(119906 119906 119906)

lt

1

119904

119866120590119887

(119906 119891119906 119891119906)

(40)

which is a contradiction Hence we have 119891119906 = 119906 Thus 119906 is afixed point of 119891

We replace condition (ii) in Theorem 26 by regularity ofthe space119883

Theorem 27 Under the same hypotheses of Theorem 26instead of condition (119894119894) assume that whenever 119909

119899 in 119883 is a

sequence such that 120572(119909119899 119909119899+1

119909119899+1

) ge 1 for all 119899 isin Ncup0 and119909119899rarr 119909 as 119899 rarr infin one has 120572(119909

119899 119909 119909) ge 1 for all 119899 isin Ncup0

Then 119891 has a fixed point

Proof Repeating the proof of Theorem 26 we can constructa sequence 119909

119899 in 119883 such that 120572(119909

119899 119909119899+1

119909119899+1

) ge 1 for all119899 isin N cup 0 and 119909

119899rarr 119906 isin 119883 for some 119906 isin 119883 Using the

assumption on 119883 we have 120572(119909119899 119906 119906) ge 1 for all 119899 isin N cup 0

Now we show that 119906 = 119891119906 By Lemma 21 and (35)

119904 [

1

119904

119866120590119887

(119906 119891119906 119891119906) minus 119866120590119887

(119906 119906 119906)]

le 119904lim sup119899rarrinfin

119866120590119887

(119909119899+1

119891119906 119891119906)

le lim sup119899rarrinfin

(120573 (119866120590119887

(119909119899 119906 119906))119872

119904(119909119899 119906 119906))

le

1

119904

lim sup119899rarrinfin

119872119904(119909119899 119906 119906)

(41)

wherelim119899rarrinfin

119872119904(119909119899 119906 119906)

= lim119899rarrinfin

max

119866120590119887

(119909119899 119906 119906)

[119866120590119887

(119909119899 119891119909119899 119891119909119899)] [119866120590119887

(119906 119891119906 119891119906)]

2

1 + 11990421198662

120590119887

(119891119909119899 119891119906 119891119906)

= lim119899rarrinfin

max

119866120590119887

(119909119899 119906 119906)

[119866120590119887

(119909119899 119909119899+1

119909119899+1

)] [119866120590119887

(119906 119891119906 119891119906)]

2

1 + 11990421198662

120590119887

(119909119899+1

119891119906 119891119906)

= max 119866120590119887

(119906 119906 119906) 0 = 0 (see (25) and (35))

(42)

Therefore we deduce that119866120590119887(119906 119891119906 119891119906) le 0 Hence we have

119906 = 119891119906

The Scientific World Journal 7

A mapping 120593 [0infin) rarr [0infin) is called a comparisonfunction if it is increasing and 120593

119899(119905) rarr 0 as 119899 rarr infin for any

119905 isin [0infin) (see eg [23 24] for more details and examples)

Definition 28 A function 120593 [0infin) rarr [0infin) is said to bea (119888)-comparison function if

(1198881) 120593 is increasing

(1198882) there exists 119896

0isin N 119886 isin (0 1) and a convergent series

of nonnegative terms suminfin

119896=1V119896such that 120593

119896+1(119905) le

119886120593119896(119905) + V

119896for 119896 ge 119896

0and any 119905 isin [0infin)

Later Berinde [5] introduced the notion of (119887)-comparison function as a generalization of (119888)-comparisonfunction

Definition 29 (see [5]) Let 119904 ge 1 be a real number Amapping120593 [0infin) rarr [0infin) is called a (119887)-comparison function ifthe following conditions are fulfilled

(1) 120593 is increasing

(2) there exist 1198960isin N 119886 isin (0 1) and a convergent series

of nonnegative terms suminfin119896=1

V119896such that 119904119896+1120593119896+1(119905) le

119886119904119896120593119896(119905) + V

119896for 119896 ge 119896

0and any 119905 isin [0infin)

Let Ψ119887be the class of all (119887)-comparison functions 120593

[0infin) rarr [0infin) It is clear that the notion of (119887)-comparison function coincideswith (119888)-comparison functionfor 119904 = 1

Lemma 30 (see [25]) If 120593 [0infin) rarr [0infin) is a (119887)-comparison function then we have the following

(1) the series suminfin119896=0

119904119896120593119896(119905) converges for any 119905 isin R

+

(2) the function 119887119904 [0infin) rarr [0infin) defined by 119887

119904(119905) =

suminfin

119896=0119904119896120593119896(119905) 119905 isin [0infin) is increasing and continuous

at 0

Remark 31 It is easy to see that if 120593 isin Ψ119887 then we have 120593(0) =

0 and 120593(119905) lt 119905 for each 119905 gt 0 and 120593 is continuous at 0

In the next example we present a class of (119887)-comparisonfunctions

Example 32 Any function of the form120595(119905) = ln((119886119904)119905+1) forall 119905 isin [0infin) where 0 lt 119886 lt 1 is a (119887)-comparison function

Proof From the part (1) of Lemma 30 the necessary condi-tion is that the series suminfin

119896=0119904119896120593119896(119905) converges for any 119905 isin R

+

But for each 119905 gt 0 and 119896 ge 1 we have

119904119896

120593119896

(119905) = 119904119896

120593 (120593119896minus1

(119905))

= 119904119896 ln(

119886

119904

120593119896minus1

(119905) + 1) le 119904119896 119886

119904

120593119896minus1

(119905)

le sdot sdot sdot le 119904119896

(

119886

119904

)

119896

119905 = 119886119896

119905

(43)

So according to the comparison test of the series we shouldhave 119886 lt 1 On the other hand we have

119904119896+1

120593119896+1

(119905) = 119904119896+1

120593 (120593119896

(119905))

= 119904119896+1 ln(

119886

119904

120593119896

(119905) + 1)

le 119904119896+1 119886

119904

120593119896

(119905) = 119904119896

119886120593119896

(119905)

(44)

Therefore for any convergent series of nonnegative termssuminfin

119896=1V119896and each 119896 ge 119896

0= 1 we have

119904119896+1

120593119896+1

(119905) le 119886119904119896

120593119896

(119905) le 119886119904119896

120593119896

(119905) + V119896 (45)

For example for 119904 = 2 and 119886 = 12 the function 120595(119905) =

ln(1199054 + 1) is a (119887)-comparison function

Theorem 33 Let (119883 119866120590119887) be a 119866

120590119887-complete generalized 119887-

metric-like space and let 119891 119883 rarr 119883 be a 119866-120572-admissiblemapping Suppose that

119904120572 (119909 119910 119911) 119866120590119887

(119891119909 119891119910 119891119911) le 120595 (119872119904(119909 119910 119911)) (46)

for all 119909 119910 119911 isin 119883 where 120595 isin Ψ119887and

119872119904(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119909 119891119909)119866120590119887

(119910 119910 119891119910)119866120590119887

(119911 119911 119891119911)

1 + 211990421198662

120590119887

(119891119909 119891119910 119891119911)

(47)

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 120572(119909

0 1198911199090 11989121199090) ge 1

(ii)

(a) 119891 is continuous and for any sequence 119909119899 in119883

with 120572(119909119899 119909119899+1

119909119899+2

) ge 1 for all 119899 isin Ncup0 suchthat 119909

119899rarr 119909 as 119899 rarr infin one has 120572(119909 119909 119909) ge 1

for all 119899 isin N cup 0(b) assume that whenever 119909

119899 in 119883 is a sequence

such that 120572(119909119899 119909119899+1

119909119899+2

) ge 1 for all 119899 isin Ncup0

and 119909119899rarr 119909 as 119899 rarr infin one has 120572(119909

119899 119909 119909) ge 1

for all 119899 isin N cup 0

Then 119891 has a fixed point

Proof Let 1199090isin 119883 be such that 120572(119909

0 1198911199090 11989121199090) ge 1 Define

a sequence 119909119899 by 119909

119899= 1198911198991199090for all 119899 isin N Since 119891 is a 119866-120572-

admissible mapping and 120572(1199090 1199091 1199092) = 120572(119909

0 1198911199090 11989121199090) ge

1 we deduce that 120572(1199091 1199092 1199093) = 120572(119891119909

0 1198911199091 1198911199092) ge 1

Continuing this process we get 120572(119909119899 119909119899+1

119909119899+2

) ge 1 for all119899 isin N cup 0

8 The Scientific World Journal

If there exists 1198990isin N such that 119909

1198990= 1199091198990+1

then 1199091198990

=

1198911199091198990and so we have nothing to prove Hence for all 119899 isin N

we assume that 119909119899

= 119909119899+1

Step I (Cauchyness of 119909119899) As 120572(119909

119899 119909119899+1

119909119899+2

) ge 1 for all119899 ge 0 using condition (46) we obtain

119904119866120590119887

(119909119899 119909119899+1

119909119899+2

)

le 119904120572 (119909119899minus1

119909119899 119909119899+1

) 119866120590119887

(119909119899 119909119899+1

119909119899+2

)

= 119904120572 (119909119899minus1

119909119899 119909119899+1

) 119866120590119887

(119891119909119899minus1

119891119909119899 119891119909119899+1

)

le 120595 (119872119904(119909119899minus1

119909119899 119909119899+1

))

(48)

Using Proposition 15(2) as 119909119899

= 119909119899+1

we get

119872119904(119909119899minus1

119909119899 119909119899+1

)

= max 119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

(119866120590119887

(119909119899minus1

119909119899minus1

119891119909119899minus1

)

times 119866120590119887

(119909119899 119909119899 119891119909119899)

times119866120590119887

(119909119899+1

119909119899+1

119891119909119899+1

))

times(1 + 21199042

1198662

120590119887

(119891119909119899minus1

119891119909119899 119891119909119899+1

))

minus1

= max 119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

(119866120590119887

(119909119899minus1

119909119899minus1

119909119899)

times 119866120590119887

(119909119899 119909119899 119909119899+1

)

times119866120590119887

(119909119899+1

119909119899+1

119909119899+2

))

times(1 + 21199042

1198662

120590119887

(119909119899 119909119899+1

119909119899+2

))

minus1

le max119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

2119904119866120590119887

(119909119899minus1

119909119899 119909119899+1

) 1198662

120590119887

(119909119899 119909119899+1

119909119899+2

)

1 + 2s21198662120590119887

(119909119899 119909119899+1

119909119899+2

)

= 119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

(49)

Hence119866120590119887

(119909119899 119909119899+1

119909119899+2

)

le 119904119866120590119887

(119909119899 119909119899+1

119909119899+2

) le 120595 (119866120590119887

(119909119899minus1

119909119899 119909119899+1

))

(50)

By induction since 119909119899

= 119909119899+1

we get that

119866120590119887

(119909119899 119909119899+1

119909119899+2

)

le 120595 (119866120590119887

(119909119899minus1

119909119899 119909119899+1

))

le 1205952

(119866120590119887

(119909119899minus2

119909119899minus1

119909119899))

le sdot sdot sdot le 120595119899

(119866120590119887

(1199090 1199091 1199092))

(51)

Let 120576 gt 0 be arbitrary Then there exists a natural number119873such that

infin

sum

119899=119873

119904119899

120595119899

(119866120590119887

(1199090 1199091 1199092)) lt

120576

2119904

(52)

Let 119898 gt 119899 ge 119873 Then by the rectangular inequality andProposition 15(2) as 119909

119899= 119909119899+1

we get

119866120590119887

(119909119899 119909119898 119909119898)

le 119904119866120590119887

(119909119899 119909119899+1

119909119899+1

) + 1199042

119866120590119887

(119909119899+1

119909119899+2

119909119899+2

)

+ sdot sdot sdot + 119904119898minus119899minus1

119866120590119887

(119909119898minus1

119909119898 119909119898)

le 2119904 (119904119866120590119887

(119909119899 119909119899 119909119899+1

) + 1199042

119866120590119887

(119909119899+1

119909119899+1

119909119899+2

)

+ sdot sdot sdot + 119904119898minus119899minus1

119866120590119887

(119909119898minus1

119909119898minus1

119909119898))

le 2119904 (119904119866120590119887

(119909119899 119909119899+1

119909119899+2

) + 1199042

119866120590119887

(119909119899+1

119909119899+2

119909119899+3

)

+ sdot sdot sdot + 119904119898minus119899minus1

119866120590119887

(119909119898minus1

119909119898 119909119898+1

))

le 2119904

119898minus2

sum

119896=119899

119904119896minus119899+1

120595119896

(119866120590119887

(1199090 1199091 1199092))

le 2119904

infin

sum

119896=119899

119904119896

120595119896

(119866120590119887

(1199090 1199091 1199092)) lt 120576

(53)

Consequently 119909119899 is a 119866

120590119887-Cauchy sequence in 119883 Since

(119883 119866120590119887) is 119866120590119887-complete so there exists 119906 isin 119883 such that

lim119899rarrinfin

119866120590119887

(119906 119909119899 119909119899) = lim119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

= 119866120590119887

(119906 119906 119906) = 0

(54)

Step II Now we show that 119906 is a fixed point of 119891 Suppose tothe contrary that is 119891119906 = 119906 then we have 119866

120590119887(119906 119906 119891119906) gt 0

Let the part (a) of (ii) holdsUsing the rectangle inequality we get

119866120590119887

(119906 119906 119891119906) le 119904119866120590119887

(119891119906 119891119909119899 119891119909119899) + 119904119866

120590119887(119891119909119899 119906 119906)

(55)

Letting 119899 rarr infin and using the continuity of 119891 we get

119866120590119887

(119906 119906 119891119906) le 119904119866120590119887

(119891119906 119891119906 119891119906) (56)

From (46) and part (a) of condition (ii) we have

119904119866120590119887

(119891119906 119891119906 119891119906) le 120595 (119872119904(119906 119906 119906)) (57)

where by using (56) we have

119872119904(119906 119906 119906)

= max119866120590119887

(119906 119906 119906)

119866120590119887

(119906 119906 119891119906)119866120590119887

(119906 119906 119891119906)119866120590119887

(119906 119906 119891119906)

1 + 211990421198662

120590119887

(119891119906 119891119906 119891119906)

le 119866120590119887

(119906 119906 119891119906)

(58)

Hence from properties of 120595 119904119866120590119887(119891119906 119891119906 119891119906) le

119866120590119887(119906 119906 119891119906) Thus by (56) we obtain that

119866120590119887

(119906 119906 119891119906) = 119904119866120590119887

(119891119906 119891119906 119891119906) (59)

The Scientific World Journal 9

Moreover (57) yields that 119866120590119887(119906 119906 119891119906) le 120595(119872

119904(119906 119906 119906)) le

120595(119866120590119887(119906 119906 119891119906))This is impossible according to our assump-

tions on 120595 Hence we have 119891119906 = 119906 Thus 119906 is a fixed point of119891

Now let part (b) of (ii) holdsAs 119909119899 is a sequence such that 120572(119909

119899 119909119899+1

119909119899+2

) ge 1 for all119899 isin N cup 0 and 119909

119899rarr 119909 as 119899 rarr infin we have 120572(119909

119899 119909 119909) ge 1

for all 119899 isin N cup 0Now we show that 119906 = 119891119906 By (46) we have

119904119866120590119887

(119891119906 119891119906 119909119899) le 119904120572 (119906 119906 119909

119899minus1) 119866120590119887

(119891119906 119891119906 119891119909119899minus1

)

le 120595 (119872119904(119906 119906 119909

119899minus1))

(60)

where

119872119904(119906 119906 119909

119899minus1)

= max

119866120590119887

(119906 119906 119909119899minus1

)

[119866120590119887

(119906 119906 119891119906)]

2

119866120590119887

(119909119899minus1

119909119899minus1

119891119909119899minus1

)

1 + [119904119866120590119887

(119891119906 119891119906 119891119909119899minus1

)]

2

(61)

Letting 119899 rarr infin in the above inequality and using (54) andLemma 21 we get

lim119899rarrinfin

119872119904(119906 119906 119909

119899minus1) = 0 (62)

Again taking the upper limit as 119899 rarr infin in (60) and using(62) and Lemma 21 we obtain

119904 [

1

119904

119866120590119887

(119906 119891119906 119891119906) minus 119866120590119887

(119906 119906 119906)]

le 119904 lim sup119899rarrinfin

119866120590119887

(119909119899 119891119906 119891119906)

le lim supnrarrinfin

120595 (119872119904(119906 119906 119909

119899minus1))

= 120595(lim sup119899rarrinfin

119872119904(119906 119906 119909

119899minus1))

= 120595 (0) = 0

(63)

So we get 119866120590119887(119906 119891119906 119891119906) = 0 That is 119891119906 = 119906

Let (119883 119866120590119887 ⪯) be a partially ordered 119866

120590119887-metric-like

space We say that 119879 is an increasing mapping on 119883 ifforall119909 119910 isin 119883 119909 ⪯ 119910 rArr 119879(119909) ⪯ 119879(119910) [26] Fixed pointtheorems for monotone operators in ordered metric spacesarewidely investigated andhave found various applications indifferential and integral equations (see [27ndash30] and referencestherein) From the results proved above we derive thefollowing new results in partially ordered 119866

120590119887-metric-like

space

Theorem 34 Let (119883 119866120590119887 ⪯) be a partially ordered 119866

120590119887-

complete generalized 119887-metric-like space and let 119891 119883 rarr 119883

be an increasing mapping Suppose that

119904119866120590119887

(119891119909 119891119910 119891119911) le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911) (64)

for all 119909 119910 119911 isin 119883 with 119909 ⪯ 119910 ⪯ 119911 where

119872119904(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119891119909 119891119909)119866120590119887

(119910 119891119910 119891119910)119866120590119887

(119911 119891119911 119891119911)

1 + 11990421198662

120590119887

(119891119909 119891119910 119891119911)

(65)

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) 119891 is continuous or assume that whenever 119909119899 in 119883 is

an increasing sequence such that 119909119899rarr 119909 as 119899 rarr infin

one has 119909119899⪯ 119909 for all 119899 isin N cup 0

Then 119891 has a fixed point

Proof Define 120572 119883 times 119883 times 119883 rarr [0 +infin) by

120572 (119909 119910 119911) =

1 if119909 ⪯ 119910 ⪯ 119911

0 otherwise(66)

First we prove that 119891 is a triangular 120572-admissible mappingHence we assume that 120572(119909 119910 119911) ge 1 Therefore we have 119909 ⪯

119910 ⪯ 119911 Since 119891 is increasing we get 119891119909 ⪯ 119891119910 ⪯ 119891119911 that is120572(119891119909 119891119910 119891119911) ge 1 Also let 120572(119909 119910 119910) ge 1 and 120572(119910 119911 119911) ge 1then 119909 ⪯ 119910 and 119910 ⪯ 119911 Consequently we deduce that 119909 ⪯ 119911that is 120572(119909 119911 119911) ge 1 Thus 119891 is a triangular 120572-admissiblemapping Since 119891 satisfies (64) so by the definition of 120572 wehave

119904120572 (119909 119910 119911) 119866120590119887

(119891119909 119891119910 119891119911)

le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911)

(67)

for all 119909 119910 119911 isin 119883 Therefore 119891 satisfies the contractivecondition (21) From (i) there exists 119909

0isin 119883 such that

1199090⪯ 1198911199090 that is 120572(119909

0 1198911199090 1198911199090) ge 1 According to (ii) we

conclude that all the conditions of Theorems 26 and 27 aresatisfied and so 119891 has a fixed point

Similarly using Theorem 33 we can prove followingresult

Theorem 35 Let (119883 119866120590119887 ⪯) be an ordered 119866

120590119887-complete

generalized 119887-metric-like space and let 119891 119883 rarr 119883 be anincreasing mapping Suppose that

119904119866120590119887

(119891119909 119891119910 119891119911) le 120595 (119872119904(119909 119910 119911)) (68)

for all 119909 119910 119911 isin 119883 with 119909 ⪯ 119910 ⪯ 119911 where 120595 isin Ψ119887and

119872119904(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119909 119891119909)119866120590119887

(119910 119910 119891119910)119866120590119887

(119911 119911 119891119911)

1 + 211990421198662

120590119887

(119891119909 119891119910 119891119911)

(69)

10 The Scientific World Journal

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii)

(a) 119891 is continuous(b) assume that whenever x

119899 in 119883 is an increasing

sequence such that 119909119899rarr 119909 as 119899 rarr infin one has

119909119899⪯ 119909 for all 119899 isin N cup 0

Then 119891 has a fixed point

We conclude this section by presenting some examplesthat illustrate our results

Example 36 Let 119883 = [0 (minus1 + radic5)2] be endowed withthe usual ordering onR Define the generalized 119887-metric-likefunction 119866

120590119887given by

119866120590119887

(119909 119910 119911) = max (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

(70)

with 119904 = 2 Consider the mapping 119891 119883 rarr 119883 defined by119891(119909) = (14)119909

radic119890minus(119909+((minus1+radic5)2))

2 and the function 120573 isin Fgiven by 120573(119905) = (12)119890

minus119905 119905 gt 0 and 120573(0) isin [0 12) Itis easy to see that 119891 is an increasing function on 119883 Weshow that 119891 is 119866

120590119887-continuous on 119883 By Proposition 20 it is

sufficient to show that119891 is119866120590119887-sequentially continuous on119883

Let 119909119899 be a sequence in119883 such that lim

119899rarrinfin119866120590119887(119909119899 119909 119909) =

119866120590119887(119909 119909 119909) so we have max(lim

119899rarrinfin119909119899

+ 119909)2

41199092 =

41199092 and equivalently lim

119899rarrinfin119909119899

= 120572 le 119909 On the otherhand we have

lim119899rarrinfin

119866120590119887

(119891119909119899 119891119909 119891119909)

= lim119899rarrinfin

max (119891119909119899+ 119891119909)

2

41198912

119909

= lim119899rarrinfin

max 1

16

(119909119899

radic119890minus(119909119899+(minus1+

radic5)2)2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= max 1

16

( lim119899rarrinfin

119909119899

radic119890minus(lim119899rarrinfin119909119899+(minus1+radic5)2)

2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= max 1

16

(120572radic119890minus(120572+(minus1+radic5)2)

2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

=

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= 119866120590119887

(119891119909 119891119909 119891119909)

(71)

So 119891 is 119866120590119887-sequentially continuous on119883

For all elements 119909 119910 119911 isin 119883 and the fact that 119892(119909) =

1199092119890minus1199092

is an increasing function on119883 we have

119904119866120590119887

(119891119909 119891119910 119891119911)

= 2max (119891119909 + 119891119910)2

(119891119910 + 119891119911)2

(119891119911 + 119891119909)2

= 2max(

1

4

119909radic119890minus(119909+(minus1+radic5)2)

2

+

1

4

119910radic119890minus(119910+(minus1+radic5)2)

2

)

2

(

1

4

119910radic119890minus(119910+(minus1+radic5)2)

2

+

1

4

119911radic119890minus(119911+(minus1+radic5)2)

2

)

2

(

1

4

119911radic119890minus(119911+(minus1+radic5)2)

2

+

1

4

119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

le 2max(

1

4

119909radic119890minus(119909+119910)

2

+

1

4

119910radic119890minus(119910+119909)

2

)

2

(

1

4

119910radic119890minus(119910+119911)

2

+

1

4

119911radic119890minus(119911+119910)

2

)

2

(

1

4

119911radic119890minus(119911+119909)

2

+

1

4

119909radic119890minus(119909+119911)

2

)

2

=

1

8

max (119909 + 119910)2

119890minus(119909+119910)

2

(119910 + 119911)2

119890minus(119910+119911)

2

(119911 + 119909)2

119890minus(119911+119909)

2

=

1

8

119890minusmax(119909+119910)2 (119910+119911)2 (119911+119909)2

timesmax (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

le

1

2

eminusmax(119909+119910)2 (119910+119911)2(119911+119909)2

timesmax (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

= 120573 (119866120590119887

(119909 119910 119911)) 119866120590119887

(119909 119910 119911)

le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911)

(72)

Hence119891 satisfies all the assumptions ofTheorem 34 and thusit has a fixed point (which is 119906 = 0)

Example 37 Let 119883 = [0 1] with the usual ordering on RDefine the generalized 119887-metric-like function 119866

120590119887given by

119866120590119887(119909 119910 119911) = max(119909 + 119910)

2

(119910 + 119911)2

(119911 + 119909)2

with 119904 = 2Consider the mapping 119891 119883 rarr 119883 defined by 119891(119909) =

The Scientific World Journal 11

ln(1 + 119909119890minus119909

4) and the function 120595 isin Ψ119887given by 120595(119905) =

(18)119905 119905 ge 0 It is easy to see that 119891 is increasing functionNow we show that 119891 is a 119866

120590119887-continuous function on119883

Let 119909119899 be a sequence in 119883 such that lim

119899rarrinfin119866120590119887(119909119899

119909 119909) = 119866120590119887(119909 119909 119909) so we havemax(lim

119899rarrinfin119909119899+119909)2

41199092 =

41199092 and equally lim

119899rarrinfin119909119899= 120572 le 119909 On the other hand we

have

lim119899rarrinfin

119866120590119887

(119891119909119899 119891119909 119891119909)

= lim119899rarrinfin

max (119891119909119899+ 119891119909)

2

41198912

119909

= max(ln(1 +

lim119899rarrinfin

119909119899119890minuslim119899rarrinfin119909119899

4

)

+ ln(1 +

119909119890minus119909

4

))

2

4(ln(1 +

119909119890minusx

4

))

2

= max(ln(1 +

120572119890minus120572

4

) + ln(1 +

119909119890minus119909

4

))

2

4(ln(1 +

119909119890minus119909

4

))

2

= 4(ln(1 +

119909119890minus119909

4

))

2

= 119866120590119887

(119891119909 119891119909 119891119909)

(73)

So 119891 is 119866120590119887-sequentially continuous on119883

For all elements 119909 119910 119911 isin 119883 we have

119904119866120590119887

(119891119909 119891119910 119891119911)

= 2max(ln(1 +

119909119890minus119909

4

) + ln(1 +

119910119890minus119910

4

))

2

(ln(1 +

119910119890minus119910

4

) + ln(1 +

119911119890minus119911

4

))

2

(ln(1 +

119911119890minus119911

4

) + ln(1 +

119909119890minus119909

4

))

2

le 2max(

119909119890minus119909

4

+

119910119890minus119910

4

)

2

(

119910119890minus119910

4

+

119911119890minus119911

4

)

2

(

119911119890minus119911

4

+

119909119890minus119909

4

)

2

le

1

8

max (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

= 120595 (119866120590119887

(119909 119910 119911))

le 120595 (119872119904(119909 119910 119911))

(74)

Hence119891 satisfies all the assumptions ofTheorem 35 and thusit has a fixed point (which is 119906 = 0)

3 Application

In this section we present an application of our results toestablish the existence of a solution to a periodic boundaryvalue problem (see [30 31])

Let 119883 = 119862([0 119879]) be the set of all real continuousfunctions on [0T] We first endow119883 with the 119887-metric-like

120590119887(119906 V) = sup

119905isin[0119879]

(|119906 (119905)| + |V (119905)|)119901 (75)

for all 119906 V isin 119883 where 119901 gt 1 and then we endow it with thegeneralized 119887-metric-like 119866

120590119887defined by

119866120590119887

(119906 V 119908)

= max sup119905isin[0119879]

(|119906 (119905)| + |V (119905)|)119901

sup119905isin[0119879]

(|V (119905)| + |119908 (119905)|)119901

sup119905isin[0119879]

(|119908 (119905)| + |119906 (119905)|)119901

(76)

Clearly (119883 119866120590119887) is a complete generalized 119887-metric-like space

with parameter 119870 = 2119901minus1 We equip 119862([0 119879]) with a partial

order given by119909 ⪯ 119910 iff119909 (119905) le 119910 (119905) forall119905 isin [0 119879] (77)

Moreover as in [30] it is proved that (119862([0 119879]) ⪯) is regularthat is whenever 119909

119899 in119883 is an increasing sequence such that

119909119899rarr 119909 as 119899 rarr infin we have 119909

119899⪯ 119909 for all 119899 isin N cup 0

Consider the first-order periodic boundary value prob-lem

1199091015840

(119905) = 119891 (119905 119909 (119905)) 119909 (0) = 119909 (119879) (78)

where 119905 isin 119868 = [0 119879] with 119879 gt 0 and 119891 119868 times R rarr R is acontinuous function

A lower solution for (78) is a function 120572 isin 1198621[0 119879] such

that1205721015840

(119905) le 119891 (119905 120572 (119905))

120572 (0) le 120572 (119879)

(79)

where 119905 isin 119868 = [0 119879]Assume that there exists 120582 gt 0 such that for all 119909 119910 isin

119862[0 119879] we have1003816100381610038161003816119891 (119905 119909 (119905)) + 120582119909 (119905)

1003816100381610038161003816+1003816100381610038161003816119891 (119905 119910 (119905)) + 120582119910 (119905)

1003816100381610038161003816

le

120582

2119901minus1

119901radicln(

119886

2119901minus1

(|119909 (119905)| +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)119901

+ 1)

(80)

where 0 le 119886 lt 1 Then the existence of a lower solution for(78) provides the existence of a solution of (78)

Problem (78) can be rewritten as

1199091015840

(119905) + 120582119909 (119905) = 119891 (119905 119909 (119905)) + 120582119909 (119905)

119909 (0) = 119909 (119879)

(81)

Consider1199091015840

(119905) + 120582119909 (119905) = 120575119909(119905) = 119865 (119905 119909 (119905))

119909 (0) = 119909 (119879)

(82)

where 119905 isin 119868

12 The Scientific World Journal

Using variation of parameters formula we get

119909 (119905) = 119909 (0) 119890minus120582119905

+ int

119905

0

119890minus120582(119905minus119904)

120575119909(119904) 119889119904 (83)

which yields

119909 (119879) = 119909 (0) 119890minus120582119879

+ int

119879

0

119890minus120582(119879minus119904)

120575119909(119904) 119889119904 (84)

Since 119909(0) = 119909(119879) we get

119909 (0) [1 minus 119890minus120582119879

] = 119890minus120582119879

int

119879

0

119890120582(119904)

120575119909(119904) 119889119904 (85)

or

119909 (0) =

1

119890120582119879

minus 1

int

119879

0

119890120582119904

120575119909(119904) 119889119904 (86)

Substituting the value of 119909(0) in (83) we arrive at

119909 (119905) = int

119879

0

119866 (119905 119904) 120575119909(119904) 119889119904 (87)

where

119866 (119905 119904) =

119890120582(119879+119904minus119905)

119890120582119879

minus 1

0 le 119904 le 119905 le 119879

119890120582(119904minus119905)

119890120582119879

minus 1

0 le 119905 le 119904 le 119879

(88)

Now define the operator 119878 119862[0 119879] rarr 119862[0 119879] as

119878119909 (119905) = int

119879

0

119866 (119905 119904) 119865 (119904 119909 (119904)) 119889119904 (89)

The mapping 119878 is increasing [31] Note that if 119906 isin 119862[0 119879] is afixed point of 119878 then 119906 isin 119862

1[0 119879] is a solution of (78)

Let 119909 119910 119911 isin 119862[0 119879] Then we have

2119901minus1

[|119878119909 (119905)| +1003816100381610038161003816119878119910 (119905)

1003816100381610038161003816]

= 2119901minus1

[

100381610038161003816100381610038161003816100381610038161003816

int

119879

0

119866 (119905 119904) 119865 (119904 119909 (119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

119879

0

119866 (119905 119904) 119865 (119904 119910 (119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

]

le 2119901minus1

int

119879

0

|119866 (119905 119904)|

times [|119865 (119904 119909 (119904))| +1003816100381610038161003816119865 (119904 119910 (119904))

1003816100381610038161003816] 119889119904

le 2119901minus1

int

119879

0

|119866 (119905 119904)|

120582

2119901minus1

times119901radicln(

119886

2119901minus1

(|119909 (119905)| +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)119901

+ 1)119889119904

le 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [int

119905

0

119890120582(119879+119904minus119905)

119890120582119879

minus 1

119889119904 + int

119879

119905

119890120582(119904minus119905)

119890120582119879

minus 1

119889119904]

= 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [

1

120582 (119890120582119879

minus 1)

(119890120582(119879+119904minus119905)10038161003816

100381610038161003816

119905

0

+ 119890120582(119904minus119905)10038161003816

100381610038161003816

119879

119905

)]

= 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [

1

120582 (119890120582119879

minus 1)

(119890120582119879

minus 119890120582(119879minus119905)

+ 119890120582(119879minus119905)

minus 1)]

=119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

le119901radicln(

119886

2119901minus1

119866120590119887

(119909 119910 119911) + 1)

le119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(90)

Similarly

2119901minus1 1003816

100381610038161003816119878119910 (119905)

1003816100381610038161003816+ |119878119911 (119905)| le

119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

|119878119909 (119905)| + |119878119911 (119905)| le119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(91)

where

119872(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119878119909 119878119909) 119866120590119887

(119910 119878119910 119878119910)119866120590119887

(119911 119878119911 119878119911)

1 + 22119901minus1

1198662

120590119887

(119878119909 119878119910 119878119911)

(92)

Equivalently

(2119901minus1

|119878119909 (119905)| +1003816100381610038161003816119878119910 (119905)

1003816100381610038161003816)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(2119901minus1 1003816

100381610038161003816119878119910 (119905)

1003816100381610038161003816+ |119878119911 (119905)|)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(2119901minus1

|119878119909 (119905)| + |119878119911 (119905)|)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(93)

which yields that

2119901minus1

120590119887(119878119909 119878119910) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

120590119887(119878119910 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

120590119887(119878119909 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(94)

The Scientific World Journal 13

Finally it is easy to obtain that

2119901minus1G120590119887

(119878119909 119878119910 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1) (95)

Finally since120572 is a lower solution for (78) so it is easy to showthat 120572 ⪯ 119878(120572) [31]

Hence the hypotheses of Theorem 35 are satisfied with120595(119905) = ln((1198862119901minus1)119905 + 1) where 0 le 119886 lt 1 Hence there existsa fixed point 119909 isin 119862[0 119879] such that 119878119909 = 119909 which is a solutionto periodic boundary value problem (78)

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) KingAbdulaziz University JeddahTherefore the firstauthor acknowledges with thanks DSR KAU for financialsupport

References

[1] S Czerwik ldquoContraction mappings in b-metric spacesrdquo ActaMathematica et Informatica Universitatis Ostraviensis vol 1 pp5ndash11 1993

[2] V Parvaneh J R Roshan and S Radenovic ldquoExistence of tri-pled coincidence points in ordered b-metric spaces and an app-lication to a system of integral equationsrdquo Fixed Point Theoryand Applications vol 2013 article 130 2013

[3] J R Roshan V Parvaneh S Sedghi N Shobkolaei and WShatanawi ldquoCommon fixed points of almost generalized con-tractive mappings in ordered b-metric spacesrdquo Fixed PointTheory and Applications vol 2013 article 159 2013

[4] H Aydi M F Bota E Karapınar and S Moradi ldquoA commonfixed point for weak 120601-contractions on b-metric spacesrdquo FixedPoint Theory vol 13 no 2 pp 337ndash346 2012

[5] V Berinde ldquoSequences of operators and fixed points in quasi-metric spacesrdquo Studia Universitatis ldquoBabes-Bolyairdquo Mathemat-ica vol 16 no 4 pp 23ndash27 1996

[6] N Hussain D ETHoric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 article 126 2012

[7] N Hussain and M H Shah ldquoKKMmappings in cone b-metricspacesrdquo Computers and Mathematics with Applications vol 62no 4 pp 1677ndash1684 2011

[8] N Hussain J R Roshan V Parvaneh and Z Kadelburg ldquoFixedpoints of contractive mappings in b-metric-like spacesrdquo TheScientific World Journal vol 2014 Article ID 471827 15 pages2014

[9] M A Khamsi and N Hussain ldquoKKMmappings in metric typespacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 73 no 9 pp 3123ndash3129 2010

[10] M A Khamsi ldquoRemarks on cone metric spaces and fixed pointtheorems of contractive mappingsrdquo Fixed Point Theory andApplications vol 2010 Article ID 315398 7 pages 2010

[11] Z Mustafa J R Roshan V Parvaneh and Z Kadelburg ldquoSomecommon fixed point results in ordered partial b-metric spacesrdquoJournal of Inequalities and Applications vol 2013 article 5622013

[12] Z Mustafa and B Sims ldquoA new approach to generalized metricspacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash297 2006

[13] A Amini-Harandi ldquoMetric-like spaces partial metric spacesand fixed pointsrdquo Fixed PointTheory and Applications vol 2012article 204 2012

[14] N Shobkolaei S Sedghi J R Roshan and N Hussain ldquoSuzukitype fixed point results in metric-like spacesrdquo Journal of Func-tion Spaces and Applications vol 2013 Article ID 143686 9pages 2013

[15] M A Alghamdi N Hussain and P Salimi ldquoFixed point andcoupled fixed point theorems on b-metric-like spacesrdquo Journalof Inequalities and Applications vol 2013 article 402 2013

[16] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572 minus 120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

[17] P Salimi and E Karapınar ldquoSuzuki-Edelstein type contractionsvia auxiliary functionsrdquoMathematical Problems in Engineeringvol 2013 Article ID 648528 8 pages 2013

[18] P Salimi C Vetro and P Vetro ldquoFixed point theorems fortwisted (120572 120573)-120595-contractive type mappings and applicationsrdquoFilomat vol 27 no 4 pp 605ndash615 2013

[19] P Salimi C Vetro and P Vetro ldquoSome new fixed point resultsin non-Archimedean fuzzy metric spacesrdquo Nonlinear AnalysisModelling and Control vol 18 no 3 pp 344ndash358 2013

[20] N Hussain P Salimi and A Latif ldquoFixed point results forsingle and set-valued 120572-120578-120595-contractive mappingsrdquo Fixed PointTheory and Applications vol 2013 article 212 2013

[21] M A Alghamdi and E Karapınar ldquoG-120573-120595 contractive-typemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 article 70 2013

[22] M Geraghty ldquoOn contractive mappingsrdquo Proceedings of theAmerican Mathematical Society vol 40 pp 604ndash608 1973

[23] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001

[24] N Hussain Z Kadelburg S Radenovic and F R Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012

[25] V Berinde ldquoGeneralized contractions in quasimetric spacesrdquo inSeminar on Fixed Point Theory Preprint no 3 pp 3ndash9 1993

[26] L Ciric M Abbas R Saadati and N Hussain ldquoCommon fixedpoints of almost generalized contractive mappings in orderedmetric spacesrdquo Applied Mathematics and Computation vol 217no 12 pp 5784ndash5789 2011

[27] R P Agarwal N Hussain and M A Taoudi ldquoFixed point the-orems in ordered Banach spaces and applications to nonlinearintegral equationsrdquo Abstract and Applied Analysis vol 2012Article ID 245872 15 pages 2012

[28] N Hussain A R Khan and R P Agarwal ldquoKrasnoselrsquoskii andKy Fan type fixed point theorems in ordered Banach spacesrdquoJournal of Nonlinear and Convex Analysis vol 11 no 3 pp 475ndash489 2010

[29] N Hussain and M A Taoudi ldquoKrasnoselrsquoskii-type fixed pointtheorems with applications to Volterra integral equationsrdquoFixed Point Theory and Applications vol 2013 article 196 2013

14 The Scientific World Journal

[30] J J Nieto and R Rodrıguez-Lopez ldquoContractivemapping theo-rems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[31] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysisTheoryMethods andApplications vol 71 no 7-8 pp 3403ndash3410 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 The Scientific World Journal

Letting119898 119899 rarr infin in the above inequality we get

lim sup119898119899rarrinfin

119872119904(119909119899 119909119898 119909119898) = lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898) (31)

Hence from (29) and (31) we obtain

lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

le 119904lim sup119898119899rarrinfin

120573 (119866120590119887

(119909119899 119909119898 119909119898)) lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

(32)

If lim sup119898119899rarrinfin

119866120590119887(119909119899 119909119898 119909119898) = 0 then we get

1

119904

le lim sup119898119899rarrinfin

120573 (119866120590119887

(119909119899 119909119898 119909119898)) (33)

Since 120573 isin F we deduce that

lim sup119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898) = 0 (34)

which is a contradiction Consequently 119909119899 is a 119866

120590119887-Cauchy

sequence in119883 Since (119883 119866120590119887) is119866120590119887-complete there exists 119906 isin

119883 such that 119909119899

rarr 119906 as 119899 rarr infin Now from (34) and 119866120590119887-

completeness of119883

lim119899rarrinfin

119866120590119887

(119906 119909119899 119909119899) = lim119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

= 119866120590119887

(119906 119906 119906) = 0

(35)

Step III Now we show that 119906 is a fixed point of 119891Using the rectangle inequality we get

119866120590119887

(119906 119891119906 119891119906) le 119904119866120590119887

(119906 119891119909119899 119891119909119899) + 119904119866

120590119887(119891119909119899 119891119906 119891119906)

(36)

Letting 119899 rarr infin and using the continuity of 119891 and (35) weobtain

119866120590119887

(119906 119891119906 119891119906) le 119904 lim119899rarrinfin

G120590119887

(119906 119891119909119899 119891119909119899)

+ 119904 lim119899rarrinfin

119866120590119887

(119891119909119899 119891119906 119891119906)

= 119904119866120590119887

(119891119906 119891119906 119891119906)

(37)

Note that from (21) as 120572(119906 119906 119906) ge 1 we have

119904119866120590119887

(119891119906 119891119906 119891119906) le 120573 (119866120590119887

(119906 119906 119906))119872119904(119906 119906 119906) (38)

where by (37)

119872119904(119906 119906 119906)

= max119866120590119887

(119906 119906 119906)

119866120590119887

(119906 119891119906 119891119906)119866120590119887

(119906 119891119906 119891119906)119866120590119887

(119906 119891119906 119891119906)

1 + 11990421198662

120590119887

(119891119906 119891119906 119891119906)

lt 119866120590119887

(119906 119891119906 119891119906)

(39)

Hence as 120573(119905) le 1 for all 119905 isin [0infin) we have 119904119866120590119887(119891119906

119891119906 119891119906) le 119866120590119887(119906 119891119906 119891119906) Thus by (37) we obtain that

119904119866120590119887(119891119906 119891119906 119891119906) = 119866

120590119887(119906 119891119906 119891119906) But then using (38) we

get that

119866120590119887

(119906 119891119906 119891119906) = 119904119866120590119887

(119891119906 119891119906 119891119906)

le 120573 (119866120590119887

(119906 119906 119906))119872119904(119906 119906 119906)

lt

1

119904

119866120590119887

(119906 119891119906 119891119906)

(40)

which is a contradiction Hence we have 119891119906 = 119906 Thus 119906 is afixed point of 119891

We replace condition (ii) in Theorem 26 by regularity ofthe space119883

Theorem 27 Under the same hypotheses of Theorem 26instead of condition (119894119894) assume that whenever 119909

119899 in 119883 is a

sequence such that 120572(119909119899 119909119899+1

119909119899+1

) ge 1 for all 119899 isin Ncup0 and119909119899rarr 119909 as 119899 rarr infin one has 120572(119909

119899 119909 119909) ge 1 for all 119899 isin Ncup0

Then 119891 has a fixed point

Proof Repeating the proof of Theorem 26 we can constructa sequence 119909

119899 in 119883 such that 120572(119909

119899 119909119899+1

119909119899+1

) ge 1 for all119899 isin N cup 0 and 119909

119899rarr 119906 isin 119883 for some 119906 isin 119883 Using the

assumption on 119883 we have 120572(119909119899 119906 119906) ge 1 for all 119899 isin N cup 0

Now we show that 119906 = 119891119906 By Lemma 21 and (35)

119904 [

1

119904

119866120590119887

(119906 119891119906 119891119906) minus 119866120590119887

(119906 119906 119906)]

le 119904lim sup119899rarrinfin

119866120590119887

(119909119899+1

119891119906 119891119906)

le lim sup119899rarrinfin

(120573 (119866120590119887

(119909119899 119906 119906))119872

119904(119909119899 119906 119906))

le

1

119904

lim sup119899rarrinfin

119872119904(119909119899 119906 119906)

(41)

wherelim119899rarrinfin

119872119904(119909119899 119906 119906)

= lim119899rarrinfin

max

119866120590119887

(119909119899 119906 119906)

[119866120590119887

(119909119899 119891119909119899 119891119909119899)] [119866120590119887

(119906 119891119906 119891119906)]

2

1 + 11990421198662

120590119887

(119891119909119899 119891119906 119891119906)

= lim119899rarrinfin

max

119866120590119887

(119909119899 119906 119906)

[119866120590119887

(119909119899 119909119899+1

119909119899+1

)] [119866120590119887

(119906 119891119906 119891119906)]

2

1 + 11990421198662

120590119887

(119909119899+1

119891119906 119891119906)

= max 119866120590119887

(119906 119906 119906) 0 = 0 (see (25) and (35))

(42)

Therefore we deduce that119866120590119887(119906 119891119906 119891119906) le 0 Hence we have

119906 = 119891119906

The Scientific World Journal 7

A mapping 120593 [0infin) rarr [0infin) is called a comparisonfunction if it is increasing and 120593

119899(119905) rarr 0 as 119899 rarr infin for any

119905 isin [0infin) (see eg [23 24] for more details and examples)

Definition 28 A function 120593 [0infin) rarr [0infin) is said to bea (119888)-comparison function if

(1198881) 120593 is increasing

(1198882) there exists 119896

0isin N 119886 isin (0 1) and a convergent series

of nonnegative terms suminfin

119896=1V119896such that 120593

119896+1(119905) le

119886120593119896(119905) + V

119896for 119896 ge 119896

0and any 119905 isin [0infin)

Later Berinde [5] introduced the notion of (119887)-comparison function as a generalization of (119888)-comparisonfunction

Definition 29 (see [5]) Let 119904 ge 1 be a real number Amapping120593 [0infin) rarr [0infin) is called a (119887)-comparison function ifthe following conditions are fulfilled

(1) 120593 is increasing

(2) there exist 1198960isin N 119886 isin (0 1) and a convergent series

of nonnegative terms suminfin119896=1

V119896such that 119904119896+1120593119896+1(119905) le

119886119904119896120593119896(119905) + V

119896for 119896 ge 119896

0and any 119905 isin [0infin)

Let Ψ119887be the class of all (119887)-comparison functions 120593

[0infin) rarr [0infin) It is clear that the notion of (119887)-comparison function coincideswith (119888)-comparison functionfor 119904 = 1

Lemma 30 (see [25]) If 120593 [0infin) rarr [0infin) is a (119887)-comparison function then we have the following

(1) the series suminfin119896=0

119904119896120593119896(119905) converges for any 119905 isin R

+

(2) the function 119887119904 [0infin) rarr [0infin) defined by 119887

119904(119905) =

suminfin

119896=0119904119896120593119896(119905) 119905 isin [0infin) is increasing and continuous

at 0

Remark 31 It is easy to see that if 120593 isin Ψ119887 then we have 120593(0) =

0 and 120593(119905) lt 119905 for each 119905 gt 0 and 120593 is continuous at 0

In the next example we present a class of (119887)-comparisonfunctions

Example 32 Any function of the form120595(119905) = ln((119886119904)119905+1) forall 119905 isin [0infin) where 0 lt 119886 lt 1 is a (119887)-comparison function

Proof From the part (1) of Lemma 30 the necessary condi-tion is that the series suminfin

119896=0119904119896120593119896(119905) converges for any 119905 isin R

+

But for each 119905 gt 0 and 119896 ge 1 we have

119904119896

120593119896

(119905) = 119904119896

120593 (120593119896minus1

(119905))

= 119904119896 ln(

119886

119904

120593119896minus1

(119905) + 1) le 119904119896 119886

119904

120593119896minus1

(119905)

le sdot sdot sdot le 119904119896

(

119886

119904

)

119896

119905 = 119886119896

119905

(43)

So according to the comparison test of the series we shouldhave 119886 lt 1 On the other hand we have

119904119896+1

120593119896+1

(119905) = 119904119896+1

120593 (120593119896

(119905))

= 119904119896+1 ln(

119886

119904

120593119896

(119905) + 1)

le 119904119896+1 119886

119904

120593119896

(119905) = 119904119896

119886120593119896

(119905)

(44)

Therefore for any convergent series of nonnegative termssuminfin

119896=1V119896and each 119896 ge 119896

0= 1 we have

119904119896+1

120593119896+1

(119905) le 119886119904119896

120593119896

(119905) le 119886119904119896

120593119896

(119905) + V119896 (45)

For example for 119904 = 2 and 119886 = 12 the function 120595(119905) =

ln(1199054 + 1) is a (119887)-comparison function

Theorem 33 Let (119883 119866120590119887) be a 119866

120590119887-complete generalized 119887-

metric-like space and let 119891 119883 rarr 119883 be a 119866-120572-admissiblemapping Suppose that

119904120572 (119909 119910 119911) 119866120590119887

(119891119909 119891119910 119891119911) le 120595 (119872119904(119909 119910 119911)) (46)

for all 119909 119910 119911 isin 119883 where 120595 isin Ψ119887and

119872119904(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119909 119891119909)119866120590119887

(119910 119910 119891119910)119866120590119887

(119911 119911 119891119911)

1 + 211990421198662

120590119887

(119891119909 119891119910 119891119911)

(47)

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 120572(119909

0 1198911199090 11989121199090) ge 1

(ii)

(a) 119891 is continuous and for any sequence 119909119899 in119883

with 120572(119909119899 119909119899+1

119909119899+2

) ge 1 for all 119899 isin Ncup0 suchthat 119909

119899rarr 119909 as 119899 rarr infin one has 120572(119909 119909 119909) ge 1

for all 119899 isin N cup 0(b) assume that whenever 119909

119899 in 119883 is a sequence

such that 120572(119909119899 119909119899+1

119909119899+2

) ge 1 for all 119899 isin Ncup0

and 119909119899rarr 119909 as 119899 rarr infin one has 120572(119909

119899 119909 119909) ge 1

for all 119899 isin N cup 0

Then 119891 has a fixed point

Proof Let 1199090isin 119883 be such that 120572(119909

0 1198911199090 11989121199090) ge 1 Define

a sequence 119909119899 by 119909

119899= 1198911198991199090for all 119899 isin N Since 119891 is a 119866-120572-

admissible mapping and 120572(1199090 1199091 1199092) = 120572(119909

0 1198911199090 11989121199090) ge

1 we deduce that 120572(1199091 1199092 1199093) = 120572(119891119909

0 1198911199091 1198911199092) ge 1

Continuing this process we get 120572(119909119899 119909119899+1

119909119899+2

) ge 1 for all119899 isin N cup 0

8 The Scientific World Journal

If there exists 1198990isin N such that 119909

1198990= 1199091198990+1

then 1199091198990

=

1198911199091198990and so we have nothing to prove Hence for all 119899 isin N

we assume that 119909119899

= 119909119899+1

Step I (Cauchyness of 119909119899) As 120572(119909

119899 119909119899+1

119909119899+2

) ge 1 for all119899 ge 0 using condition (46) we obtain

119904119866120590119887

(119909119899 119909119899+1

119909119899+2

)

le 119904120572 (119909119899minus1

119909119899 119909119899+1

) 119866120590119887

(119909119899 119909119899+1

119909119899+2

)

= 119904120572 (119909119899minus1

119909119899 119909119899+1

) 119866120590119887

(119891119909119899minus1

119891119909119899 119891119909119899+1

)

le 120595 (119872119904(119909119899minus1

119909119899 119909119899+1

))

(48)

Using Proposition 15(2) as 119909119899

= 119909119899+1

we get

119872119904(119909119899minus1

119909119899 119909119899+1

)

= max 119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

(119866120590119887

(119909119899minus1

119909119899minus1

119891119909119899minus1

)

times 119866120590119887

(119909119899 119909119899 119891119909119899)

times119866120590119887

(119909119899+1

119909119899+1

119891119909119899+1

))

times(1 + 21199042

1198662

120590119887

(119891119909119899minus1

119891119909119899 119891119909119899+1

))

minus1

= max 119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

(119866120590119887

(119909119899minus1

119909119899minus1

119909119899)

times 119866120590119887

(119909119899 119909119899 119909119899+1

)

times119866120590119887

(119909119899+1

119909119899+1

119909119899+2

))

times(1 + 21199042

1198662

120590119887

(119909119899 119909119899+1

119909119899+2

))

minus1

le max119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

2119904119866120590119887

(119909119899minus1

119909119899 119909119899+1

) 1198662

120590119887

(119909119899 119909119899+1

119909119899+2

)

1 + 2s21198662120590119887

(119909119899 119909119899+1

119909119899+2

)

= 119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

(49)

Hence119866120590119887

(119909119899 119909119899+1

119909119899+2

)

le 119904119866120590119887

(119909119899 119909119899+1

119909119899+2

) le 120595 (119866120590119887

(119909119899minus1

119909119899 119909119899+1

))

(50)

By induction since 119909119899

= 119909119899+1

we get that

119866120590119887

(119909119899 119909119899+1

119909119899+2

)

le 120595 (119866120590119887

(119909119899minus1

119909119899 119909119899+1

))

le 1205952

(119866120590119887

(119909119899minus2

119909119899minus1

119909119899))

le sdot sdot sdot le 120595119899

(119866120590119887

(1199090 1199091 1199092))

(51)

Let 120576 gt 0 be arbitrary Then there exists a natural number119873such that

infin

sum

119899=119873

119904119899

120595119899

(119866120590119887

(1199090 1199091 1199092)) lt

120576

2119904

(52)

Let 119898 gt 119899 ge 119873 Then by the rectangular inequality andProposition 15(2) as 119909

119899= 119909119899+1

we get

119866120590119887

(119909119899 119909119898 119909119898)

le 119904119866120590119887

(119909119899 119909119899+1

119909119899+1

) + 1199042

119866120590119887

(119909119899+1

119909119899+2

119909119899+2

)

+ sdot sdot sdot + 119904119898minus119899minus1

119866120590119887

(119909119898minus1

119909119898 119909119898)

le 2119904 (119904119866120590119887

(119909119899 119909119899 119909119899+1

) + 1199042

119866120590119887

(119909119899+1

119909119899+1

119909119899+2

)

+ sdot sdot sdot + 119904119898minus119899minus1

119866120590119887

(119909119898minus1

119909119898minus1

119909119898))

le 2119904 (119904119866120590119887

(119909119899 119909119899+1

119909119899+2

) + 1199042

119866120590119887

(119909119899+1

119909119899+2

119909119899+3

)

+ sdot sdot sdot + 119904119898minus119899minus1

119866120590119887

(119909119898minus1

119909119898 119909119898+1

))

le 2119904

119898minus2

sum

119896=119899

119904119896minus119899+1

120595119896

(119866120590119887

(1199090 1199091 1199092))

le 2119904

infin

sum

119896=119899

119904119896

120595119896

(119866120590119887

(1199090 1199091 1199092)) lt 120576

(53)

Consequently 119909119899 is a 119866

120590119887-Cauchy sequence in 119883 Since

(119883 119866120590119887) is 119866120590119887-complete so there exists 119906 isin 119883 such that

lim119899rarrinfin

119866120590119887

(119906 119909119899 119909119899) = lim119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

= 119866120590119887

(119906 119906 119906) = 0

(54)

Step II Now we show that 119906 is a fixed point of 119891 Suppose tothe contrary that is 119891119906 = 119906 then we have 119866

120590119887(119906 119906 119891119906) gt 0

Let the part (a) of (ii) holdsUsing the rectangle inequality we get

119866120590119887

(119906 119906 119891119906) le 119904119866120590119887

(119891119906 119891119909119899 119891119909119899) + 119904119866

120590119887(119891119909119899 119906 119906)

(55)

Letting 119899 rarr infin and using the continuity of 119891 we get

119866120590119887

(119906 119906 119891119906) le 119904119866120590119887

(119891119906 119891119906 119891119906) (56)

From (46) and part (a) of condition (ii) we have

119904119866120590119887

(119891119906 119891119906 119891119906) le 120595 (119872119904(119906 119906 119906)) (57)

where by using (56) we have

119872119904(119906 119906 119906)

= max119866120590119887

(119906 119906 119906)

119866120590119887

(119906 119906 119891119906)119866120590119887

(119906 119906 119891119906)119866120590119887

(119906 119906 119891119906)

1 + 211990421198662

120590119887

(119891119906 119891119906 119891119906)

le 119866120590119887

(119906 119906 119891119906)

(58)

Hence from properties of 120595 119904119866120590119887(119891119906 119891119906 119891119906) le

119866120590119887(119906 119906 119891119906) Thus by (56) we obtain that

119866120590119887

(119906 119906 119891119906) = 119904119866120590119887

(119891119906 119891119906 119891119906) (59)

The Scientific World Journal 9

Moreover (57) yields that 119866120590119887(119906 119906 119891119906) le 120595(119872

119904(119906 119906 119906)) le

120595(119866120590119887(119906 119906 119891119906))This is impossible according to our assump-

tions on 120595 Hence we have 119891119906 = 119906 Thus 119906 is a fixed point of119891

Now let part (b) of (ii) holdsAs 119909119899 is a sequence such that 120572(119909

119899 119909119899+1

119909119899+2

) ge 1 for all119899 isin N cup 0 and 119909

119899rarr 119909 as 119899 rarr infin we have 120572(119909

119899 119909 119909) ge 1

for all 119899 isin N cup 0Now we show that 119906 = 119891119906 By (46) we have

119904119866120590119887

(119891119906 119891119906 119909119899) le 119904120572 (119906 119906 119909

119899minus1) 119866120590119887

(119891119906 119891119906 119891119909119899minus1

)

le 120595 (119872119904(119906 119906 119909

119899minus1))

(60)

where

119872119904(119906 119906 119909

119899minus1)

= max

119866120590119887

(119906 119906 119909119899minus1

)

[119866120590119887

(119906 119906 119891119906)]

2

119866120590119887

(119909119899minus1

119909119899minus1

119891119909119899minus1

)

1 + [119904119866120590119887

(119891119906 119891119906 119891119909119899minus1

)]

2

(61)

Letting 119899 rarr infin in the above inequality and using (54) andLemma 21 we get

lim119899rarrinfin

119872119904(119906 119906 119909

119899minus1) = 0 (62)

Again taking the upper limit as 119899 rarr infin in (60) and using(62) and Lemma 21 we obtain

119904 [

1

119904

119866120590119887

(119906 119891119906 119891119906) minus 119866120590119887

(119906 119906 119906)]

le 119904 lim sup119899rarrinfin

119866120590119887

(119909119899 119891119906 119891119906)

le lim supnrarrinfin

120595 (119872119904(119906 119906 119909

119899minus1))

= 120595(lim sup119899rarrinfin

119872119904(119906 119906 119909

119899minus1))

= 120595 (0) = 0

(63)

So we get 119866120590119887(119906 119891119906 119891119906) = 0 That is 119891119906 = 119906

Let (119883 119866120590119887 ⪯) be a partially ordered 119866

120590119887-metric-like

space We say that 119879 is an increasing mapping on 119883 ifforall119909 119910 isin 119883 119909 ⪯ 119910 rArr 119879(119909) ⪯ 119879(119910) [26] Fixed pointtheorems for monotone operators in ordered metric spacesarewidely investigated andhave found various applications indifferential and integral equations (see [27ndash30] and referencestherein) From the results proved above we derive thefollowing new results in partially ordered 119866

120590119887-metric-like

space

Theorem 34 Let (119883 119866120590119887 ⪯) be a partially ordered 119866

120590119887-

complete generalized 119887-metric-like space and let 119891 119883 rarr 119883

be an increasing mapping Suppose that

119904119866120590119887

(119891119909 119891119910 119891119911) le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911) (64)

for all 119909 119910 119911 isin 119883 with 119909 ⪯ 119910 ⪯ 119911 where

119872119904(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119891119909 119891119909)119866120590119887

(119910 119891119910 119891119910)119866120590119887

(119911 119891119911 119891119911)

1 + 11990421198662

120590119887

(119891119909 119891119910 119891119911)

(65)

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) 119891 is continuous or assume that whenever 119909119899 in 119883 is

an increasing sequence such that 119909119899rarr 119909 as 119899 rarr infin

one has 119909119899⪯ 119909 for all 119899 isin N cup 0

Then 119891 has a fixed point

Proof Define 120572 119883 times 119883 times 119883 rarr [0 +infin) by

120572 (119909 119910 119911) =

1 if119909 ⪯ 119910 ⪯ 119911

0 otherwise(66)

First we prove that 119891 is a triangular 120572-admissible mappingHence we assume that 120572(119909 119910 119911) ge 1 Therefore we have 119909 ⪯

119910 ⪯ 119911 Since 119891 is increasing we get 119891119909 ⪯ 119891119910 ⪯ 119891119911 that is120572(119891119909 119891119910 119891119911) ge 1 Also let 120572(119909 119910 119910) ge 1 and 120572(119910 119911 119911) ge 1then 119909 ⪯ 119910 and 119910 ⪯ 119911 Consequently we deduce that 119909 ⪯ 119911that is 120572(119909 119911 119911) ge 1 Thus 119891 is a triangular 120572-admissiblemapping Since 119891 satisfies (64) so by the definition of 120572 wehave

119904120572 (119909 119910 119911) 119866120590119887

(119891119909 119891119910 119891119911)

le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911)

(67)

for all 119909 119910 119911 isin 119883 Therefore 119891 satisfies the contractivecondition (21) From (i) there exists 119909

0isin 119883 such that

1199090⪯ 1198911199090 that is 120572(119909

0 1198911199090 1198911199090) ge 1 According to (ii) we

conclude that all the conditions of Theorems 26 and 27 aresatisfied and so 119891 has a fixed point

Similarly using Theorem 33 we can prove followingresult

Theorem 35 Let (119883 119866120590119887 ⪯) be an ordered 119866

120590119887-complete

generalized 119887-metric-like space and let 119891 119883 rarr 119883 be anincreasing mapping Suppose that

119904119866120590119887

(119891119909 119891119910 119891119911) le 120595 (119872119904(119909 119910 119911)) (68)

for all 119909 119910 119911 isin 119883 with 119909 ⪯ 119910 ⪯ 119911 where 120595 isin Ψ119887and

119872119904(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119909 119891119909)119866120590119887

(119910 119910 119891119910)119866120590119887

(119911 119911 119891119911)

1 + 211990421198662

120590119887

(119891119909 119891119910 119891119911)

(69)

10 The Scientific World Journal

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii)

(a) 119891 is continuous(b) assume that whenever x

119899 in 119883 is an increasing

sequence such that 119909119899rarr 119909 as 119899 rarr infin one has

119909119899⪯ 119909 for all 119899 isin N cup 0

Then 119891 has a fixed point

We conclude this section by presenting some examplesthat illustrate our results

Example 36 Let 119883 = [0 (minus1 + radic5)2] be endowed withthe usual ordering onR Define the generalized 119887-metric-likefunction 119866

120590119887given by

119866120590119887

(119909 119910 119911) = max (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

(70)

with 119904 = 2 Consider the mapping 119891 119883 rarr 119883 defined by119891(119909) = (14)119909

radic119890minus(119909+((minus1+radic5)2))

2 and the function 120573 isin Fgiven by 120573(119905) = (12)119890

minus119905 119905 gt 0 and 120573(0) isin [0 12) Itis easy to see that 119891 is an increasing function on 119883 Weshow that 119891 is 119866

120590119887-continuous on 119883 By Proposition 20 it is

sufficient to show that119891 is119866120590119887-sequentially continuous on119883

Let 119909119899 be a sequence in119883 such that lim

119899rarrinfin119866120590119887(119909119899 119909 119909) =

119866120590119887(119909 119909 119909) so we have max(lim

119899rarrinfin119909119899

+ 119909)2

41199092 =

41199092 and equivalently lim

119899rarrinfin119909119899

= 120572 le 119909 On the otherhand we have

lim119899rarrinfin

119866120590119887

(119891119909119899 119891119909 119891119909)

= lim119899rarrinfin

max (119891119909119899+ 119891119909)

2

41198912

119909

= lim119899rarrinfin

max 1

16

(119909119899

radic119890minus(119909119899+(minus1+

radic5)2)2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= max 1

16

( lim119899rarrinfin

119909119899

radic119890minus(lim119899rarrinfin119909119899+(minus1+radic5)2)

2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= max 1

16

(120572radic119890minus(120572+(minus1+radic5)2)

2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

=

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= 119866120590119887

(119891119909 119891119909 119891119909)

(71)

So 119891 is 119866120590119887-sequentially continuous on119883

For all elements 119909 119910 119911 isin 119883 and the fact that 119892(119909) =

1199092119890minus1199092

is an increasing function on119883 we have

119904119866120590119887

(119891119909 119891119910 119891119911)

= 2max (119891119909 + 119891119910)2

(119891119910 + 119891119911)2

(119891119911 + 119891119909)2

= 2max(

1

4

119909radic119890minus(119909+(minus1+radic5)2)

2

+

1

4

119910radic119890minus(119910+(minus1+radic5)2)

2

)

2

(

1

4

119910radic119890minus(119910+(minus1+radic5)2)

2

+

1

4

119911radic119890minus(119911+(minus1+radic5)2)

2

)

2

(

1

4

119911radic119890minus(119911+(minus1+radic5)2)

2

+

1

4

119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

le 2max(

1

4

119909radic119890minus(119909+119910)

2

+

1

4

119910radic119890minus(119910+119909)

2

)

2

(

1

4

119910radic119890minus(119910+119911)

2

+

1

4

119911radic119890minus(119911+119910)

2

)

2

(

1

4

119911radic119890minus(119911+119909)

2

+

1

4

119909radic119890minus(119909+119911)

2

)

2

=

1

8

max (119909 + 119910)2

119890minus(119909+119910)

2

(119910 + 119911)2

119890minus(119910+119911)

2

(119911 + 119909)2

119890minus(119911+119909)

2

=

1

8

119890minusmax(119909+119910)2 (119910+119911)2 (119911+119909)2

timesmax (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

le

1

2

eminusmax(119909+119910)2 (119910+119911)2(119911+119909)2

timesmax (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

= 120573 (119866120590119887

(119909 119910 119911)) 119866120590119887

(119909 119910 119911)

le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911)

(72)

Hence119891 satisfies all the assumptions ofTheorem 34 and thusit has a fixed point (which is 119906 = 0)

Example 37 Let 119883 = [0 1] with the usual ordering on RDefine the generalized 119887-metric-like function 119866

120590119887given by

119866120590119887(119909 119910 119911) = max(119909 + 119910)

2

(119910 + 119911)2

(119911 + 119909)2

with 119904 = 2Consider the mapping 119891 119883 rarr 119883 defined by 119891(119909) =

The Scientific World Journal 11

ln(1 + 119909119890minus119909

4) and the function 120595 isin Ψ119887given by 120595(119905) =

(18)119905 119905 ge 0 It is easy to see that 119891 is increasing functionNow we show that 119891 is a 119866

120590119887-continuous function on119883

Let 119909119899 be a sequence in 119883 such that lim

119899rarrinfin119866120590119887(119909119899

119909 119909) = 119866120590119887(119909 119909 119909) so we havemax(lim

119899rarrinfin119909119899+119909)2

41199092 =

41199092 and equally lim

119899rarrinfin119909119899= 120572 le 119909 On the other hand we

have

lim119899rarrinfin

119866120590119887

(119891119909119899 119891119909 119891119909)

= lim119899rarrinfin

max (119891119909119899+ 119891119909)

2

41198912

119909

= max(ln(1 +

lim119899rarrinfin

119909119899119890minuslim119899rarrinfin119909119899

4

)

+ ln(1 +

119909119890minus119909

4

))

2

4(ln(1 +

119909119890minusx

4

))

2

= max(ln(1 +

120572119890minus120572

4

) + ln(1 +

119909119890minus119909

4

))

2

4(ln(1 +

119909119890minus119909

4

))

2

= 4(ln(1 +

119909119890minus119909

4

))

2

= 119866120590119887

(119891119909 119891119909 119891119909)

(73)

So 119891 is 119866120590119887-sequentially continuous on119883

For all elements 119909 119910 119911 isin 119883 we have

119904119866120590119887

(119891119909 119891119910 119891119911)

= 2max(ln(1 +

119909119890minus119909

4

) + ln(1 +

119910119890minus119910

4

))

2

(ln(1 +

119910119890minus119910

4

) + ln(1 +

119911119890minus119911

4

))

2

(ln(1 +

119911119890minus119911

4

) + ln(1 +

119909119890minus119909

4

))

2

le 2max(

119909119890minus119909

4

+

119910119890minus119910

4

)

2

(

119910119890minus119910

4

+

119911119890minus119911

4

)

2

(

119911119890minus119911

4

+

119909119890minus119909

4

)

2

le

1

8

max (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

= 120595 (119866120590119887

(119909 119910 119911))

le 120595 (119872119904(119909 119910 119911))

(74)

Hence119891 satisfies all the assumptions ofTheorem 35 and thusit has a fixed point (which is 119906 = 0)

3 Application

In this section we present an application of our results toestablish the existence of a solution to a periodic boundaryvalue problem (see [30 31])

Let 119883 = 119862([0 119879]) be the set of all real continuousfunctions on [0T] We first endow119883 with the 119887-metric-like

120590119887(119906 V) = sup

119905isin[0119879]

(|119906 (119905)| + |V (119905)|)119901 (75)

for all 119906 V isin 119883 where 119901 gt 1 and then we endow it with thegeneralized 119887-metric-like 119866

120590119887defined by

119866120590119887

(119906 V 119908)

= max sup119905isin[0119879]

(|119906 (119905)| + |V (119905)|)119901

sup119905isin[0119879]

(|V (119905)| + |119908 (119905)|)119901

sup119905isin[0119879]

(|119908 (119905)| + |119906 (119905)|)119901

(76)

Clearly (119883 119866120590119887) is a complete generalized 119887-metric-like space

with parameter 119870 = 2119901minus1 We equip 119862([0 119879]) with a partial

order given by119909 ⪯ 119910 iff119909 (119905) le 119910 (119905) forall119905 isin [0 119879] (77)

Moreover as in [30] it is proved that (119862([0 119879]) ⪯) is regularthat is whenever 119909

119899 in119883 is an increasing sequence such that

119909119899rarr 119909 as 119899 rarr infin we have 119909

119899⪯ 119909 for all 119899 isin N cup 0

Consider the first-order periodic boundary value prob-lem

1199091015840

(119905) = 119891 (119905 119909 (119905)) 119909 (0) = 119909 (119879) (78)

where 119905 isin 119868 = [0 119879] with 119879 gt 0 and 119891 119868 times R rarr R is acontinuous function

A lower solution for (78) is a function 120572 isin 1198621[0 119879] such

that1205721015840

(119905) le 119891 (119905 120572 (119905))

120572 (0) le 120572 (119879)

(79)

where 119905 isin 119868 = [0 119879]Assume that there exists 120582 gt 0 such that for all 119909 119910 isin

119862[0 119879] we have1003816100381610038161003816119891 (119905 119909 (119905)) + 120582119909 (119905)

1003816100381610038161003816+1003816100381610038161003816119891 (119905 119910 (119905)) + 120582119910 (119905)

1003816100381610038161003816

le

120582

2119901minus1

119901radicln(

119886

2119901minus1

(|119909 (119905)| +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)119901

+ 1)

(80)

where 0 le 119886 lt 1 Then the existence of a lower solution for(78) provides the existence of a solution of (78)

Problem (78) can be rewritten as

1199091015840

(119905) + 120582119909 (119905) = 119891 (119905 119909 (119905)) + 120582119909 (119905)

119909 (0) = 119909 (119879)

(81)

Consider1199091015840

(119905) + 120582119909 (119905) = 120575119909(119905) = 119865 (119905 119909 (119905))

119909 (0) = 119909 (119879)

(82)

where 119905 isin 119868

12 The Scientific World Journal

Using variation of parameters formula we get

119909 (119905) = 119909 (0) 119890minus120582119905

+ int

119905

0

119890minus120582(119905minus119904)

120575119909(119904) 119889119904 (83)

which yields

119909 (119879) = 119909 (0) 119890minus120582119879

+ int

119879

0

119890minus120582(119879minus119904)

120575119909(119904) 119889119904 (84)

Since 119909(0) = 119909(119879) we get

119909 (0) [1 minus 119890minus120582119879

] = 119890minus120582119879

int

119879

0

119890120582(119904)

120575119909(119904) 119889119904 (85)

or

119909 (0) =

1

119890120582119879

minus 1

int

119879

0

119890120582119904

120575119909(119904) 119889119904 (86)

Substituting the value of 119909(0) in (83) we arrive at

119909 (119905) = int

119879

0

119866 (119905 119904) 120575119909(119904) 119889119904 (87)

where

119866 (119905 119904) =

119890120582(119879+119904minus119905)

119890120582119879

minus 1

0 le 119904 le 119905 le 119879

119890120582(119904minus119905)

119890120582119879

minus 1

0 le 119905 le 119904 le 119879

(88)

Now define the operator 119878 119862[0 119879] rarr 119862[0 119879] as

119878119909 (119905) = int

119879

0

119866 (119905 119904) 119865 (119904 119909 (119904)) 119889119904 (89)

The mapping 119878 is increasing [31] Note that if 119906 isin 119862[0 119879] is afixed point of 119878 then 119906 isin 119862

1[0 119879] is a solution of (78)

Let 119909 119910 119911 isin 119862[0 119879] Then we have

2119901minus1

[|119878119909 (119905)| +1003816100381610038161003816119878119910 (119905)

1003816100381610038161003816]

= 2119901minus1

[

100381610038161003816100381610038161003816100381610038161003816

int

119879

0

119866 (119905 119904) 119865 (119904 119909 (119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

119879

0

119866 (119905 119904) 119865 (119904 119910 (119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

]

le 2119901minus1

int

119879

0

|119866 (119905 119904)|

times [|119865 (119904 119909 (119904))| +1003816100381610038161003816119865 (119904 119910 (119904))

1003816100381610038161003816] 119889119904

le 2119901minus1

int

119879

0

|119866 (119905 119904)|

120582

2119901minus1

times119901radicln(

119886

2119901minus1

(|119909 (119905)| +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)119901

+ 1)119889119904

le 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [int

119905

0

119890120582(119879+119904minus119905)

119890120582119879

minus 1

119889119904 + int

119879

119905

119890120582(119904minus119905)

119890120582119879

minus 1

119889119904]

= 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [

1

120582 (119890120582119879

minus 1)

(119890120582(119879+119904minus119905)10038161003816

100381610038161003816

119905

0

+ 119890120582(119904minus119905)10038161003816

100381610038161003816

119879

119905

)]

= 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [

1

120582 (119890120582119879

minus 1)

(119890120582119879

minus 119890120582(119879minus119905)

+ 119890120582(119879minus119905)

minus 1)]

=119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

le119901radicln(

119886

2119901minus1

119866120590119887

(119909 119910 119911) + 1)

le119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(90)

Similarly

2119901minus1 1003816

100381610038161003816119878119910 (119905)

1003816100381610038161003816+ |119878119911 (119905)| le

119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

|119878119909 (119905)| + |119878119911 (119905)| le119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(91)

where

119872(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119878119909 119878119909) 119866120590119887

(119910 119878119910 119878119910)119866120590119887

(119911 119878119911 119878119911)

1 + 22119901minus1

1198662

120590119887

(119878119909 119878119910 119878119911)

(92)

Equivalently

(2119901minus1

|119878119909 (119905)| +1003816100381610038161003816119878119910 (119905)

1003816100381610038161003816)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(2119901minus1 1003816

100381610038161003816119878119910 (119905)

1003816100381610038161003816+ |119878119911 (119905)|)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(2119901minus1

|119878119909 (119905)| + |119878119911 (119905)|)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(93)

which yields that

2119901minus1

120590119887(119878119909 119878119910) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

120590119887(119878119910 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

120590119887(119878119909 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(94)

The Scientific World Journal 13

Finally it is easy to obtain that

2119901minus1G120590119887

(119878119909 119878119910 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1) (95)

Finally since120572 is a lower solution for (78) so it is easy to showthat 120572 ⪯ 119878(120572) [31]

Hence the hypotheses of Theorem 35 are satisfied with120595(119905) = ln((1198862119901minus1)119905 + 1) where 0 le 119886 lt 1 Hence there existsa fixed point 119909 isin 119862[0 119879] such that 119878119909 = 119909 which is a solutionto periodic boundary value problem (78)

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) KingAbdulaziz University JeddahTherefore the firstauthor acknowledges with thanks DSR KAU for financialsupport

References

[1] S Czerwik ldquoContraction mappings in b-metric spacesrdquo ActaMathematica et Informatica Universitatis Ostraviensis vol 1 pp5ndash11 1993

[2] V Parvaneh J R Roshan and S Radenovic ldquoExistence of tri-pled coincidence points in ordered b-metric spaces and an app-lication to a system of integral equationsrdquo Fixed Point Theoryand Applications vol 2013 article 130 2013

[3] J R Roshan V Parvaneh S Sedghi N Shobkolaei and WShatanawi ldquoCommon fixed points of almost generalized con-tractive mappings in ordered b-metric spacesrdquo Fixed PointTheory and Applications vol 2013 article 159 2013

[4] H Aydi M F Bota E Karapınar and S Moradi ldquoA commonfixed point for weak 120601-contractions on b-metric spacesrdquo FixedPoint Theory vol 13 no 2 pp 337ndash346 2012

[5] V Berinde ldquoSequences of operators and fixed points in quasi-metric spacesrdquo Studia Universitatis ldquoBabes-Bolyairdquo Mathemat-ica vol 16 no 4 pp 23ndash27 1996

[6] N Hussain D ETHoric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 article 126 2012

[7] N Hussain and M H Shah ldquoKKMmappings in cone b-metricspacesrdquo Computers and Mathematics with Applications vol 62no 4 pp 1677ndash1684 2011

[8] N Hussain J R Roshan V Parvaneh and Z Kadelburg ldquoFixedpoints of contractive mappings in b-metric-like spacesrdquo TheScientific World Journal vol 2014 Article ID 471827 15 pages2014

[9] M A Khamsi and N Hussain ldquoKKMmappings in metric typespacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 73 no 9 pp 3123ndash3129 2010

[10] M A Khamsi ldquoRemarks on cone metric spaces and fixed pointtheorems of contractive mappingsrdquo Fixed Point Theory andApplications vol 2010 Article ID 315398 7 pages 2010

[11] Z Mustafa J R Roshan V Parvaneh and Z Kadelburg ldquoSomecommon fixed point results in ordered partial b-metric spacesrdquoJournal of Inequalities and Applications vol 2013 article 5622013

[12] Z Mustafa and B Sims ldquoA new approach to generalized metricspacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash297 2006

[13] A Amini-Harandi ldquoMetric-like spaces partial metric spacesand fixed pointsrdquo Fixed PointTheory and Applications vol 2012article 204 2012

[14] N Shobkolaei S Sedghi J R Roshan and N Hussain ldquoSuzukitype fixed point results in metric-like spacesrdquo Journal of Func-tion Spaces and Applications vol 2013 Article ID 143686 9pages 2013

[15] M A Alghamdi N Hussain and P Salimi ldquoFixed point andcoupled fixed point theorems on b-metric-like spacesrdquo Journalof Inequalities and Applications vol 2013 article 402 2013

[16] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572 minus 120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

[17] P Salimi and E Karapınar ldquoSuzuki-Edelstein type contractionsvia auxiliary functionsrdquoMathematical Problems in Engineeringvol 2013 Article ID 648528 8 pages 2013

[18] P Salimi C Vetro and P Vetro ldquoFixed point theorems fortwisted (120572 120573)-120595-contractive type mappings and applicationsrdquoFilomat vol 27 no 4 pp 605ndash615 2013

[19] P Salimi C Vetro and P Vetro ldquoSome new fixed point resultsin non-Archimedean fuzzy metric spacesrdquo Nonlinear AnalysisModelling and Control vol 18 no 3 pp 344ndash358 2013

[20] N Hussain P Salimi and A Latif ldquoFixed point results forsingle and set-valued 120572-120578-120595-contractive mappingsrdquo Fixed PointTheory and Applications vol 2013 article 212 2013

[21] M A Alghamdi and E Karapınar ldquoG-120573-120595 contractive-typemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 article 70 2013

[22] M Geraghty ldquoOn contractive mappingsrdquo Proceedings of theAmerican Mathematical Society vol 40 pp 604ndash608 1973

[23] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001

[24] N Hussain Z Kadelburg S Radenovic and F R Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012

[25] V Berinde ldquoGeneralized contractions in quasimetric spacesrdquo inSeminar on Fixed Point Theory Preprint no 3 pp 3ndash9 1993

[26] L Ciric M Abbas R Saadati and N Hussain ldquoCommon fixedpoints of almost generalized contractive mappings in orderedmetric spacesrdquo Applied Mathematics and Computation vol 217no 12 pp 5784ndash5789 2011

[27] R P Agarwal N Hussain and M A Taoudi ldquoFixed point the-orems in ordered Banach spaces and applications to nonlinearintegral equationsrdquo Abstract and Applied Analysis vol 2012Article ID 245872 15 pages 2012

[28] N Hussain A R Khan and R P Agarwal ldquoKrasnoselrsquoskii andKy Fan type fixed point theorems in ordered Banach spacesrdquoJournal of Nonlinear and Convex Analysis vol 11 no 3 pp 475ndash489 2010

[29] N Hussain and M A Taoudi ldquoKrasnoselrsquoskii-type fixed pointtheorems with applications to Volterra integral equationsrdquoFixed Point Theory and Applications vol 2013 article 196 2013

14 The Scientific World Journal

[30] J J Nieto and R Rodrıguez-Lopez ldquoContractivemapping theo-rems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[31] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysisTheoryMethods andApplications vol 71 no 7-8 pp 3403ndash3410 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 7

A mapping 120593 [0infin) rarr [0infin) is called a comparisonfunction if it is increasing and 120593

119899(119905) rarr 0 as 119899 rarr infin for any

119905 isin [0infin) (see eg [23 24] for more details and examples)

Definition 28 A function 120593 [0infin) rarr [0infin) is said to bea (119888)-comparison function if

(1198881) 120593 is increasing

(1198882) there exists 119896

0isin N 119886 isin (0 1) and a convergent series

of nonnegative terms suminfin

119896=1V119896such that 120593

119896+1(119905) le

119886120593119896(119905) + V

119896for 119896 ge 119896

0and any 119905 isin [0infin)

Later Berinde [5] introduced the notion of (119887)-comparison function as a generalization of (119888)-comparisonfunction

Definition 29 (see [5]) Let 119904 ge 1 be a real number Amapping120593 [0infin) rarr [0infin) is called a (119887)-comparison function ifthe following conditions are fulfilled

(1) 120593 is increasing

(2) there exist 1198960isin N 119886 isin (0 1) and a convergent series

of nonnegative terms suminfin119896=1

V119896such that 119904119896+1120593119896+1(119905) le

119886119904119896120593119896(119905) + V

119896for 119896 ge 119896

0and any 119905 isin [0infin)

Let Ψ119887be the class of all (119887)-comparison functions 120593

[0infin) rarr [0infin) It is clear that the notion of (119887)-comparison function coincideswith (119888)-comparison functionfor 119904 = 1

Lemma 30 (see [25]) If 120593 [0infin) rarr [0infin) is a (119887)-comparison function then we have the following

(1) the series suminfin119896=0

119904119896120593119896(119905) converges for any 119905 isin R

+

(2) the function 119887119904 [0infin) rarr [0infin) defined by 119887

119904(119905) =

suminfin

119896=0119904119896120593119896(119905) 119905 isin [0infin) is increasing and continuous

at 0

Remark 31 It is easy to see that if 120593 isin Ψ119887 then we have 120593(0) =

0 and 120593(119905) lt 119905 for each 119905 gt 0 and 120593 is continuous at 0

In the next example we present a class of (119887)-comparisonfunctions

Example 32 Any function of the form120595(119905) = ln((119886119904)119905+1) forall 119905 isin [0infin) where 0 lt 119886 lt 1 is a (119887)-comparison function

Proof From the part (1) of Lemma 30 the necessary condi-tion is that the series suminfin

119896=0119904119896120593119896(119905) converges for any 119905 isin R

+

But for each 119905 gt 0 and 119896 ge 1 we have

119904119896

120593119896

(119905) = 119904119896

120593 (120593119896minus1

(119905))

= 119904119896 ln(

119886

119904

120593119896minus1

(119905) + 1) le 119904119896 119886

119904

120593119896minus1

(119905)

le sdot sdot sdot le 119904119896

(

119886

119904

)

119896

119905 = 119886119896

119905

(43)

So according to the comparison test of the series we shouldhave 119886 lt 1 On the other hand we have

119904119896+1

120593119896+1

(119905) = 119904119896+1

120593 (120593119896

(119905))

= 119904119896+1 ln(

119886

119904

120593119896

(119905) + 1)

le 119904119896+1 119886

119904

120593119896

(119905) = 119904119896

119886120593119896

(119905)

(44)

Therefore for any convergent series of nonnegative termssuminfin

119896=1V119896and each 119896 ge 119896

0= 1 we have

119904119896+1

120593119896+1

(119905) le 119886119904119896

120593119896

(119905) le 119886119904119896

120593119896

(119905) + V119896 (45)

For example for 119904 = 2 and 119886 = 12 the function 120595(119905) =

ln(1199054 + 1) is a (119887)-comparison function

Theorem 33 Let (119883 119866120590119887) be a 119866

120590119887-complete generalized 119887-

metric-like space and let 119891 119883 rarr 119883 be a 119866-120572-admissiblemapping Suppose that

119904120572 (119909 119910 119911) 119866120590119887

(119891119909 119891119910 119891119911) le 120595 (119872119904(119909 119910 119911)) (46)

for all 119909 119910 119911 isin 119883 where 120595 isin Ψ119887and

119872119904(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119909 119891119909)119866120590119887

(119910 119910 119891119910)119866120590119887

(119911 119911 119891119911)

1 + 211990421198662

120590119887

(119891119909 119891119910 119891119911)

(47)

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 120572(119909

0 1198911199090 11989121199090) ge 1

(ii)

(a) 119891 is continuous and for any sequence 119909119899 in119883

with 120572(119909119899 119909119899+1

119909119899+2

) ge 1 for all 119899 isin Ncup0 suchthat 119909

119899rarr 119909 as 119899 rarr infin one has 120572(119909 119909 119909) ge 1

for all 119899 isin N cup 0(b) assume that whenever 119909

119899 in 119883 is a sequence

such that 120572(119909119899 119909119899+1

119909119899+2

) ge 1 for all 119899 isin Ncup0

and 119909119899rarr 119909 as 119899 rarr infin one has 120572(119909

119899 119909 119909) ge 1

for all 119899 isin N cup 0

Then 119891 has a fixed point

Proof Let 1199090isin 119883 be such that 120572(119909

0 1198911199090 11989121199090) ge 1 Define

a sequence 119909119899 by 119909

119899= 1198911198991199090for all 119899 isin N Since 119891 is a 119866-120572-

admissible mapping and 120572(1199090 1199091 1199092) = 120572(119909

0 1198911199090 11989121199090) ge

1 we deduce that 120572(1199091 1199092 1199093) = 120572(119891119909

0 1198911199091 1198911199092) ge 1

Continuing this process we get 120572(119909119899 119909119899+1

119909119899+2

) ge 1 for all119899 isin N cup 0

8 The Scientific World Journal

If there exists 1198990isin N such that 119909

1198990= 1199091198990+1

then 1199091198990

=

1198911199091198990and so we have nothing to prove Hence for all 119899 isin N

we assume that 119909119899

= 119909119899+1

Step I (Cauchyness of 119909119899) As 120572(119909

119899 119909119899+1

119909119899+2

) ge 1 for all119899 ge 0 using condition (46) we obtain

119904119866120590119887

(119909119899 119909119899+1

119909119899+2

)

le 119904120572 (119909119899minus1

119909119899 119909119899+1

) 119866120590119887

(119909119899 119909119899+1

119909119899+2

)

= 119904120572 (119909119899minus1

119909119899 119909119899+1

) 119866120590119887

(119891119909119899minus1

119891119909119899 119891119909119899+1

)

le 120595 (119872119904(119909119899minus1

119909119899 119909119899+1

))

(48)

Using Proposition 15(2) as 119909119899

= 119909119899+1

we get

119872119904(119909119899minus1

119909119899 119909119899+1

)

= max 119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

(119866120590119887

(119909119899minus1

119909119899minus1

119891119909119899minus1

)

times 119866120590119887

(119909119899 119909119899 119891119909119899)

times119866120590119887

(119909119899+1

119909119899+1

119891119909119899+1

))

times(1 + 21199042

1198662

120590119887

(119891119909119899minus1

119891119909119899 119891119909119899+1

))

minus1

= max 119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

(119866120590119887

(119909119899minus1

119909119899minus1

119909119899)

times 119866120590119887

(119909119899 119909119899 119909119899+1

)

times119866120590119887

(119909119899+1

119909119899+1

119909119899+2

))

times(1 + 21199042

1198662

120590119887

(119909119899 119909119899+1

119909119899+2

))

minus1

le max119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

2119904119866120590119887

(119909119899minus1

119909119899 119909119899+1

) 1198662

120590119887

(119909119899 119909119899+1

119909119899+2

)

1 + 2s21198662120590119887

(119909119899 119909119899+1

119909119899+2

)

= 119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

(49)

Hence119866120590119887

(119909119899 119909119899+1

119909119899+2

)

le 119904119866120590119887

(119909119899 119909119899+1

119909119899+2

) le 120595 (119866120590119887

(119909119899minus1

119909119899 119909119899+1

))

(50)

By induction since 119909119899

= 119909119899+1

we get that

119866120590119887

(119909119899 119909119899+1

119909119899+2

)

le 120595 (119866120590119887

(119909119899minus1

119909119899 119909119899+1

))

le 1205952

(119866120590119887

(119909119899minus2

119909119899minus1

119909119899))

le sdot sdot sdot le 120595119899

(119866120590119887

(1199090 1199091 1199092))

(51)

Let 120576 gt 0 be arbitrary Then there exists a natural number119873such that

infin

sum

119899=119873

119904119899

120595119899

(119866120590119887

(1199090 1199091 1199092)) lt

120576

2119904

(52)

Let 119898 gt 119899 ge 119873 Then by the rectangular inequality andProposition 15(2) as 119909

119899= 119909119899+1

we get

119866120590119887

(119909119899 119909119898 119909119898)

le 119904119866120590119887

(119909119899 119909119899+1

119909119899+1

) + 1199042

119866120590119887

(119909119899+1

119909119899+2

119909119899+2

)

+ sdot sdot sdot + 119904119898minus119899minus1

119866120590119887

(119909119898minus1

119909119898 119909119898)

le 2119904 (119904119866120590119887

(119909119899 119909119899 119909119899+1

) + 1199042

119866120590119887

(119909119899+1

119909119899+1

119909119899+2

)

+ sdot sdot sdot + 119904119898minus119899minus1

119866120590119887

(119909119898minus1

119909119898minus1

119909119898))

le 2119904 (119904119866120590119887

(119909119899 119909119899+1

119909119899+2

) + 1199042

119866120590119887

(119909119899+1

119909119899+2

119909119899+3

)

+ sdot sdot sdot + 119904119898minus119899minus1

119866120590119887

(119909119898minus1

119909119898 119909119898+1

))

le 2119904

119898minus2

sum

119896=119899

119904119896minus119899+1

120595119896

(119866120590119887

(1199090 1199091 1199092))

le 2119904

infin

sum

119896=119899

119904119896

120595119896

(119866120590119887

(1199090 1199091 1199092)) lt 120576

(53)

Consequently 119909119899 is a 119866

120590119887-Cauchy sequence in 119883 Since

(119883 119866120590119887) is 119866120590119887-complete so there exists 119906 isin 119883 such that

lim119899rarrinfin

119866120590119887

(119906 119909119899 119909119899) = lim119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

= 119866120590119887

(119906 119906 119906) = 0

(54)

Step II Now we show that 119906 is a fixed point of 119891 Suppose tothe contrary that is 119891119906 = 119906 then we have 119866

120590119887(119906 119906 119891119906) gt 0

Let the part (a) of (ii) holdsUsing the rectangle inequality we get

119866120590119887

(119906 119906 119891119906) le 119904119866120590119887

(119891119906 119891119909119899 119891119909119899) + 119904119866

120590119887(119891119909119899 119906 119906)

(55)

Letting 119899 rarr infin and using the continuity of 119891 we get

119866120590119887

(119906 119906 119891119906) le 119904119866120590119887

(119891119906 119891119906 119891119906) (56)

From (46) and part (a) of condition (ii) we have

119904119866120590119887

(119891119906 119891119906 119891119906) le 120595 (119872119904(119906 119906 119906)) (57)

where by using (56) we have

119872119904(119906 119906 119906)

= max119866120590119887

(119906 119906 119906)

119866120590119887

(119906 119906 119891119906)119866120590119887

(119906 119906 119891119906)119866120590119887

(119906 119906 119891119906)

1 + 211990421198662

120590119887

(119891119906 119891119906 119891119906)

le 119866120590119887

(119906 119906 119891119906)

(58)

Hence from properties of 120595 119904119866120590119887(119891119906 119891119906 119891119906) le

119866120590119887(119906 119906 119891119906) Thus by (56) we obtain that

119866120590119887

(119906 119906 119891119906) = 119904119866120590119887

(119891119906 119891119906 119891119906) (59)

The Scientific World Journal 9

Moreover (57) yields that 119866120590119887(119906 119906 119891119906) le 120595(119872

119904(119906 119906 119906)) le

120595(119866120590119887(119906 119906 119891119906))This is impossible according to our assump-

tions on 120595 Hence we have 119891119906 = 119906 Thus 119906 is a fixed point of119891

Now let part (b) of (ii) holdsAs 119909119899 is a sequence such that 120572(119909

119899 119909119899+1

119909119899+2

) ge 1 for all119899 isin N cup 0 and 119909

119899rarr 119909 as 119899 rarr infin we have 120572(119909

119899 119909 119909) ge 1

for all 119899 isin N cup 0Now we show that 119906 = 119891119906 By (46) we have

119904119866120590119887

(119891119906 119891119906 119909119899) le 119904120572 (119906 119906 119909

119899minus1) 119866120590119887

(119891119906 119891119906 119891119909119899minus1

)

le 120595 (119872119904(119906 119906 119909

119899minus1))

(60)

where

119872119904(119906 119906 119909

119899minus1)

= max

119866120590119887

(119906 119906 119909119899minus1

)

[119866120590119887

(119906 119906 119891119906)]

2

119866120590119887

(119909119899minus1

119909119899minus1

119891119909119899minus1

)

1 + [119904119866120590119887

(119891119906 119891119906 119891119909119899minus1

)]

2

(61)

Letting 119899 rarr infin in the above inequality and using (54) andLemma 21 we get

lim119899rarrinfin

119872119904(119906 119906 119909

119899minus1) = 0 (62)

Again taking the upper limit as 119899 rarr infin in (60) and using(62) and Lemma 21 we obtain

119904 [

1

119904

119866120590119887

(119906 119891119906 119891119906) minus 119866120590119887

(119906 119906 119906)]

le 119904 lim sup119899rarrinfin

119866120590119887

(119909119899 119891119906 119891119906)

le lim supnrarrinfin

120595 (119872119904(119906 119906 119909

119899minus1))

= 120595(lim sup119899rarrinfin

119872119904(119906 119906 119909

119899minus1))

= 120595 (0) = 0

(63)

So we get 119866120590119887(119906 119891119906 119891119906) = 0 That is 119891119906 = 119906

Let (119883 119866120590119887 ⪯) be a partially ordered 119866

120590119887-metric-like

space We say that 119879 is an increasing mapping on 119883 ifforall119909 119910 isin 119883 119909 ⪯ 119910 rArr 119879(119909) ⪯ 119879(119910) [26] Fixed pointtheorems for monotone operators in ordered metric spacesarewidely investigated andhave found various applications indifferential and integral equations (see [27ndash30] and referencestherein) From the results proved above we derive thefollowing new results in partially ordered 119866

120590119887-metric-like

space

Theorem 34 Let (119883 119866120590119887 ⪯) be a partially ordered 119866

120590119887-

complete generalized 119887-metric-like space and let 119891 119883 rarr 119883

be an increasing mapping Suppose that

119904119866120590119887

(119891119909 119891119910 119891119911) le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911) (64)

for all 119909 119910 119911 isin 119883 with 119909 ⪯ 119910 ⪯ 119911 where

119872119904(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119891119909 119891119909)119866120590119887

(119910 119891119910 119891119910)119866120590119887

(119911 119891119911 119891119911)

1 + 11990421198662

120590119887

(119891119909 119891119910 119891119911)

(65)

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) 119891 is continuous or assume that whenever 119909119899 in 119883 is

an increasing sequence such that 119909119899rarr 119909 as 119899 rarr infin

one has 119909119899⪯ 119909 for all 119899 isin N cup 0

Then 119891 has a fixed point

Proof Define 120572 119883 times 119883 times 119883 rarr [0 +infin) by

120572 (119909 119910 119911) =

1 if119909 ⪯ 119910 ⪯ 119911

0 otherwise(66)

First we prove that 119891 is a triangular 120572-admissible mappingHence we assume that 120572(119909 119910 119911) ge 1 Therefore we have 119909 ⪯

119910 ⪯ 119911 Since 119891 is increasing we get 119891119909 ⪯ 119891119910 ⪯ 119891119911 that is120572(119891119909 119891119910 119891119911) ge 1 Also let 120572(119909 119910 119910) ge 1 and 120572(119910 119911 119911) ge 1then 119909 ⪯ 119910 and 119910 ⪯ 119911 Consequently we deduce that 119909 ⪯ 119911that is 120572(119909 119911 119911) ge 1 Thus 119891 is a triangular 120572-admissiblemapping Since 119891 satisfies (64) so by the definition of 120572 wehave

119904120572 (119909 119910 119911) 119866120590119887

(119891119909 119891119910 119891119911)

le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911)

(67)

for all 119909 119910 119911 isin 119883 Therefore 119891 satisfies the contractivecondition (21) From (i) there exists 119909

0isin 119883 such that

1199090⪯ 1198911199090 that is 120572(119909

0 1198911199090 1198911199090) ge 1 According to (ii) we

conclude that all the conditions of Theorems 26 and 27 aresatisfied and so 119891 has a fixed point

Similarly using Theorem 33 we can prove followingresult

Theorem 35 Let (119883 119866120590119887 ⪯) be an ordered 119866

120590119887-complete

generalized 119887-metric-like space and let 119891 119883 rarr 119883 be anincreasing mapping Suppose that

119904119866120590119887

(119891119909 119891119910 119891119911) le 120595 (119872119904(119909 119910 119911)) (68)

for all 119909 119910 119911 isin 119883 with 119909 ⪯ 119910 ⪯ 119911 where 120595 isin Ψ119887and

119872119904(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119909 119891119909)119866120590119887

(119910 119910 119891119910)119866120590119887

(119911 119911 119891119911)

1 + 211990421198662

120590119887

(119891119909 119891119910 119891119911)

(69)

10 The Scientific World Journal

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii)

(a) 119891 is continuous(b) assume that whenever x

119899 in 119883 is an increasing

sequence such that 119909119899rarr 119909 as 119899 rarr infin one has

119909119899⪯ 119909 for all 119899 isin N cup 0

Then 119891 has a fixed point

We conclude this section by presenting some examplesthat illustrate our results

Example 36 Let 119883 = [0 (minus1 + radic5)2] be endowed withthe usual ordering onR Define the generalized 119887-metric-likefunction 119866

120590119887given by

119866120590119887

(119909 119910 119911) = max (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

(70)

with 119904 = 2 Consider the mapping 119891 119883 rarr 119883 defined by119891(119909) = (14)119909

radic119890minus(119909+((minus1+radic5)2))

2 and the function 120573 isin Fgiven by 120573(119905) = (12)119890

minus119905 119905 gt 0 and 120573(0) isin [0 12) Itis easy to see that 119891 is an increasing function on 119883 Weshow that 119891 is 119866

120590119887-continuous on 119883 By Proposition 20 it is

sufficient to show that119891 is119866120590119887-sequentially continuous on119883

Let 119909119899 be a sequence in119883 such that lim

119899rarrinfin119866120590119887(119909119899 119909 119909) =

119866120590119887(119909 119909 119909) so we have max(lim

119899rarrinfin119909119899

+ 119909)2

41199092 =

41199092 and equivalently lim

119899rarrinfin119909119899

= 120572 le 119909 On the otherhand we have

lim119899rarrinfin

119866120590119887

(119891119909119899 119891119909 119891119909)

= lim119899rarrinfin

max (119891119909119899+ 119891119909)

2

41198912

119909

= lim119899rarrinfin

max 1

16

(119909119899

radic119890minus(119909119899+(minus1+

radic5)2)2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= max 1

16

( lim119899rarrinfin

119909119899

radic119890minus(lim119899rarrinfin119909119899+(minus1+radic5)2)

2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= max 1

16

(120572radic119890minus(120572+(minus1+radic5)2)

2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

=

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= 119866120590119887

(119891119909 119891119909 119891119909)

(71)

So 119891 is 119866120590119887-sequentially continuous on119883

For all elements 119909 119910 119911 isin 119883 and the fact that 119892(119909) =

1199092119890minus1199092

is an increasing function on119883 we have

119904119866120590119887

(119891119909 119891119910 119891119911)

= 2max (119891119909 + 119891119910)2

(119891119910 + 119891119911)2

(119891119911 + 119891119909)2

= 2max(

1

4

119909radic119890minus(119909+(minus1+radic5)2)

2

+

1

4

119910radic119890minus(119910+(minus1+radic5)2)

2

)

2

(

1

4

119910radic119890minus(119910+(minus1+radic5)2)

2

+

1

4

119911radic119890minus(119911+(minus1+radic5)2)

2

)

2

(

1

4

119911radic119890minus(119911+(minus1+radic5)2)

2

+

1

4

119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

le 2max(

1

4

119909radic119890minus(119909+119910)

2

+

1

4

119910radic119890minus(119910+119909)

2

)

2

(

1

4

119910radic119890minus(119910+119911)

2

+

1

4

119911radic119890minus(119911+119910)

2

)

2

(

1

4

119911radic119890minus(119911+119909)

2

+

1

4

119909radic119890minus(119909+119911)

2

)

2

=

1

8

max (119909 + 119910)2

119890minus(119909+119910)

2

(119910 + 119911)2

119890minus(119910+119911)

2

(119911 + 119909)2

119890minus(119911+119909)

2

=

1

8

119890minusmax(119909+119910)2 (119910+119911)2 (119911+119909)2

timesmax (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

le

1

2

eminusmax(119909+119910)2 (119910+119911)2(119911+119909)2

timesmax (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

= 120573 (119866120590119887

(119909 119910 119911)) 119866120590119887

(119909 119910 119911)

le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911)

(72)

Hence119891 satisfies all the assumptions ofTheorem 34 and thusit has a fixed point (which is 119906 = 0)

Example 37 Let 119883 = [0 1] with the usual ordering on RDefine the generalized 119887-metric-like function 119866

120590119887given by

119866120590119887(119909 119910 119911) = max(119909 + 119910)

2

(119910 + 119911)2

(119911 + 119909)2

with 119904 = 2Consider the mapping 119891 119883 rarr 119883 defined by 119891(119909) =

The Scientific World Journal 11

ln(1 + 119909119890minus119909

4) and the function 120595 isin Ψ119887given by 120595(119905) =

(18)119905 119905 ge 0 It is easy to see that 119891 is increasing functionNow we show that 119891 is a 119866

120590119887-continuous function on119883

Let 119909119899 be a sequence in 119883 such that lim

119899rarrinfin119866120590119887(119909119899

119909 119909) = 119866120590119887(119909 119909 119909) so we havemax(lim

119899rarrinfin119909119899+119909)2

41199092 =

41199092 and equally lim

119899rarrinfin119909119899= 120572 le 119909 On the other hand we

have

lim119899rarrinfin

119866120590119887

(119891119909119899 119891119909 119891119909)

= lim119899rarrinfin

max (119891119909119899+ 119891119909)

2

41198912

119909

= max(ln(1 +

lim119899rarrinfin

119909119899119890minuslim119899rarrinfin119909119899

4

)

+ ln(1 +

119909119890minus119909

4

))

2

4(ln(1 +

119909119890minusx

4

))

2

= max(ln(1 +

120572119890minus120572

4

) + ln(1 +

119909119890minus119909

4

))

2

4(ln(1 +

119909119890minus119909

4

))

2

= 4(ln(1 +

119909119890minus119909

4

))

2

= 119866120590119887

(119891119909 119891119909 119891119909)

(73)

So 119891 is 119866120590119887-sequentially continuous on119883

For all elements 119909 119910 119911 isin 119883 we have

119904119866120590119887

(119891119909 119891119910 119891119911)

= 2max(ln(1 +

119909119890minus119909

4

) + ln(1 +

119910119890minus119910

4

))

2

(ln(1 +

119910119890minus119910

4

) + ln(1 +

119911119890minus119911

4

))

2

(ln(1 +

119911119890minus119911

4

) + ln(1 +

119909119890minus119909

4

))

2

le 2max(

119909119890minus119909

4

+

119910119890minus119910

4

)

2

(

119910119890minus119910

4

+

119911119890minus119911

4

)

2

(

119911119890minus119911

4

+

119909119890minus119909

4

)

2

le

1

8

max (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

= 120595 (119866120590119887

(119909 119910 119911))

le 120595 (119872119904(119909 119910 119911))

(74)

Hence119891 satisfies all the assumptions ofTheorem 35 and thusit has a fixed point (which is 119906 = 0)

3 Application

In this section we present an application of our results toestablish the existence of a solution to a periodic boundaryvalue problem (see [30 31])

Let 119883 = 119862([0 119879]) be the set of all real continuousfunctions on [0T] We first endow119883 with the 119887-metric-like

120590119887(119906 V) = sup

119905isin[0119879]

(|119906 (119905)| + |V (119905)|)119901 (75)

for all 119906 V isin 119883 where 119901 gt 1 and then we endow it with thegeneralized 119887-metric-like 119866

120590119887defined by

119866120590119887

(119906 V 119908)

= max sup119905isin[0119879]

(|119906 (119905)| + |V (119905)|)119901

sup119905isin[0119879]

(|V (119905)| + |119908 (119905)|)119901

sup119905isin[0119879]

(|119908 (119905)| + |119906 (119905)|)119901

(76)

Clearly (119883 119866120590119887) is a complete generalized 119887-metric-like space

with parameter 119870 = 2119901minus1 We equip 119862([0 119879]) with a partial

order given by119909 ⪯ 119910 iff119909 (119905) le 119910 (119905) forall119905 isin [0 119879] (77)

Moreover as in [30] it is proved that (119862([0 119879]) ⪯) is regularthat is whenever 119909

119899 in119883 is an increasing sequence such that

119909119899rarr 119909 as 119899 rarr infin we have 119909

119899⪯ 119909 for all 119899 isin N cup 0

Consider the first-order periodic boundary value prob-lem

1199091015840

(119905) = 119891 (119905 119909 (119905)) 119909 (0) = 119909 (119879) (78)

where 119905 isin 119868 = [0 119879] with 119879 gt 0 and 119891 119868 times R rarr R is acontinuous function

A lower solution for (78) is a function 120572 isin 1198621[0 119879] such

that1205721015840

(119905) le 119891 (119905 120572 (119905))

120572 (0) le 120572 (119879)

(79)

where 119905 isin 119868 = [0 119879]Assume that there exists 120582 gt 0 such that for all 119909 119910 isin

119862[0 119879] we have1003816100381610038161003816119891 (119905 119909 (119905)) + 120582119909 (119905)

1003816100381610038161003816+1003816100381610038161003816119891 (119905 119910 (119905)) + 120582119910 (119905)

1003816100381610038161003816

le

120582

2119901minus1

119901radicln(

119886

2119901minus1

(|119909 (119905)| +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)119901

+ 1)

(80)

where 0 le 119886 lt 1 Then the existence of a lower solution for(78) provides the existence of a solution of (78)

Problem (78) can be rewritten as

1199091015840

(119905) + 120582119909 (119905) = 119891 (119905 119909 (119905)) + 120582119909 (119905)

119909 (0) = 119909 (119879)

(81)

Consider1199091015840

(119905) + 120582119909 (119905) = 120575119909(119905) = 119865 (119905 119909 (119905))

119909 (0) = 119909 (119879)

(82)

where 119905 isin 119868

12 The Scientific World Journal

Using variation of parameters formula we get

119909 (119905) = 119909 (0) 119890minus120582119905

+ int

119905

0

119890minus120582(119905minus119904)

120575119909(119904) 119889119904 (83)

which yields

119909 (119879) = 119909 (0) 119890minus120582119879

+ int

119879

0

119890minus120582(119879minus119904)

120575119909(119904) 119889119904 (84)

Since 119909(0) = 119909(119879) we get

119909 (0) [1 minus 119890minus120582119879

] = 119890minus120582119879

int

119879

0

119890120582(119904)

120575119909(119904) 119889119904 (85)

or

119909 (0) =

1

119890120582119879

minus 1

int

119879

0

119890120582119904

120575119909(119904) 119889119904 (86)

Substituting the value of 119909(0) in (83) we arrive at

119909 (119905) = int

119879

0

119866 (119905 119904) 120575119909(119904) 119889119904 (87)

where

119866 (119905 119904) =

119890120582(119879+119904minus119905)

119890120582119879

minus 1

0 le 119904 le 119905 le 119879

119890120582(119904minus119905)

119890120582119879

minus 1

0 le 119905 le 119904 le 119879

(88)

Now define the operator 119878 119862[0 119879] rarr 119862[0 119879] as

119878119909 (119905) = int

119879

0

119866 (119905 119904) 119865 (119904 119909 (119904)) 119889119904 (89)

The mapping 119878 is increasing [31] Note that if 119906 isin 119862[0 119879] is afixed point of 119878 then 119906 isin 119862

1[0 119879] is a solution of (78)

Let 119909 119910 119911 isin 119862[0 119879] Then we have

2119901minus1

[|119878119909 (119905)| +1003816100381610038161003816119878119910 (119905)

1003816100381610038161003816]

= 2119901minus1

[

100381610038161003816100381610038161003816100381610038161003816

int

119879

0

119866 (119905 119904) 119865 (119904 119909 (119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

119879

0

119866 (119905 119904) 119865 (119904 119910 (119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

]

le 2119901minus1

int

119879

0

|119866 (119905 119904)|

times [|119865 (119904 119909 (119904))| +1003816100381610038161003816119865 (119904 119910 (119904))

1003816100381610038161003816] 119889119904

le 2119901minus1

int

119879

0

|119866 (119905 119904)|

120582

2119901minus1

times119901radicln(

119886

2119901minus1

(|119909 (119905)| +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)119901

+ 1)119889119904

le 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [int

119905

0

119890120582(119879+119904minus119905)

119890120582119879

minus 1

119889119904 + int

119879

119905

119890120582(119904minus119905)

119890120582119879

minus 1

119889119904]

= 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [

1

120582 (119890120582119879

minus 1)

(119890120582(119879+119904minus119905)10038161003816

100381610038161003816

119905

0

+ 119890120582(119904minus119905)10038161003816

100381610038161003816

119879

119905

)]

= 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [

1

120582 (119890120582119879

minus 1)

(119890120582119879

minus 119890120582(119879minus119905)

+ 119890120582(119879minus119905)

minus 1)]

=119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

le119901radicln(

119886

2119901minus1

119866120590119887

(119909 119910 119911) + 1)

le119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(90)

Similarly

2119901minus1 1003816

100381610038161003816119878119910 (119905)

1003816100381610038161003816+ |119878119911 (119905)| le

119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

|119878119909 (119905)| + |119878119911 (119905)| le119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(91)

where

119872(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119878119909 119878119909) 119866120590119887

(119910 119878119910 119878119910)119866120590119887

(119911 119878119911 119878119911)

1 + 22119901minus1

1198662

120590119887

(119878119909 119878119910 119878119911)

(92)

Equivalently

(2119901minus1

|119878119909 (119905)| +1003816100381610038161003816119878119910 (119905)

1003816100381610038161003816)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(2119901minus1 1003816

100381610038161003816119878119910 (119905)

1003816100381610038161003816+ |119878119911 (119905)|)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(2119901minus1

|119878119909 (119905)| + |119878119911 (119905)|)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(93)

which yields that

2119901minus1

120590119887(119878119909 119878119910) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

120590119887(119878119910 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

120590119887(119878119909 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(94)

The Scientific World Journal 13

Finally it is easy to obtain that

2119901minus1G120590119887

(119878119909 119878119910 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1) (95)

Finally since120572 is a lower solution for (78) so it is easy to showthat 120572 ⪯ 119878(120572) [31]

Hence the hypotheses of Theorem 35 are satisfied with120595(119905) = ln((1198862119901minus1)119905 + 1) where 0 le 119886 lt 1 Hence there existsa fixed point 119909 isin 119862[0 119879] such that 119878119909 = 119909 which is a solutionto periodic boundary value problem (78)

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) KingAbdulaziz University JeddahTherefore the firstauthor acknowledges with thanks DSR KAU for financialsupport

References

[1] S Czerwik ldquoContraction mappings in b-metric spacesrdquo ActaMathematica et Informatica Universitatis Ostraviensis vol 1 pp5ndash11 1993

[2] V Parvaneh J R Roshan and S Radenovic ldquoExistence of tri-pled coincidence points in ordered b-metric spaces and an app-lication to a system of integral equationsrdquo Fixed Point Theoryand Applications vol 2013 article 130 2013

[3] J R Roshan V Parvaneh S Sedghi N Shobkolaei and WShatanawi ldquoCommon fixed points of almost generalized con-tractive mappings in ordered b-metric spacesrdquo Fixed PointTheory and Applications vol 2013 article 159 2013

[4] H Aydi M F Bota E Karapınar and S Moradi ldquoA commonfixed point for weak 120601-contractions on b-metric spacesrdquo FixedPoint Theory vol 13 no 2 pp 337ndash346 2012

[5] V Berinde ldquoSequences of operators and fixed points in quasi-metric spacesrdquo Studia Universitatis ldquoBabes-Bolyairdquo Mathemat-ica vol 16 no 4 pp 23ndash27 1996

[6] N Hussain D ETHoric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 article 126 2012

[7] N Hussain and M H Shah ldquoKKMmappings in cone b-metricspacesrdquo Computers and Mathematics with Applications vol 62no 4 pp 1677ndash1684 2011

[8] N Hussain J R Roshan V Parvaneh and Z Kadelburg ldquoFixedpoints of contractive mappings in b-metric-like spacesrdquo TheScientific World Journal vol 2014 Article ID 471827 15 pages2014

[9] M A Khamsi and N Hussain ldquoKKMmappings in metric typespacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 73 no 9 pp 3123ndash3129 2010

[10] M A Khamsi ldquoRemarks on cone metric spaces and fixed pointtheorems of contractive mappingsrdquo Fixed Point Theory andApplications vol 2010 Article ID 315398 7 pages 2010

[11] Z Mustafa J R Roshan V Parvaneh and Z Kadelburg ldquoSomecommon fixed point results in ordered partial b-metric spacesrdquoJournal of Inequalities and Applications vol 2013 article 5622013

[12] Z Mustafa and B Sims ldquoA new approach to generalized metricspacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash297 2006

[13] A Amini-Harandi ldquoMetric-like spaces partial metric spacesand fixed pointsrdquo Fixed PointTheory and Applications vol 2012article 204 2012

[14] N Shobkolaei S Sedghi J R Roshan and N Hussain ldquoSuzukitype fixed point results in metric-like spacesrdquo Journal of Func-tion Spaces and Applications vol 2013 Article ID 143686 9pages 2013

[15] M A Alghamdi N Hussain and P Salimi ldquoFixed point andcoupled fixed point theorems on b-metric-like spacesrdquo Journalof Inequalities and Applications vol 2013 article 402 2013

[16] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572 minus 120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

[17] P Salimi and E Karapınar ldquoSuzuki-Edelstein type contractionsvia auxiliary functionsrdquoMathematical Problems in Engineeringvol 2013 Article ID 648528 8 pages 2013

[18] P Salimi C Vetro and P Vetro ldquoFixed point theorems fortwisted (120572 120573)-120595-contractive type mappings and applicationsrdquoFilomat vol 27 no 4 pp 605ndash615 2013

[19] P Salimi C Vetro and P Vetro ldquoSome new fixed point resultsin non-Archimedean fuzzy metric spacesrdquo Nonlinear AnalysisModelling and Control vol 18 no 3 pp 344ndash358 2013

[20] N Hussain P Salimi and A Latif ldquoFixed point results forsingle and set-valued 120572-120578-120595-contractive mappingsrdquo Fixed PointTheory and Applications vol 2013 article 212 2013

[21] M A Alghamdi and E Karapınar ldquoG-120573-120595 contractive-typemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 article 70 2013

[22] M Geraghty ldquoOn contractive mappingsrdquo Proceedings of theAmerican Mathematical Society vol 40 pp 604ndash608 1973

[23] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001

[24] N Hussain Z Kadelburg S Radenovic and F R Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012

[25] V Berinde ldquoGeneralized contractions in quasimetric spacesrdquo inSeminar on Fixed Point Theory Preprint no 3 pp 3ndash9 1993

[26] L Ciric M Abbas R Saadati and N Hussain ldquoCommon fixedpoints of almost generalized contractive mappings in orderedmetric spacesrdquo Applied Mathematics and Computation vol 217no 12 pp 5784ndash5789 2011

[27] R P Agarwal N Hussain and M A Taoudi ldquoFixed point the-orems in ordered Banach spaces and applications to nonlinearintegral equationsrdquo Abstract and Applied Analysis vol 2012Article ID 245872 15 pages 2012

[28] N Hussain A R Khan and R P Agarwal ldquoKrasnoselrsquoskii andKy Fan type fixed point theorems in ordered Banach spacesrdquoJournal of Nonlinear and Convex Analysis vol 11 no 3 pp 475ndash489 2010

[29] N Hussain and M A Taoudi ldquoKrasnoselrsquoskii-type fixed pointtheorems with applications to Volterra integral equationsrdquoFixed Point Theory and Applications vol 2013 article 196 2013

14 The Scientific World Journal

[30] J J Nieto and R Rodrıguez-Lopez ldquoContractivemapping theo-rems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[31] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysisTheoryMethods andApplications vol 71 no 7-8 pp 3403ndash3410 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 The Scientific World Journal

If there exists 1198990isin N such that 119909

1198990= 1199091198990+1

then 1199091198990

=

1198911199091198990and so we have nothing to prove Hence for all 119899 isin N

we assume that 119909119899

= 119909119899+1

Step I (Cauchyness of 119909119899) As 120572(119909

119899 119909119899+1

119909119899+2

) ge 1 for all119899 ge 0 using condition (46) we obtain

119904119866120590119887

(119909119899 119909119899+1

119909119899+2

)

le 119904120572 (119909119899minus1

119909119899 119909119899+1

) 119866120590119887

(119909119899 119909119899+1

119909119899+2

)

= 119904120572 (119909119899minus1

119909119899 119909119899+1

) 119866120590119887

(119891119909119899minus1

119891119909119899 119891119909119899+1

)

le 120595 (119872119904(119909119899minus1

119909119899 119909119899+1

))

(48)

Using Proposition 15(2) as 119909119899

= 119909119899+1

we get

119872119904(119909119899minus1

119909119899 119909119899+1

)

= max 119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

(119866120590119887

(119909119899minus1

119909119899minus1

119891119909119899minus1

)

times 119866120590119887

(119909119899 119909119899 119891119909119899)

times119866120590119887

(119909119899+1

119909119899+1

119891119909119899+1

))

times(1 + 21199042

1198662

120590119887

(119891119909119899minus1

119891119909119899 119891119909119899+1

))

minus1

= max 119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

(119866120590119887

(119909119899minus1

119909119899minus1

119909119899)

times 119866120590119887

(119909119899 119909119899 119909119899+1

)

times119866120590119887

(119909119899+1

119909119899+1

119909119899+2

))

times(1 + 21199042

1198662

120590119887

(119909119899 119909119899+1

119909119899+2

))

minus1

le max119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

2119904119866120590119887

(119909119899minus1

119909119899 119909119899+1

) 1198662

120590119887

(119909119899 119909119899+1

119909119899+2

)

1 + 2s21198662120590119887

(119909119899 119909119899+1

119909119899+2

)

= 119866120590119887

(119909119899minus1

119909119899 119909119899+1

)

(49)

Hence119866120590119887

(119909119899 119909119899+1

119909119899+2

)

le 119904119866120590119887

(119909119899 119909119899+1

119909119899+2

) le 120595 (119866120590119887

(119909119899minus1

119909119899 119909119899+1

))

(50)

By induction since 119909119899

= 119909119899+1

we get that

119866120590119887

(119909119899 119909119899+1

119909119899+2

)

le 120595 (119866120590119887

(119909119899minus1

119909119899 119909119899+1

))

le 1205952

(119866120590119887

(119909119899minus2

119909119899minus1

119909119899))

le sdot sdot sdot le 120595119899

(119866120590119887

(1199090 1199091 1199092))

(51)

Let 120576 gt 0 be arbitrary Then there exists a natural number119873such that

infin

sum

119899=119873

119904119899

120595119899

(119866120590119887

(1199090 1199091 1199092)) lt

120576

2119904

(52)

Let 119898 gt 119899 ge 119873 Then by the rectangular inequality andProposition 15(2) as 119909

119899= 119909119899+1

we get

119866120590119887

(119909119899 119909119898 119909119898)

le 119904119866120590119887

(119909119899 119909119899+1

119909119899+1

) + 1199042

119866120590119887

(119909119899+1

119909119899+2

119909119899+2

)

+ sdot sdot sdot + 119904119898minus119899minus1

119866120590119887

(119909119898minus1

119909119898 119909119898)

le 2119904 (119904119866120590119887

(119909119899 119909119899 119909119899+1

) + 1199042

119866120590119887

(119909119899+1

119909119899+1

119909119899+2

)

+ sdot sdot sdot + 119904119898minus119899minus1

119866120590119887

(119909119898minus1

119909119898minus1

119909119898))

le 2119904 (119904119866120590119887

(119909119899 119909119899+1

119909119899+2

) + 1199042

119866120590119887

(119909119899+1

119909119899+2

119909119899+3

)

+ sdot sdot sdot + 119904119898minus119899minus1

119866120590119887

(119909119898minus1

119909119898 119909119898+1

))

le 2119904

119898minus2

sum

119896=119899

119904119896minus119899+1

120595119896

(119866120590119887

(1199090 1199091 1199092))

le 2119904

infin

sum

119896=119899

119904119896

120595119896

(119866120590119887

(1199090 1199091 1199092)) lt 120576

(53)

Consequently 119909119899 is a 119866

120590119887-Cauchy sequence in 119883 Since

(119883 119866120590119887) is 119866120590119887-complete so there exists 119906 isin 119883 such that

lim119899rarrinfin

119866120590119887

(119906 119909119899 119909119899) = lim119898119899rarrinfin

119866120590119887

(119909119899 119909119898 119909119898)

= 119866120590119887

(119906 119906 119906) = 0

(54)

Step II Now we show that 119906 is a fixed point of 119891 Suppose tothe contrary that is 119891119906 = 119906 then we have 119866

120590119887(119906 119906 119891119906) gt 0

Let the part (a) of (ii) holdsUsing the rectangle inequality we get

119866120590119887

(119906 119906 119891119906) le 119904119866120590119887

(119891119906 119891119909119899 119891119909119899) + 119904119866

120590119887(119891119909119899 119906 119906)

(55)

Letting 119899 rarr infin and using the continuity of 119891 we get

119866120590119887

(119906 119906 119891119906) le 119904119866120590119887

(119891119906 119891119906 119891119906) (56)

From (46) and part (a) of condition (ii) we have

119904119866120590119887

(119891119906 119891119906 119891119906) le 120595 (119872119904(119906 119906 119906)) (57)

where by using (56) we have

119872119904(119906 119906 119906)

= max119866120590119887

(119906 119906 119906)

119866120590119887

(119906 119906 119891119906)119866120590119887

(119906 119906 119891119906)119866120590119887

(119906 119906 119891119906)

1 + 211990421198662

120590119887

(119891119906 119891119906 119891119906)

le 119866120590119887

(119906 119906 119891119906)

(58)

Hence from properties of 120595 119904119866120590119887(119891119906 119891119906 119891119906) le

119866120590119887(119906 119906 119891119906) Thus by (56) we obtain that

119866120590119887

(119906 119906 119891119906) = 119904119866120590119887

(119891119906 119891119906 119891119906) (59)

The Scientific World Journal 9

Moreover (57) yields that 119866120590119887(119906 119906 119891119906) le 120595(119872

119904(119906 119906 119906)) le

120595(119866120590119887(119906 119906 119891119906))This is impossible according to our assump-

tions on 120595 Hence we have 119891119906 = 119906 Thus 119906 is a fixed point of119891

Now let part (b) of (ii) holdsAs 119909119899 is a sequence such that 120572(119909

119899 119909119899+1

119909119899+2

) ge 1 for all119899 isin N cup 0 and 119909

119899rarr 119909 as 119899 rarr infin we have 120572(119909

119899 119909 119909) ge 1

for all 119899 isin N cup 0Now we show that 119906 = 119891119906 By (46) we have

119904119866120590119887

(119891119906 119891119906 119909119899) le 119904120572 (119906 119906 119909

119899minus1) 119866120590119887

(119891119906 119891119906 119891119909119899minus1

)

le 120595 (119872119904(119906 119906 119909

119899minus1))

(60)

where

119872119904(119906 119906 119909

119899minus1)

= max

119866120590119887

(119906 119906 119909119899minus1

)

[119866120590119887

(119906 119906 119891119906)]

2

119866120590119887

(119909119899minus1

119909119899minus1

119891119909119899minus1

)

1 + [119904119866120590119887

(119891119906 119891119906 119891119909119899minus1

)]

2

(61)

Letting 119899 rarr infin in the above inequality and using (54) andLemma 21 we get

lim119899rarrinfin

119872119904(119906 119906 119909

119899minus1) = 0 (62)

Again taking the upper limit as 119899 rarr infin in (60) and using(62) and Lemma 21 we obtain

119904 [

1

119904

119866120590119887

(119906 119891119906 119891119906) minus 119866120590119887

(119906 119906 119906)]

le 119904 lim sup119899rarrinfin

119866120590119887

(119909119899 119891119906 119891119906)

le lim supnrarrinfin

120595 (119872119904(119906 119906 119909

119899minus1))

= 120595(lim sup119899rarrinfin

119872119904(119906 119906 119909

119899minus1))

= 120595 (0) = 0

(63)

So we get 119866120590119887(119906 119891119906 119891119906) = 0 That is 119891119906 = 119906

Let (119883 119866120590119887 ⪯) be a partially ordered 119866

120590119887-metric-like

space We say that 119879 is an increasing mapping on 119883 ifforall119909 119910 isin 119883 119909 ⪯ 119910 rArr 119879(119909) ⪯ 119879(119910) [26] Fixed pointtheorems for monotone operators in ordered metric spacesarewidely investigated andhave found various applications indifferential and integral equations (see [27ndash30] and referencestherein) From the results proved above we derive thefollowing new results in partially ordered 119866

120590119887-metric-like

space

Theorem 34 Let (119883 119866120590119887 ⪯) be a partially ordered 119866

120590119887-

complete generalized 119887-metric-like space and let 119891 119883 rarr 119883

be an increasing mapping Suppose that

119904119866120590119887

(119891119909 119891119910 119891119911) le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911) (64)

for all 119909 119910 119911 isin 119883 with 119909 ⪯ 119910 ⪯ 119911 where

119872119904(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119891119909 119891119909)119866120590119887

(119910 119891119910 119891119910)119866120590119887

(119911 119891119911 119891119911)

1 + 11990421198662

120590119887

(119891119909 119891119910 119891119911)

(65)

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) 119891 is continuous or assume that whenever 119909119899 in 119883 is

an increasing sequence such that 119909119899rarr 119909 as 119899 rarr infin

one has 119909119899⪯ 119909 for all 119899 isin N cup 0

Then 119891 has a fixed point

Proof Define 120572 119883 times 119883 times 119883 rarr [0 +infin) by

120572 (119909 119910 119911) =

1 if119909 ⪯ 119910 ⪯ 119911

0 otherwise(66)

First we prove that 119891 is a triangular 120572-admissible mappingHence we assume that 120572(119909 119910 119911) ge 1 Therefore we have 119909 ⪯

119910 ⪯ 119911 Since 119891 is increasing we get 119891119909 ⪯ 119891119910 ⪯ 119891119911 that is120572(119891119909 119891119910 119891119911) ge 1 Also let 120572(119909 119910 119910) ge 1 and 120572(119910 119911 119911) ge 1then 119909 ⪯ 119910 and 119910 ⪯ 119911 Consequently we deduce that 119909 ⪯ 119911that is 120572(119909 119911 119911) ge 1 Thus 119891 is a triangular 120572-admissiblemapping Since 119891 satisfies (64) so by the definition of 120572 wehave

119904120572 (119909 119910 119911) 119866120590119887

(119891119909 119891119910 119891119911)

le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911)

(67)

for all 119909 119910 119911 isin 119883 Therefore 119891 satisfies the contractivecondition (21) From (i) there exists 119909

0isin 119883 such that

1199090⪯ 1198911199090 that is 120572(119909

0 1198911199090 1198911199090) ge 1 According to (ii) we

conclude that all the conditions of Theorems 26 and 27 aresatisfied and so 119891 has a fixed point

Similarly using Theorem 33 we can prove followingresult

Theorem 35 Let (119883 119866120590119887 ⪯) be an ordered 119866

120590119887-complete

generalized 119887-metric-like space and let 119891 119883 rarr 119883 be anincreasing mapping Suppose that

119904119866120590119887

(119891119909 119891119910 119891119911) le 120595 (119872119904(119909 119910 119911)) (68)

for all 119909 119910 119911 isin 119883 with 119909 ⪯ 119910 ⪯ 119911 where 120595 isin Ψ119887and

119872119904(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119909 119891119909)119866120590119887

(119910 119910 119891119910)119866120590119887

(119911 119911 119891119911)

1 + 211990421198662

120590119887

(119891119909 119891119910 119891119911)

(69)

10 The Scientific World Journal

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii)

(a) 119891 is continuous(b) assume that whenever x

119899 in 119883 is an increasing

sequence such that 119909119899rarr 119909 as 119899 rarr infin one has

119909119899⪯ 119909 for all 119899 isin N cup 0

Then 119891 has a fixed point

We conclude this section by presenting some examplesthat illustrate our results

Example 36 Let 119883 = [0 (minus1 + radic5)2] be endowed withthe usual ordering onR Define the generalized 119887-metric-likefunction 119866

120590119887given by

119866120590119887

(119909 119910 119911) = max (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

(70)

with 119904 = 2 Consider the mapping 119891 119883 rarr 119883 defined by119891(119909) = (14)119909

radic119890minus(119909+((minus1+radic5)2))

2 and the function 120573 isin Fgiven by 120573(119905) = (12)119890

minus119905 119905 gt 0 and 120573(0) isin [0 12) Itis easy to see that 119891 is an increasing function on 119883 Weshow that 119891 is 119866

120590119887-continuous on 119883 By Proposition 20 it is

sufficient to show that119891 is119866120590119887-sequentially continuous on119883

Let 119909119899 be a sequence in119883 such that lim

119899rarrinfin119866120590119887(119909119899 119909 119909) =

119866120590119887(119909 119909 119909) so we have max(lim

119899rarrinfin119909119899

+ 119909)2

41199092 =

41199092 and equivalently lim

119899rarrinfin119909119899

= 120572 le 119909 On the otherhand we have

lim119899rarrinfin

119866120590119887

(119891119909119899 119891119909 119891119909)

= lim119899rarrinfin

max (119891119909119899+ 119891119909)

2

41198912

119909

= lim119899rarrinfin

max 1

16

(119909119899

radic119890minus(119909119899+(minus1+

radic5)2)2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= max 1

16

( lim119899rarrinfin

119909119899

radic119890minus(lim119899rarrinfin119909119899+(minus1+radic5)2)

2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= max 1

16

(120572radic119890minus(120572+(minus1+radic5)2)

2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

=

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= 119866120590119887

(119891119909 119891119909 119891119909)

(71)

So 119891 is 119866120590119887-sequentially continuous on119883

For all elements 119909 119910 119911 isin 119883 and the fact that 119892(119909) =

1199092119890minus1199092

is an increasing function on119883 we have

119904119866120590119887

(119891119909 119891119910 119891119911)

= 2max (119891119909 + 119891119910)2

(119891119910 + 119891119911)2

(119891119911 + 119891119909)2

= 2max(

1

4

119909radic119890minus(119909+(minus1+radic5)2)

2

+

1

4

119910radic119890minus(119910+(minus1+radic5)2)

2

)

2

(

1

4

119910radic119890minus(119910+(minus1+radic5)2)

2

+

1

4

119911radic119890minus(119911+(minus1+radic5)2)

2

)

2

(

1

4

119911radic119890minus(119911+(minus1+radic5)2)

2

+

1

4

119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

le 2max(

1

4

119909radic119890minus(119909+119910)

2

+

1

4

119910radic119890minus(119910+119909)

2

)

2

(

1

4

119910radic119890minus(119910+119911)

2

+

1

4

119911radic119890minus(119911+119910)

2

)

2

(

1

4

119911radic119890minus(119911+119909)

2

+

1

4

119909radic119890minus(119909+119911)

2

)

2

=

1

8

max (119909 + 119910)2

119890minus(119909+119910)

2

(119910 + 119911)2

119890minus(119910+119911)

2

(119911 + 119909)2

119890minus(119911+119909)

2

=

1

8

119890minusmax(119909+119910)2 (119910+119911)2 (119911+119909)2

timesmax (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

le

1

2

eminusmax(119909+119910)2 (119910+119911)2(119911+119909)2

timesmax (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

= 120573 (119866120590119887

(119909 119910 119911)) 119866120590119887

(119909 119910 119911)

le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911)

(72)

Hence119891 satisfies all the assumptions ofTheorem 34 and thusit has a fixed point (which is 119906 = 0)

Example 37 Let 119883 = [0 1] with the usual ordering on RDefine the generalized 119887-metric-like function 119866

120590119887given by

119866120590119887(119909 119910 119911) = max(119909 + 119910)

2

(119910 + 119911)2

(119911 + 119909)2

with 119904 = 2Consider the mapping 119891 119883 rarr 119883 defined by 119891(119909) =

The Scientific World Journal 11

ln(1 + 119909119890minus119909

4) and the function 120595 isin Ψ119887given by 120595(119905) =

(18)119905 119905 ge 0 It is easy to see that 119891 is increasing functionNow we show that 119891 is a 119866

120590119887-continuous function on119883

Let 119909119899 be a sequence in 119883 such that lim

119899rarrinfin119866120590119887(119909119899

119909 119909) = 119866120590119887(119909 119909 119909) so we havemax(lim

119899rarrinfin119909119899+119909)2

41199092 =

41199092 and equally lim

119899rarrinfin119909119899= 120572 le 119909 On the other hand we

have

lim119899rarrinfin

119866120590119887

(119891119909119899 119891119909 119891119909)

= lim119899rarrinfin

max (119891119909119899+ 119891119909)

2

41198912

119909

= max(ln(1 +

lim119899rarrinfin

119909119899119890minuslim119899rarrinfin119909119899

4

)

+ ln(1 +

119909119890minus119909

4

))

2

4(ln(1 +

119909119890minusx

4

))

2

= max(ln(1 +

120572119890minus120572

4

) + ln(1 +

119909119890minus119909

4

))

2

4(ln(1 +

119909119890minus119909

4

))

2

= 4(ln(1 +

119909119890minus119909

4

))

2

= 119866120590119887

(119891119909 119891119909 119891119909)

(73)

So 119891 is 119866120590119887-sequentially continuous on119883

For all elements 119909 119910 119911 isin 119883 we have

119904119866120590119887

(119891119909 119891119910 119891119911)

= 2max(ln(1 +

119909119890minus119909

4

) + ln(1 +

119910119890minus119910

4

))

2

(ln(1 +

119910119890minus119910

4

) + ln(1 +

119911119890minus119911

4

))

2

(ln(1 +

119911119890minus119911

4

) + ln(1 +

119909119890minus119909

4

))

2

le 2max(

119909119890minus119909

4

+

119910119890minus119910

4

)

2

(

119910119890minus119910

4

+

119911119890minus119911

4

)

2

(

119911119890minus119911

4

+

119909119890minus119909

4

)

2

le

1

8

max (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

= 120595 (119866120590119887

(119909 119910 119911))

le 120595 (119872119904(119909 119910 119911))

(74)

Hence119891 satisfies all the assumptions ofTheorem 35 and thusit has a fixed point (which is 119906 = 0)

3 Application

In this section we present an application of our results toestablish the existence of a solution to a periodic boundaryvalue problem (see [30 31])

Let 119883 = 119862([0 119879]) be the set of all real continuousfunctions on [0T] We first endow119883 with the 119887-metric-like

120590119887(119906 V) = sup

119905isin[0119879]

(|119906 (119905)| + |V (119905)|)119901 (75)

for all 119906 V isin 119883 where 119901 gt 1 and then we endow it with thegeneralized 119887-metric-like 119866

120590119887defined by

119866120590119887

(119906 V 119908)

= max sup119905isin[0119879]

(|119906 (119905)| + |V (119905)|)119901

sup119905isin[0119879]

(|V (119905)| + |119908 (119905)|)119901

sup119905isin[0119879]

(|119908 (119905)| + |119906 (119905)|)119901

(76)

Clearly (119883 119866120590119887) is a complete generalized 119887-metric-like space

with parameter 119870 = 2119901minus1 We equip 119862([0 119879]) with a partial

order given by119909 ⪯ 119910 iff119909 (119905) le 119910 (119905) forall119905 isin [0 119879] (77)

Moreover as in [30] it is proved that (119862([0 119879]) ⪯) is regularthat is whenever 119909

119899 in119883 is an increasing sequence such that

119909119899rarr 119909 as 119899 rarr infin we have 119909

119899⪯ 119909 for all 119899 isin N cup 0

Consider the first-order periodic boundary value prob-lem

1199091015840

(119905) = 119891 (119905 119909 (119905)) 119909 (0) = 119909 (119879) (78)

where 119905 isin 119868 = [0 119879] with 119879 gt 0 and 119891 119868 times R rarr R is acontinuous function

A lower solution for (78) is a function 120572 isin 1198621[0 119879] such

that1205721015840

(119905) le 119891 (119905 120572 (119905))

120572 (0) le 120572 (119879)

(79)

where 119905 isin 119868 = [0 119879]Assume that there exists 120582 gt 0 such that for all 119909 119910 isin

119862[0 119879] we have1003816100381610038161003816119891 (119905 119909 (119905)) + 120582119909 (119905)

1003816100381610038161003816+1003816100381610038161003816119891 (119905 119910 (119905)) + 120582119910 (119905)

1003816100381610038161003816

le

120582

2119901minus1

119901radicln(

119886

2119901minus1

(|119909 (119905)| +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)119901

+ 1)

(80)

where 0 le 119886 lt 1 Then the existence of a lower solution for(78) provides the existence of a solution of (78)

Problem (78) can be rewritten as

1199091015840

(119905) + 120582119909 (119905) = 119891 (119905 119909 (119905)) + 120582119909 (119905)

119909 (0) = 119909 (119879)

(81)

Consider1199091015840

(119905) + 120582119909 (119905) = 120575119909(119905) = 119865 (119905 119909 (119905))

119909 (0) = 119909 (119879)

(82)

where 119905 isin 119868

12 The Scientific World Journal

Using variation of parameters formula we get

119909 (119905) = 119909 (0) 119890minus120582119905

+ int

119905

0

119890minus120582(119905minus119904)

120575119909(119904) 119889119904 (83)

which yields

119909 (119879) = 119909 (0) 119890minus120582119879

+ int

119879

0

119890minus120582(119879minus119904)

120575119909(119904) 119889119904 (84)

Since 119909(0) = 119909(119879) we get

119909 (0) [1 minus 119890minus120582119879

] = 119890minus120582119879

int

119879

0

119890120582(119904)

120575119909(119904) 119889119904 (85)

or

119909 (0) =

1

119890120582119879

minus 1

int

119879

0

119890120582119904

120575119909(119904) 119889119904 (86)

Substituting the value of 119909(0) in (83) we arrive at

119909 (119905) = int

119879

0

119866 (119905 119904) 120575119909(119904) 119889119904 (87)

where

119866 (119905 119904) =

119890120582(119879+119904minus119905)

119890120582119879

minus 1

0 le 119904 le 119905 le 119879

119890120582(119904minus119905)

119890120582119879

minus 1

0 le 119905 le 119904 le 119879

(88)

Now define the operator 119878 119862[0 119879] rarr 119862[0 119879] as

119878119909 (119905) = int

119879

0

119866 (119905 119904) 119865 (119904 119909 (119904)) 119889119904 (89)

The mapping 119878 is increasing [31] Note that if 119906 isin 119862[0 119879] is afixed point of 119878 then 119906 isin 119862

1[0 119879] is a solution of (78)

Let 119909 119910 119911 isin 119862[0 119879] Then we have

2119901minus1

[|119878119909 (119905)| +1003816100381610038161003816119878119910 (119905)

1003816100381610038161003816]

= 2119901minus1

[

100381610038161003816100381610038161003816100381610038161003816

int

119879

0

119866 (119905 119904) 119865 (119904 119909 (119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

119879

0

119866 (119905 119904) 119865 (119904 119910 (119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

]

le 2119901minus1

int

119879

0

|119866 (119905 119904)|

times [|119865 (119904 119909 (119904))| +1003816100381610038161003816119865 (119904 119910 (119904))

1003816100381610038161003816] 119889119904

le 2119901minus1

int

119879

0

|119866 (119905 119904)|

120582

2119901minus1

times119901radicln(

119886

2119901minus1

(|119909 (119905)| +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)119901

+ 1)119889119904

le 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [int

119905

0

119890120582(119879+119904minus119905)

119890120582119879

minus 1

119889119904 + int

119879

119905

119890120582(119904minus119905)

119890120582119879

minus 1

119889119904]

= 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [

1

120582 (119890120582119879

minus 1)

(119890120582(119879+119904minus119905)10038161003816

100381610038161003816

119905

0

+ 119890120582(119904minus119905)10038161003816

100381610038161003816

119879

119905

)]

= 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [

1

120582 (119890120582119879

minus 1)

(119890120582119879

minus 119890120582(119879minus119905)

+ 119890120582(119879minus119905)

minus 1)]

=119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

le119901radicln(

119886

2119901minus1

119866120590119887

(119909 119910 119911) + 1)

le119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(90)

Similarly

2119901minus1 1003816

100381610038161003816119878119910 (119905)

1003816100381610038161003816+ |119878119911 (119905)| le

119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

|119878119909 (119905)| + |119878119911 (119905)| le119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(91)

where

119872(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119878119909 119878119909) 119866120590119887

(119910 119878119910 119878119910)119866120590119887

(119911 119878119911 119878119911)

1 + 22119901minus1

1198662

120590119887

(119878119909 119878119910 119878119911)

(92)

Equivalently

(2119901minus1

|119878119909 (119905)| +1003816100381610038161003816119878119910 (119905)

1003816100381610038161003816)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(2119901minus1 1003816

100381610038161003816119878119910 (119905)

1003816100381610038161003816+ |119878119911 (119905)|)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(2119901minus1

|119878119909 (119905)| + |119878119911 (119905)|)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(93)

which yields that

2119901minus1

120590119887(119878119909 119878119910) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

120590119887(119878119910 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

120590119887(119878119909 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(94)

The Scientific World Journal 13

Finally it is easy to obtain that

2119901minus1G120590119887

(119878119909 119878119910 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1) (95)

Finally since120572 is a lower solution for (78) so it is easy to showthat 120572 ⪯ 119878(120572) [31]

Hence the hypotheses of Theorem 35 are satisfied with120595(119905) = ln((1198862119901minus1)119905 + 1) where 0 le 119886 lt 1 Hence there existsa fixed point 119909 isin 119862[0 119879] such that 119878119909 = 119909 which is a solutionto periodic boundary value problem (78)

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) KingAbdulaziz University JeddahTherefore the firstauthor acknowledges with thanks DSR KAU for financialsupport

References

[1] S Czerwik ldquoContraction mappings in b-metric spacesrdquo ActaMathematica et Informatica Universitatis Ostraviensis vol 1 pp5ndash11 1993

[2] V Parvaneh J R Roshan and S Radenovic ldquoExistence of tri-pled coincidence points in ordered b-metric spaces and an app-lication to a system of integral equationsrdquo Fixed Point Theoryand Applications vol 2013 article 130 2013

[3] J R Roshan V Parvaneh S Sedghi N Shobkolaei and WShatanawi ldquoCommon fixed points of almost generalized con-tractive mappings in ordered b-metric spacesrdquo Fixed PointTheory and Applications vol 2013 article 159 2013

[4] H Aydi M F Bota E Karapınar and S Moradi ldquoA commonfixed point for weak 120601-contractions on b-metric spacesrdquo FixedPoint Theory vol 13 no 2 pp 337ndash346 2012

[5] V Berinde ldquoSequences of operators and fixed points in quasi-metric spacesrdquo Studia Universitatis ldquoBabes-Bolyairdquo Mathemat-ica vol 16 no 4 pp 23ndash27 1996

[6] N Hussain D ETHoric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 article 126 2012

[7] N Hussain and M H Shah ldquoKKMmappings in cone b-metricspacesrdquo Computers and Mathematics with Applications vol 62no 4 pp 1677ndash1684 2011

[8] N Hussain J R Roshan V Parvaneh and Z Kadelburg ldquoFixedpoints of contractive mappings in b-metric-like spacesrdquo TheScientific World Journal vol 2014 Article ID 471827 15 pages2014

[9] M A Khamsi and N Hussain ldquoKKMmappings in metric typespacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 73 no 9 pp 3123ndash3129 2010

[10] M A Khamsi ldquoRemarks on cone metric spaces and fixed pointtheorems of contractive mappingsrdquo Fixed Point Theory andApplications vol 2010 Article ID 315398 7 pages 2010

[11] Z Mustafa J R Roshan V Parvaneh and Z Kadelburg ldquoSomecommon fixed point results in ordered partial b-metric spacesrdquoJournal of Inequalities and Applications vol 2013 article 5622013

[12] Z Mustafa and B Sims ldquoA new approach to generalized metricspacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash297 2006

[13] A Amini-Harandi ldquoMetric-like spaces partial metric spacesand fixed pointsrdquo Fixed PointTheory and Applications vol 2012article 204 2012

[14] N Shobkolaei S Sedghi J R Roshan and N Hussain ldquoSuzukitype fixed point results in metric-like spacesrdquo Journal of Func-tion Spaces and Applications vol 2013 Article ID 143686 9pages 2013

[15] M A Alghamdi N Hussain and P Salimi ldquoFixed point andcoupled fixed point theorems on b-metric-like spacesrdquo Journalof Inequalities and Applications vol 2013 article 402 2013

[16] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572 minus 120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

[17] P Salimi and E Karapınar ldquoSuzuki-Edelstein type contractionsvia auxiliary functionsrdquoMathematical Problems in Engineeringvol 2013 Article ID 648528 8 pages 2013

[18] P Salimi C Vetro and P Vetro ldquoFixed point theorems fortwisted (120572 120573)-120595-contractive type mappings and applicationsrdquoFilomat vol 27 no 4 pp 605ndash615 2013

[19] P Salimi C Vetro and P Vetro ldquoSome new fixed point resultsin non-Archimedean fuzzy metric spacesrdquo Nonlinear AnalysisModelling and Control vol 18 no 3 pp 344ndash358 2013

[20] N Hussain P Salimi and A Latif ldquoFixed point results forsingle and set-valued 120572-120578-120595-contractive mappingsrdquo Fixed PointTheory and Applications vol 2013 article 212 2013

[21] M A Alghamdi and E Karapınar ldquoG-120573-120595 contractive-typemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 article 70 2013

[22] M Geraghty ldquoOn contractive mappingsrdquo Proceedings of theAmerican Mathematical Society vol 40 pp 604ndash608 1973

[23] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001

[24] N Hussain Z Kadelburg S Radenovic and F R Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012

[25] V Berinde ldquoGeneralized contractions in quasimetric spacesrdquo inSeminar on Fixed Point Theory Preprint no 3 pp 3ndash9 1993

[26] L Ciric M Abbas R Saadati and N Hussain ldquoCommon fixedpoints of almost generalized contractive mappings in orderedmetric spacesrdquo Applied Mathematics and Computation vol 217no 12 pp 5784ndash5789 2011

[27] R P Agarwal N Hussain and M A Taoudi ldquoFixed point the-orems in ordered Banach spaces and applications to nonlinearintegral equationsrdquo Abstract and Applied Analysis vol 2012Article ID 245872 15 pages 2012

[28] N Hussain A R Khan and R P Agarwal ldquoKrasnoselrsquoskii andKy Fan type fixed point theorems in ordered Banach spacesrdquoJournal of Nonlinear and Convex Analysis vol 11 no 3 pp 475ndash489 2010

[29] N Hussain and M A Taoudi ldquoKrasnoselrsquoskii-type fixed pointtheorems with applications to Volterra integral equationsrdquoFixed Point Theory and Applications vol 2013 article 196 2013

14 The Scientific World Journal

[30] J J Nieto and R Rodrıguez-Lopez ldquoContractivemapping theo-rems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[31] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysisTheoryMethods andApplications vol 71 no 7-8 pp 3403ndash3410 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 9

Moreover (57) yields that 119866120590119887(119906 119906 119891119906) le 120595(119872

119904(119906 119906 119906)) le

120595(119866120590119887(119906 119906 119891119906))This is impossible according to our assump-

tions on 120595 Hence we have 119891119906 = 119906 Thus 119906 is a fixed point of119891

Now let part (b) of (ii) holdsAs 119909119899 is a sequence such that 120572(119909

119899 119909119899+1

119909119899+2

) ge 1 for all119899 isin N cup 0 and 119909

119899rarr 119909 as 119899 rarr infin we have 120572(119909

119899 119909 119909) ge 1

for all 119899 isin N cup 0Now we show that 119906 = 119891119906 By (46) we have

119904119866120590119887

(119891119906 119891119906 119909119899) le 119904120572 (119906 119906 119909

119899minus1) 119866120590119887

(119891119906 119891119906 119891119909119899minus1

)

le 120595 (119872119904(119906 119906 119909

119899minus1))

(60)

where

119872119904(119906 119906 119909

119899minus1)

= max

119866120590119887

(119906 119906 119909119899minus1

)

[119866120590119887

(119906 119906 119891119906)]

2

119866120590119887

(119909119899minus1

119909119899minus1

119891119909119899minus1

)

1 + [119904119866120590119887

(119891119906 119891119906 119891119909119899minus1

)]

2

(61)

Letting 119899 rarr infin in the above inequality and using (54) andLemma 21 we get

lim119899rarrinfin

119872119904(119906 119906 119909

119899minus1) = 0 (62)

Again taking the upper limit as 119899 rarr infin in (60) and using(62) and Lemma 21 we obtain

119904 [

1

119904

119866120590119887

(119906 119891119906 119891119906) minus 119866120590119887

(119906 119906 119906)]

le 119904 lim sup119899rarrinfin

119866120590119887

(119909119899 119891119906 119891119906)

le lim supnrarrinfin

120595 (119872119904(119906 119906 119909

119899minus1))

= 120595(lim sup119899rarrinfin

119872119904(119906 119906 119909

119899minus1))

= 120595 (0) = 0

(63)

So we get 119866120590119887(119906 119891119906 119891119906) = 0 That is 119891119906 = 119906

Let (119883 119866120590119887 ⪯) be a partially ordered 119866

120590119887-metric-like

space We say that 119879 is an increasing mapping on 119883 ifforall119909 119910 isin 119883 119909 ⪯ 119910 rArr 119879(119909) ⪯ 119879(119910) [26] Fixed pointtheorems for monotone operators in ordered metric spacesarewidely investigated andhave found various applications indifferential and integral equations (see [27ndash30] and referencestherein) From the results proved above we derive thefollowing new results in partially ordered 119866

120590119887-metric-like

space

Theorem 34 Let (119883 119866120590119887 ⪯) be a partially ordered 119866

120590119887-

complete generalized 119887-metric-like space and let 119891 119883 rarr 119883

be an increasing mapping Suppose that

119904119866120590119887

(119891119909 119891119910 119891119911) le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911) (64)

for all 119909 119910 119911 isin 119883 with 119909 ⪯ 119910 ⪯ 119911 where

119872119904(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119891119909 119891119909)119866120590119887

(119910 119891119910 119891119910)119866120590119887

(119911 119891119911 119891119911)

1 + 11990421198662

120590119887

(119891119909 119891119910 119891119911)

(65)

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii) 119891 is continuous or assume that whenever 119909119899 in 119883 is

an increasing sequence such that 119909119899rarr 119909 as 119899 rarr infin

one has 119909119899⪯ 119909 for all 119899 isin N cup 0

Then 119891 has a fixed point

Proof Define 120572 119883 times 119883 times 119883 rarr [0 +infin) by

120572 (119909 119910 119911) =

1 if119909 ⪯ 119910 ⪯ 119911

0 otherwise(66)

First we prove that 119891 is a triangular 120572-admissible mappingHence we assume that 120572(119909 119910 119911) ge 1 Therefore we have 119909 ⪯

119910 ⪯ 119911 Since 119891 is increasing we get 119891119909 ⪯ 119891119910 ⪯ 119891119911 that is120572(119891119909 119891119910 119891119911) ge 1 Also let 120572(119909 119910 119910) ge 1 and 120572(119910 119911 119911) ge 1then 119909 ⪯ 119910 and 119910 ⪯ 119911 Consequently we deduce that 119909 ⪯ 119911that is 120572(119909 119911 119911) ge 1 Thus 119891 is a triangular 120572-admissiblemapping Since 119891 satisfies (64) so by the definition of 120572 wehave

119904120572 (119909 119910 119911) 119866120590119887

(119891119909 119891119910 119891119911)

le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911)

(67)

for all 119909 119910 119911 isin 119883 Therefore 119891 satisfies the contractivecondition (21) From (i) there exists 119909

0isin 119883 such that

1199090⪯ 1198911199090 that is 120572(119909

0 1198911199090 1198911199090) ge 1 According to (ii) we

conclude that all the conditions of Theorems 26 and 27 aresatisfied and so 119891 has a fixed point

Similarly using Theorem 33 we can prove followingresult

Theorem 35 Let (119883 119866120590119887 ⪯) be an ordered 119866

120590119887-complete

generalized 119887-metric-like space and let 119891 119883 rarr 119883 be anincreasing mapping Suppose that

119904119866120590119887

(119891119909 119891119910 119891119911) le 120595 (119872119904(119909 119910 119911)) (68)

for all 119909 119910 119911 isin 119883 with 119909 ⪯ 119910 ⪯ 119911 where 120595 isin Ψ119887and

119872119904(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119909 119891119909)119866120590119887

(119910 119910 119891119910)119866120590119887

(119911 119911 119891119911)

1 + 211990421198662

120590119887

(119891119909 119891119910 119891119911)

(69)

10 The Scientific World Journal

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii)

(a) 119891 is continuous(b) assume that whenever x

119899 in 119883 is an increasing

sequence such that 119909119899rarr 119909 as 119899 rarr infin one has

119909119899⪯ 119909 for all 119899 isin N cup 0

Then 119891 has a fixed point

We conclude this section by presenting some examplesthat illustrate our results

Example 36 Let 119883 = [0 (minus1 + radic5)2] be endowed withthe usual ordering onR Define the generalized 119887-metric-likefunction 119866

120590119887given by

119866120590119887

(119909 119910 119911) = max (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

(70)

with 119904 = 2 Consider the mapping 119891 119883 rarr 119883 defined by119891(119909) = (14)119909

radic119890minus(119909+((minus1+radic5)2))

2 and the function 120573 isin Fgiven by 120573(119905) = (12)119890

minus119905 119905 gt 0 and 120573(0) isin [0 12) Itis easy to see that 119891 is an increasing function on 119883 Weshow that 119891 is 119866

120590119887-continuous on 119883 By Proposition 20 it is

sufficient to show that119891 is119866120590119887-sequentially continuous on119883

Let 119909119899 be a sequence in119883 such that lim

119899rarrinfin119866120590119887(119909119899 119909 119909) =

119866120590119887(119909 119909 119909) so we have max(lim

119899rarrinfin119909119899

+ 119909)2

41199092 =

41199092 and equivalently lim

119899rarrinfin119909119899

= 120572 le 119909 On the otherhand we have

lim119899rarrinfin

119866120590119887

(119891119909119899 119891119909 119891119909)

= lim119899rarrinfin

max (119891119909119899+ 119891119909)

2

41198912

119909

= lim119899rarrinfin

max 1

16

(119909119899

radic119890minus(119909119899+(minus1+

radic5)2)2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= max 1

16

( lim119899rarrinfin

119909119899

radic119890minus(lim119899rarrinfin119909119899+(minus1+radic5)2)

2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= max 1

16

(120572radic119890minus(120572+(minus1+radic5)2)

2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

=

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= 119866120590119887

(119891119909 119891119909 119891119909)

(71)

So 119891 is 119866120590119887-sequentially continuous on119883

For all elements 119909 119910 119911 isin 119883 and the fact that 119892(119909) =

1199092119890minus1199092

is an increasing function on119883 we have

119904119866120590119887

(119891119909 119891119910 119891119911)

= 2max (119891119909 + 119891119910)2

(119891119910 + 119891119911)2

(119891119911 + 119891119909)2

= 2max(

1

4

119909radic119890minus(119909+(minus1+radic5)2)

2

+

1

4

119910radic119890minus(119910+(minus1+radic5)2)

2

)

2

(

1

4

119910radic119890minus(119910+(minus1+radic5)2)

2

+

1

4

119911radic119890minus(119911+(minus1+radic5)2)

2

)

2

(

1

4

119911radic119890minus(119911+(minus1+radic5)2)

2

+

1

4

119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

le 2max(

1

4

119909radic119890minus(119909+119910)

2

+

1

4

119910radic119890minus(119910+119909)

2

)

2

(

1

4

119910radic119890minus(119910+119911)

2

+

1

4

119911radic119890minus(119911+119910)

2

)

2

(

1

4

119911radic119890minus(119911+119909)

2

+

1

4

119909radic119890minus(119909+119911)

2

)

2

=

1

8

max (119909 + 119910)2

119890minus(119909+119910)

2

(119910 + 119911)2

119890minus(119910+119911)

2

(119911 + 119909)2

119890minus(119911+119909)

2

=

1

8

119890minusmax(119909+119910)2 (119910+119911)2 (119911+119909)2

timesmax (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

le

1

2

eminusmax(119909+119910)2 (119910+119911)2(119911+119909)2

timesmax (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

= 120573 (119866120590119887

(119909 119910 119911)) 119866120590119887

(119909 119910 119911)

le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911)

(72)

Hence119891 satisfies all the assumptions ofTheorem 34 and thusit has a fixed point (which is 119906 = 0)

Example 37 Let 119883 = [0 1] with the usual ordering on RDefine the generalized 119887-metric-like function 119866

120590119887given by

119866120590119887(119909 119910 119911) = max(119909 + 119910)

2

(119910 + 119911)2

(119911 + 119909)2

with 119904 = 2Consider the mapping 119891 119883 rarr 119883 defined by 119891(119909) =

The Scientific World Journal 11

ln(1 + 119909119890minus119909

4) and the function 120595 isin Ψ119887given by 120595(119905) =

(18)119905 119905 ge 0 It is easy to see that 119891 is increasing functionNow we show that 119891 is a 119866

120590119887-continuous function on119883

Let 119909119899 be a sequence in 119883 such that lim

119899rarrinfin119866120590119887(119909119899

119909 119909) = 119866120590119887(119909 119909 119909) so we havemax(lim

119899rarrinfin119909119899+119909)2

41199092 =

41199092 and equally lim

119899rarrinfin119909119899= 120572 le 119909 On the other hand we

have

lim119899rarrinfin

119866120590119887

(119891119909119899 119891119909 119891119909)

= lim119899rarrinfin

max (119891119909119899+ 119891119909)

2

41198912

119909

= max(ln(1 +

lim119899rarrinfin

119909119899119890minuslim119899rarrinfin119909119899

4

)

+ ln(1 +

119909119890minus119909

4

))

2

4(ln(1 +

119909119890minusx

4

))

2

= max(ln(1 +

120572119890minus120572

4

) + ln(1 +

119909119890minus119909

4

))

2

4(ln(1 +

119909119890minus119909

4

))

2

= 4(ln(1 +

119909119890minus119909

4

))

2

= 119866120590119887

(119891119909 119891119909 119891119909)

(73)

So 119891 is 119866120590119887-sequentially continuous on119883

For all elements 119909 119910 119911 isin 119883 we have

119904119866120590119887

(119891119909 119891119910 119891119911)

= 2max(ln(1 +

119909119890minus119909

4

) + ln(1 +

119910119890minus119910

4

))

2

(ln(1 +

119910119890minus119910

4

) + ln(1 +

119911119890minus119911

4

))

2

(ln(1 +

119911119890minus119911

4

) + ln(1 +

119909119890minus119909

4

))

2

le 2max(

119909119890minus119909

4

+

119910119890minus119910

4

)

2

(

119910119890minus119910

4

+

119911119890minus119911

4

)

2

(

119911119890minus119911

4

+

119909119890minus119909

4

)

2

le

1

8

max (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

= 120595 (119866120590119887

(119909 119910 119911))

le 120595 (119872119904(119909 119910 119911))

(74)

Hence119891 satisfies all the assumptions ofTheorem 35 and thusit has a fixed point (which is 119906 = 0)

3 Application

In this section we present an application of our results toestablish the existence of a solution to a periodic boundaryvalue problem (see [30 31])

Let 119883 = 119862([0 119879]) be the set of all real continuousfunctions on [0T] We first endow119883 with the 119887-metric-like

120590119887(119906 V) = sup

119905isin[0119879]

(|119906 (119905)| + |V (119905)|)119901 (75)

for all 119906 V isin 119883 where 119901 gt 1 and then we endow it with thegeneralized 119887-metric-like 119866

120590119887defined by

119866120590119887

(119906 V 119908)

= max sup119905isin[0119879]

(|119906 (119905)| + |V (119905)|)119901

sup119905isin[0119879]

(|V (119905)| + |119908 (119905)|)119901

sup119905isin[0119879]

(|119908 (119905)| + |119906 (119905)|)119901

(76)

Clearly (119883 119866120590119887) is a complete generalized 119887-metric-like space

with parameter 119870 = 2119901minus1 We equip 119862([0 119879]) with a partial

order given by119909 ⪯ 119910 iff119909 (119905) le 119910 (119905) forall119905 isin [0 119879] (77)

Moreover as in [30] it is proved that (119862([0 119879]) ⪯) is regularthat is whenever 119909

119899 in119883 is an increasing sequence such that

119909119899rarr 119909 as 119899 rarr infin we have 119909

119899⪯ 119909 for all 119899 isin N cup 0

Consider the first-order periodic boundary value prob-lem

1199091015840

(119905) = 119891 (119905 119909 (119905)) 119909 (0) = 119909 (119879) (78)

where 119905 isin 119868 = [0 119879] with 119879 gt 0 and 119891 119868 times R rarr R is acontinuous function

A lower solution for (78) is a function 120572 isin 1198621[0 119879] such

that1205721015840

(119905) le 119891 (119905 120572 (119905))

120572 (0) le 120572 (119879)

(79)

where 119905 isin 119868 = [0 119879]Assume that there exists 120582 gt 0 such that for all 119909 119910 isin

119862[0 119879] we have1003816100381610038161003816119891 (119905 119909 (119905)) + 120582119909 (119905)

1003816100381610038161003816+1003816100381610038161003816119891 (119905 119910 (119905)) + 120582119910 (119905)

1003816100381610038161003816

le

120582

2119901minus1

119901radicln(

119886

2119901minus1

(|119909 (119905)| +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)119901

+ 1)

(80)

where 0 le 119886 lt 1 Then the existence of a lower solution for(78) provides the existence of a solution of (78)

Problem (78) can be rewritten as

1199091015840

(119905) + 120582119909 (119905) = 119891 (119905 119909 (119905)) + 120582119909 (119905)

119909 (0) = 119909 (119879)

(81)

Consider1199091015840

(119905) + 120582119909 (119905) = 120575119909(119905) = 119865 (119905 119909 (119905))

119909 (0) = 119909 (119879)

(82)

where 119905 isin 119868

12 The Scientific World Journal

Using variation of parameters formula we get

119909 (119905) = 119909 (0) 119890minus120582119905

+ int

119905

0

119890minus120582(119905minus119904)

120575119909(119904) 119889119904 (83)

which yields

119909 (119879) = 119909 (0) 119890minus120582119879

+ int

119879

0

119890minus120582(119879minus119904)

120575119909(119904) 119889119904 (84)

Since 119909(0) = 119909(119879) we get

119909 (0) [1 minus 119890minus120582119879

] = 119890minus120582119879

int

119879

0

119890120582(119904)

120575119909(119904) 119889119904 (85)

or

119909 (0) =

1

119890120582119879

minus 1

int

119879

0

119890120582119904

120575119909(119904) 119889119904 (86)

Substituting the value of 119909(0) in (83) we arrive at

119909 (119905) = int

119879

0

119866 (119905 119904) 120575119909(119904) 119889119904 (87)

where

119866 (119905 119904) =

119890120582(119879+119904minus119905)

119890120582119879

minus 1

0 le 119904 le 119905 le 119879

119890120582(119904minus119905)

119890120582119879

minus 1

0 le 119905 le 119904 le 119879

(88)

Now define the operator 119878 119862[0 119879] rarr 119862[0 119879] as

119878119909 (119905) = int

119879

0

119866 (119905 119904) 119865 (119904 119909 (119904)) 119889119904 (89)

The mapping 119878 is increasing [31] Note that if 119906 isin 119862[0 119879] is afixed point of 119878 then 119906 isin 119862

1[0 119879] is a solution of (78)

Let 119909 119910 119911 isin 119862[0 119879] Then we have

2119901minus1

[|119878119909 (119905)| +1003816100381610038161003816119878119910 (119905)

1003816100381610038161003816]

= 2119901minus1

[

100381610038161003816100381610038161003816100381610038161003816

int

119879

0

119866 (119905 119904) 119865 (119904 119909 (119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

119879

0

119866 (119905 119904) 119865 (119904 119910 (119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

]

le 2119901minus1

int

119879

0

|119866 (119905 119904)|

times [|119865 (119904 119909 (119904))| +1003816100381610038161003816119865 (119904 119910 (119904))

1003816100381610038161003816] 119889119904

le 2119901minus1

int

119879

0

|119866 (119905 119904)|

120582

2119901minus1

times119901radicln(

119886

2119901minus1

(|119909 (119905)| +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)119901

+ 1)119889119904

le 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [int

119905

0

119890120582(119879+119904minus119905)

119890120582119879

minus 1

119889119904 + int

119879

119905

119890120582(119904minus119905)

119890120582119879

minus 1

119889119904]

= 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [

1

120582 (119890120582119879

minus 1)

(119890120582(119879+119904minus119905)10038161003816

100381610038161003816

119905

0

+ 119890120582(119904minus119905)10038161003816

100381610038161003816

119879

119905

)]

= 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [

1

120582 (119890120582119879

minus 1)

(119890120582119879

minus 119890120582(119879minus119905)

+ 119890120582(119879minus119905)

minus 1)]

=119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

le119901radicln(

119886

2119901minus1

119866120590119887

(119909 119910 119911) + 1)

le119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(90)

Similarly

2119901minus1 1003816

100381610038161003816119878119910 (119905)

1003816100381610038161003816+ |119878119911 (119905)| le

119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

|119878119909 (119905)| + |119878119911 (119905)| le119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(91)

where

119872(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119878119909 119878119909) 119866120590119887

(119910 119878119910 119878119910)119866120590119887

(119911 119878119911 119878119911)

1 + 22119901minus1

1198662

120590119887

(119878119909 119878119910 119878119911)

(92)

Equivalently

(2119901minus1

|119878119909 (119905)| +1003816100381610038161003816119878119910 (119905)

1003816100381610038161003816)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(2119901minus1 1003816

100381610038161003816119878119910 (119905)

1003816100381610038161003816+ |119878119911 (119905)|)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(2119901minus1

|119878119909 (119905)| + |119878119911 (119905)|)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(93)

which yields that

2119901minus1

120590119887(119878119909 119878119910) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

120590119887(119878119910 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

120590119887(119878119909 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(94)

The Scientific World Journal 13

Finally it is easy to obtain that

2119901minus1G120590119887

(119878119909 119878119910 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1) (95)

Finally since120572 is a lower solution for (78) so it is easy to showthat 120572 ⪯ 119878(120572) [31]

Hence the hypotheses of Theorem 35 are satisfied with120595(119905) = ln((1198862119901minus1)119905 + 1) where 0 le 119886 lt 1 Hence there existsa fixed point 119909 isin 119862[0 119879] such that 119878119909 = 119909 which is a solutionto periodic boundary value problem (78)

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) KingAbdulaziz University JeddahTherefore the firstauthor acknowledges with thanks DSR KAU for financialsupport

References

[1] S Czerwik ldquoContraction mappings in b-metric spacesrdquo ActaMathematica et Informatica Universitatis Ostraviensis vol 1 pp5ndash11 1993

[2] V Parvaneh J R Roshan and S Radenovic ldquoExistence of tri-pled coincidence points in ordered b-metric spaces and an app-lication to a system of integral equationsrdquo Fixed Point Theoryand Applications vol 2013 article 130 2013

[3] J R Roshan V Parvaneh S Sedghi N Shobkolaei and WShatanawi ldquoCommon fixed points of almost generalized con-tractive mappings in ordered b-metric spacesrdquo Fixed PointTheory and Applications vol 2013 article 159 2013

[4] H Aydi M F Bota E Karapınar and S Moradi ldquoA commonfixed point for weak 120601-contractions on b-metric spacesrdquo FixedPoint Theory vol 13 no 2 pp 337ndash346 2012

[5] V Berinde ldquoSequences of operators and fixed points in quasi-metric spacesrdquo Studia Universitatis ldquoBabes-Bolyairdquo Mathemat-ica vol 16 no 4 pp 23ndash27 1996

[6] N Hussain D ETHoric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 article 126 2012

[7] N Hussain and M H Shah ldquoKKMmappings in cone b-metricspacesrdquo Computers and Mathematics with Applications vol 62no 4 pp 1677ndash1684 2011

[8] N Hussain J R Roshan V Parvaneh and Z Kadelburg ldquoFixedpoints of contractive mappings in b-metric-like spacesrdquo TheScientific World Journal vol 2014 Article ID 471827 15 pages2014

[9] M A Khamsi and N Hussain ldquoKKMmappings in metric typespacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 73 no 9 pp 3123ndash3129 2010

[10] M A Khamsi ldquoRemarks on cone metric spaces and fixed pointtheorems of contractive mappingsrdquo Fixed Point Theory andApplications vol 2010 Article ID 315398 7 pages 2010

[11] Z Mustafa J R Roshan V Parvaneh and Z Kadelburg ldquoSomecommon fixed point results in ordered partial b-metric spacesrdquoJournal of Inequalities and Applications vol 2013 article 5622013

[12] Z Mustafa and B Sims ldquoA new approach to generalized metricspacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash297 2006

[13] A Amini-Harandi ldquoMetric-like spaces partial metric spacesand fixed pointsrdquo Fixed PointTheory and Applications vol 2012article 204 2012

[14] N Shobkolaei S Sedghi J R Roshan and N Hussain ldquoSuzukitype fixed point results in metric-like spacesrdquo Journal of Func-tion Spaces and Applications vol 2013 Article ID 143686 9pages 2013

[15] M A Alghamdi N Hussain and P Salimi ldquoFixed point andcoupled fixed point theorems on b-metric-like spacesrdquo Journalof Inequalities and Applications vol 2013 article 402 2013

[16] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572 minus 120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

[17] P Salimi and E Karapınar ldquoSuzuki-Edelstein type contractionsvia auxiliary functionsrdquoMathematical Problems in Engineeringvol 2013 Article ID 648528 8 pages 2013

[18] P Salimi C Vetro and P Vetro ldquoFixed point theorems fortwisted (120572 120573)-120595-contractive type mappings and applicationsrdquoFilomat vol 27 no 4 pp 605ndash615 2013

[19] P Salimi C Vetro and P Vetro ldquoSome new fixed point resultsin non-Archimedean fuzzy metric spacesrdquo Nonlinear AnalysisModelling and Control vol 18 no 3 pp 344ndash358 2013

[20] N Hussain P Salimi and A Latif ldquoFixed point results forsingle and set-valued 120572-120578-120595-contractive mappingsrdquo Fixed PointTheory and Applications vol 2013 article 212 2013

[21] M A Alghamdi and E Karapınar ldquoG-120573-120595 contractive-typemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 article 70 2013

[22] M Geraghty ldquoOn contractive mappingsrdquo Proceedings of theAmerican Mathematical Society vol 40 pp 604ndash608 1973

[23] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001

[24] N Hussain Z Kadelburg S Radenovic and F R Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012

[25] V Berinde ldquoGeneralized contractions in quasimetric spacesrdquo inSeminar on Fixed Point Theory Preprint no 3 pp 3ndash9 1993

[26] L Ciric M Abbas R Saadati and N Hussain ldquoCommon fixedpoints of almost generalized contractive mappings in orderedmetric spacesrdquo Applied Mathematics and Computation vol 217no 12 pp 5784ndash5789 2011

[27] R P Agarwal N Hussain and M A Taoudi ldquoFixed point the-orems in ordered Banach spaces and applications to nonlinearintegral equationsrdquo Abstract and Applied Analysis vol 2012Article ID 245872 15 pages 2012

[28] N Hussain A R Khan and R P Agarwal ldquoKrasnoselrsquoskii andKy Fan type fixed point theorems in ordered Banach spacesrdquoJournal of Nonlinear and Convex Analysis vol 11 no 3 pp 475ndash489 2010

[29] N Hussain and M A Taoudi ldquoKrasnoselrsquoskii-type fixed pointtheorems with applications to Volterra integral equationsrdquoFixed Point Theory and Applications vol 2013 article 196 2013

14 The Scientific World Journal

[30] J J Nieto and R Rodrıguez-Lopez ldquoContractivemapping theo-rems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[31] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysisTheoryMethods andApplications vol 71 no 7-8 pp 3403ndash3410 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 The Scientific World Journal

Also suppose that the following assertions hold

(i) there exists 1199090isin 119883 such that 119909

0⪯ 1198911199090

(ii)

(a) 119891 is continuous(b) assume that whenever x

119899 in 119883 is an increasing

sequence such that 119909119899rarr 119909 as 119899 rarr infin one has

119909119899⪯ 119909 for all 119899 isin N cup 0

Then 119891 has a fixed point

We conclude this section by presenting some examplesthat illustrate our results

Example 36 Let 119883 = [0 (minus1 + radic5)2] be endowed withthe usual ordering onR Define the generalized 119887-metric-likefunction 119866

120590119887given by

119866120590119887

(119909 119910 119911) = max (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

(70)

with 119904 = 2 Consider the mapping 119891 119883 rarr 119883 defined by119891(119909) = (14)119909

radic119890minus(119909+((minus1+radic5)2))

2 and the function 120573 isin Fgiven by 120573(119905) = (12)119890

minus119905 119905 gt 0 and 120573(0) isin [0 12) Itis easy to see that 119891 is an increasing function on 119883 Weshow that 119891 is 119866

120590119887-continuous on 119883 By Proposition 20 it is

sufficient to show that119891 is119866120590119887-sequentially continuous on119883

Let 119909119899 be a sequence in119883 such that lim

119899rarrinfin119866120590119887(119909119899 119909 119909) =

119866120590119887(119909 119909 119909) so we have max(lim

119899rarrinfin119909119899

+ 119909)2

41199092 =

41199092 and equivalently lim

119899rarrinfin119909119899

= 120572 le 119909 On the otherhand we have

lim119899rarrinfin

119866120590119887

(119891119909119899 119891119909 119891119909)

= lim119899rarrinfin

max (119891119909119899+ 119891119909)

2

41198912

119909

= lim119899rarrinfin

max 1

16

(119909119899

radic119890minus(119909119899+(minus1+

radic5)2)2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= max 1

16

( lim119899rarrinfin

119909119899

radic119890minus(lim119899rarrinfin119909119899+(minus1+radic5)2)

2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= max 1

16

(120572radic119890minus(120572+(minus1+radic5)2)

2

+119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

=

1

4

1199092

119890minus(119909+(minus1+radic5)2)

2

= 119866120590119887

(119891119909 119891119909 119891119909)

(71)

So 119891 is 119866120590119887-sequentially continuous on119883

For all elements 119909 119910 119911 isin 119883 and the fact that 119892(119909) =

1199092119890minus1199092

is an increasing function on119883 we have

119904119866120590119887

(119891119909 119891119910 119891119911)

= 2max (119891119909 + 119891119910)2

(119891119910 + 119891119911)2

(119891119911 + 119891119909)2

= 2max(

1

4

119909radic119890minus(119909+(minus1+radic5)2)

2

+

1

4

119910radic119890minus(119910+(minus1+radic5)2)

2

)

2

(

1

4

119910radic119890minus(119910+(minus1+radic5)2)

2

+

1

4

119911radic119890minus(119911+(minus1+radic5)2)

2

)

2

(

1

4

119911radic119890minus(119911+(minus1+radic5)2)

2

+

1

4

119909radic119890minus(119909+(minus1+radic5)2)

2

)

2

le 2max(

1

4

119909radic119890minus(119909+119910)

2

+

1

4

119910radic119890minus(119910+119909)

2

)

2

(

1

4

119910radic119890minus(119910+119911)

2

+

1

4

119911radic119890minus(119911+119910)

2

)

2

(

1

4

119911radic119890minus(119911+119909)

2

+

1

4

119909radic119890minus(119909+119911)

2

)

2

=

1

8

max (119909 + 119910)2

119890minus(119909+119910)

2

(119910 + 119911)2

119890minus(119910+119911)

2

(119911 + 119909)2

119890minus(119911+119909)

2

=

1

8

119890minusmax(119909+119910)2 (119910+119911)2 (119911+119909)2

timesmax (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

le

1

2

eminusmax(119909+119910)2 (119910+119911)2(119911+119909)2

timesmax (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

= 120573 (119866120590119887

(119909 119910 119911)) 119866120590119887

(119909 119910 119911)

le 120573 (119866120590119887

(119909 119910 119911))119872119904(119909 119910 119911)

(72)

Hence119891 satisfies all the assumptions ofTheorem 34 and thusit has a fixed point (which is 119906 = 0)

Example 37 Let 119883 = [0 1] with the usual ordering on RDefine the generalized 119887-metric-like function 119866

120590119887given by

119866120590119887(119909 119910 119911) = max(119909 + 119910)

2

(119910 + 119911)2

(119911 + 119909)2

with 119904 = 2Consider the mapping 119891 119883 rarr 119883 defined by 119891(119909) =

The Scientific World Journal 11

ln(1 + 119909119890minus119909

4) and the function 120595 isin Ψ119887given by 120595(119905) =

(18)119905 119905 ge 0 It is easy to see that 119891 is increasing functionNow we show that 119891 is a 119866

120590119887-continuous function on119883

Let 119909119899 be a sequence in 119883 such that lim

119899rarrinfin119866120590119887(119909119899

119909 119909) = 119866120590119887(119909 119909 119909) so we havemax(lim

119899rarrinfin119909119899+119909)2

41199092 =

41199092 and equally lim

119899rarrinfin119909119899= 120572 le 119909 On the other hand we

have

lim119899rarrinfin

119866120590119887

(119891119909119899 119891119909 119891119909)

= lim119899rarrinfin

max (119891119909119899+ 119891119909)

2

41198912

119909

= max(ln(1 +

lim119899rarrinfin

119909119899119890minuslim119899rarrinfin119909119899

4

)

+ ln(1 +

119909119890minus119909

4

))

2

4(ln(1 +

119909119890minusx

4

))

2

= max(ln(1 +

120572119890minus120572

4

) + ln(1 +

119909119890minus119909

4

))

2

4(ln(1 +

119909119890minus119909

4

))

2

= 4(ln(1 +

119909119890minus119909

4

))

2

= 119866120590119887

(119891119909 119891119909 119891119909)

(73)

So 119891 is 119866120590119887-sequentially continuous on119883

For all elements 119909 119910 119911 isin 119883 we have

119904119866120590119887

(119891119909 119891119910 119891119911)

= 2max(ln(1 +

119909119890minus119909

4

) + ln(1 +

119910119890minus119910

4

))

2

(ln(1 +

119910119890minus119910

4

) + ln(1 +

119911119890minus119911

4

))

2

(ln(1 +

119911119890minus119911

4

) + ln(1 +

119909119890minus119909

4

))

2

le 2max(

119909119890minus119909

4

+

119910119890minus119910

4

)

2

(

119910119890minus119910

4

+

119911119890minus119911

4

)

2

(

119911119890minus119911

4

+

119909119890minus119909

4

)

2

le

1

8

max (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

= 120595 (119866120590119887

(119909 119910 119911))

le 120595 (119872119904(119909 119910 119911))

(74)

Hence119891 satisfies all the assumptions ofTheorem 35 and thusit has a fixed point (which is 119906 = 0)

3 Application

In this section we present an application of our results toestablish the existence of a solution to a periodic boundaryvalue problem (see [30 31])

Let 119883 = 119862([0 119879]) be the set of all real continuousfunctions on [0T] We first endow119883 with the 119887-metric-like

120590119887(119906 V) = sup

119905isin[0119879]

(|119906 (119905)| + |V (119905)|)119901 (75)

for all 119906 V isin 119883 where 119901 gt 1 and then we endow it with thegeneralized 119887-metric-like 119866

120590119887defined by

119866120590119887

(119906 V 119908)

= max sup119905isin[0119879]

(|119906 (119905)| + |V (119905)|)119901

sup119905isin[0119879]

(|V (119905)| + |119908 (119905)|)119901

sup119905isin[0119879]

(|119908 (119905)| + |119906 (119905)|)119901

(76)

Clearly (119883 119866120590119887) is a complete generalized 119887-metric-like space

with parameter 119870 = 2119901minus1 We equip 119862([0 119879]) with a partial

order given by119909 ⪯ 119910 iff119909 (119905) le 119910 (119905) forall119905 isin [0 119879] (77)

Moreover as in [30] it is proved that (119862([0 119879]) ⪯) is regularthat is whenever 119909

119899 in119883 is an increasing sequence such that

119909119899rarr 119909 as 119899 rarr infin we have 119909

119899⪯ 119909 for all 119899 isin N cup 0

Consider the first-order periodic boundary value prob-lem

1199091015840

(119905) = 119891 (119905 119909 (119905)) 119909 (0) = 119909 (119879) (78)

where 119905 isin 119868 = [0 119879] with 119879 gt 0 and 119891 119868 times R rarr R is acontinuous function

A lower solution for (78) is a function 120572 isin 1198621[0 119879] such

that1205721015840

(119905) le 119891 (119905 120572 (119905))

120572 (0) le 120572 (119879)

(79)

where 119905 isin 119868 = [0 119879]Assume that there exists 120582 gt 0 such that for all 119909 119910 isin

119862[0 119879] we have1003816100381610038161003816119891 (119905 119909 (119905)) + 120582119909 (119905)

1003816100381610038161003816+1003816100381610038161003816119891 (119905 119910 (119905)) + 120582119910 (119905)

1003816100381610038161003816

le

120582

2119901minus1

119901radicln(

119886

2119901minus1

(|119909 (119905)| +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)119901

+ 1)

(80)

where 0 le 119886 lt 1 Then the existence of a lower solution for(78) provides the existence of a solution of (78)

Problem (78) can be rewritten as

1199091015840

(119905) + 120582119909 (119905) = 119891 (119905 119909 (119905)) + 120582119909 (119905)

119909 (0) = 119909 (119879)

(81)

Consider1199091015840

(119905) + 120582119909 (119905) = 120575119909(119905) = 119865 (119905 119909 (119905))

119909 (0) = 119909 (119879)

(82)

where 119905 isin 119868

12 The Scientific World Journal

Using variation of parameters formula we get

119909 (119905) = 119909 (0) 119890minus120582119905

+ int

119905

0

119890minus120582(119905minus119904)

120575119909(119904) 119889119904 (83)

which yields

119909 (119879) = 119909 (0) 119890minus120582119879

+ int

119879

0

119890minus120582(119879minus119904)

120575119909(119904) 119889119904 (84)

Since 119909(0) = 119909(119879) we get

119909 (0) [1 minus 119890minus120582119879

] = 119890minus120582119879

int

119879

0

119890120582(119904)

120575119909(119904) 119889119904 (85)

or

119909 (0) =

1

119890120582119879

minus 1

int

119879

0

119890120582119904

120575119909(119904) 119889119904 (86)

Substituting the value of 119909(0) in (83) we arrive at

119909 (119905) = int

119879

0

119866 (119905 119904) 120575119909(119904) 119889119904 (87)

where

119866 (119905 119904) =

119890120582(119879+119904minus119905)

119890120582119879

minus 1

0 le 119904 le 119905 le 119879

119890120582(119904minus119905)

119890120582119879

minus 1

0 le 119905 le 119904 le 119879

(88)

Now define the operator 119878 119862[0 119879] rarr 119862[0 119879] as

119878119909 (119905) = int

119879

0

119866 (119905 119904) 119865 (119904 119909 (119904)) 119889119904 (89)

The mapping 119878 is increasing [31] Note that if 119906 isin 119862[0 119879] is afixed point of 119878 then 119906 isin 119862

1[0 119879] is a solution of (78)

Let 119909 119910 119911 isin 119862[0 119879] Then we have

2119901minus1

[|119878119909 (119905)| +1003816100381610038161003816119878119910 (119905)

1003816100381610038161003816]

= 2119901minus1

[

100381610038161003816100381610038161003816100381610038161003816

int

119879

0

119866 (119905 119904) 119865 (119904 119909 (119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

119879

0

119866 (119905 119904) 119865 (119904 119910 (119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

]

le 2119901minus1

int

119879

0

|119866 (119905 119904)|

times [|119865 (119904 119909 (119904))| +1003816100381610038161003816119865 (119904 119910 (119904))

1003816100381610038161003816] 119889119904

le 2119901minus1

int

119879

0

|119866 (119905 119904)|

120582

2119901minus1

times119901radicln(

119886

2119901minus1

(|119909 (119905)| +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)119901

+ 1)119889119904

le 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [int

119905

0

119890120582(119879+119904minus119905)

119890120582119879

minus 1

119889119904 + int

119879

119905

119890120582(119904minus119905)

119890120582119879

minus 1

119889119904]

= 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [

1

120582 (119890120582119879

minus 1)

(119890120582(119879+119904minus119905)10038161003816

100381610038161003816

119905

0

+ 119890120582(119904minus119905)10038161003816

100381610038161003816

119879

119905

)]

= 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [

1

120582 (119890120582119879

minus 1)

(119890120582119879

minus 119890120582(119879minus119905)

+ 119890120582(119879minus119905)

minus 1)]

=119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

le119901radicln(

119886

2119901minus1

119866120590119887

(119909 119910 119911) + 1)

le119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(90)

Similarly

2119901minus1 1003816

100381610038161003816119878119910 (119905)

1003816100381610038161003816+ |119878119911 (119905)| le

119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

|119878119909 (119905)| + |119878119911 (119905)| le119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(91)

where

119872(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119878119909 119878119909) 119866120590119887

(119910 119878119910 119878119910)119866120590119887

(119911 119878119911 119878119911)

1 + 22119901minus1

1198662

120590119887

(119878119909 119878119910 119878119911)

(92)

Equivalently

(2119901minus1

|119878119909 (119905)| +1003816100381610038161003816119878119910 (119905)

1003816100381610038161003816)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(2119901minus1 1003816

100381610038161003816119878119910 (119905)

1003816100381610038161003816+ |119878119911 (119905)|)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(2119901minus1

|119878119909 (119905)| + |119878119911 (119905)|)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(93)

which yields that

2119901minus1

120590119887(119878119909 119878119910) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

120590119887(119878119910 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

120590119887(119878119909 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(94)

The Scientific World Journal 13

Finally it is easy to obtain that

2119901minus1G120590119887

(119878119909 119878119910 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1) (95)

Finally since120572 is a lower solution for (78) so it is easy to showthat 120572 ⪯ 119878(120572) [31]

Hence the hypotheses of Theorem 35 are satisfied with120595(119905) = ln((1198862119901minus1)119905 + 1) where 0 le 119886 lt 1 Hence there existsa fixed point 119909 isin 119862[0 119879] such that 119878119909 = 119909 which is a solutionto periodic boundary value problem (78)

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) KingAbdulaziz University JeddahTherefore the firstauthor acknowledges with thanks DSR KAU for financialsupport

References

[1] S Czerwik ldquoContraction mappings in b-metric spacesrdquo ActaMathematica et Informatica Universitatis Ostraviensis vol 1 pp5ndash11 1993

[2] V Parvaneh J R Roshan and S Radenovic ldquoExistence of tri-pled coincidence points in ordered b-metric spaces and an app-lication to a system of integral equationsrdquo Fixed Point Theoryand Applications vol 2013 article 130 2013

[3] J R Roshan V Parvaneh S Sedghi N Shobkolaei and WShatanawi ldquoCommon fixed points of almost generalized con-tractive mappings in ordered b-metric spacesrdquo Fixed PointTheory and Applications vol 2013 article 159 2013

[4] H Aydi M F Bota E Karapınar and S Moradi ldquoA commonfixed point for weak 120601-contractions on b-metric spacesrdquo FixedPoint Theory vol 13 no 2 pp 337ndash346 2012

[5] V Berinde ldquoSequences of operators and fixed points in quasi-metric spacesrdquo Studia Universitatis ldquoBabes-Bolyairdquo Mathemat-ica vol 16 no 4 pp 23ndash27 1996

[6] N Hussain D ETHoric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 article 126 2012

[7] N Hussain and M H Shah ldquoKKMmappings in cone b-metricspacesrdquo Computers and Mathematics with Applications vol 62no 4 pp 1677ndash1684 2011

[8] N Hussain J R Roshan V Parvaneh and Z Kadelburg ldquoFixedpoints of contractive mappings in b-metric-like spacesrdquo TheScientific World Journal vol 2014 Article ID 471827 15 pages2014

[9] M A Khamsi and N Hussain ldquoKKMmappings in metric typespacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 73 no 9 pp 3123ndash3129 2010

[10] M A Khamsi ldquoRemarks on cone metric spaces and fixed pointtheorems of contractive mappingsrdquo Fixed Point Theory andApplications vol 2010 Article ID 315398 7 pages 2010

[11] Z Mustafa J R Roshan V Parvaneh and Z Kadelburg ldquoSomecommon fixed point results in ordered partial b-metric spacesrdquoJournal of Inequalities and Applications vol 2013 article 5622013

[12] Z Mustafa and B Sims ldquoA new approach to generalized metricspacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash297 2006

[13] A Amini-Harandi ldquoMetric-like spaces partial metric spacesand fixed pointsrdquo Fixed PointTheory and Applications vol 2012article 204 2012

[14] N Shobkolaei S Sedghi J R Roshan and N Hussain ldquoSuzukitype fixed point results in metric-like spacesrdquo Journal of Func-tion Spaces and Applications vol 2013 Article ID 143686 9pages 2013

[15] M A Alghamdi N Hussain and P Salimi ldquoFixed point andcoupled fixed point theorems on b-metric-like spacesrdquo Journalof Inequalities and Applications vol 2013 article 402 2013

[16] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572 minus 120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

[17] P Salimi and E Karapınar ldquoSuzuki-Edelstein type contractionsvia auxiliary functionsrdquoMathematical Problems in Engineeringvol 2013 Article ID 648528 8 pages 2013

[18] P Salimi C Vetro and P Vetro ldquoFixed point theorems fortwisted (120572 120573)-120595-contractive type mappings and applicationsrdquoFilomat vol 27 no 4 pp 605ndash615 2013

[19] P Salimi C Vetro and P Vetro ldquoSome new fixed point resultsin non-Archimedean fuzzy metric spacesrdquo Nonlinear AnalysisModelling and Control vol 18 no 3 pp 344ndash358 2013

[20] N Hussain P Salimi and A Latif ldquoFixed point results forsingle and set-valued 120572-120578-120595-contractive mappingsrdquo Fixed PointTheory and Applications vol 2013 article 212 2013

[21] M A Alghamdi and E Karapınar ldquoG-120573-120595 contractive-typemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 article 70 2013

[22] M Geraghty ldquoOn contractive mappingsrdquo Proceedings of theAmerican Mathematical Society vol 40 pp 604ndash608 1973

[23] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001

[24] N Hussain Z Kadelburg S Radenovic and F R Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012

[25] V Berinde ldquoGeneralized contractions in quasimetric spacesrdquo inSeminar on Fixed Point Theory Preprint no 3 pp 3ndash9 1993

[26] L Ciric M Abbas R Saadati and N Hussain ldquoCommon fixedpoints of almost generalized contractive mappings in orderedmetric spacesrdquo Applied Mathematics and Computation vol 217no 12 pp 5784ndash5789 2011

[27] R P Agarwal N Hussain and M A Taoudi ldquoFixed point the-orems in ordered Banach spaces and applications to nonlinearintegral equationsrdquo Abstract and Applied Analysis vol 2012Article ID 245872 15 pages 2012

[28] N Hussain A R Khan and R P Agarwal ldquoKrasnoselrsquoskii andKy Fan type fixed point theorems in ordered Banach spacesrdquoJournal of Nonlinear and Convex Analysis vol 11 no 3 pp 475ndash489 2010

[29] N Hussain and M A Taoudi ldquoKrasnoselrsquoskii-type fixed pointtheorems with applications to Volterra integral equationsrdquoFixed Point Theory and Applications vol 2013 article 196 2013

14 The Scientific World Journal

[30] J J Nieto and R Rodrıguez-Lopez ldquoContractivemapping theo-rems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[31] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysisTheoryMethods andApplications vol 71 no 7-8 pp 3403ndash3410 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 11

ln(1 + 119909119890minus119909

4) and the function 120595 isin Ψ119887given by 120595(119905) =

(18)119905 119905 ge 0 It is easy to see that 119891 is increasing functionNow we show that 119891 is a 119866

120590119887-continuous function on119883

Let 119909119899 be a sequence in 119883 such that lim

119899rarrinfin119866120590119887(119909119899

119909 119909) = 119866120590119887(119909 119909 119909) so we havemax(lim

119899rarrinfin119909119899+119909)2

41199092 =

41199092 and equally lim

119899rarrinfin119909119899= 120572 le 119909 On the other hand we

have

lim119899rarrinfin

119866120590119887

(119891119909119899 119891119909 119891119909)

= lim119899rarrinfin

max (119891119909119899+ 119891119909)

2

41198912

119909

= max(ln(1 +

lim119899rarrinfin

119909119899119890minuslim119899rarrinfin119909119899

4

)

+ ln(1 +

119909119890minus119909

4

))

2

4(ln(1 +

119909119890minusx

4

))

2

= max(ln(1 +

120572119890minus120572

4

) + ln(1 +

119909119890minus119909

4

))

2

4(ln(1 +

119909119890minus119909

4

))

2

= 4(ln(1 +

119909119890minus119909

4

))

2

= 119866120590119887

(119891119909 119891119909 119891119909)

(73)

So 119891 is 119866120590119887-sequentially continuous on119883

For all elements 119909 119910 119911 isin 119883 we have

119904119866120590119887

(119891119909 119891119910 119891119911)

= 2max(ln(1 +

119909119890minus119909

4

) + ln(1 +

119910119890minus119910

4

))

2

(ln(1 +

119910119890minus119910

4

) + ln(1 +

119911119890minus119911

4

))

2

(ln(1 +

119911119890minus119911

4

) + ln(1 +

119909119890minus119909

4

))

2

le 2max(

119909119890minus119909

4

+

119910119890minus119910

4

)

2

(

119910119890minus119910

4

+

119911119890minus119911

4

)

2

(

119911119890minus119911

4

+

119909119890minus119909

4

)

2

le

1

8

max (119909 + 119910)2

(119910 + 119911)2

(119911 + 119909)2

= 120595 (119866120590119887

(119909 119910 119911))

le 120595 (119872119904(119909 119910 119911))

(74)

Hence119891 satisfies all the assumptions ofTheorem 35 and thusit has a fixed point (which is 119906 = 0)

3 Application

In this section we present an application of our results toestablish the existence of a solution to a periodic boundaryvalue problem (see [30 31])

Let 119883 = 119862([0 119879]) be the set of all real continuousfunctions on [0T] We first endow119883 with the 119887-metric-like

120590119887(119906 V) = sup

119905isin[0119879]

(|119906 (119905)| + |V (119905)|)119901 (75)

for all 119906 V isin 119883 where 119901 gt 1 and then we endow it with thegeneralized 119887-metric-like 119866

120590119887defined by

119866120590119887

(119906 V 119908)

= max sup119905isin[0119879]

(|119906 (119905)| + |V (119905)|)119901

sup119905isin[0119879]

(|V (119905)| + |119908 (119905)|)119901

sup119905isin[0119879]

(|119908 (119905)| + |119906 (119905)|)119901

(76)

Clearly (119883 119866120590119887) is a complete generalized 119887-metric-like space

with parameter 119870 = 2119901minus1 We equip 119862([0 119879]) with a partial

order given by119909 ⪯ 119910 iff119909 (119905) le 119910 (119905) forall119905 isin [0 119879] (77)

Moreover as in [30] it is proved that (119862([0 119879]) ⪯) is regularthat is whenever 119909

119899 in119883 is an increasing sequence such that

119909119899rarr 119909 as 119899 rarr infin we have 119909

119899⪯ 119909 for all 119899 isin N cup 0

Consider the first-order periodic boundary value prob-lem

1199091015840

(119905) = 119891 (119905 119909 (119905)) 119909 (0) = 119909 (119879) (78)

where 119905 isin 119868 = [0 119879] with 119879 gt 0 and 119891 119868 times R rarr R is acontinuous function

A lower solution for (78) is a function 120572 isin 1198621[0 119879] such

that1205721015840

(119905) le 119891 (119905 120572 (119905))

120572 (0) le 120572 (119879)

(79)

where 119905 isin 119868 = [0 119879]Assume that there exists 120582 gt 0 such that for all 119909 119910 isin

119862[0 119879] we have1003816100381610038161003816119891 (119905 119909 (119905)) + 120582119909 (119905)

1003816100381610038161003816+1003816100381610038161003816119891 (119905 119910 (119905)) + 120582119910 (119905)

1003816100381610038161003816

le

120582

2119901minus1

119901radicln(

119886

2119901minus1

(|119909 (119905)| +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)119901

+ 1)

(80)

where 0 le 119886 lt 1 Then the existence of a lower solution for(78) provides the existence of a solution of (78)

Problem (78) can be rewritten as

1199091015840

(119905) + 120582119909 (119905) = 119891 (119905 119909 (119905)) + 120582119909 (119905)

119909 (0) = 119909 (119879)

(81)

Consider1199091015840

(119905) + 120582119909 (119905) = 120575119909(119905) = 119865 (119905 119909 (119905))

119909 (0) = 119909 (119879)

(82)

where 119905 isin 119868

12 The Scientific World Journal

Using variation of parameters formula we get

119909 (119905) = 119909 (0) 119890minus120582119905

+ int

119905

0

119890minus120582(119905minus119904)

120575119909(119904) 119889119904 (83)

which yields

119909 (119879) = 119909 (0) 119890minus120582119879

+ int

119879

0

119890minus120582(119879minus119904)

120575119909(119904) 119889119904 (84)

Since 119909(0) = 119909(119879) we get

119909 (0) [1 minus 119890minus120582119879

] = 119890minus120582119879

int

119879

0

119890120582(119904)

120575119909(119904) 119889119904 (85)

or

119909 (0) =

1

119890120582119879

minus 1

int

119879

0

119890120582119904

120575119909(119904) 119889119904 (86)

Substituting the value of 119909(0) in (83) we arrive at

119909 (119905) = int

119879

0

119866 (119905 119904) 120575119909(119904) 119889119904 (87)

where

119866 (119905 119904) =

119890120582(119879+119904minus119905)

119890120582119879

minus 1

0 le 119904 le 119905 le 119879

119890120582(119904minus119905)

119890120582119879

minus 1

0 le 119905 le 119904 le 119879

(88)

Now define the operator 119878 119862[0 119879] rarr 119862[0 119879] as

119878119909 (119905) = int

119879

0

119866 (119905 119904) 119865 (119904 119909 (119904)) 119889119904 (89)

The mapping 119878 is increasing [31] Note that if 119906 isin 119862[0 119879] is afixed point of 119878 then 119906 isin 119862

1[0 119879] is a solution of (78)

Let 119909 119910 119911 isin 119862[0 119879] Then we have

2119901minus1

[|119878119909 (119905)| +1003816100381610038161003816119878119910 (119905)

1003816100381610038161003816]

= 2119901minus1

[

100381610038161003816100381610038161003816100381610038161003816

int

119879

0

119866 (119905 119904) 119865 (119904 119909 (119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

119879

0

119866 (119905 119904) 119865 (119904 119910 (119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

]

le 2119901minus1

int

119879

0

|119866 (119905 119904)|

times [|119865 (119904 119909 (119904))| +1003816100381610038161003816119865 (119904 119910 (119904))

1003816100381610038161003816] 119889119904

le 2119901minus1

int

119879

0

|119866 (119905 119904)|

120582

2119901minus1

times119901radicln(

119886

2119901minus1

(|119909 (119905)| +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)119901

+ 1)119889119904

le 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [int

119905

0

119890120582(119879+119904minus119905)

119890120582119879

minus 1

119889119904 + int

119879

119905

119890120582(119904minus119905)

119890120582119879

minus 1

119889119904]

= 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [

1

120582 (119890120582119879

minus 1)

(119890120582(119879+119904minus119905)10038161003816

100381610038161003816

119905

0

+ 119890120582(119904minus119905)10038161003816

100381610038161003816

119879

119905

)]

= 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [

1

120582 (119890120582119879

minus 1)

(119890120582119879

minus 119890120582(119879minus119905)

+ 119890120582(119879minus119905)

minus 1)]

=119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

le119901radicln(

119886

2119901minus1

119866120590119887

(119909 119910 119911) + 1)

le119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(90)

Similarly

2119901minus1 1003816

100381610038161003816119878119910 (119905)

1003816100381610038161003816+ |119878119911 (119905)| le

119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

|119878119909 (119905)| + |119878119911 (119905)| le119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(91)

where

119872(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119878119909 119878119909) 119866120590119887

(119910 119878119910 119878119910)119866120590119887

(119911 119878119911 119878119911)

1 + 22119901minus1

1198662

120590119887

(119878119909 119878119910 119878119911)

(92)

Equivalently

(2119901minus1

|119878119909 (119905)| +1003816100381610038161003816119878119910 (119905)

1003816100381610038161003816)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(2119901minus1 1003816

100381610038161003816119878119910 (119905)

1003816100381610038161003816+ |119878119911 (119905)|)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(2119901minus1

|119878119909 (119905)| + |119878119911 (119905)|)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(93)

which yields that

2119901minus1

120590119887(119878119909 119878119910) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

120590119887(119878119910 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

120590119887(119878119909 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(94)

The Scientific World Journal 13

Finally it is easy to obtain that

2119901minus1G120590119887

(119878119909 119878119910 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1) (95)

Finally since120572 is a lower solution for (78) so it is easy to showthat 120572 ⪯ 119878(120572) [31]

Hence the hypotheses of Theorem 35 are satisfied with120595(119905) = ln((1198862119901minus1)119905 + 1) where 0 le 119886 lt 1 Hence there existsa fixed point 119909 isin 119862[0 119879] such that 119878119909 = 119909 which is a solutionto periodic boundary value problem (78)

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) KingAbdulaziz University JeddahTherefore the firstauthor acknowledges with thanks DSR KAU for financialsupport

References

[1] S Czerwik ldquoContraction mappings in b-metric spacesrdquo ActaMathematica et Informatica Universitatis Ostraviensis vol 1 pp5ndash11 1993

[2] V Parvaneh J R Roshan and S Radenovic ldquoExistence of tri-pled coincidence points in ordered b-metric spaces and an app-lication to a system of integral equationsrdquo Fixed Point Theoryand Applications vol 2013 article 130 2013

[3] J R Roshan V Parvaneh S Sedghi N Shobkolaei and WShatanawi ldquoCommon fixed points of almost generalized con-tractive mappings in ordered b-metric spacesrdquo Fixed PointTheory and Applications vol 2013 article 159 2013

[4] H Aydi M F Bota E Karapınar and S Moradi ldquoA commonfixed point for weak 120601-contractions on b-metric spacesrdquo FixedPoint Theory vol 13 no 2 pp 337ndash346 2012

[5] V Berinde ldquoSequences of operators and fixed points in quasi-metric spacesrdquo Studia Universitatis ldquoBabes-Bolyairdquo Mathemat-ica vol 16 no 4 pp 23ndash27 1996

[6] N Hussain D ETHoric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 article 126 2012

[7] N Hussain and M H Shah ldquoKKMmappings in cone b-metricspacesrdquo Computers and Mathematics with Applications vol 62no 4 pp 1677ndash1684 2011

[8] N Hussain J R Roshan V Parvaneh and Z Kadelburg ldquoFixedpoints of contractive mappings in b-metric-like spacesrdquo TheScientific World Journal vol 2014 Article ID 471827 15 pages2014

[9] M A Khamsi and N Hussain ldquoKKMmappings in metric typespacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 73 no 9 pp 3123ndash3129 2010

[10] M A Khamsi ldquoRemarks on cone metric spaces and fixed pointtheorems of contractive mappingsrdquo Fixed Point Theory andApplications vol 2010 Article ID 315398 7 pages 2010

[11] Z Mustafa J R Roshan V Parvaneh and Z Kadelburg ldquoSomecommon fixed point results in ordered partial b-metric spacesrdquoJournal of Inequalities and Applications vol 2013 article 5622013

[12] Z Mustafa and B Sims ldquoA new approach to generalized metricspacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash297 2006

[13] A Amini-Harandi ldquoMetric-like spaces partial metric spacesand fixed pointsrdquo Fixed PointTheory and Applications vol 2012article 204 2012

[14] N Shobkolaei S Sedghi J R Roshan and N Hussain ldquoSuzukitype fixed point results in metric-like spacesrdquo Journal of Func-tion Spaces and Applications vol 2013 Article ID 143686 9pages 2013

[15] M A Alghamdi N Hussain and P Salimi ldquoFixed point andcoupled fixed point theorems on b-metric-like spacesrdquo Journalof Inequalities and Applications vol 2013 article 402 2013

[16] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572 minus 120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

[17] P Salimi and E Karapınar ldquoSuzuki-Edelstein type contractionsvia auxiliary functionsrdquoMathematical Problems in Engineeringvol 2013 Article ID 648528 8 pages 2013

[18] P Salimi C Vetro and P Vetro ldquoFixed point theorems fortwisted (120572 120573)-120595-contractive type mappings and applicationsrdquoFilomat vol 27 no 4 pp 605ndash615 2013

[19] P Salimi C Vetro and P Vetro ldquoSome new fixed point resultsin non-Archimedean fuzzy metric spacesrdquo Nonlinear AnalysisModelling and Control vol 18 no 3 pp 344ndash358 2013

[20] N Hussain P Salimi and A Latif ldquoFixed point results forsingle and set-valued 120572-120578-120595-contractive mappingsrdquo Fixed PointTheory and Applications vol 2013 article 212 2013

[21] M A Alghamdi and E Karapınar ldquoG-120573-120595 contractive-typemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 article 70 2013

[22] M Geraghty ldquoOn contractive mappingsrdquo Proceedings of theAmerican Mathematical Society vol 40 pp 604ndash608 1973

[23] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001

[24] N Hussain Z Kadelburg S Radenovic and F R Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012

[25] V Berinde ldquoGeneralized contractions in quasimetric spacesrdquo inSeminar on Fixed Point Theory Preprint no 3 pp 3ndash9 1993

[26] L Ciric M Abbas R Saadati and N Hussain ldquoCommon fixedpoints of almost generalized contractive mappings in orderedmetric spacesrdquo Applied Mathematics and Computation vol 217no 12 pp 5784ndash5789 2011

[27] R P Agarwal N Hussain and M A Taoudi ldquoFixed point the-orems in ordered Banach spaces and applications to nonlinearintegral equationsrdquo Abstract and Applied Analysis vol 2012Article ID 245872 15 pages 2012

[28] N Hussain A R Khan and R P Agarwal ldquoKrasnoselrsquoskii andKy Fan type fixed point theorems in ordered Banach spacesrdquoJournal of Nonlinear and Convex Analysis vol 11 no 3 pp 475ndash489 2010

[29] N Hussain and M A Taoudi ldquoKrasnoselrsquoskii-type fixed pointtheorems with applications to Volterra integral equationsrdquoFixed Point Theory and Applications vol 2013 article 196 2013

14 The Scientific World Journal

[30] J J Nieto and R Rodrıguez-Lopez ldquoContractivemapping theo-rems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[31] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysisTheoryMethods andApplications vol 71 no 7-8 pp 3403ndash3410 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 The Scientific World Journal

Using variation of parameters formula we get

119909 (119905) = 119909 (0) 119890minus120582119905

+ int

119905

0

119890minus120582(119905minus119904)

120575119909(119904) 119889119904 (83)

which yields

119909 (119879) = 119909 (0) 119890minus120582119879

+ int

119879

0

119890minus120582(119879minus119904)

120575119909(119904) 119889119904 (84)

Since 119909(0) = 119909(119879) we get

119909 (0) [1 minus 119890minus120582119879

] = 119890minus120582119879

int

119879

0

119890120582(119904)

120575119909(119904) 119889119904 (85)

or

119909 (0) =

1

119890120582119879

minus 1

int

119879

0

119890120582119904

120575119909(119904) 119889119904 (86)

Substituting the value of 119909(0) in (83) we arrive at

119909 (119905) = int

119879

0

119866 (119905 119904) 120575119909(119904) 119889119904 (87)

where

119866 (119905 119904) =

119890120582(119879+119904minus119905)

119890120582119879

minus 1

0 le 119904 le 119905 le 119879

119890120582(119904minus119905)

119890120582119879

minus 1

0 le 119905 le 119904 le 119879

(88)

Now define the operator 119878 119862[0 119879] rarr 119862[0 119879] as

119878119909 (119905) = int

119879

0

119866 (119905 119904) 119865 (119904 119909 (119904)) 119889119904 (89)

The mapping 119878 is increasing [31] Note that if 119906 isin 119862[0 119879] is afixed point of 119878 then 119906 isin 119862

1[0 119879] is a solution of (78)

Let 119909 119910 119911 isin 119862[0 119879] Then we have

2119901minus1

[|119878119909 (119905)| +1003816100381610038161003816119878119910 (119905)

1003816100381610038161003816]

= 2119901minus1

[

100381610038161003816100381610038161003816100381610038161003816

int

119879

0

119866 (119905 119904) 119865 (119904 119909 (119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

119879

0

119866 (119905 119904) 119865 (119904 119910 (119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

]

le 2119901minus1

int

119879

0

|119866 (119905 119904)|

times [|119865 (119904 119909 (119904))| +1003816100381610038161003816119865 (119904 119910 (119904))

1003816100381610038161003816] 119889119904

le 2119901minus1

int

119879

0

|119866 (119905 119904)|

120582

2119901minus1

times119901radicln(

119886

2119901minus1

(|119909 (119905)| +1003816100381610038161003816119910 (119905)

1003816100381610038161003816)119901

+ 1)119889119904

le 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [int

119905

0

119890120582(119879+119904minus119905)

119890120582119879

minus 1

119889119904 + int

119879

119905

119890120582(119904minus119905)

119890120582119879

minus 1

119889119904]

= 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [

1

120582 (119890120582119879

minus 1)

(119890120582(119879+119904minus119905)10038161003816

100381610038161003816

119905

0

+ 119890120582(119904minus119905)10038161003816

100381610038161003816

119879

119905

)]

= 120582119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

times [

1

120582 (119890120582119879

minus 1)

(119890120582119879

minus 119890120582(119879minus119905)

+ 119890120582(119879minus119905)

minus 1)]

=119901radicln(

119886

2119901minus1

120590119887(119909 119910) + 1)

le119901radicln(

119886

2119901minus1

119866120590119887

(119909 119910 119911) + 1)

le119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(90)

Similarly

2119901minus1 1003816

100381610038161003816119878119910 (119905)

1003816100381610038161003816+ |119878119911 (119905)| le

119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

|119878119909 (119905)| + |119878119911 (119905)| le119901radicln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(91)

where

119872(119909 119910 119911)

= max119866120590119887

(119909 119910 119911)

119866120590119887

(119909 119878119909 119878119909) 119866120590119887

(119910 119878119910 119878119910)119866120590119887

(119911 119878119911 119878119911)

1 + 22119901minus1

1198662

120590119887

(119878119909 119878119910 119878119911)

(92)

Equivalently

(2119901minus1

|119878119909 (119905)| +1003816100381610038161003816119878119910 (119905)

1003816100381610038161003816)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(2119901minus1 1003816

100381610038161003816119878119910 (119905)

1003816100381610038161003816+ |119878119911 (119905)|)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(2119901minus1

|119878119909 (119905)| + |119878119911 (119905)|)

119901

le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(93)

which yields that

2119901minus1

120590119887(119878119909 119878119910) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

120590119887(119878119910 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

2119901minus1

120590119887(119878119909 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1)

(94)

The Scientific World Journal 13

Finally it is easy to obtain that

2119901minus1G120590119887

(119878119909 119878119910 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1) (95)

Finally since120572 is a lower solution for (78) so it is easy to showthat 120572 ⪯ 119878(120572) [31]

Hence the hypotheses of Theorem 35 are satisfied with120595(119905) = ln((1198862119901minus1)119905 + 1) where 0 le 119886 lt 1 Hence there existsa fixed point 119909 isin 119862[0 119879] such that 119878119909 = 119909 which is a solutionto periodic boundary value problem (78)

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) KingAbdulaziz University JeddahTherefore the firstauthor acknowledges with thanks DSR KAU for financialsupport

References

[1] S Czerwik ldquoContraction mappings in b-metric spacesrdquo ActaMathematica et Informatica Universitatis Ostraviensis vol 1 pp5ndash11 1993

[2] V Parvaneh J R Roshan and S Radenovic ldquoExistence of tri-pled coincidence points in ordered b-metric spaces and an app-lication to a system of integral equationsrdquo Fixed Point Theoryand Applications vol 2013 article 130 2013

[3] J R Roshan V Parvaneh S Sedghi N Shobkolaei and WShatanawi ldquoCommon fixed points of almost generalized con-tractive mappings in ordered b-metric spacesrdquo Fixed PointTheory and Applications vol 2013 article 159 2013

[4] H Aydi M F Bota E Karapınar and S Moradi ldquoA commonfixed point for weak 120601-contractions on b-metric spacesrdquo FixedPoint Theory vol 13 no 2 pp 337ndash346 2012

[5] V Berinde ldquoSequences of operators and fixed points in quasi-metric spacesrdquo Studia Universitatis ldquoBabes-Bolyairdquo Mathemat-ica vol 16 no 4 pp 23ndash27 1996

[6] N Hussain D ETHoric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 article 126 2012

[7] N Hussain and M H Shah ldquoKKMmappings in cone b-metricspacesrdquo Computers and Mathematics with Applications vol 62no 4 pp 1677ndash1684 2011

[8] N Hussain J R Roshan V Parvaneh and Z Kadelburg ldquoFixedpoints of contractive mappings in b-metric-like spacesrdquo TheScientific World Journal vol 2014 Article ID 471827 15 pages2014

[9] M A Khamsi and N Hussain ldquoKKMmappings in metric typespacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 73 no 9 pp 3123ndash3129 2010

[10] M A Khamsi ldquoRemarks on cone metric spaces and fixed pointtheorems of contractive mappingsrdquo Fixed Point Theory andApplications vol 2010 Article ID 315398 7 pages 2010

[11] Z Mustafa J R Roshan V Parvaneh and Z Kadelburg ldquoSomecommon fixed point results in ordered partial b-metric spacesrdquoJournal of Inequalities and Applications vol 2013 article 5622013

[12] Z Mustafa and B Sims ldquoA new approach to generalized metricspacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash297 2006

[13] A Amini-Harandi ldquoMetric-like spaces partial metric spacesand fixed pointsrdquo Fixed PointTheory and Applications vol 2012article 204 2012

[14] N Shobkolaei S Sedghi J R Roshan and N Hussain ldquoSuzukitype fixed point results in metric-like spacesrdquo Journal of Func-tion Spaces and Applications vol 2013 Article ID 143686 9pages 2013

[15] M A Alghamdi N Hussain and P Salimi ldquoFixed point andcoupled fixed point theorems on b-metric-like spacesrdquo Journalof Inequalities and Applications vol 2013 article 402 2013

[16] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572 minus 120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

[17] P Salimi and E Karapınar ldquoSuzuki-Edelstein type contractionsvia auxiliary functionsrdquoMathematical Problems in Engineeringvol 2013 Article ID 648528 8 pages 2013

[18] P Salimi C Vetro and P Vetro ldquoFixed point theorems fortwisted (120572 120573)-120595-contractive type mappings and applicationsrdquoFilomat vol 27 no 4 pp 605ndash615 2013

[19] P Salimi C Vetro and P Vetro ldquoSome new fixed point resultsin non-Archimedean fuzzy metric spacesrdquo Nonlinear AnalysisModelling and Control vol 18 no 3 pp 344ndash358 2013

[20] N Hussain P Salimi and A Latif ldquoFixed point results forsingle and set-valued 120572-120578-120595-contractive mappingsrdquo Fixed PointTheory and Applications vol 2013 article 212 2013

[21] M A Alghamdi and E Karapınar ldquoG-120573-120595 contractive-typemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 article 70 2013

[22] M Geraghty ldquoOn contractive mappingsrdquo Proceedings of theAmerican Mathematical Society vol 40 pp 604ndash608 1973

[23] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001

[24] N Hussain Z Kadelburg S Radenovic and F R Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012

[25] V Berinde ldquoGeneralized contractions in quasimetric spacesrdquo inSeminar on Fixed Point Theory Preprint no 3 pp 3ndash9 1993

[26] L Ciric M Abbas R Saadati and N Hussain ldquoCommon fixedpoints of almost generalized contractive mappings in orderedmetric spacesrdquo Applied Mathematics and Computation vol 217no 12 pp 5784ndash5789 2011

[27] R P Agarwal N Hussain and M A Taoudi ldquoFixed point the-orems in ordered Banach spaces and applications to nonlinearintegral equationsrdquo Abstract and Applied Analysis vol 2012Article ID 245872 15 pages 2012

[28] N Hussain A R Khan and R P Agarwal ldquoKrasnoselrsquoskii andKy Fan type fixed point theorems in ordered Banach spacesrdquoJournal of Nonlinear and Convex Analysis vol 11 no 3 pp 475ndash489 2010

[29] N Hussain and M A Taoudi ldquoKrasnoselrsquoskii-type fixed pointtheorems with applications to Volterra integral equationsrdquoFixed Point Theory and Applications vol 2013 article 196 2013

14 The Scientific World Journal

[30] J J Nieto and R Rodrıguez-Lopez ldquoContractivemapping theo-rems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[31] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysisTheoryMethods andApplications vol 71 no 7-8 pp 3403ndash3410 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 13

Finally it is easy to obtain that

2119901minus1G120590119887

(119878119909 119878119910 119878119911) le ln(

119886

2119901minus1

119872(119909 119910 119911) + 1) (95)

Finally since120572 is a lower solution for (78) so it is easy to showthat 120572 ⪯ 119878(120572) [31]

Hence the hypotheses of Theorem 35 are satisfied with120595(119905) = ln((1198862119901minus1)119905 + 1) where 0 le 119886 lt 1 Hence there existsa fixed point 119909 isin 119862[0 119879] such that 119878119909 = 119909 which is a solutionto periodic boundary value problem (78)

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) KingAbdulaziz University JeddahTherefore the firstauthor acknowledges with thanks DSR KAU for financialsupport

References

[1] S Czerwik ldquoContraction mappings in b-metric spacesrdquo ActaMathematica et Informatica Universitatis Ostraviensis vol 1 pp5ndash11 1993

[2] V Parvaneh J R Roshan and S Radenovic ldquoExistence of tri-pled coincidence points in ordered b-metric spaces and an app-lication to a system of integral equationsrdquo Fixed Point Theoryand Applications vol 2013 article 130 2013

[3] J R Roshan V Parvaneh S Sedghi N Shobkolaei and WShatanawi ldquoCommon fixed points of almost generalized con-tractive mappings in ordered b-metric spacesrdquo Fixed PointTheory and Applications vol 2013 article 159 2013

[4] H Aydi M F Bota E Karapınar and S Moradi ldquoA commonfixed point for weak 120601-contractions on b-metric spacesrdquo FixedPoint Theory vol 13 no 2 pp 337ndash346 2012

[5] V Berinde ldquoSequences of operators and fixed points in quasi-metric spacesrdquo Studia Universitatis ldquoBabes-Bolyairdquo Mathemat-ica vol 16 no 4 pp 23ndash27 1996

[6] N Hussain D ETHoric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 article 126 2012

[7] N Hussain and M H Shah ldquoKKMmappings in cone b-metricspacesrdquo Computers and Mathematics with Applications vol 62no 4 pp 1677ndash1684 2011

[8] N Hussain J R Roshan V Parvaneh and Z Kadelburg ldquoFixedpoints of contractive mappings in b-metric-like spacesrdquo TheScientific World Journal vol 2014 Article ID 471827 15 pages2014

[9] M A Khamsi and N Hussain ldquoKKMmappings in metric typespacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 73 no 9 pp 3123ndash3129 2010

[10] M A Khamsi ldquoRemarks on cone metric spaces and fixed pointtheorems of contractive mappingsrdquo Fixed Point Theory andApplications vol 2010 Article ID 315398 7 pages 2010

[11] Z Mustafa J R Roshan V Parvaneh and Z Kadelburg ldquoSomecommon fixed point results in ordered partial b-metric spacesrdquoJournal of Inequalities and Applications vol 2013 article 5622013

[12] Z Mustafa and B Sims ldquoA new approach to generalized metricspacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash297 2006

[13] A Amini-Harandi ldquoMetric-like spaces partial metric spacesand fixed pointsrdquo Fixed PointTheory and Applications vol 2012article 204 2012

[14] N Shobkolaei S Sedghi J R Roshan and N Hussain ldquoSuzukitype fixed point results in metric-like spacesrdquo Journal of Func-tion Spaces and Applications vol 2013 Article ID 143686 9pages 2013

[15] M A Alghamdi N Hussain and P Salimi ldquoFixed point andcoupled fixed point theorems on b-metric-like spacesrdquo Journalof Inequalities and Applications vol 2013 article 402 2013

[16] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572 minus 120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

[17] P Salimi and E Karapınar ldquoSuzuki-Edelstein type contractionsvia auxiliary functionsrdquoMathematical Problems in Engineeringvol 2013 Article ID 648528 8 pages 2013

[18] P Salimi C Vetro and P Vetro ldquoFixed point theorems fortwisted (120572 120573)-120595-contractive type mappings and applicationsrdquoFilomat vol 27 no 4 pp 605ndash615 2013

[19] P Salimi C Vetro and P Vetro ldquoSome new fixed point resultsin non-Archimedean fuzzy metric spacesrdquo Nonlinear AnalysisModelling and Control vol 18 no 3 pp 344ndash358 2013

[20] N Hussain P Salimi and A Latif ldquoFixed point results forsingle and set-valued 120572-120578-120595-contractive mappingsrdquo Fixed PointTheory and Applications vol 2013 article 212 2013

[21] M A Alghamdi and E Karapınar ldquoG-120573-120595 contractive-typemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 article 70 2013

[22] M Geraghty ldquoOn contractive mappingsrdquo Proceedings of theAmerican Mathematical Society vol 40 pp 604ndash608 1973

[23] I A Rus Generalized Contractions and Applications ClujUniversity Press Cluj-Napoca Romania 2001

[24] N Hussain Z Kadelburg S Radenovic and F R Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012

[25] V Berinde ldquoGeneralized contractions in quasimetric spacesrdquo inSeminar on Fixed Point Theory Preprint no 3 pp 3ndash9 1993

[26] L Ciric M Abbas R Saadati and N Hussain ldquoCommon fixedpoints of almost generalized contractive mappings in orderedmetric spacesrdquo Applied Mathematics and Computation vol 217no 12 pp 5784ndash5789 2011

[27] R P Agarwal N Hussain and M A Taoudi ldquoFixed point the-orems in ordered Banach spaces and applications to nonlinearintegral equationsrdquo Abstract and Applied Analysis vol 2012Article ID 245872 15 pages 2012

[28] N Hussain A R Khan and R P Agarwal ldquoKrasnoselrsquoskii andKy Fan type fixed point theorems in ordered Banach spacesrdquoJournal of Nonlinear and Convex Analysis vol 11 no 3 pp 475ndash489 2010

[29] N Hussain and M A Taoudi ldquoKrasnoselrsquoskii-type fixed pointtheorems with applications to Volterra integral equationsrdquoFixed Point Theory and Applications vol 2013 article 196 2013

14 The Scientific World Journal

[30] J J Nieto and R Rodrıguez-Lopez ldquoContractivemapping theo-rems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[31] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysisTheoryMethods andApplications vol 71 no 7-8 pp 3403ndash3410 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 The Scientific World Journal

[30] J J Nieto and R Rodrıguez-Lopez ldquoContractivemapping theo-rems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[31] J Harjani andK Sadarangani ldquoFixed point theorems for weaklycontractive mappings in partially ordered setsrdquoNonlinear Anal-ysisTheoryMethods andApplications vol 71 no 7-8 pp 3403ndash3410 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of