Research Article(120572 minus120595)-Contractive Mappings on Generalized Quasimetric Spaces

Erdal KarapJnar12 and Hosein Lakzian3

1 Department of Mathematics Atilim University Incek 06836 Ankara Turkey2Nonlinear Analysis and Applied Mathematics Research Group (NAAM) King Abdulaziz University Jeddah Saudi Arabia3 Department of Mathematics Payame Noor University Tehran 19395-4697 Iran

Correspondence should be addressed to Hosein Lakzian h lakzianpnuacir

Received 26 June 2014 Revised 30 July 2014 Accepted 30 July 2014 Published 18 August 2014

Academic Editor Nawab Hussain

Copyright copy 2014 E Karapınar and H LakzianThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We consider the existence of a fixed point of (120572minus120595)-contractive mappings in the context of generalized quasimetric spaces withoutHausdorff assumption The obtained results extend several results on the topic in the literature

1 Introduction and Preliminaries

In the last decade quasimetric spaces have been one of theinteresting topics for the researchers in the field of fixedpoint theory due to two reasons The first reason is thatthe assumptions of quasimetric are weaker than the moregeneralmetric Consequently the obtained fixed point resultsin this space are more general and hence the correspondingresults in metric space are covered The second reason is thefact that fixed point problems in 119866-metric space (introducedby Mustafa and Sims [1]) can be reduced to related fixedpoint problems in the context of quasimetric space (see eg[2 3]) Very recently Lin et al [4] introduced the notion ofgeneralized quasimetric spaces and investigated the existenceof a certain operator on such spaces In this paper [4] theauthors assumed that the generalized quasimetric space isHausdorff to get a fixed point

In this paper we examine the existence of (120572-120595)-contract-ive mappings in the context of generalized quasimetric spacewithout the Hausdorffness assumption Consequently ourresults extend improve and generalize several results in theliterature

In what follows we recall the basic definitions and resultson the topics for the sake of completeness Throughoutthe paper the symbols R N and N

0denote the real

numbers the natural numbers and the positive integersrespectively

Let 119883 be a nonempty set and let 119889 119883 times 119883 rarr [0infin)Then 119889 is called a distance function if for every 119909 119910 119911 isin 119883it satisfies

(1198891) 119889(119909 119909) = 0

(1198892) 119889(119909 119910) = 119889(119910 119909) = 0 rArr 119909 = 119910

(1198893) 119889(119909 119910) = 119889(119910 119909)

(1198894) 119889(119909 119911) le 119889(119909 119910) + 119889(119910 119911)

Notice that if 119889 satisfies the conditions (1198892) (1198893) and (119889

4)

then 119889 is called a dislocated metric on 119883 If 119889 satisfies theconditions (119889

1) (1198892) and (119889

4) then 119889 is called a quasimetric

on 119883 On the other hand if 119889 satisfies the conditions(1198891)ndash(1198894) then 119889 is called a metric on 119883

One of the very natural generalizations of the notion of ametric was introduced by Branciari [5] in 2000 by replacingthe triangle inequality assumption of a metric with a weakercondition quadrilateral inequality

Definition 1 (see [5]) Let 119883 be a nonempty set and let 119889

119883times119883 rarr [0infin) be a mapping such that for all 119909 119910 isin 119883 andfor all distinct points 119906 V isin 119883 each of them different from 119909

and 119910 one has

(i) 119889(119909 119910) = 0 if and only if 119909 = 119910(ii) 119889(119909 119910) = 119889(119910 119909)

Hindawi Publishing CorporationJournal of Function SpacesVolume 2014 Article ID 914398 7 pageshttpdxdoiorg1011552014914398

2 Journal of Function Spaces

(iii) 119889(119909 119910) le 119889(119909 119906) + 119889(119906 V) + 119889(V 119910) (quadrilateralinequality)

Then (119883 119889) is called a generalized metric space (or shortlygms)

We present an example to show that not every generalizedmetric on a set 119883 is a metric on 119883

Example 2 (see eg [4]) Let 119883 = 119905 2119905 3119905 4119905 5119905 with 119905 gt 0

be a constant and we define 119889 119883 times 119883 rarr [0infin) by

(1) 119889(119909 119909) = 0 for all 119909 isin 119883(2) 119889(119909 119910) = 119889(119910 119909) for all 119909 119910 isin 119883(3) 119889(119905 2119905) = 3119904(4) 119889(119905 3119905) = 119889(2119905 3119905) = 119904(5) 119889(119905 4119905) = 119889(2119905 4119905) = 119889(3119905 4119905) = 2119904(6) 119889(119905 5119905) = 119889(2119905 5119905) = 119889(3119905 5119905) = 119889(4119905 5119905) = (32)119904

where 119904 gt 0 is a constant Then (119883 119889) is a generalized metricspace but it is not a metric space because

119889 (119905 2119905) = 3119904 gt 119889 (119905 3119905) + 119889 (3119905 2119905) = 2119904 (1)

Despite the analogy between the definitions ofmetric andgeneralized metric their topological properties differ fromeach other For example for a generalizedmetric space (119883 119889)we have the following

(P1) a generalized metric need not be continuous(P2) a convergent sequence in generalized metric space

need not be Cauchy(P3) an open ball (119861

119903(119909) = 119910 isin 119883 119889(119909 119910) lt 119903 119903 gt 0)

need not be open set(P4) a generalized metric space need not be Hausdorff

and hence the uniqueness of limits cannot be guar-anteed

Example 3 (see [6] Example 11) Let 119883 = 119860 cup 119861 where 119860 =

0 2 and 119860 = 1119899 119899 isin N Define 119889 119883 times 119883 rarr [0infin) inthe following way

119889 (119909 119910) =

0 if 119909 = 119910

1 if 119909 = 119910 [119909 119910 sub 119860 or 119909 119910 sub 119861]

119910 if 119909 isin 119860 119910 isin 119861

(2)

Notice that 119889(119886 119887) = 119889(119887 119886) = 119887 whenever 119886 isin 119860 and 119887 isin 119861Furthermore (119883 119889) is a complete generalized metric spaceClearly we have (1198751)ndash(1198754) Indeed the sequence 1119899 119899 isin

N converges to both 0 and 2 There is no 119903 gt 0 such that119861119903(0) cap119861

119903(2) = 0 and hence it is not Hausdorff It is clear that

the ball11986123

(13) = 0 13 2 since there is no 119903 gt 0 such that119861119903(0) sub 119861

23(13) that is open balls may not be an open set

The function 119889 is not continuous since lim119899rarrinfin

119889(1119899 12) =

119889(0 12) although lim119899rarrinfin

1119899 = 0 Formore details see eg[6 7]

Regarding the weakness of the topology of generalizedmetric space mentioned above the authors add some addi-tional conditions to get the analog of existing fixed pointresults in the literature see eg [8ndash15] Very recently Suzuki[16] underlined the importance of generalized metric spaceby emphasizing that generalized metric space and metricspace have no compatible topology

The following is the definition of the notion of generalizedquasimetric space defined by Lin et al [4]

Definition 4 Let 119883 be a nonempty set and let 119889 119883 times 119883 rarr

[0infin) be a mapping such that for all 119909 119910 isin 119883 and for alldistinct points 119906 V isin 119883 each of them different from 119909 and 119910one has

(i) 119889(119909 119910) = 0 if and only if 119909 = 119910(ii) 119889(119909 119910) le 119889(119909 119906) + 119889(119906 V) + 119889(V 119910)

Then (119883 119889) is called a generalized quasimetric space (orshortly gqms)

It is evident that any generalized metric space is ageneralized quasimetric space but the converse is not truein general We give an example to show that not everygeneralized quasimetric on a set119883 is a generalized metric on119883

Example 5 (see [4]) Let 119883 = 119905 2119905 3119905 4119905 5119905 with 119905 gt 0 be aconstant and we define 119889 119883 times 119883 rarr [0infin) by

(1) 119889(119909 119909) = 0 for all 119909 isin 119883(2) 119889(119905 2119905) = 119889(2119905 119905) = 3119904(3) 119889(119905 3119905) = 119889(2119905 3119905) = 119889(3119905 119905) = 119889(3119905 2119905) = 119904(4) 119889(119905 4119905) = 119889(2119905 4119905) = 119889(3119905 4119905) = 119889(4119905 119905) = 119889(4119905 2119905) =

119889(4119905 3119905) = 2119904(5) 119889(119905 5119905) = 119889(2119905 5119905) = 119889(3119905 5119905) = 119889(4119905 5119905) = (32)119904(6) 119889(5119905 119905) = 119889(5119905 2119905) = 119889(5119905 3119905) = 119889(5119905 4119905) = (54)119904

where 119904 gt 0 is a constant Then (119883 119889) is a generalized quasi-metric space but it is not a generalized metric space because

119889 (119905 5119905) =3

2119904 = 119889 (5119905 119905) =

5

4119904 (3)

We next give the definitions of convergence and com-pleteness on generalized quasimetric spaces

Definition 6 (see [4]) Let (119883 119889) be a gqms let 119909119899 be

a sequence in 119883 and 119909 isin 119883 We say that 119909119899 is gqms

convergent to 119909 if and only if

lim119899rarrinfin

119889 (119909119899 119909) = lim

119899rarrinfin119889 (119909 119909

119899) = 0 (4)

Definition 7 (see [4]) Let (119883 119889) be a gqms and let 119909119899 be

a sequence in119883 We say that 119909119899 is left-Cauchy if and only if

for every 120576 gt 0 there exists 119896 isin N such that 119889(119909119899 119909119898) lt 120576 for

all 119899 ge 119898 gt 119896 We say that 119909119899 is right-Cauchy if and only if

for every 120576 gt 0 there exists 119896 isin N such that 119889(119909119899 119909119898) lt 120576 for

all 119898 ge 119899 gt 119896 We say that 119909119899 is Cauchy if and only if for

every 120576 gt 0 there exists 119896 isin N such that 119889(119909119899 119909119898) lt 120576 for all

119898 119899 gt 119896

Journal of Function Spaces 3

Remark 8 A sequence 119909119899 in a gqms is Cauchy if and only

if it is left-Cauchy and right-Cauchy

Definition 9 (see [4]) Let (119883 119889) be a gqms We say that

(1) (119883 119889) is left-complete if and only if each left-Cauchysequence in 119883 is convergent

(2) (119883 119889) is right-complete if and only if each right-Cauchy sequence in 119883 is convergent

(3) (119883 119889) is complete if and only if each Cauchy sequencein 119883 is convergent

Notice that in the literature in several reports for fixedpoint results in generalized metric space an additionalbut superfluous condition ldquoHausdorffnessrdquo was assumedRecently Jleli and Samet [17] Kirk and Shahzad [18] Kara-pınar [19] Kadeburg and Radenovic [7] and Aydi et al[20] reported new some fixed point results by removing theassumption of Hausdorffness in the context of generalizedmetric spaces The following crucial lemma is inspired from[7 17]

Lemma 10 Let (119883 119889) be a generalized quasimetric space andlet 119909119899 be a Cauchy sequence in119883 such that 119909

119898= 119909119899whenever

119898 = 119899 Then the sequence 119909119899 can converge to at most one

point

Proof Given 120576 gt 0 since 119909119899 is a Cauchy sequence there

exists 1198960isin N such that

119889 (119909119899 119909119898) lt 120576 forall119898 119899 gt 119896

0 (5)

We use themethod of Reductio ad absurdum Suppose on thecontrary that there exist two distinct points119909 and119910 in119883 suchthat the sequence 119909

119899 converges to 119909 and 119910 that is

lim119899rarrinfin

119889 (119909119899 119909) = lim

119899rarrinfin119889 (119909 119909

119899) = 0

lim119899rarrinfin

119889 (119909119899 119910) = lim

119899rarrinfin119889 (119910 119909

119899) = 0

(6)

By assumption for any 119899 isin N 119909119899

= 119909119898 and since 119909 = 119910 there

exists 1198961isin N such that 119909

119899= 119909 and 119909

119899= 119910 for any 119899 gt 119896

1ge 1198960

Due to quadrilateral inequality we have

119889 (119909 119910) le 119889 (119909 119909119899) + 119889 (119909

119899 119909119898) + 119889 (119909

119898 119910) (7)

Letting 119899119898 rarr infin we can obtain that 119889(119909 119910) = 0 by regard-ing (5) and (6) Hence we get 119909 = 119910which is a contradiction

2 Main Results

In this section we state and prove the main result of thispaper We start by introducing the following family of func-tions

Let Ψ be the family of functions 120595 [0infin) rarr [0infin)

satisfying the following conditions

(Ψ1) 120595 is nondecreasing

(Ψ2) sum+infin

119899=1120595119899(119905) lt infin for all 119905 gt 0 where120595119899 is the 119899th

iterate of 120595

These functions are known in the literature as (c)-comparisonfunctions It is easily proved that if 120595 is a (c)-comparisonfunction then 120595(119905) lt 119905 for any 119905 gt 0 For more details aboutsuch function we refer the reader to [21 22] In this study wediscuss the notion of 120572-admissible mappings see eg [23ndash27] The following definition was introduced in [23]

Definition 11 Let 119891 119883 rarr 119883 be a self-mapping of a set 119883and 120572 119883 times 119883 rarr R+ Then 119891 is called a 120572-admissible if

119909 119910 isin 119883 120572 (119909 119910) ge 1 997904rArr 120572 (119891119909 119891119910) ge 1 (8)

Inwhat followswe define the (120572minus120595)-contractivemappingin the setting of generalized quasimetric space

Definition 12 Let (119883 119889) be a gqms and let 119891 119883 rarr 119883

be a given mapping We say that 119891 is an (120572 minus 120595)-contractivemapping if there exist two functions 120572 119883 times 119883 rarr [0infin)

and 120595 isin Ψ such that

120572 (119909 119910) 119889 (119891119909 119891119910) le 120595 (119889 (119909 119910)) forall119909 119910 isin 119883 (9)

Now we state the following fixed point theorem

Theorem 13 Let (119883 119889) be a complete 119892119902119898119904 and let 119891

119883 rarr 119883 be an (120572 minus 120595)-contractive mapping Suppose that

(i) 119891 is 120572-admissible(ii) there exists 119909

0isin 119883 such that 120572(119909

0 1198911199090) ge 1

120572(1198911199090 1199090) ge 1 120572(119909

0 11989121199090) ge 1 and 120572(1198912119909

0 1199090) ge 1

(iii) 119891 is continuous

Then 119891 has a periodic point

Proof Due to statement (ii) of theorem there exists 1199090isin 119883

which is an arbitrary point such that 120572(1199090 1198911199090) ge 1 and

120572(1199090 1198911199090) ge 1 We will construct a sequence 119909

119899 in 119883 by

119909119899+1

= 119891119909119899= 119891119899+1119909

0for all 119899 ge 0 If we have 119909

1198990= 1199091198990+1

forsome 119899

0 then 119906 = 119909

1198990is a fixed point of 119891 Hence for the rest

of the proof we presume that

119909119899

= 119909119899+1

forall119899 (10)

Since 119891 is 120572-admissible we have

120572 (1199090 1199091) = 120572 (119909

0 1198911199090) ge 1

997904rArr 120572 (1198911199090 1198911199091) = 120572 (119909

1 1199092) ge 1

(11)

Utilizing the expression above we obtain that

120572 (119909119899 119909119899+1

) ge 1 forall119899 = 0 1 (12)

By repeating the same stepswith startingwith the assumption120572(1199091 1199090) = 120572(119891119909

0 1199090) ge 1 we conclude that

120572 (119909119899+1

119909119899) ge 1 forall119899 = 0 1 (13)

4 Journal of Function Spaces

In a similar way we derive that

120572 (1199090 1199092) = 120572 (119909

0 11989121199090) ge 1

997904rArr 120572 (1198911199090 1198911199092) = 120572 (119909

1 1199093) ge 1

(14)

Recursively we get that

120572 (119909119899 119909119899+2

) ge 1 forall119899 = 0 1 (15)

Analogously we can easily derive that

120572 (119909119899+2

119909119899) ge 1 forall119899 = 0 1 (16)

Step 1 We will show that lim119899rarrinfin

119889(119909119899 119909119899+1

) = 0 andlim119899rarrinfin

119889(119909119899 119909119899+2

) = 0 Regarding (8) and (12) we deducethat

119889 (119909119899 119909119899+1

) = 119889 (119891119909119899minus1

119891119909119899)

le 120572 (119909119899minus1

119909119899) 119889 (119891119909

119899minus1 119891119909119899)

le 120595 (119889 (119909119899minus1

119909119899))

(17)

for all 119899 ge 1Iteratively we find that

119889 (119909119899 119909119899+1

) le 120595119899(119889 (1199090 1199091)) forall119899 ge 1 (18)

Similarly

119889 (119909119899 119909119899+2

) le 120595119899(119889 (1199090 1199092)) forall119899 ge 1 (19)

By the properties of 120595 we can conclude thatlim119899rarrinfin

120595119899(119889(1199090 1199091)) = 0 that is

lim119899rarrinfin

119889 (119909119899 119909119899+1

) = 0 (20)

Similarly lim119899rarrinfin

120595119899(119889(1199090 1199092)) = 0 that is

lim119899rarrinfin

119889 (119909119899 119909119899+2

) = 0 (21)

Step 2 We will prove that 119909119899 is a right-Cauchy sequence

that islim119899rarrinfin

119889 (119909119899 119909119899+119896

) = 0 forall119896 isin N (22)

The cases 119896 = 1 and 119896 = 2 are proved respectively by (20)and (21) Now take 119896 ge 3 arbitrary It is sufficient to examinetwo cases

Case (I) Suppose that 119896 = 2119898+1 where119898 ge 1Then by usingStep 1 and the quadrilateral inequality together with (18) wefind119889 (119909119899 119909119899+119896

) = 119889 (119909119899 119909119899+2119898+1

)

le 119889 (119909119899 119909119899+1

) + 119889 (119909119899+1

119909119899+2

)

+ sdot sdot sdot + 119889 (119909119899+2119898

119909119899+2119898+1

)

le

119899+2119898

sum119901=119899

120595119901(119889 (1199090 1199091))

le

+infin

sum119901=119899

120595119901(119889 (1199090 1199091)) 997888rarr 0 as 119899 997888rarr infin

(23)

Case (II) Suppose that 119896 = 2119898 where 119898 ge 2 Again byapplying the quadrilateral inequality and Step 1 together with(18) and (19) we find

119889 (119909119899 119909119899+119896

) = 119889 (119909119899 119909119899+2119898

)

le 119889 (119909119899 119909119899+2

) + 119889 (119909119899+2

119909119899+3

)

+ sdot sdot sdot + 119889 (119909119899+2119898minus1

119909119899+2119898

)

le 119889 (119909119899 119909119899+2

) +

119899+2119898minus1

sum119901=119899+2

120595119901(119889 (1199090 1199091))

le 119889 (119909119899 119909119899+2

)

+

+infin

sum119901=119899

120595119901(119889 (1199090 1199091)) 997888rarr 0 as 119899 997888rarr infin

(24)

By combining the expressions (23) and (24) we have

lim119899rarrinfin

119889 (119909119899 119909119899+119896

) = 0 forall119896 ge 3 (25)

We conclude that 119909119899 is a right-Cauchy sequence in (119883 119889)

In the same way 119909119899 is a left-Cauchy sequence in (119883 119889)

So it is aCauchy sequence Since119883 is a complete gqms thereexists 119906 isin 119883 such that

lim119899rarrinfin

119889 (119909119899 119906) = lim

119899rarrinfin119889 (119906 119909

119899) = 0 (26)

Also we can easily see that 119909119899

= 119909119898for whenever 119898 = 119899

Indeed if 119909119899= 119909119898 for some 119898 119899 isin N with 119899 lt 119898 then

119889 (119909119899 119909119899+1

) = 119889 (119909119898 119909119898+1

)

le 120595119898minus119899

119889 (119909119899 119909119899+1

) lt 119889 (119909119899 119909119899+1

) (27)

which is a contradiction Analogously we derive the sameconclusion for the case 119899 gt 119898 Therefore we conclude thatthe sequence 119909

119899 cannot have two limits due to Lemma 10

Step 3 We claim that 119891 has a periodic point in 119883 Supposeon the contrary that 119891 has no periodic point Since 119891 iscontinuous from Step 2 we have 119906 = 119891119906 (119901 = 1) whichcontradicts the assumption that 119891 has no periodic pointTherefore there exists 119906 isin 119883 such that 119906 = 119891

119901(119906) for some119901 isin N So 119891 has a periodic point in 119883

Now we state the following fixed point theorem

Theorem 14 Let (119883 119889) be a complete 119892119902119898119904 and let 119891

119883 rarr 119883 be an (120572 minus 120595)-contractive mapping Suppose that

(i) 119891 is 120572-admissible(ii) there exists 119909

0isin 119883 such that 120572(119909

0 1198911199090) ge 1

120572(1198911199090 1199090) ge 1 120572(119909

0 11989121199090) ge 1 and 120572(1198912119909

0 1199090) ge 1

(iii) if 119909119899 is a sequence in 119883 such that 120572(119909

119899 119909119899+1

) ge 1 forall 119899 and 119909

119899rarr 119909 isin 119883 as 119899 rarr infin then 120572(119909

119899 119909) ge 1

for all 119899

Then 119891 has a periodic point

Journal of Function Spaces 5

Proof Following the proof of Theorem 13 we know that thesequence 119909

119899 defined by 119909

119899+1= 119891119909119899for all 119899 ge 0 converges

for some 119906 isin 119883 It is sufficient to show that 119891 admitsa periodic point Suppose on the contrary that 119891 has noperiodic point Notice that119909

119899= 119906 and119909

119899= 119891119906 for sufficiently

large 119899 By the quadrilateral inequality for this 119899 we have

119889 (119906 119891119906) le 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 119889 (119909119899+1

119891119906) (28)

On account of the fact that 120595(119905) lt 119905 for all 119905 gt 0 andregarding the assumption (iii) we get that

119889 (119906 119891119906) le 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 119889 (119909119899+1

119891119906)

le 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 119889 (119891119909119899 119891119906)

le 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 120572 (119909119899 119906) 119889 (119891119909

119899 119891119906)

le 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 120595 (119889 (119909119899 119906))

lt 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 119889 (119909119899 119906)

(29)

Letting 119899 rarr infin in the above equality from (20) we find that

lim119899rarrinfin

119889 (119909119899 119906) = 0 and so lim

119899rarrinfin119889 (119909119899+1

119891119906) = 0 (30)

Again from (20) (26) and (30) we can obtain 119889(119906 119891119906) = 0

and hence 119906 is a periodic point of 119891

In what follows we give an example to illustrateTheorem 13

Example 15 In Example 5 define the mapping 119891 119883 rarr 119883 as

119891119909def=

5119905 if 119909 = 4119905 or 119905

119905 if 119909 = 4119905 or 119905(31)

First we can see easily that the classic Branciari contrac-tion [5] cannot be applied in this case since

119889 (1198913119905 119891119905) =5

4119904 gt 119904 = 119889 (3119905 119905) (32)

Now we define the mapping 120572 from 119883 times 119883 rarr [0infin) by

120572 (119909 119909) = 120572 (119905 4119905) = 120572 (2119905 3119905) = 120572 (2119905 5119905)

= 120572 (3119905 5119905) = 1 otherwise 120572 (119909 119910) = 0(33)

for all 119909 119910 isin 119883 For 120595(119903) = 1199032 where 119903 ge 0 we have

120572 (119909 119910) 119889 (119891119909 119891119910) le1

2119889 (119909 119910) (34)

Obviously 119891 is 120572-admissible and also for 1199090= 119905 we have

120572 (119905 119891119905) = 120572 (119891119905 119905) = 120572 (119905 1198912119905) = 120572 (119891

2119905 119905) = 120572 (119905 119905) = 1

(35)

Finally 119891 is continuous Therefore 119891 satisfies in Theorem 13and we can see that 119891 has two fixed points 119905 and 3119905

21 Existence Theorem Let (119883 119889) be a gqms and Per(119891119883)

denote the set of periodic points of 119891 119883 rarr 119883

Property (E) For all 119886 isin Per(119891119883) we have 120572(119891119901119886 119891119901+1119886) ge 1

and 120572(119891119901+1119886 119891119901119886) ge 1 for any 119901 ge 1

Theorem 16 Adding Property (E) to the hypotheses ofTheorem 13 (res Theorem 14) one obtains existence of a fixedpoint of 119891

Proof Suppose that 119886 is a periodic point of119891 that is119891119901119886 = 119886If 119901 = 1 then 119886 is a fixed point of 119891 that is 119891119901119886 = 119891119886 = 119886Assume that 119901 gt 1 We will show that 119911 = 119891119901minus1119886 is a fixedpoint of 119891

Suppose on the contrary that 119891119901minus1119886 = 119891119901119886 Then119889(119891119901minus1119886 119891119901119886) gt 0 119889(119891119901119886 119891119901minus1119886) gt 0120595(119889(119891119901minus1119886 119891119901119886)) gt 0and 120595(119889(119891119901119886 119891119901minus1119886)) gt 0

By Property (119864) we have 120572(119886 119891119901119886) ge 1 and 120572(119891119901119886 119886) ge

1 Due to (9) we can obtain

119889 (119886 119891119886) = 119889 (119891119901119886 119891119901+1

119886)

le 120572 (119891119901119886 119891119901+1

119886) 119889 (119891119901119886 119891119901+1

119886)

le 120595 (119889 (119891119901minus1

119886 119891119901119886))

(36)

Due to property 120595(119905) lt 119905 we get that

119889 (119886 119891119886) lt 119889 (119891119901minus1

119886 119891119901119886) (37)

Again by (9) we have

119889 (119891119901minus1

119886 119891119901119886) le 120572 (119891

119901minus1119886 119891119901119886) 119889 (119891

119901minus1119886 119891119901119886)

le 120595 (119889 (119891119901minus2

119886 119891119901minus1

119886))

le 120595119901(119889 (119886 119891119886))

lt 119889 (119886 119891119886)

(38)

Consequently we get the following contradiction119889(119886 119891119886) lt 119889(119886 119891119886) Hence the assumption that 119911 is not afixed point of 119891 is not true and thus 119911 = 119891

119901minus1119886 is a fixedpoint of 119891

To assure the uniqueness of the fixed point we willconsider the following properties

Property (U) For all 119909 119910 isin Fix(119891) we have 120572(119909 119910) ge 1

Theorem 17 Adding property (U) to the hypotheses ofTheorem 16 one obtains uniqueness of the fixed point of 119891

Proof Suppose that 119906 and V are two distinct fixed points of119891By property (119880) 120572(119879119906 119879V) = 120572(119906 V) ge 1

6 Journal of Function Spaces

Thus by 120572-admissibility of 119891 and the above relation wecan obtain

119889 (119906 V) le 120572 (119906 V) 119889 (119906 V)

= 120572 (119891119906 119891V) 119889 (119891119906 119891V)

le 120595119889 (119906 V) lt 119889 (119906 V)

(39)

which is a contradiction Therefore 119906 = V

3 Consequences

Definition 18 Let (119883 119889) be a gqms and let 119891 119883 rarr 119883 bea function satisfying

119889 (119891119909 119891119910) le 120595 (119889 (119909 119910)) (40)

for all 119909 119910 isin 119883Then119891 is said to be a120595-contractivemapping

Theorem 19 Let (119883 119889) be a complete 119892119902119898119904 and let 119891 be acontinuous 120595-contractive mapping Then 119891 is a unique point 119906in 119883

Proof It is sufficient to take 120572(119909 119910) = 1 for all 119909 119910 isin 119883

Theorem 20 Let (119883 119889) be a complete 119892119902119898119904 and let 119891 be acontinuous mapping Suppose that there exists 119896 isin [0 1) suchthat

119889 (119891119909 119891119910) le 119896119889 (119909 119910) forall119909 119910 isin 119883 (41)

Then 119891 is a unique point 119906 in 119883

Proof Take 120595(119905) = 119896119905 where 119896 isin [0 1) It is clear that allconditions of Theorem 19 are satisfied

4 Conclusion

It is evident that almost all fixed point theorems in thecontext of generalized metric spaces can be represented inthe setting of generalized quasimetric spaces Thus all fixedpoint results obtained in Section 2 infer the analog of thefixed point theorems in the context of generalized metricspaces Consequently several results in the literature (see eg[5 8 9 19 28]) can be derived from our main results

Conflict of Interests

The authors declare that they have no conflict of interests

Authorsrsquo Contribution

All authors contributed equally and significantly in writingthis paper All authors read and approved the final paper

Acknowledgment

The authors thank the anonymous referees for their remark-able comments suggestion and ideas that helped to improvethis paper

References

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

[2] M Jleli and B Samet ldquoRemarks on G-metric spaces and fixedpoint theoremsrdquo Fixed Point Theory and Applications vol 2012article 210 2012

[3] N Bilgili E Karapınar andB Samet ldquoGeneralized120572-120595 contrac-tive mappings in quasi-metric spaces and related fixed-pointtheoremsrdquo Journal of Inequalities and Applications vol 2014article 36 2014

[4] I Lin C Chen and E Karapınar ldquoPeriodic points of weakermeir-keeler contractive mappings on generalized quasimetricspacesrdquo Abstract and Applied Analysis vol 2014 Article ID490450 6 pages 2014

[5] A Branciari ldquoA fixed point theorem of Banach-Caccioppolitype on a class of generalized metric spacesrdquo PublicationesMathematicae Debrecen vol 57 no 1-2 pp 31ndash37 2000

[6] I R Sarma J M Rao and S S Rao ldquoContractions over gener-alized metric spacesrdquo Journal of Nonlinear Science and ItsApplications vol 2 no 3 pp 180ndash182 2009

[7] Z Kadeburg and S Radenovic ldquoOn generalized metric spacesa surveyrdquo TWMS Journal of Pure and Applied Math vol 5 no1 pp 3ndash13 2014

[8] A Azam and M Arshad ldquoKannan fixed point theorem ongeneralized metric spacesrdquo Journal of Nonlinear Sciences and ItsApplications vol 1 no 1 pp 45ndash48 2008

[9] P Das ldquoA fixed point theorem on a class of generalized metricspacesrdquo Korean Journal of Mathematical Sciences vol 9 pp 29ndash33 2002

[10] M Erhan E Karapınar and T Sekulic ldquoFixed points of (120595 120601)

contractions on rectangular metric spacesrdquo Fixed Point Theoryand Applications vol 2012 article 138 2012

[11] D Mihet ldquoOn Kannan fixed point principle in generalizedmetric spacesrdquo Journal of Nonlinear Science and its ApplicationsTJNSA vol 2 no 2 pp 92ndash96 2009

[12] B Samet ldquoA fixed point theorem in a generalized metric spaceformappings satisfying a contractive condition of integral typerdquoInternational Journal ofMathematical Analysis vol 3 no 25ndash28pp 1265ndash1271 2009

[13] B Samet ldquoDisscussion on a fixed point theorem of Banach-Caccioppli type on a class of generalized metric spacesrdquo Pub-licationes Mathematicae vol 76 no 4 pp 493ndash494 2010

[14] H Lakzian and B Samet ldquoFixed points for (120595 120593)-weaklycontractive mappings in generalized metric spacesrdquo AppliedMathematics Letters vol 25 no 5 pp 902ndash906 2012

[15] H Aydi E Karapınar and H Lakzian ldquoFixed point results on aclass of generalized metric spacesrdquo Mathematical Sciences vol6 article 46 2012

[16] T Suzuki ldquoGeneralized metric spaces do not have the compat-ible topologyrdquo Abstract and Applied Analysis vol 2014 ArticleID 458098 2014

[17] M Jleli and B Samet ldquoThe Kannanrsquos fixed point theorem in acone rectangularmetric spacerdquo Journal of Nonlinear Science andIts Applications vol 2 no 3 pp 161ndash167 2009

[18] W A Kirk and N Shahzad ldquoGeneralized metrics and Caristirsquostheoremrdquo Fixed Point Theory and Applications vol 2013 article129 2013

[19] E Karapınar ldquoDiscussion on 120572-120595 contractions on generalizedmetric spacesrdquo Abstract and Applied Analysis vol 2014 ArticleID 962784 7 pages 2014

Journal of Function Spaces 7

[20] H Aydi E Karapınar and B Samet ldquoFixed points for gener-alized (alphapsi)-contractions on generalized metric spacesrdquoJournal of Inequalities and Applications vol 2014 article 2292014

[21] I A Rus Generalized Contractions and Applications Cluj Uni-versity Press Cluj-Napoca Romania 2001

[22] R M Bianchini and M Grandolfi ldquoTrasformazioni di tipocontrattivo generalizzato in uno spazio metricordquo Atti dellaAccademia Nazionale dei Lincei Rendiconti Classe di ScienzeFisiche Matematiche e Naturali vol 45 pp 212ndash216 1968

[23] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572minusminus120595-contractive type mappingsrdquoNonlinear Analysis TheoryMethods amp Applications vol 75 no 4 pp 2154ndash2165 2012

[24] N Hussain E Karapınar P Salimi and P Vetro ldquoFixedpoint results for119866119898-Meir-Keeler contractive andG-(120572120595)-Meir-Keeler contractive mappingsrdquo Fixed Point Theory and Applica-tions vol 2013 article 34 2013

[25] N Hussain E Karapınar P Salimi and F Akbar ldquo120572-admissiblemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 article 114 11 pages 2013

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

[27] L B Ciric S M Alsulami P Salimi and P Vetro ldquoPPF depen-dent fixed point results for triangular 120572

119888-admissible mappingsrdquo

The Scientific World Journal vol 2014 Article ID 673647 10pages 2014

[28] M A Alghamdi C-M Chen and E Karapınar ldquoA generalizedweaker (alpha-phi-varphi)-contractive mappings and relatedfixed point results in complete generalized metric spacesrdquoAbstract and Applied Analysis vol 2014 Article ID 985080 10pages 2014

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

2 Journal of Function Spaces

(iii) 119889(119909 119910) le 119889(119909 119906) + 119889(119906 V) + 119889(V 119910) (quadrilateralinequality)

Then (119883 119889) is called a generalized metric space (or shortlygms)

We present an example to show that not every generalizedmetric on a set 119883 is a metric on 119883

Example 2 (see eg [4]) Let 119883 = 119905 2119905 3119905 4119905 5119905 with 119905 gt 0

be a constant and we define 119889 119883 times 119883 rarr [0infin) by

(1) 119889(119909 119909) = 0 for all 119909 isin 119883(2) 119889(119909 119910) = 119889(119910 119909) for all 119909 119910 isin 119883(3) 119889(119905 2119905) = 3119904(4) 119889(119905 3119905) = 119889(2119905 3119905) = 119904(5) 119889(119905 4119905) = 119889(2119905 4119905) = 119889(3119905 4119905) = 2119904(6) 119889(119905 5119905) = 119889(2119905 5119905) = 119889(3119905 5119905) = 119889(4119905 5119905) = (32)119904

where 119904 gt 0 is a constant Then (119883 119889) is a generalized metricspace but it is not a metric space because

119889 (119905 2119905) = 3119904 gt 119889 (119905 3119905) + 119889 (3119905 2119905) = 2119904 (1)

Despite the analogy between the definitions ofmetric andgeneralized metric their topological properties differ fromeach other For example for a generalizedmetric space (119883 119889)we have the following

(P1) a generalized metric need not be continuous(P2) a convergent sequence in generalized metric space

need not be Cauchy(P3) an open ball (119861

119903(119909) = 119910 isin 119883 119889(119909 119910) lt 119903 119903 gt 0)

need not be open set(P4) a generalized metric space need not be Hausdorff

and hence the uniqueness of limits cannot be guar-anteed

Example 3 (see [6] Example 11) Let 119883 = 119860 cup 119861 where 119860 =

0 2 and 119860 = 1119899 119899 isin N Define 119889 119883 times 119883 rarr [0infin) inthe following way

119889 (119909 119910) =

0 if 119909 = 119910

1 if 119909 = 119910 [119909 119910 sub 119860 or 119909 119910 sub 119861]

119910 if 119909 isin 119860 119910 isin 119861

(2)

Notice that 119889(119886 119887) = 119889(119887 119886) = 119887 whenever 119886 isin 119860 and 119887 isin 119861Furthermore (119883 119889) is a complete generalized metric spaceClearly we have (1198751)ndash(1198754) Indeed the sequence 1119899 119899 isin

N converges to both 0 and 2 There is no 119903 gt 0 such that119861119903(0) cap119861

119903(2) = 0 and hence it is not Hausdorff It is clear that

the ball11986123

(13) = 0 13 2 since there is no 119903 gt 0 such that119861119903(0) sub 119861

23(13) that is open balls may not be an open set

The function 119889 is not continuous since lim119899rarrinfin

119889(1119899 12) =

119889(0 12) although lim119899rarrinfin

1119899 = 0 Formore details see eg[6 7]

Regarding the weakness of the topology of generalizedmetric space mentioned above the authors add some addi-tional conditions to get the analog of existing fixed pointresults in the literature see eg [8ndash15] Very recently Suzuki[16] underlined the importance of generalized metric spaceby emphasizing that generalized metric space and metricspace have no compatible topology

The following is the definition of the notion of generalizedquasimetric space defined by Lin et al [4]

Definition 4 Let 119883 be a nonempty set and let 119889 119883 times 119883 rarr

[0infin) be a mapping such that for all 119909 119910 isin 119883 and for alldistinct points 119906 V isin 119883 each of them different from 119909 and 119910one has

(i) 119889(119909 119910) = 0 if and only if 119909 = 119910(ii) 119889(119909 119910) le 119889(119909 119906) + 119889(119906 V) + 119889(V 119910)

Then (119883 119889) is called a generalized quasimetric space (orshortly gqms)

It is evident that any generalized metric space is ageneralized quasimetric space but the converse is not truein general We give an example to show that not everygeneralized quasimetric on a set119883 is a generalized metric on119883

Example 5 (see [4]) Let 119883 = 119905 2119905 3119905 4119905 5119905 with 119905 gt 0 be aconstant and we define 119889 119883 times 119883 rarr [0infin) by

(1) 119889(119909 119909) = 0 for all 119909 isin 119883(2) 119889(119905 2119905) = 119889(2119905 119905) = 3119904(3) 119889(119905 3119905) = 119889(2119905 3119905) = 119889(3119905 119905) = 119889(3119905 2119905) = 119904(4) 119889(119905 4119905) = 119889(2119905 4119905) = 119889(3119905 4119905) = 119889(4119905 119905) = 119889(4119905 2119905) =

119889(4119905 3119905) = 2119904(5) 119889(119905 5119905) = 119889(2119905 5119905) = 119889(3119905 5119905) = 119889(4119905 5119905) = (32)119904(6) 119889(5119905 119905) = 119889(5119905 2119905) = 119889(5119905 3119905) = 119889(5119905 4119905) = (54)119904

where 119904 gt 0 is a constant Then (119883 119889) is a generalized quasi-metric space but it is not a generalized metric space because

119889 (119905 5119905) =3

2119904 = 119889 (5119905 119905) =

5

4119904 (3)

We next give the definitions of convergence and com-pleteness on generalized quasimetric spaces

Definition 6 (see [4]) Let (119883 119889) be a gqms let 119909119899 be

a sequence in 119883 and 119909 isin 119883 We say that 119909119899 is gqms

convergent to 119909 if and only if

lim119899rarrinfin

119889 (119909119899 119909) = lim

119899rarrinfin119889 (119909 119909

119899) = 0 (4)

Definition 7 (see [4]) Let (119883 119889) be a gqms and let 119909119899 be

a sequence in119883 We say that 119909119899 is left-Cauchy if and only if

for every 120576 gt 0 there exists 119896 isin N such that 119889(119909119899 119909119898) lt 120576 for

all 119899 ge 119898 gt 119896 We say that 119909119899 is right-Cauchy if and only if

for every 120576 gt 0 there exists 119896 isin N such that 119889(119909119899 119909119898) lt 120576 for

all 119898 ge 119899 gt 119896 We say that 119909119899 is Cauchy if and only if for

every 120576 gt 0 there exists 119896 isin N such that 119889(119909119899 119909119898) lt 120576 for all

119898 119899 gt 119896

Journal of Function Spaces 3

Remark 8 A sequence 119909119899 in a gqms is Cauchy if and only

if it is left-Cauchy and right-Cauchy

Definition 9 (see [4]) Let (119883 119889) be a gqms We say that

(1) (119883 119889) is left-complete if and only if each left-Cauchysequence in 119883 is convergent

(2) (119883 119889) is right-complete if and only if each right-Cauchy sequence in 119883 is convergent

(3) (119883 119889) is complete if and only if each Cauchy sequencein 119883 is convergent

Notice that in the literature in several reports for fixedpoint results in generalized metric space an additionalbut superfluous condition ldquoHausdorffnessrdquo was assumedRecently Jleli and Samet [17] Kirk and Shahzad [18] Kara-pınar [19] Kadeburg and Radenovic [7] and Aydi et al[20] reported new some fixed point results by removing theassumption of Hausdorffness in the context of generalizedmetric spaces The following crucial lemma is inspired from[7 17]

Lemma 10 Let (119883 119889) be a generalized quasimetric space andlet 119909119899 be a Cauchy sequence in119883 such that 119909

119898= 119909119899whenever

119898 = 119899 Then the sequence 119909119899 can converge to at most one

point

Proof Given 120576 gt 0 since 119909119899 is a Cauchy sequence there

exists 1198960isin N such that

119889 (119909119899 119909119898) lt 120576 forall119898 119899 gt 119896

0 (5)

We use themethod of Reductio ad absurdum Suppose on thecontrary that there exist two distinct points119909 and119910 in119883 suchthat the sequence 119909

119899 converges to 119909 and 119910 that is

lim119899rarrinfin

119889 (119909119899 119909) = lim

119899rarrinfin119889 (119909 119909

119899) = 0

lim119899rarrinfin

119889 (119909119899 119910) = lim

119899rarrinfin119889 (119910 119909

119899) = 0

(6)

By assumption for any 119899 isin N 119909119899

= 119909119898 and since 119909 = 119910 there

exists 1198961isin N such that 119909

119899= 119909 and 119909

119899= 119910 for any 119899 gt 119896

1ge 1198960

Due to quadrilateral inequality we have

119889 (119909 119910) le 119889 (119909 119909119899) + 119889 (119909

119899 119909119898) + 119889 (119909

119898 119910) (7)

Letting 119899119898 rarr infin we can obtain that 119889(119909 119910) = 0 by regard-ing (5) and (6) Hence we get 119909 = 119910which is a contradiction

2 Main Results

In this section we state and prove the main result of thispaper We start by introducing the following family of func-tions

Let Ψ be the family of functions 120595 [0infin) rarr [0infin)

satisfying the following conditions

(Ψ1) 120595 is nondecreasing

(Ψ2) sum+infin

119899=1120595119899(119905) lt infin for all 119905 gt 0 where120595119899 is the 119899th

iterate of 120595

These functions are known in the literature as (c)-comparisonfunctions It is easily proved that if 120595 is a (c)-comparisonfunction then 120595(119905) lt 119905 for any 119905 gt 0 For more details aboutsuch function we refer the reader to [21 22] In this study wediscuss the notion of 120572-admissible mappings see eg [23ndash27] The following definition was introduced in [23]

Definition 11 Let 119891 119883 rarr 119883 be a self-mapping of a set 119883and 120572 119883 times 119883 rarr R+ Then 119891 is called a 120572-admissible if

119909 119910 isin 119883 120572 (119909 119910) ge 1 997904rArr 120572 (119891119909 119891119910) ge 1 (8)

Inwhat followswe define the (120572minus120595)-contractivemappingin the setting of generalized quasimetric space

Definition 12 Let (119883 119889) be a gqms and let 119891 119883 rarr 119883

be a given mapping We say that 119891 is an (120572 minus 120595)-contractivemapping if there exist two functions 120572 119883 times 119883 rarr [0infin)

and 120595 isin Ψ such that

120572 (119909 119910) 119889 (119891119909 119891119910) le 120595 (119889 (119909 119910)) forall119909 119910 isin 119883 (9)

Now we state the following fixed point theorem

Theorem 13 Let (119883 119889) be a complete 119892119902119898119904 and let 119891

119883 rarr 119883 be an (120572 minus 120595)-contractive mapping Suppose that

(i) 119891 is 120572-admissible(ii) there exists 119909

0isin 119883 such that 120572(119909

0 1198911199090) ge 1

120572(1198911199090 1199090) ge 1 120572(119909

0 11989121199090) ge 1 and 120572(1198912119909

0 1199090) ge 1

(iii) 119891 is continuous

Then 119891 has a periodic point

Proof Due to statement (ii) of theorem there exists 1199090isin 119883

which is an arbitrary point such that 120572(1199090 1198911199090) ge 1 and

120572(1199090 1198911199090) ge 1 We will construct a sequence 119909

119899 in 119883 by

119909119899+1

= 119891119909119899= 119891119899+1119909

0for all 119899 ge 0 If we have 119909

1198990= 1199091198990+1

forsome 119899

0 then 119906 = 119909

1198990is a fixed point of 119891 Hence for the rest

of the proof we presume that

119909119899

= 119909119899+1

forall119899 (10)

Since 119891 is 120572-admissible we have

120572 (1199090 1199091) = 120572 (119909

0 1198911199090) ge 1

997904rArr 120572 (1198911199090 1198911199091) = 120572 (119909

1 1199092) ge 1

(11)

Utilizing the expression above we obtain that

120572 (119909119899 119909119899+1

) ge 1 forall119899 = 0 1 (12)

By repeating the same stepswith startingwith the assumption120572(1199091 1199090) = 120572(119891119909

0 1199090) ge 1 we conclude that

120572 (119909119899+1

119909119899) ge 1 forall119899 = 0 1 (13)

4 Journal of Function Spaces

In a similar way we derive that

120572 (1199090 1199092) = 120572 (119909

0 11989121199090) ge 1

997904rArr 120572 (1198911199090 1198911199092) = 120572 (119909

1 1199093) ge 1

(14)

Recursively we get that

120572 (119909119899 119909119899+2

) ge 1 forall119899 = 0 1 (15)

Analogously we can easily derive that

120572 (119909119899+2

119909119899) ge 1 forall119899 = 0 1 (16)

Step 1 We will show that lim119899rarrinfin

119889(119909119899 119909119899+1

) = 0 andlim119899rarrinfin

119889(119909119899 119909119899+2

) = 0 Regarding (8) and (12) we deducethat

119889 (119909119899 119909119899+1

) = 119889 (119891119909119899minus1

119891119909119899)

le 120572 (119909119899minus1

119909119899) 119889 (119891119909

119899minus1 119891119909119899)

le 120595 (119889 (119909119899minus1

119909119899))

(17)

for all 119899 ge 1Iteratively we find that

119889 (119909119899 119909119899+1

) le 120595119899(119889 (1199090 1199091)) forall119899 ge 1 (18)

Similarly

119889 (119909119899 119909119899+2

) le 120595119899(119889 (1199090 1199092)) forall119899 ge 1 (19)

By the properties of 120595 we can conclude thatlim119899rarrinfin

120595119899(119889(1199090 1199091)) = 0 that is

lim119899rarrinfin

119889 (119909119899 119909119899+1

) = 0 (20)

Similarly lim119899rarrinfin

120595119899(119889(1199090 1199092)) = 0 that is

lim119899rarrinfin

119889 (119909119899 119909119899+2

) = 0 (21)

Step 2 We will prove that 119909119899 is a right-Cauchy sequence

that islim119899rarrinfin

119889 (119909119899 119909119899+119896

) = 0 forall119896 isin N (22)

The cases 119896 = 1 and 119896 = 2 are proved respectively by (20)and (21) Now take 119896 ge 3 arbitrary It is sufficient to examinetwo cases

Case (I) Suppose that 119896 = 2119898+1 where119898 ge 1Then by usingStep 1 and the quadrilateral inequality together with (18) wefind119889 (119909119899 119909119899+119896

) = 119889 (119909119899 119909119899+2119898+1

)

le 119889 (119909119899 119909119899+1

) + 119889 (119909119899+1

119909119899+2

)

+ sdot sdot sdot + 119889 (119909119899+2119898

119909119899+2119898+1

)

le

119899+2119898

sum119901=119899

120595119901(119889 (1199090 1199091))

le

+infin

sum119901=119899

120595119901(119889 (1199090 1199091)) 997888rarr 0 as 119899 997888rarr infin

(23)

Case (II) Suppose that 119896 = 2119898 where 119898 ge 2 Again byapplying the quadrilateral inequality and Step 1 together with(18) and (19) we find

119889 (119909119899 119909119899+119896

) = 119889 (119909119899 119909119899+2119898

)

le 119889 (119909119899 119909119899+2

) + 119889 (119909119899+2

119909119899+3

)

+ sdot sdot sdot + 119889 (119909119899+2119898minus1

119909119899+2119898

)

le 119889 (119909119899 119909119899+2

) +

119899+2119898minus1

sum119901=119899+2

120595119901(119889 (1199090 1199091))

le 119889 (119909119899 119909119899+2

)

+

+infin

sum119901=119899

120595119901(119889 (1199090 1199091)) 997888rarr 0 as 119899 997888rarr infin

(24)

By combining the expressions (23) and (24) we have

lim119899rarrinfin

119889 (119909119899 119909119899+119896

) = 0 forall119896 ge 3 (25)

We conclude that 119909119899 is a right-Cauchy sequence in (119883 119889)

In the same way 119909119899 is a left-Cauchy sequence in (119883 119889)

So it is aCauchy sequence Since119883 is a complete gqms thereexists 119906 isin 119883 such that

lim119899rarrinfin

119889 (119909119899 119906) = lim

119899rarrinfin119889 (119906 119909

119899) = 0 (26)

Also we can easily see that 119909119899

= 119909119898for whenever 119898 = 119899

Indeed if 119909119899= 119909119898 for some 119898 119899 isin N with 119899 lt 119898 then

119889 (119909119899 119909119899+1

) = 119889 (119909119898 119909119898+1

)

le 120595119898minus119899

119889 (119909119899 119909119899+1

) lt 119889 (119909119899 119909119899+1

) (27)

which is a contradiction Analogously we derive the sameconclusion for the case 119899 gt 119898 Therefore we conclude thatthe sequence 119909

119899 cannot have two limits due to Lemma 10

Step 3 We claim that 119891 has a periodic point in 119883 Supposeon the contrary that 119891 has no periodic point Since 119891 iscontinuous from Step 2 we have 119906 = 119891119906 (119901 = 1) whichcontradicts the assumption that 119891 has no periodic pointTherefore there exists 119906 isin 119883 such that 119906 = 119891

119901(119906) for some119901 isin N So 119891 has a periodic point in 119883

Now we state the following fixed point theorem

Theorem 14 Let (119883 119889) be a complete 119892119902119898119904 and let 119891

119883 rarr 119883 be an (120572 minus 120595)-contractive mapping Suppose that

(i) 119891 is 120572-admissible(ii) there exists 119909

0isin 119883 such that 120572(119909

0 1198911199090) ge 1

120572(1198911199090 1199090) ge 1 120572(119909

0 11989121199090) ge 1 and 120572(1198912119909

0 1199090) ge 1

(iii) if 119909119899 is a sequence in 119883 such that 120572(119909

119899 119909119899+1

) ge 1 forall 119899 and 119909

119899rarr 119909 isin 119883 as 119899 rarr infin then 120572(119909

119899 119909) ge 1

for all 119899

Then 119891 has a periodic point

Journal of Function Spaces 5

Proof Following the proof of Theorem 13 we know that thesequence 119909

119899 defined by 119909

119899+1= 119891119909119899for all 119899 ge 0 converges

for some 119906 isin 119883 It is sufficient to show that 119891 admitsa periodic point Suppose on the contrary that 119891 has noperiodic point Notice that119909

119899= 119906 and119909

119899= 119891119906 for sufficiently

large 119899 By the quadrilateral inequality for this 119899 we have

119889 (119906 119891119906) le 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 119889 (119909119899+1

119891119906) (28)

On account of the fact that 120595(119905) lt 119905 for all 119905 gt 0 andregarding the assumption (iii) we get that

119889 (119906 119891119906) le 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 119889 (119909119899+1

119891119906)

le 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 119889 (119891119909119899 119891119906)

le 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 120572 (119909119899 119906) 119889 (119891119909

119899 119891119906)

le 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 120595 (119889 (119909119899 119906))

lt 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 119889 (119909119899 119906)

(29)

Letting 119899 rarr infin in the above equality from (20) we find that

lim119899rarrinfin

119889 (119909119899 119906) = 0 and so lim

119899rarrinfin119889 (119909119899+1

119891119906) = 0 (30)

Again from (20) (26) and (30) we can obtain 119889(119906 119891119906) = 0

and hence 119906 is a periodic point of 119891

In what follows we give an example to illustrateTheorem 13

Example 15 In Example 5 define the mapping 119891 119883 rarr 119883 as

119891119909def=

5119905 if 119909 = 4119905 or 119905

119905 if 119909 = 4119905 or 119905(31)

First we can see easily that the classic Branciari contrac-tion [5] cannot be applied in this case since

119889 (1198913119905 119891119905) =5

4119904 gt 119904 = 119889 (3119905 119905) (32)

Now we define the mapping 120572 from 119883 times 119883 rarr [0infin) by

120572 (119909 119909) = 120572 (119905 4119905) = 120572 (2119905 3119905) = 120572 (2119905 5119905)

= 120572 (3119905 5119905) = 1 otherwise 120572 (119909 119910) = 0(33)

for all 119909 119910 isin 119883 For 120595(119903) = 1199032 where 119903 ge 0 we have

120572 (119909 119910) 119889 (119891119909 119891119910) le1

2119889 (119909 119910) (34)

Obviously 119891 is 120572-admissible and also for 1199090= 119905 we have

120572 (119905 119891119905) = 120572 (119891119905 119905) = 120572 (119905 1198912119905) = 120572 (119891

2119905 119905) = 120572 (119905 119905) = 1

(35)

Finally 119891 is continuous Therefore 119891 satisfies in Theorem 13and we can see that 119891 has two fixed points 119905 and 3119905

21 Existence Theorem Let (119883 119889) be a gqms and Per(119891119883)

denote the set of periodic points of 119891 119883 rarr 119883

Property (E) For all 119886 isin Per(119891119883) we have 120572(119891119901119886 119891119901+1119886) ge 1

and 120572(119891119901+1119886 119891119901119886) ge 1 for any 119901 ge 1

Theorem 16 Adding Property (E) to the hypotheses ofTheorem 13 (res Theorem 14) one obtains existence of a fixedpoint of 119891

Proof Suppose that 119886 is a periodic point of119891 that is119891119901119886 = 119886If 119901 = 1 then 119886 is a fixed point of 119891 that is 119891119901119886 = 119891119886 = 119886Assume that 119901 gt 1 We will show that 119911 = 119891119901minus1119886 is a fixedpoint of 119891

Suppose on the contrary that 119891119901minus1119886 = 119891119901119886 Then119889(119891119901minus1119886 119891119901119886) gt 0 119889(119891119901119886 119891119901minus1119886) gt 0120595(119889(119891119901minus1119886 119891119901119886)) gt 0and 120595(119889(119891119901119886 119891119901minus1119886)) gt 0

By Property (119864) we have 120572(119886 119891119901119886) ge 1 and 120572(119891119901119886 119886) ge

1 Due to (9) we can obtain

119889 (119886 119891119886) = 119889 (119891119901119886 119891119901+1

119886)

le 120572 (119891119901119886 119891119901+1

119886) 119889 (119891119901119886 119891119901+1

119886)

le 120595 (119889 (119891119901minus1

119886 119891119901119886))

(36)

Due to property 120595(119905) lt 119905 we get that

119889 (119886 119891119886) lt 119889 (119891119901minus1

119886 119891119901119886) (37)

Again by (9) we have

119889 (119891119901minus1

119886 119891119901119886) le 120572 (119891

119901minus1119886 119891119901119886) 119889 (119891

119901minus1119886 119891119901119886)

le 120595 (119889 (119891119901minus2

119886 119891119901minus1

119886))

le 120595119901(119889 (119886 119891119886))

lt 119889 (119886 119891119886)

(38)

Consequently we get the following contradiction119889(119886 119891119886) lt 119889(119886 119891119886) Hence the assumption that 119911 is not afixed point of 119891 is not true and thus 119911 = 119891

119901minus1119886 is a fixedpoint of 119891

To assure the uniqueness of the fixed point we willconsider the following properties

Property (U) For all 119909 119910 isin Fix(119891) we have 120572(119909 119910) ge 1

Theorem 17 Adding property (U) to the hypotheses ofTheorem 16 one obtains uniqueness of the fixed point of 119891

Proof Suppose that 119906 and V are two distinct fixed points of119891By property (119880) 120572(119879119906 119879V) = 120572(119906 V) ge 1

6 Journal of Function Spaces

Thus by 120572-admissibility of 119891 and the above relation wecan obtain

119889 (119906 V) le 120572 (119906 V) 119889 (119906 V)

= 120572 (119891119906 119891V) 119889 (119891119906 119891V)

le 120595119889 (119906 V) lt 119889 (119906 V)

(39)

which is a contradiction Therefore 119906 = V

3 Consequences

Definition 18 Let (119883 119889) be a gqms and let 119891 119883 rarr 119883 bea function satisfying

119889 (119891119909 119891119910) le 120595 (119889 (119909 119910)) (40)

for all 119909 119910 isin 119883Then119891 is said to be a120595-contractivemapping

Theorem 19 Let (119883 119889) be a complete 119892119902119898119904 and let 119891 be acontinuous 120595-contractive mapping Then 119891 is a unique point 119906in 119883

Proof It is sufficient to take 120572(119909 119910) = 1 for all 119909 119910 isin 119883

Theorem 20 Let (119883 119889) be a complete 119892119902119898119904 and let 119891 be acontinuous mapping Suppose that there exists 119896 isin [0 1) suchthat

119889 (119891119909 119891119910) le 119896119889 (119909 119910) forall119909 119910 isin 119883 (41)

Then 119891 is a unique point 119906 in 119883

Proof Take 120595(119905) = 119896119905 where 119896 isin [0 1) It is clear that allconditions of Theorem 19 are satisfied

4 Conclusion

It is evident that almost all fixed point theorems in thecontext of generalized metric spaces can be represented inthe setting of generalized quasimetric spaces Thus all fixedpoint results obtained in Section 2 infer the analog of thefixed point theorems in the context of generalized metricspaces Consequently several results in the literature (see eg[5 8 9 19 28]) can be derived from our main results

Conflict of Interests

The authors declare that they have no conflict of interests

Authorsrsquo Contribution

All authors contributed equally and significantly in writingthis paper All authors read and approved the final paper

Acknowledgment

The authors thank the anonymous referees for their remark-able comments suggestion and ideas that helped to improvethis paper

References

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

[2] M Jleli and B Samet ldquoRemarks on G-metric spaces and fixedpoint theoremsrdquo Fixed Point Theory and Applications vol 2012article 210 2012

[3] N Bilgili E Karapınar andB Samet ldquoGeneralized120572-120595 contrac-tive mappings in quasi-metric spaces and related fixed-pointtheoremsrdquo Journal of Inequalities and Applications vol 2014article 36 2014

[4] I Lin C Chen and E Karapınar ldquoPeriodic points of weakermeir-keeler contractive mappings on generalized quasimetricspacesrdquo Abstract and Applied Analysis vol 2014 Article ID490450 6 pages 2014

[5] A Branciari ldquoA fixed point theorem of Banach-Caccioppolitype on a class of generalized metric spacesrdquo PublicationesMathematicae Debrecen vol 57 no 1-2 pp 31ndash37 2000

[6] I R Sarma J M Rao and S S Rao ldquoContractions over gener-alized metric spacesrdquo Journal of Nonlinear Science and ItsApplications vol 2 no 3 pp 180ndash182 2009

[7] Z Kadeburg and S Radenovic ldquoOn generalized metric spacesa surveyrdquo TWMS Journal of Pure and Applied Math vol 5 no1 pp 3ndash13 2014

[8] A Azam and M Arshad ldquoKannan fixed point theorem ongeneralized metric spacesrdquo Journal of Nonlinear Sciences and ItsApplications vol 1 no 1 pp 45ndash48 2008

[9] P Das ldquoA fixed point theorem on a class of generalized metricspacesrdquo Korean Journal of Mathematical Sciences vol 9 pp 29ndash33 2002

[10] M Erhan E Karapınar and T Sekulic ldquoFixed points of (120595 120601)

contractions on rectangular metric spacesrdquo Fixed Point Theoryand Applications vol 2012 article 138 2012

[11] D Mihet ldquoOn Kannan fixed point principle in generalizedmetric spacesrdquo Journal of Nonlinear Science and its ApplicationsTJNSA vol 2 no 2 pp 92ndash96 2009

[12] B Samet ldquoA fixed point theorem in a generalized metric spaceformappings satisfying a contractive condition of integral typerdquoInternational Journal ofMathematical Analysis vol 3 no 25ndash28pp 1265ndash1271 2009

[13] B Samet ldquoDisscussion on a fixed point theorem of Banach-Caccioppli type on a class of generalized metric spacesrdquo Pub-licationes Mathematicae vol 76 no 4 pp 493ndash494 2010

[14] H Lakzian and B Samet ldquoFixed points for (120595 120593)-weaklycontractive mappings in generalized metric spacesrdquo AppliedMathematics Letters vol 25 no 5 pp 902ndash906 2012

[15] H Aydi E Karapınar and H Lakzian ldquoFixed point results on aclass of generalized metric spacesrdquo Mathematical Sciences vol6 article 46 2012

[16] T Suzuki ldquoGeneralized metric spaces do not have the compat-ible topologyrdquo Abstract and Applied Analysis vol 2014 ArticleID 458098 2014

[17] M Jleli and B Samet ldquoThe Kannanrsquos fixed point theorem in acone rectangularmetric spacerdquo Journal of Nonlinear Science andIts Applications vol 2 no 3 pp 161ndash167 2009

[18] W A Kirk and N Shahzad ldquoGeneralized metrics and Caristirsquostheoremrdquo Fixed Point Theory and Applications vol 2013 article129 2013

[19] E Karapınar ldquoDiscussion on 120572-120595 contractions on generalizedmetric spacesrdquo Abstract and Applied Analysis vol 2014 ArticleID 962784 7 pages 2014

Journal of Function Spaces 7

[20] H Aydi E Karapınar and B Samet ldquoFixed points for gener-alized (alphapsi)-contractions on generalized metric spacesrdquoJournal of Inequalities and Applications vol 2014 article 2292014

[21] I A Rus Generalized Contractions and Applications Cluj Uni-versity Press Cluj-Napoca Romania 2001

[22] R M Bianchini and M Grandolfi ldquoTrasformazioni di tipocontrattivo generalizzato in uno spazio metricordquo Atti dellaAccademia Nazionale dei Lincei Rendiconti Classe di ScienzeFisiche Matematiche e Naturali vol 45 pp 212ndash216 1968

[23] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572minusminus120595-contractive type mappingsrdquoNonlinear Analysis TheoryMethods amp Applications vol 75 no 4 pp 2154ndash2165 2012

[24] N Hussain E Karapınar P Salimi and P Vetro ldquoFixedpoint results for119866119898-Meir-Keeler contractive andG-(120572120595)-Meir-Keeler contractive mappingsrdquo Fixed Point Theory and Applica-tions vol 2013 article 34 2013

[25] N Hussain E Karapınar P Salimi and F Akbar ldquo120572-admissiblemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 article 114 11 pages 2013

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

[27] L B Ciric S M Alsulami P Salimi and P Vetro ldquoPPF depen-dent fixed point results for triangular 120572

119888-admissible mappingsrdquo

The Scientific World Journal vol 2014 Article ID 673647 10pages 2014

[28] M A Alghamdi C-M Chen and E Karapınar ldquoA generalizedweaker (alpha-phi-varphi)-contractive mappings and relatedfixed point results in complete generalized metric spacesrdquoAbstract and Applied Analysis vol 2014 Article ID 985080 10pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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

Journal of Function Spaces 3

Remark 8 A sequence 119909119899 in a gqms is Cauchy if and only

if it is left-Cauchy and right-Cauchy

Definition 9 (see [4]) Let (119883 119889) be a gqms We say that

(1) (119883 119889) is left-complete if and only if each left-Cauchysequence in 119883 is convergent

(2) (119883 119889) is right-complete if and only if each right-Cauchy sequence in 119883 is convergent

(3) (119883 119889) is complete if and only if each Cauchy sequencein 119883 is convergent

Notice that in the literature in several reports for fixedpoint results in generalized metric space an additionalbut superfluous condition ldquoHausdorffnessrdquo was assumedRecently Jleli and Samet [17] Kirk and Shahzad [18] Kara-pınar [19] Kadeburg and Radenovic [7] and Aydi et al[20] reported new some fixed point results by removing theassumption of Hausdorffness in the context of generalizedmetric spaces The following crucial lemma is inspired from[7 17]

Lemma 10 Let (119883 119889) be a generalized quasimetric space andlet 119909119899 be a Cauchy sequence in119883 such that 119909

119898= 119909119899whenever

119898 = 119899 Then the sequence 119909119899 can converge to at most one

point

Proof Given 120576 gt 0 since 119909119899 is a Cauchy sequence there

exists 1198960isin N such that

119889 (119909119899 119909119898) lt 120576 forall119898 119899 gt 119896

0 (5)

We use themethod of Reductio ad absurdum Suppose on thecontrary that there exist two distinct points119909 and119910 in119883 suchthat the sequence 119909

119899 converges to 119909 and 119910 that is

lim119899rarrinfin

119889 (119909119899 119909) = lim

119899rarrinfin119889 (119909 119909

119899) = 0

lim119899rarrinfin

119889 (119909119899 119910) = lim

119899rarrinfin119889 (119910 119909

119899) = 0

(6)

By assumption for any 119899 isin N 119909119899

= 119909119898 and since 119909 = 119910 there

exists 1198961isin N such that 119909

119899= 119909 and 119909

119899= 119910 for any 119899 gt 119896

1ge 1198960

Due to quadrilateral inequality we have

119889 (119909 119910) le 119889 (119909 119909119899) + 119889 (119909

119899 119909119898) + 119889 (119909

119898 119910) (7)

Letting 119899119898 rarr infin we can obtain that 119889(119909 119910) = 0 by regard-ing (5) and (6) Hence we get 119909 = 119910which is a contradiction

2 Main Results

In this section we state and prove the main result of thispaper We start by introducing the following family of func-tions

Let Ψ be the family of functions 120595 [0infin) rarr [0infin)

satisfying the following conditions

(Ψ1) 120595 is nondecreasing

(Ψ2) sum+infin

119899=1120595119899(119905) lt infin for all 119905 gt 0 where120595119899 is the 119899th

iterate of 120595

These functions are known in the literature as (c)-comparisonfunctions It is easily proved that if 120595 is a (c)-comparisonfunction then 120595(119905) lt 119905 for any 119905 gt 0 For more details aboutsuch function we refer the reader to [21 22] In this study wediscuss the notion of 120572-admissible mappings see eg [23ndash27] The following definition was introduced in [23]

Definition 11 Let 119891 119883 rarr 119883 be a self-mapping of a set 119883and 120572 119883 times 119883 rarr R+ Then 119891 is called a 120572-admissible if

119909 119910 isin 119883 120572 (119909 119910) ge 1 997904rArr 120572 (119891119909 119891119910) ge 1 (8)

Inwhat followswe define the (120572minus120595)-contractivemappingin the setting of generalized quasimetric space

Definition 12 Let (119883 119889) be a gqms and let 119891 119883 rarr 119883

be a given mapping We say that 119891 is an (120572 minus 120595)-contractivemapping if there exist two functions 120572 119883 times 119883 rarr [0infin)

and 120595 isin Ψ such that

120572 (119909 119910) 119889 (119891119909 119891119910) le 120595 (119889 (119909 119910)) forall119909 119910 isin 119883 (9)

Now we state the following fixed point theorem

Theorem 13 Let (119883 119889) be a complete 119892119902119898119904 and let 119891

119883 rarr 119883 be an (120572 minus 120595)-contractive mapping Suppose that

(i) 119891 is 120572-admissible(ii) there exists 119909

0isin 119883 such that 120572(119909

0 1198911199090) ge 1

120572(1198911199090 1199090) ge 1 120572(119909

0 11989121199090) ge 1 and 120572(1198912119909

0 1199090) ge 1

(iii) 119891 is continuous

Then 119891 has a periodic point

Proof Due to statement (ii) of theorem there exists 1199090isin 119883

which is an arbitrary point such that 120572(1199090 1198911199090) ge 1 and

120572(1199090 1198911199090) ge 1 We will construct a sequence 119909

119899 in 119883 by

119909119899+1

= 119891119909119899= 119891119899+1119909

0for all 119899 ge 0 If we have 119909

1198990= 1199091198990+1

forsome 119899

0 then 119906 = 119909

1198990is a fixed point of 119891 Hence for the rest

of the proof we presume that

119909119899

= 119909119899+1

forall119899 (10)

Since 119891 is 120572-admissible we have

120572 (1199090 1199091) = 120572 (119909

0 1198911199090) ge 1

997904rArr 120572 (1198911199090 1198911199091) = 120572 (119909

1 1199092) ge 1

(11)

Utilizing the expression above we obtain that

120572 (119909119899 119909119899+1

) ge 1 forall119899 = 0 1 (12)

By repeating the same stepswith startingwith the assumption120572(1199091 1199090) = 120572(119891119909

0 1199090) ge 1 we conclude that

120572 (119909119899+1

119909119899) ge 1 forall119899 = 0 1 (13)

4 Journal of Function Spaces

In a similar way we derive that

120572 (1199090 1199092) = 120572 (119909

0 11989121199090) ge 1

997904rArr 120572 (1198911199090 1198911199092) = 120572 (119909

1 1199093) ge 1

(14)

Recursively we get that

120572 (119909119899 119909119899+2

) ge 1 forall119899 = 0 1 (15)

Analogously we can easily derive that

120572 (119909119899+2

119909119899) ge 1 forall119899 = 0 1 (16)

Step 1 We will show that lim119899rarrinfin

119889(119909119899 119909119899+1

) = 0 andlim119899rarrinfin

119889(119909119899 119909119899+2

) = 0 Regarding (8) and (12) we deducethat

119889 (119909119899 119909119899+1

) = 119889 (119891119909119899minus1

119891119909119899)

le 120572 (119909119899minus1

119909119899) 119889 (119891119909

119899minus1 119891119909119899)

le 120595 (119889 (119909119899minus1

119909119899))

(17)

for all 119899 ge 1Iteratively we find that

119889 (119909119899 119909119899+1

) le 120595119899(119889 (1199090 1199091)) forall119899 ge 1 (18)

Similarly

119889 (119909119899 119909119899+2

) le 120595119899(119889 (1199090 1199092)) forall119899 ge 1 (19)

By the properties of 120595 we can conclude thatlim119899rarrinfin

120595119899(119889(1199090 1199091)) = 0 that is

lim119899rarrinfin

119889 (119909119899 119909119899+1

) = 0 (20)

Similarly lim119899rarrinfin

120595119899(119889(1199090 1199092)) = 0 that is

lim119899rarrinfin

119889 (119909119899 119909119899+2

) = 0 (21)

Step 2 We will prove that 119909119899 is a right-Cauchy sequence

that islim119899rarrinfin

119889 (119909119899 119909119899+119896

) = 0 forall119896 isin N (22)

The cases 119896 = 1 and 119896 = 2 are proved respectively by (20)and (21) Now take 119896 ge 3 arbitrary It is sufficient to examinetwo cases

Case (I) Suppose that 119896 = 2119898+1 where119898 ge 1Then by usingStep 1 and the quadrilateral inequality together with (18) wefind119889 (119909119899 119909119899+119896

) = 119889 (119909119899 119909119899+2119898+1

)

le 119889 (119909119899 119909119899+1

) + 119889 (119909119899+1

119909119899+2

)

+ sdot sdot sdot + 119889 (119909119899+2119898

119909119899+2119898+1

)

le

119899+2119898

sum119901=119899

120595119901(119889 (1199090 1199091))

le

+infin

sum119901=119899

120595119901(119889 (1199090 1199091)) 997888rarr 0 as 119899 997888rarr infin

(23)

Case (II) Suppose that 119896 = 2119898 where 119898 ge 2 Again byapplying the quadrilateral inequality and Step 1 together with(18) and (19) we find

119889 (119909119899 119909119899+119896

) = 119889 (119909119899 119909119899+2119898

)

le 119889 (119909119899 119909119899+2

) + 119889 (119909119899+2

119909119899+3

)

+ sdot sdot sdot + 119889 (119909119899+2119898minus1

119909119899+2119898

)

le 119889 (119909119899 119909119899+2

) +

119899+2119898minus1

sum119901=119899+2

120595119901(119889 (1199090 1199091))

le 119889 (119909119899 119909119899+2

)

+

+infin

sum119901=119899

120595119901(119889 (1199090 1199091)) 997888rarr 0 as 119899 997888rarr infin

(24)

By combining the expressions (23) and (24) we have

lim119899rarrinfin

119889 (119909119899 119909119899+119896

) = 0 forall119896 ge 3 (25)

We conclude that 119909119899 is a right-Cauchy sequence in (119883 119889)

In the same way 119909119899 is a left-Cauchy sequence in (119883 119889)

So it is aCauchy sequence Since119883 is a complete gqms thereexists 119906 isin 119883 such that

lim119899rarrinfin

119889 (119909119899 119906) = lim

119899rarrinfin119889 (119906 119909

119899) = 0 (26)

Also we can easily see that 119909119899

= 119909119898for whenever 119898 = 119899

Indeed if 119909119899= 119909119898 for some 119898 119899 isin N with 119899 lt 119898 then

119889 (119909119899 119909119899+1

) = 119889 (119909119898 119909119898+1

)

le 120595119898minus119899

119889 (119909119899 119909119899+1

) lt 119889 (119909119899 119909119899+1

) (27)

which is a contradiction Analogously we derive the sameconclusion for the case 119899 gt 119898 Therefore we conclude thatthe sequence 119909

119899 cannot have two limits due to Lemma 10

Step 3 We claim that 119891 has a periodic point in 119883 Supposeon the contrary that 119891 has no periodic point Since 119891 iscontinuous from Step 2 we have 119906 = 119891119906 (119901 = 1) whichcontradicts the assumption that 119891 has no periodic pointTherefore there exists 119906 isin 119883 such that 119906 = 119891

119901(119906) for some119901 isin N So 119891 has a periodic point in 119883

Now we state the following fixed point theorem

Theorem 14 Let (119883 119889) be a complete 119892119902119898119904 and let 119891

119883 rarr 119883 be an (120572 minus 120595)-contractive mapping Suppose that

(i) 119891 is 120572-admissible(ii) there exists 119909

0isin 119883 such that 120572(119909

0 1198911199090) ge 1

120572(1198911199090 1199090) ge 1 120572(119909

0 11989121199090) ge 1 and 120572(1198912119909

0 1199090) ge 1

(iii) if 119909119899 is a sequence in 119883 such that 120572(119909

119899 119909119899+1

) ge 1 forall 119899 and 119909

119899rarr 119909 isin 119883 as 119899 rarr infin then 120572(119909

119899 119909) ge 1

for all 119899

Then 119891 has a periodic point

Journal of Function Spaces 5

Proof Following the proof of Theorem 13 we know that thesequence 119909

119899 defined by 119909

119899+1= 119891119909119899for all 119899 ge 0 converges

for some 119906 isin 119883 It is sufficient to show that 119891 admitsa periodic point Suppose on the contrary that 119891 has noperiodic point Notice that119909

119899= 119906 and119909

119899= 119891119906 for sufficiently

large 119899 By the quadrilateral inequality for this 119899 we have

119889 (119906 119891119906) le 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 119889 (119909119899+1

119891119906) (28)

On account of the fact that 120595(119905) lt 119905 for all 119905 gt 0 andregarding the assumption (iii) we get that

119889 (119906 119891119906) le 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 119889 (119909119899+1

119891119906)

le 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 119889 (119891119909119899 119891119906)

le 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 120572 (119909119899 119906) 119889 (119891119909

119899 119891119906)

le 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 120595 (119889 (119909119899 119906))

lt 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 119889 (119909119899 119906)

(29)

Letting 119899 rarr infin in the above equality from (20) we find that

lim119899rarrinfin

119889 (119909119899 119906) = 0 and so lim

119899rarrinfin119889 (119909119899+1

119891119906) = 0 (30)

Again from (20) (26) and (30) we can obtain 119889(119906 119891119906) = 0

and hence 119906 is a periodic point of 119891

In what follows we give an example to illustrateTheorem 13

Example 15 In Example 5 define the mapping 119891 119883 rarr 119883 as

119891119909def=

5119905 if 119909 = 4119905 or 119905

119905 if 119909 = 4119905 or 119905(31)

First we can see easily that the classic Branciari contrac-tion [5] cannot be applied in this case since

119889 (1198913119905 119891119905) =5

4119904 gt 119904 = 119889 (3119905 119905) (32)

Now we define the mapping 120572 from 119883 times 119883 rarr [0infin) by

120572 (119909 119909) = 120572 (119905 4119905) = 120572 (2119905 3119905) = 120572 (2119905 5119905)

= 120572 (3119905 5119905) = 1 otherwise 120572 (119909 119910) = 0(33)

for all 119909 119910 isin 119883 For 120595(119903) = 1199032 where 119903 ge 0 we have

120572 (119909 119910) 119889 (119891119909 119891119910) le1

2119889 (119909 119910) (34)

Obviously 119891 is 120572-admissible and also for 1199090= 119905 we have

120572 (119905 119891119905) = 120572 (119891119905 119905) = 120572 (119905 1198912119905) = 120572 (119891

2119905 119905) = 120572 (119905 119905) = 1

(35)

Finally 119891 is continuous Therefore 119891 satisfies in Theorem 13and we can see that 119891 has two fixed points 119905 and 3119905

21 Existence Theorem Let (119883 119889) be a gqms and Per(119891119883)

denote the set of periodic points of 119891 119883 rarr 119883

Property (E) For all 119886 isin Per(119891119883) we have 120572(119891119901119886 119891119901+1119886) ge 1

and 120572(119891119901+1119886 119891119901119886) ge 1 for any 119901 ge 1

Theorem 16 Adding Property (E) to the hypotheses ofTheorem 13 (res Theorem 14) one obtains existence of a fixedpoint of 119891

Proof Suppose that 119886 is a periodic point of119891 that is119891119901119886 = 119886If 119901 = 1 then 119886 is a fixed point of 119891 that is 119891119901119886 = 119891119886 = 119886Assume that 119901 gt 1 We will show that 119911 = 119891119901minus1119886 is a fixedpoint of 119891

Suppose on the contrary that 119891119901minus1119886 = 119891119901119886 Then119889(119891119901minus1119886 119891119901119886) gt 0 119889(119891119901119886 119891119901minus1119886) gt 0120595(119889(119891119901minus1119886 119891119901119886)) gt 0and 120595(119889(119891119901119886 119891119901minus1119886)) gt 0

By Property (119864) we have 120572(119886 119891119901119886) ge 1 and 120572(119891119901119886 119886) ge

1 Due to (9) we can obtain

119889 (119886 119891119886) = 119889 (119891119901119886 119891119901+1

119886)

le 120572 (119891119901119886 119891119901+1

119886) 119889 (119891119901119886 119891119901+1

119886)

le 120595 (119889 (119891119901minus1

119886 119891119901119886))

(36)

Due to property 120595(119905) lt 119905 we get that

119889 (119886 119891119886) lt 119889 (119891119901minus1

119886 119891119901119886) (37)

Again by (9) we have

119889 (119891119901minus1

119886 119891119901119886) le 120572 (119891

119901minus1119886 119891119901119886) 119889 (119891

119901minus1119886 119891119901119886)

le 120595 (119889 (119891119901minus2

119886 119891119901minus1

119886))

le 120595119901(119889 (119886 119891119886))

lt 119889 (119886 119891119886)

(38)

Consequently we get the following contradiction119889(119886 119891119886) lt 119889(119886 119891119886) Hence the assumption that 119911 is not afixed point of 119891 is not true and thus 119911 = 119891

119901minus1119886 is a fixedpoint of 119891

To assure the uniqueness of the fixed point we willconsider the following properties

Property (U) For all 119909 119910 isin Fix(119891) we have 120572(119909 119910) ge 1

Theorem 17 Adding property (U) to the hypotheses ofTheorem 16 one obtains uniqueness of the fixed point of 119891

Proof Suppose that 119906 and V are two distinct fixed points of119891By property (119880) 120572(119879119906 119879V) = 120572(119906 V) ge 1

6 Journal of Function Spaces

Thus by 120572-admissibility of 119891 and the above relation wecan obtain

119889 (119906 V) le 120572 (119906 V) 119889 (119906 V)

= 120572 (119891119906 119891V) 119889 (119891119906 119891V)

le 120595119889 (119906 V) lt 119889 (119906 V)

(39)

which is a contradiction Therefore 119906 = V

3 Consequences

Definition 18 Let (119883 119889) be a gqms and let 119891 119883 rarr 119883 bea function satisfying

119889 (119891119909 119891119910) le 120595 (119889 (119909 119910)) (40)

for all 119909 119910 isin 119883Then119891 is said to be a120595-contractivemapping

Theorem 19 Let (119883 119889) be a complete 119892119902119898119904 and let 119891 be acontinuous 120595-contractive mapping Then 119891 is a unique point 119906in 119883

Proof It is sufficient to take 120572(119909 119910) = 1 for all 119909 119910 isin 119883

Theorem 20 Let (119883 119889) be a complete 119892119902119898119904 and let 119891 be acontinuous mapping Suppose that there exists 119896 isin [0 1) suchthat

119889 (119891119909 119891119910) le 119896119889 (119909 119910) forall119909 119910 isin 119883 (41)

Then 119891 is a unique point 119906 in 119883

Proof Take 120595(119905) = 119896119905 where 119896 isin [0 1) It is clear that allconditions of Theorem 19 are satisfied

4 Conclusion

It is evident that almost all fixed point theorems in thecontext of generalized metric spaces can be represented inthe setting of generalized quasimetric spaces Thus all fixedpoint results obtained in Section 2 infer the analog of thefixed point theorems in the context of generalized metricspaces Consequently several results in the literature (see eg[5 8 9 19 28]) can be derived from our main results

Conflict of Interests

The authors declare that they have no conflict of interests

Authorsrsquo Contribution

All authors contributed equally and significantly in writingthis paper All authors read and approved the final paper

Acknowledgment

The authors thank the anonymous referees for their remark-able comments suggestion and ideas that helped to improvethis paper

References

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

[2] M Jleli and B Samet ldquoRemarks on G-metric spaces and fixedpoint theoremsrdquo Fixed Point Theory and Applications vol 2012article 210 2012

[3] N Bilgili E Karapınar andB Samet ldquoGeneralized120572-120595 contrac-tive mappings in quasi-metric spaces and related fixed-pointtheoremsrdquo Journal of Inequalities and Applications vol 2014article 36 2014

[4] I Lin C Chen and E Karapınar ldquoPeriodic points of weakermeir-keeler contractive mappings on generalized quasimetricspacesrdquo Abstract and Applied Analysis vol 2014 Article ID490450 6 pages 2014

[5] A Branciari ldquoA fixed point theorem of Banach-Caccioppolitype on a class of generalized metric spacesrdquo PublicationesMathematicae Debrecen vol 57 no 1-2 pp 31ndash37 2000

[6] I R Sarma J M Rao and S S Rao ldquoContractions over gener-alized metric spacesrdquo Journal of Nonlinear Science and ItsApplications vol 2 no 3 pp 180ndash182 2009

[7] Z Kadeburg and S Radenovic ldquoOn generalized metric spacesa surveyrdquo TWMS Journal of Pure and Applied Math vol 5 no1 pp 3ndash13 2014

[8] A Azam and M Arshad ldquoKannan fixed point theorem ongeneralized metric spacesrdquo Journal of Nonlinear Sciences and ItsApplications vol 1 no 1 pp 45ndash48 2008

[9] P Das ldquoA fixed point theorem on a class of generalized metricspacesrdquo Korean Journal of Mathematical Sciences vol 9 pp 29ndash33 2002

[10] M Erhan E Karapınar and T Sekulic ldquoFixed points of (120595 120601)

contractions on rectangular metric spacesrdquo Fixed Point Theoryand Applications vol 2012 article 138 2012

[11] D Mihet ldquoOn Kannan fixed point principle in generalizedmetric spacesrdquo Journal of Nonlinear Science and its ApplicationsTJNSA vol 2 no 2 pp 92ndash96 2009

[12] B Samet ldquoA fixed point theorem in a generalized metric spaceformappings satisfying a contractive condition of integral typerdquoInternational Journal ofMathematical Analysis vol 3 no 25ndash28pp 1265ndash1271 2009

[13] B Samet ldquoDisscussion on a fixed point theorem of Banach-Caccioppli type on a class of generalized metric spacesrdquo Pub-licationes Mathematicae vol 76 no 4 pp 493ndash494 2010

[14] H Lakzian and B Samet ldquoFixed points for (120595 120593)-weaklycontractive mappings in generalized metric spacesrdquo AppliedMathematics Letters vol 25 no 5 pp 902ndash906 2012

[15] H Aydi E Karapınar and H Lakzian ldquoFixed point results on aclass of generalized metric spacesrdquo Mathematical Sciences vol6 article 46 2012

[16] T Suzuki ldquoGeneralized metric spaces do not have the compat-ible topologyrdquo Abstract and Applied Analysis vol 2014 ArticleID 458098 2014

[17] M Jleli and B Samet ldquoThe Kannanrsquos fixed point theorem in acone rectangularmetric spacerdquo Journal of Nonlinear Science andIts Applications vol 2 no 3 pp 161ndash167 2009

[18] W A Kirk and N Shahzad ldquoGeneralized metrics and Caristirsquostheoremrdquo Fixed Point Theory and Applications vol 2013 article129 2013

[19] E Karapınar ldquoDiscussion on 120572-120595 contractions on generalizedmetric spacesrdquo Abstract and Applied Analysis vol 2014 ArticleID 962784 7 pages 2014

Journal of Function Spaces 7

[20] H Aydi E Karapınar and B Samet ldquoFixed points for gener-alized (alphapsi)-contractions on generalized metric spacesrdquoJournal of Inequalities and Applications vol 2014 article 2292014

[21] I A Rus Generalized Contractions and Applications Cluj Uni-versity Press Cluj-Napoca Romania 2001

[22] R M Bianchini and M Grandolfi ldquoTrasformazioni di tipocontrattivo generalizzato in uno spazio metricordquo Atti dellaAccademia Nazionale dei Lincei Rendiconti Classe di ScienzeFisiche Matematiche e Naturali vol 45 pp 212ndash216 1968

[23] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572minusminus120595-contractive type mappingsrdquoNonlinear Analysis TheoryMethods amp Applications vol 75 no 4 pp 2154ndash2165 2012

[24] N Hussain E Karapınar P Salimi and P Vetro ldquoFixedpoint results for119866119898-Meir-Keeler contractive andG-(120572120595)-Meir-Keeler contractive mappingsrdquo Fixed Point Theory and Applica-tions vol 2013 article 34 2013

[25] N Hussain E Karapınar P Salimi and F Akbar ldquo120572-admissiblemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 article 114 11 pages 2013

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

[27] L B Ciric S M Alsulami P Salimi and P Vetro ldquoPPF depen-dent fixed point results for triangular 120572

119888-admissible mappingsrdquo

The Scientific World Journal vol 2014 Article ID 673647 10pages 2014

[28] M A Alghamdi C-M Chen and E Karapınar ldquoA generalizedweaker (alpha-phi-varphi)-contractive mappings and relatedfixed point results in complete generalized metric spacesrdquoAbstract and Applied Analysis vol 2014 Article ID 985080 10pages 2014

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

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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

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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 Journal of Function Spaces

In a similar way we derive that

120572 (1199090 1199092) = 120572 (119909

0 11989121199090) ge 1

997904rArr 120572 (1198911199090 1198911199092) = 120572 (119909

1 1199093) ge 1

(14)

Recursively we get that

120572 (119909119899 119909119899+2

) ge 1 forall119899 = 0 1 (15)

Analogously we can easily derive that

120572 (119909119899+2

119909119899) ge 1 forall119899 = 0 1 (16)

Step 1 We will show that lim119899rarrinfin

119889(119909119899 119909119899+1

) = 0 andlim119899rarrinfin

119889(119909119899 119909119899+2

) = 0 Regarding (8) and (12) we deducethat

119889 (119909119899 119909119899+1

) = 119889 (119891119909119899minus1

119891119909119899)

le 120572 (119909119899minus1

119909119899) 119889 (119891119909

119899minus1 119891119909119899)

le 120595 (119889 (119909119899minus1

119909119899))

(17)

for all 119899 ge 1Iteratively we find that

119889 (119909119899 119909119899+1

) le 120595119899(119889 (1199090 1199091)) forall119899 ge 1 (18)

Similarly

119889 (119909119899 119909119899+2

) le 120595119899(119889 (1199090 1199092)) forall119899 ge 1 (19)

By the properties of 120595 we can conclude thatlim119899rarrinfin

120595119899(119889(1199090 1199091)) = 0 that is

lim119899rarrinfin

119889 (119909119899 119909119899+1

) = 0 (20)

Similarly lim119899rarrinfin

120595119899(119889(1199090 1199092)) = 0 that is

lim119899rarrinfin

119889 (119909119899 119909119899+2

) = 0 (21)

Step 2 We will prove that 119909119899 is a right-Cauchy sequence

that islim119899rarrinfin

119889 (119909119899 119909119899+119896

) = 0 forall119896 isin N (22)

The cases 119896 = 1 and 119896 = 2 are proved respectively by (20)and (21) Now take 119896 ge 3 arbitrary It is sufficient to examinetwo cases

Case (I) Suppose that 119896 = 2119898+1 where119898 ge 1Then by usingStep 1 and the quadrilateral inequality together with (18) wefind119889 (119909119899 119909119899+119896

) = 119889 (119909119899 119909119899+2119898+1

)

le 119889 (119909119899 119909119899+1

) + 119889 (119909119899+1

119909119899+2

)

+ sdot sdot sdot + 119889 (119909119899+2119898

119909119899+2119898+1

)

le

119899+2119898

sum119901=119899

120595119901(119889 (1199090 1199091))

le

+infin

sum119901=119899

120595119901(119889 (1199090 1199091)) 997888rarr 0 as 119899 997888rarr infin

(23)

Case (II) Suppose that 119896 = 2119898 where 119898 ge 2 Again byapplying the quadrilateral inequality and Step 1 together with(18) and (19) we find

119889 (119909119899 119909119899+119896

) = 119889 (119909119899 119909119899+2119898

)

le 119889 (119909119899 119909119899+2

) + 119889 (119909119899+2

119909119899+3

)

+ sdot sdot sdot + 119889 (119909119899+2119898minus1

119909119899+2119898

)

le 119889 (119909119899 119909119899+2

) +

119899+2119898minus1

sum119901=119899+2

120595119901(119889 (1199090 1199091))

le 119889 (119909119899 119909119899+2

)

+

+infin

sum119901=119899

120595119901(119889 (1199090 1199091)) 997888rarr 0 as 119899 997888rarr infin

(24)

By combining the expressions (23) and (24) we have

lim119899rarrinfin

119889 (119909119899 119909119899+119896

) = 0 forall119896 ge 3 (25)

We conclude that 119909119899 is a right-Cauchy sequence in (119883 119889)

In the same way 119909119899 is a left-Cauchy sequence in (119883 119889)

So it is aCauchy sequence Since119883 is a complete gqms thereexists 119906 isin 119883 such that

lim119899rarrinfin

119889 (119909119899 119906) = lim

119899rarrinfin119889 (119906 119909

119899) = 0 (26)

Also we can easily see that 119909119899

= 119909119898for whenever 119898 = 119899

Indeed if 119909119899= 119909119898 for some 119898 119899 isin N with 119899 lt 119898 then

119889 (119909119899 119909119899+1

) = 119889 (119909119898 119909119898+1

)

le 120595119898minus119899

119889 (119909119899 119909119899+1

) lt 119889 (119909119899 119909119899+1

) (27)

which is a contradiction Analogously we derive the sameconclusion for the case 119899 gt 119898 Therefore we conclude thatthe sequence 119909

119899 cannot have two limits due to Lemma 10

Step 3 We claim that 119891 has a periodic point in 119883 Supposeon the contrary that 119891 has no periodic point Since 119891 iscontinuous from Step 2 we have 119906 = 119891119906 (119901 = 1) whichcontradicts the assumption that 119891 has no periodic pointTherefore there exists 119906 isin 119883 such that 119906 = 119891

119901(119906) for some119901 isin N So 119891 has a periodic point in 119883

Now we state the following fixed point theorem

Theorem 14 Let (119883 119889) be a complete 119892119902119898119904 and let 119891

119883 rarr 119883 be an (120572 minus 120595)-contractive mapping Suppose that

(i) 119891 is 120572-admissible(ii) there exists 119909

0isin 119883 such that 120572(119909

0 1198911199090) ge 1

120572(1198911199090 1199090) ge 1 120572(119909

0 11989121199090) ge 1 and 120572(1198912119909

0 1199090) ge 1

(iii) if 119909119899 is a sequence in 119883 such that 120572(119909

119899 119909119899+1

) ge 1 forall 119899 and 119909

119899rarr 119909 isin 119883 as 119899 rarr infin then 120572(119909

119899 119909) ge 1

for all 119899

Then 119891 has a periodic point

Journal of Function Spaces 5

Proof Following the proof of Theorem 13 we know that thesequence 119909

119899 defined by 119909

119899+1= 119891119909119899for all 119899 ge 0 converges

for some 119906 isin 119883 It is sufficient to show that 119891 admitsa periodic point Suppose on the contrary that 119891 has noperiodic point Notice that119909

119899= 119906 and119909

119899= 119891119906 for sufficiently

large 119899 By the quadrilateral inequality for this 119899 we have

119889 (119906 119891119906) le 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 119889 (119909119899+1

119891119906) (28)

On account of the fact that 120595(119905) lt 119905 for all 119905 gt 0 andregarding the assumption (iii) we get that

119889 (119906 119891119906) le 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 119889 (119909119899+1

119891119906)

le 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 119889 (119891119909119899 119891119906)

le 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 120572 (119909119899 119906) 119889 (119891119909

119899 119891119906)

le 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 120595 (119889 (119909119899 119906))

lt 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 119889 (119909119899 119906)

(29)

Letting 119899 rarr infin in the above equality from (20) we find that

lim119899rarrinfin

119889 (119909119899 119906) = 0 and so lim

119899rarrinfin119889 (119909119899+1

119891119906) = 0 (30)

Again from (20) (26) and (30) we can obtain 119889(119906 119891119906) = 0

and hence 119906 is a periodic point of 119891

In what follows we give an example to illustrateTheorem 13

Example 15 In Example 5 define the mapping 119891 119883 rarr 119883 as

119891119909def=

5119905 if 119909 = 4119905 or 119905

119905 if 119909 = 4119905 or 119905(31)

First we can see easily that the classic Branciari contrac-tion [5] cannot be applied in this case since

119889 (1198913119905 119891119905) =5

4119904 gt 119904 = 119889 (3119905 119905) (32)

Now we define the mapping 120572 from 119883 times 119883 rarr [0infin) by

120572 (119909 119909) = 120572 (119905 4119905) = 120572 (2119905 3119905) = 120572 (2119905 5119905)

= 120572 (3119905 5119905) = 1 otherwise 120572 (119909 119910) = 0(33)

for all 119909 119910 isin 119883 For 120595(119903) = 1199032 where 119903 ge 0 we have

120572 (119909 119910) 119889 (119891119909 119891119910) le1

2119889 (119909 119910) (34)

Obviously 119891 is 120572-admissible and also for 1199090= 119905 we have

120572 (119905 119891119905) = 120572 (119891119905 119905) = 120572 (119905 1198912119905) = 120572 (119891

2119905 119905) = 120572 (119905 119905) = 1

(35)

Finally 119891 is continuous Therefore 119891 satisfies in Theorem 13and we can see that 119891 has two fixed points 119905 and 3119905

21 Existence Theorem Let (119883 119889) be a gqms and Per(119891119883)

denote the set of periodic points of 119891 119883 rarr 119883

Property (E) For all 119886 isin Per(119891119883) we have 120572(119891119901119886 119891119901+1119886) ge 1

and 120572(119891119901+1119886 119891119901119886) ge 1 for any 119901 ge 1

Theorem 16 Adding Property (E) to the hypotheses ofTheorem 13 (res Theorem 14) one obtains existence of a fixedpoint of 119891

Proof Suppose that 119886 is a periodic point of119891 that is119891119901119886 = 119886If 119901 = 1 then 119886 is a fixed point of 119891 that is 119891119901119886 = 119891119886 = 119886Assume that 119901 gt 1 We will show that 119911 = 119891119901minus1119886 is a fixedpoint of 119891

Suppose on the contrary that 119891119901minus1119886 = 119891119901119886 Then119889(119891119901minus1119886 119891119901119886) gt 0 119889(119891119901119886 119891119901minus1119886) gt 0120595(119889(119891119901minus1119886 119891119901119886)) gt 0and 120595(119889(119891119901119886 119891119901minus1119886)) gt 0

By Property (119864) we have 120572(119886 119891119901119886) ge 1 and 120572(119891119901119886 119886) ge

1 Due to (9) we can obtain

119889 (119886 119891119886) = 119889 (119891119901119886 119891119901+1

119886)

le 120572 (119891119901119886 119891119901+1

119886) 119889 (119891119901119886 119891119901+1

119886)

le 120595 (119889 (119891119901minus1

119886 119891119901119886))

(36)

Due to property 120595(119905) lt 119905 we get that

119889 (119886 119891119886) lt 119889 (119891119901minus1

119886 119891119901119886) (37)

Again by (9) we have

119889 (119891119901minus1

119886 119891119901119886) le 120572 (119891

119901minus1119886 119891119901119886) 119889 (119891

119901minus1119886 119891119901119886)

le 120595 (119889 (119891119901minus2

119886 119891119901minus1

119886))

le 120595119901(119889 (119886 119891119886))

lt 119889 (119886 119891119886)

(38)

Consequently we get the following contradiction119889(119886 119891119886) lt 119889(119886 119891119886) Hence the assumption that 119911 is not afixed point of 119891 is not true and thus 119911 = 119891

119901minus1119886 is a fixedpoint of 119891

To assure the uniqueness of the fixed point we willconsider the following properties

Property (U) For all 119909 119910 isin Fix(119891) we have 120572(119909 119910) ge 1

Theorem 17 Adding property (U) to the hypotheses ofTheorem 16 one obtains uniqueness of the fixed point of 119891

Proof Suppose that 119906 and V are two distinct fixed points of119891By property (119880) 120572(119879119906 119879V) = 120572(119906 V) ge 1

6 Journal of Function Spaces

Thus by 120572-admissibility of 119891 and the above relation wecan obtain

119889 (119906 V) le 120572 (119906 V) 119889 (119906 V)

= 120572 (119891119906 119891V) 119889 (119891119906 119891V)

le 120595119889 (119906 V) lt 119889 (119906 V)

(39)

which is a contradiction Therefore 119906 = V

3 Consequences

Definition 18 Let (119883 119889) be a gqms and let 119891 119883 rarr 119883 bea function satisfying

119889 (119891119909 119891119910) le 120595 (119889 (119909 119910)) (40)

for all 119909 119910 isin 119883Then119891 is said to be a120595-contractivemapping

Theorem 19 Let (119883 119889) be a complete 119892119902119898119904 and let 119891 be acontinuous 120595-contractive mapping Then 119891 is a unique point 119906in 119883

Proof It is sufficient to take 120572(119909 119910) = 1 for all 119909 119910 isin 119883

Theorem 20 Let (119883 119889) be a complete 119892119902119898119904 and let 119891 be acontinuous mapping Suppose that there exists 119896 isin [0 1) suchthat

119889 (119891119909 119891119910) le 119896119889 (119909 119910) forall119909 119910 isin 119883 (41)

Then 119891 is a unique point 119906 in 119883

Proof Take 120595(119905) = 119896119905 where 119896 isin [0 1) It is clear that allconditions of Theorem 19 are satisfied

4 Conclusion

It is evident that almost all fixed point theorems in thecontext of generalized metric spaces can be represented inthe setting of generalized quasimetric spaces Thus all fixedpoint results obtained in Section 2 infer the analog of thefixed point theorems in the context of generalized metricspaces Consequently several results in the literature (see eg[5 8 9 19 28]) can be derived from our main results

Conflict of Interests

The authors declare that they have no conflict of interests

Authorsrsquo Contribution

All authors contributed equally and significantly in writingthis paper All authors read and approved the final paper

Acknowledgment

The authors thank the anonymous referees for their remark-able comments suggestion and ideas that helped to improvethis paper

References

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

[2] M Jleli and B Samet ldquoRemarks on G-metric spaces and fixedpoint theoremsrdquo Fixed Point Theory and Applications vol 2012article 210 2012

[3] N Bilgili E Karapınar andB Samet ldquoGeneralized120572-120595 contrac-tive mappings in quasi-metric spaces and related fixed-pointtheoremsrdquo Journal of Inequalities and Applications vol 2014article 36 2014

[4] I Lin C Chen and E Karapınar ldquoPeriodic points of weakermeir-keeler contractive mappings on generalized quasimetricspacesrdquo Abstract and Applied Analysis vol 2014 Article ID490450 6 pages 2014

[5] A Branciari ldquoA fixed point theorem of Banach-Caccioppolitype on a class of generalized metric spacesrdquo PublicationesMathematicae Debrecen vol 57 no 1-2 pp 31ndash37 2000

[6] I R Sarma J M Rao and S S Rao ldquoContractions over gener-alized metric spacesrdquo Journal of Nonlinear Science and ItsApplications vol 2 no 3 pp 180ndash182 2009

[7] Z Kadeburg and S Radenovic ldquoOn generalized metric spacesa surveyrdquo TWMS Journal of Pure and Applied Math vol 5 no1 pp 3ndash13 2014

[8] A Azam and M Arshad ldquoKannan fixed point theorem ongeneralized metric spacesrdquo Journal of Nonlinear Sciences and ItsApplications vol 1 no 1 pp 45ndash48 2008

[9] P Das ldquoA fixed point theorem on a class of generalized metricspacesrdquo Korean Journal of Mathematical Sciences vol 9 pp 29ndash33 2002

[10] M Erhan E Karapınar and T Sekulic ldquoFixed points of (120595 120601)

contractions on rectangular metric spacesrdquo Fixed Point Theoryand Applications vol 2012 article 138 2012

[11] D Mihet ldquoOn Kannan fixed point principle in generalizedmetric spacesrdquo Journal of Nonlinear Science and its ApplicationsTJNSA vol 2 no 2 pp 92ndash96 2009

[12] B Samet ldquoA fixed point theorem in a generalized metric spaceformappings satisfying a contractive condition of integral typerdquoInternational Journal ofMathematical Analysis vol 3 no 25ndash28pp 1265ndash1271 2009

[13] B Samet ldquoDisscussion on a fixed point theorem of Banach-Caccioppli type on a class of generalized metric spacesrdquo Pub-licationes Mathematicae vol 76 no 4 pp 493ndash494 2010

[14] H Lakzian and B Samet ldquoFixed points for (120595 120593)-weaklycontractive mappings in generalized metric spacesrdquo AppliedMathematics Letters vol 25 no 5 pp 902ndash906 2012

[15] H Aydi E Karapınar and H Lakzian ldquoFixed point results on aclass of generalized metric spacesrdquo Mathematical Sciences vol6 article 46 2012

[16] T Suzuki ldquoGeneralized metric spaces do not have the compat-ible topologyrdquo Abstract and Applied Analysis vol 2014 ArticleID 458098 2014

[17] M Jleli and B Samet ldquoThe Kannanrsquos fixed point theorem in acone rectangularmetric spacerdquo Journal of Nonlinear Science andIts Applications vol 2 no 3 pp 161ndash167 2009

[18] W A Kirk and N Shahzad ldquoGeneralized metrics and Caristirsquostheoremrdquo Fixed Point Theory and Applications vol 2013 article129 2013

[19] E Karapınar ldquoDiscussion on 120572-120595 contractions on generalizedmetric spacesrdquo Abstract and Applied Analysis vol 2014 ArticleID 962784 7 pages 2014

Journal of Function Spaces 7

[20] H Aydi E Karapınar and B Samet ldquoFixed points for gener-alized (alphapsi)-contractions on generalized metric spacesrdquoJournal of Inequalities and Applications vol 2014 article 2292014

[21] I A Rus Generalized Contractions and Applications Cluj Uni-versity Press Cluj-Napoca Romania 2001

[22] R M Bianchini and M Grandolfi ldquoTrasformazioni di tipocontrattivo generalizzato in uno spazio metricordquo Atti dellaAccademia Nazionale dei Lincei Rendiconti Classe di ScienzeFisiche Matematiche e Naturali vol 45 pp 212ndash216 1968

[23] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572minusminus120595-contractive type mappingsrdquoNonlinear Analysis TheoryMethods amp Applications vol 75 no 4 pp 2154ndash2165 2012

[24] N Hussain E Karapınar P Salimi and P Vetro ldquoFixedpoint results for119866119898-Meir-Keeler contractive andG-(120572120595)-Meir-Keeler contractive mappingsrdquo Fixed Point Theory and Applica-tions vol 2013 article 34 2013

[25] N Hussain E Karapınar P Salimi and F Akbar ldquo120572-admissiblemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 article 114 11 pages 2013

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

[27] L B Ciric S M Alsulami P Salimi and P Vetro ldquoPPF depen-dent fixed point results for triangular 120572

119888-admissible mappingsrdquo

The Scientific World Journal vol 2014 Article ID 673647 10pages 2014

[28] M A Alghamdi C-M Chen and E Karapınar ldquoA generalizedweaker (alpha-phi-varphi)-contractive mappings and relatedfixed point results in complete generalized metric spacesrdquoAbstract and Applied Analysis vol 2014 Article ID 985080 10pages 2014

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

Journal of Function Spaces 5

Proof Following the proof of Theorem 13 we know that thesequence 119909

119899 defined by 119909

119899+1= 119891119909119899for all 119899 ge 0 converges

for some 119906 isin 119883 It is sufficient to show that 119891 admitsa periodic point Suppose on the contrary that 119891 has noperiodic point Notice that119909

119899= 119906 and119909

119899= 119891119906 for sufficiently

large 119899 By the quadrilateral inequality for this 119899 we have

119889 (119906 119891119906) le 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 119889 (119909119899+1

119891119906) (28)

On account of the fact that 120595(119905) lt 119905 for all 119905 gt 0 andregarding the assumption (iii) we get that

119889 (119906 119891119906) le 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 119889 (119909119899+1

119891119906)

le 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 119889 (119891119909119899 119891119906)

le 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 120572 (119909119899 119906) 119889 (119891119909

119899 119891119906)

le 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 120595 (119889 (119909119899 119906))

lt 119889 (119906 119909119899) + 119889 (119909

119899 119909119899+1

) + 119889 (119909119899 119906)

(29)

Letting 119899 rarr infin in the above equality from (20) we find that

lim119899rarrinfin

119889 (119909119899 119906) = 0 and so lim

119899rarrinfin119889 (119909119899+1

119891119906) = 0 (30)

Again from (20) (26) and (30) we can obtain 119889(119906 119891119906) = 0

and hence 119906 is a periodic point of 119891

In what follows we give an example to illustrateTheorem 13

Example 15 In Example 5 define the mapping 119891 119883 rarr 119883 as

119891119909def=

5119905 if 119909 = 4119905 or 119905

119905 if 119909 = 4119905 or 119905(31)

First we can see easily that the classic Branciari contrac-tion [5] cannot be applied in this case since

119889 (1198913119905 119891119905) =5

4119904 gt 119904 = 119889 (3119905 119905) (32)

Now we define the mapping 120572 from 119883 times 119883 rarr [0infin) by

120572 (119909 119909) = 120572 (119905 4119905) = 120572 (2119905 3119905) = 120572 (2119905 5119905)

= 120572 (3119905 5119905) = 1 otherwise 120572 (119909 119910) = 0(33)

for all 119909 119910 isin 119883 For 120595(119903) = 1199032 where 119903 ge 0 we have

120572 (119909 119910) 119889 (119891119909 119891119910) le1

2119889 (119909 119910) (34)

Obviously 119891 is 120572-admissible and also for 1199090= 119905 we have

120572 (119905 119891119905) = 120572 (119891119905 119905) = 120572 (119905 1198912119905) = 120572 (119891

2119905 119905) = 120572 (119905 119905) = 1

(35)

Finally 119891 is continuous Therefore 119891 satisfies in Theorem 13and we can see that 119891 has two fixed points 119905 and 3119905

21 Existence Theorem Let (119883 119889) be a gqms and Per(119891119883)

denote the set of periodic points of 119891 119883 rarr 119883

Property (E) For all 119886 isin Per(119891119883) we have 120572(119891119901119886 119891119901+1119886) ge 1

and 120572(119891119901+1119886 119891119901119886) ge 1 for any 119901 ge 1

Theorem 16 Adding Property (E) to the hypotheses ofTheorem 13 (res Theorem 14) one obtains existence of a fixedpoint of 119891

Proof Suppose that 119886 is a periodic point of119891 that is119891119901119886 = 119886If 119901 = 1 then 119886 is a fixed point of 119891 that is 119891119901119886 = 119891119886 = 119886Assume that 119901 gt 1 We will show that 119911 = 119891119901minus1119886 is a fixedpoint of 119891

Suppose on the contrary that 119891119901minus1119886 = 119891119901119886 Then119889(119891119901minus1119886 119891119901119886) gt 0 119889(119891119901119886 119891119901minus1119886) gt 0120595(119889(119891119901minus1119886 119891119901119886)) gt 0and 120595(119889(119891119901119886 119891119901minus1119886)) gt 0

By Property (119864) we have 120572(119886 119891119901119886) ge 1 and 120572(119891119901119886 119886) ge

1 Due to (9) we can obtain

119889 (119886 119891119886) = 119889 (119891119901119886 119891119901+1

119886)

le 120572 (119891119901119886 119891119901+1

119886) 119889 (119891119901119886 119891119901+1

119886)

le 120595 (119889 (119891119901minus1

119886 119891119901119886))

(36)

Due to property 120595(119905) lt 119905 we get that

119889 (119886 119891119886) lt 119889 (119891119901minus1

119886 119891119901119886) (37)

Again by (9) we have

119889 (119891119901minus1

119886 119891119901119886) le 120572 (119891

119901minus1119886 119891119901119886) 119889 (119891

119901minus1119886 119891119901119886)

le 120595 (119889 (119891119901minus2

119886 119891119901minus1

119886))

le 120595119901(119889 (119886 119891119886))

lt 119889 (119886 119891119886)

(38)

Consequently we get the following contradiction119889(119886 119891119886) lt 119889(119886 119891119886) Hence the assumption that 119911 is not afixed point of 119891 is not true and thus 119911 = 119891

119901minus1119886 is a fixedpoint of 119891

To assure the uniqueness of the fixed point we willconsider the following properties

Property (U) For all 119909 119910 isin Fix(119891) we have 120572(119909 119910) ge 1

Theorem 17 Adding property (U) to the hypotheses ofTheorem 16 one obtains uniqueness of the fixed point of 119891

Proof Suppose that 119906 and V are two distinct fixed points of119891By property (119880) 120572(119879119906 119879V) = 120572(119906 V) ge 1

6 Journal of Function Spaces

Thus by 120572-admissibility of 119891 and the above relation wecan obtain

119889 (119906 V) le 120572 (119906 V) 119889 (119906 V)

= 120572 (119891119906 119891V) 119889 (119891119906 119891V)

le 120595119889 (119906 V) lt 119889 (119906 V)

(39)

which is a contradiction Therefore 119906 = V

3 Consequences

Definition 18 Let (119883 119889) be a gqms and let 119891 119883 rarr 119883 bea function satisfying

119889 (119891119909 119891119910) le 120595 (119889 (119909 119910)) (40)

for all 119909 119910 isin 119883Then119891 is said to be a120595-contractivemapping

Theorem 19 Let (119883 119889) be a complete 119892119902119898119904 and let 119891 be acontinuous 120595-contractive mapping Then 119891 is a unique point 119906in 119883

Proof It is sufficient to take 120572(119909 119910) = 1 for all 119909 119910 isin 119883

Theorem 20 Let (119883 119889) be a complete 119892119902119898119904 and let 119891 be acontinuous mapping Suppose that there exists 119896 isin [0 1) suchthat

119889 (119891119909 119891119910) le 119896119889 (119909 119910) forall119909 119910 isin 119883 (41)

Then 119891 is a unique point 119906 in 119883

Proof Take 120595(119905) = 119896119905 where 119896 isin [0 1) It is clear that allconditions of Theorem 19 are satisfied

4 Conclusion

It is evident that almost all fixed point theorems in thecontext of generalized metric spaces can be represented inthe setting of generalized quasimetric spaces Thus all fixedpoint results obtained in Section 2 infer the analog of thefixed point theorems in the context of generalized metricspaces Consequently several results in the literature (see eg[5 8 9 19 28]) can be derived from our main results

Conflict of Interests

The authors declare that they have no conflict of interests

Authorsrsquo Contribution

All authors contributed equally and significantly in writingthis paper All authors read and approved the final paper

Acknowledgment

The authors thank the anonymous referees for their remark-able comments suggestion and ideas that helped to improvethis paper

References

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

[2] M Jleli and B Samet ldquoRemarks on G-metric spaces and fixedpoint theoremsrdquo Fixed Point Theory and Applications vol 2012article 210 2012

[3] N Bilgili E Karapınar andB Samet ldquoGeneralized120572-120595 contrac-tive mappings in quasi-metric spaces and related fixed-pointtheoremsrdquo Journal of Inequalities and Applications vol 2014article 36 2014

[4] I Lin C Chen and E Karapınar ldquoPeriodic points of weakermeir-keeler contractive mappings on generalized quasimetricspacesrdquo Abstract and Applied Analysis vol 2014 Article ID490450 6 pages 2014

[5] A Branciari ldquoA fixed point theorem of Banach-Caccioppolitype on a class of generalized metric spacesrdquo PublicationesMathematicae Debrecen vol 57 no 1-2 pp 31ndash37 2000

[6] I R Sarma J M Rao and S S Rao ldquoContractions over gener-alized metric spacesrdquo Journal of Nonlinear Science and ItsApplications vol 2 no 3 pp 180ndash182 2009

[7] Z Kadeburg and S Radenovic ldquoOn generalized metric spacesa surveyrdquo TWMS Journal of Pure and Applied Math vol 5 no1 pp 3ndash13 2014

[8] A Azam and M Arshad ldquoKannan fixed point theorem ongeneralized metric spacesrdquo Journal of Nonlinear Sciences and ItsApplications vol 1 no 1 pp 45ndash48 2008

[9] P Das ldquoA fixed point theorem on a class of generalized metricspacesrdquo Korean Journal of Mathematical Sciences vol 9 pp 29ndash33 2002

[10] M Erhan E Karapınar and T Sekulic ldquoFixed points of (120595 120601)

contractions on rectangular metric spacesrdquo Fixed Point Theoryand Applications vol 2012 article 138 2012

[11] D Mihet ldquoOn Kannan fixed point principle in generalizedmetric spacesrdquo Journal of Nonlinear Science and its ApplicationsTJNSA vol 2 no 2 pp 92ndash96 2009

[12] B Samet ldquoA fixed point theorem in a generalized metric spaceformappings satisfying a contractive condition of integral typerdquoInternational Journal ofMathematical Analysis vol 3 no 25ndash28pp 1265ndash1271 2009

[13] B Samet ldquoDisscussion on a fixed point theorem of Banach-Caccioppli type on a class of generalized metric spacesrdquo Pub-licationes Mathematicae vol 76 no 4 pp 493ndash494 2010

[14] H Lakzian and B Samet ldquoFixed points for (120595 120593)-weaklycontractive mappings in generalized metric spacesrdquo AppliedMathematics Letters vol 25 no 5 pp 902ndash906 2012

[15] H Aydi E Karapınar and H Lakzian ldquoFixed point results on aclass of generalized metric spacesrdquo Mathematical Sciences vol6 article 46 2012

[16] T Suzuki ldquoGeneralized metric spaces do not have the compat-ible topologyrdquo Abstract and Applied Analysis vol 2014 ArticleID 458098 2014

[17] M Jleli and B Samet ldquoThe Kannanrsquos fixed point theorem in acone rectangularmetric spacerdquo Journal of Nonlinear Science andIts Applications vol 2 no 3 pp 161ndash167 2009

[18] W A Kirk and N Shahzad ldquoGeneralized metrics and Caristirsquostheoremrdquo Fixed Point Theory and Applications vol 2013 article129 2013

[19] E Karapınar ldquoDiscussion on 120572-120595 contractions on generalizedmetric spacesrdquo Abstract and Applied Analysis vol 2014 ArticleID 962784 7 pages 2014

Journal of Function Spaces 7

[20] H Aydi E Karapınar and B Samet ldquoFixed points for gener-alized (alphapsi)-contractions on generalized metric spacesrdquoJournal of Inequalities and Applications vol 2014 article 2292014

[21] I A Rus Generalized Contractions and Applications Cluj Uni-versity Press Cluj-Napoca Romania 2001

[22] R M Bianchini and M Grandolfi ldquoTrasformazioni di tipocontrattivo generalizzato in uno spazio metricordquo Atti dellaAccademia Nazionale dei Lincei Rendiconti Classe di ScienzeFisiche Matematiche e Naturali vol 45 pp 212ndash216 1968

[23] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572minusminus120595-contractive type mappingsrdquoNonlinear Analysis TheoryMethods amp Applications vol 75 no 4 pp 2154ndash2165 2012

[24] N Hussain E Karapınar P Salimi and P Vetro ldquoFixedpoint results for119866119898-Meir-Keeler contractive andG-(120572120595)-Meir-Keeler contractive mappingsrdquo Fixed Point Theory and Applica-tions vol 2013 article 34 2013

[25] N Hussain E Karapınar P Salimi and F Akbar ldquo120572-admissiblemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 article 114 11 pages 2013

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

[27] L B Ciric S M Alsulami P Salimi and P Vetro ldquoPPF depen-dent fixed point results for triangular 120572

119888-admissible mappingsrdquo

The Scientific World Journal vol 2014 Article ID 673647 10pages 2014

[28] M A Alghamdi C-M Chen and E Karapınar ldquoA generalizedweaker (alpha-phi-varphi)-contractive mappings and relatedfixed point results in complete generalized metric spacesrdquoAbstract and Applied Analysis vol 2014 Article ID 985080 10pages 2014

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 Journal of Function Spaces

Thus by 120572-admissibility of 119891 and the above relation wecan obtain

119889 (119906 V) le 120572 (119906 V) 119889 (119906 V)

= 120572 (119891119906 119891V) 119889 (119891119906 119891V)

le 120595119889 (119906 V) lt 119889 (119906 V)

(39)

which is a contradiction Therefore 119906 = V

3 Consequences

Definition 18 Let (119883 119889) be a gqms and let 119891 119883 rarr 119883 bea function satisfying

119889 (119891119909 119891119910) le 120595 (119889 (119909 119910)) (40)

for all 119909 119910 isin 119883Then119891 is said to be a120595-contractivemapping

Theorem 19 Let (119883 119889) be a complete 119892119902119898119904 and let 119891 be acontinuous 120595-contractive mapping Then 119891 is a unique point 119906in 119883

Proof It is sufficient to take 120572(119909 119910) = 1 for all 119909 119910 isin 119883

Theorem 20 Let (119883 119889) be a complete 119892119902119898119904 and let 119891 be acontinuous mapping Suppose that there exists 119896 isin [0 1) suchthat

119889 (119891119909 119891119910) le 119896119889 (119909 119910) forall119909 119910 isin 119883 (41)

Then 119891 is a unique point 119906 in 119883

Proof Take 120595(119905) = 119896119905 where 119896 isin [0 1) It is clear that allconditions of Theorem 19 are satisfied

4 Conclusion

It is evident that almost all fixed point theorems in thecontext of generalized metric spaces can be represented inthe setting of generalized quasimetric spaces Thus all fixedpoint results obtained in Section 2 infer the analog of thefixed point theorems in the context of generalized metricspaces Consequently several results in the literature (see eg[5 8 9 19 28]) can be derived from our main results

Conflict of Interests

The authors declare that they have no conflict of interests

Authorsrsquo Contribution

All authors contributed equally and significantly in writingthis paper All authors read and approved the final paper

Acknowledgment

The authors thank the anonymous referees for their remark-able comments suggestion and ideas that helped to improvethis paper

References

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

[2] M Jleli and B Samet ldquoRemarks on G-metric spaces and fixedpoint theoremsrdquo Fixed Point Theory and Applications vol 2012article 210 2012

[3] N Bilgili E Karapınar andB Samet ldquoGeneralized120572-120595 contrac-tive mappings in quasi-metric spaces and related fixed-pointtheoremsrdquo Journal of Inequalities and Applications vol 2014article 36 2014

[4] I Lin C Chen and E Karapınar ldquoPeriodic points of weakermeir-keeler contractive mappings on generalized quasimetricspacesrdquo Abstract and Applied Analysis vol 2014 Article ID490450 6 pages 2014

[5] A Branciari ldquoA fixed point theorem of Banach-Caccioppolitype on a class of generalized metric spacesrdquo PublicationesMathematicae Debrecen vol 57 no 1-2 pp 31ndash37 2000

[6] I R Sarma J M Rao and S S Rao ldquoContractions over gener-alized metric spacesrdquo Journal of Nonlinear Science and ItsApplications vol 2 no 3 pp 180ndash182 2009

[7] Z Kadeburg and S Radenovic ldquoOn generalized metric spacesa surveyrdquo TWMS Journal of Pure and Applied Math vol 5 no1 pp 3ndash13 2014

[8] A Azam and M Arshad ldquoKannan fixed point theorem ongeneralized metric spacesrdquo Journal of Nonlinear Sciences and ItsApplications vol 1 no 1 pp 45ndash48 2008

[9] P Das ldquoA fixed point theorem on a class of generalized metricspacesrdquo Korean Journal of Mathematical Sciences vol 9 pp 29ndash33 2002

[10] M Erhan E Karapınar and T Sekulic ldquoFixed points of (120595 120601)

contractions on rectangular metric spacesrdquo Fixed Point Theoryand Applications vol 2012 article 138 2012

[11] D Mihet ldquoOn Kannan fixed point principle in generalizedmetric spacesrdquo Journal of Nonlinear Science and its ApplicationsTJNSA vol 2 no 2 pp 92ndash96 2009

[12] B Samet ldquoA fixed point theorem in a generalized metric spaceformappings satisfying a contractive condition of integral typerdquoInternational Journal ofMathematical Analysis vol 3 no 25ndash28pp 1265ndash1271 2009

[13] B Samet ldquoDisscussion on a fixed point theorem of Banach-Caccioppli type on a class of generalized metric spacesrdquo Pub-licationes Mathematicae vol 76 no 4 pp 493ndash494 2010

[14] H Lakzian and B Samet ldquoFixed points for (120595 120593)-weaklycontractive mappings in generalized metric spacesrdquo AppliedMathematics Letters vol 25 no 5 pp 902ndash906 2012

[15] H Aydi E Karapınar and H Lakzian ldquoFixed point results on aclass of generalized metric spacesrdquo Mathematical Sciences vol6 article 46 2012

[16] T Suzuki ldquoGeneralized metric spaces do not have the compat-ible topologyrdquo Abstract and Applied Analysis vol 2014 ArticleID 458098 2014

[17] M Jleli and B Samet ldquoThe Kannanrsquos fixed point theorem in acone rectangularmetric spacerdquo Journal of Nonlinear Science andIts Applications vol 2 no 3 pp 161ndash167 2009

[18] W A Kirk and N Shahzad ldquoGeneralized metrics and Caristirsquostheoremrdquo Fixed Point Theory and Applications vol 2013 article129 2013

[19] E Karapınar ldquoDiscussion on 120572-120595 contractions on generalizedmetric spacesrdquo Abstract and Applied Analysis vol 2014 ArticleID 962784 7 pages 2014

Journal of Function Spaces 7

[20] H Aydi E Karapınar and B Samet ldquoFixed points for gener-alized (alphapsi)-contractions on generalized metric spacesrdquoJournal of Inequalities and Applications vol 2014 article 2292014

[21] I A Rus Generalized Contractions and Applications Cluj Uni-versity Press Cluj-Napoca Romania 2001

[22] R M Bianchini and M Grandolfi ldquoTrasformazioni di tipocontrattivo generalizzato in uno spazio metricordquo Atti dellaAccademia Nazionale dei Lincei Rendiconti Classe di ScienzeFisiche Matematiche e Naturali vol 45 pp 212ndash216 1968

[23] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572minusminus120595-contractive type mappingsrdquoNonlinear Analysis TheoryMethods amp Applications vol 75 no 4 pp 2154ndash2165 2012

[24] N Hussain E Karapınar P Salimi and P Vetro ldquoFixedpoint results for119866119898-Meir-Keeler contractive andG-(120572120595)-Meir-Keeler contractive mappingsrdquo Fixed Point Theory and Applica-tions vol 2013 article 34 2013

[25] N Hussain E Karapınar P Salimi and F Akbar ldquo120572-admissiblemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 article 114 11 pages 2013

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

[27] L B Ciric S M Alsulami P Salimi and P Vetro ldquoPPF depen-dent fixed point results for triangular 120572

119888-admissible mappingsrdquo

The Scientific World Journal vol 2014 Article ID 673647 10pages 2014

[28] M A Alghamdi C-M Chen and E Karapınar ldquoA generalizedweaker (alpha-phi-varphi)-contractive mappings and relatedfixed point results in complete generalized metric spacesrdquoAbstract and Applied Analysis vol 2014 Article ID 985080 10pages 2014

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

Journal of Function Spaces 7

[20] H Aydi E Karapınar and B Samet ldquoFixed points for gener-alized (alphapsi)-contractions on generalized metric spacesrdquoJournal of Inequalities and Applications vol 2014 article 2292014

[21] I A Rus Generalized Contractions and Applications Cluj Uni-versity Press Cluj-Napoca Romania 2001

[22] R M Bianchini and M Grandolfi ldquoTrasformazioni di tipocontrattivo generalizzato in uno spazio metricordquo Atti dellaAccademia Nazionale dei Lincei Rendiconti Classe di ScienzeFisiche Matematiche e Naturali vol 45 pp 212ndash216 1968

[23] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572minusminus120595-contractive type mappingsrdquoNonlinear Analysis TheoryMethods amp Applications vol 75 no 4 pp 2154ndash2165 2012

[24] N Hussain E Karapınar P Salimi and P Vetro ldquoFixedpoint results for119866119898-Meir-Keeler contractive andG-(120572120595)-Meir-Keeler contractive mappingsrdquo Fixed Point Theory and Applica-tions vol 2013 article 34 2013

[25] N Hussain E Karapınar P Salimi and F Akbar ldquo120572-admissiblemappings and related fixed point theoremsrdquo Journal of Inequal-ities and Applications vol 2013 article 114 11 pages 2013

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

[27] L B Ciric S M Alsulami P Salimi and P Vetro ldquoPPF depen-dent fixed point results for triangular 120572

119888-admissible mappingsrdquo

The Scientific World Journal vol 2014 Article ID 673647 10pages 2014

[28] M A Alghamdi C-M Chen and E Karapınar ldquoA generalizedweaker (alpha-phi-varphi)-contractive mappings and relatedfixed point results in complete generalized metric spacesrdquoAbstract and Applied Analysis vol 2014 Article ID 985080 10pages 2014

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