Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 302438 7 pageshttpdxdoiorg1011552013302438

Research ArticleFixed Point Theory of Weak Contractions inPartially Ordered Metric Spaces

Chi-Ming Chen1 Jin-Chirng Lee2 and Chao-Hung Chen2

1 Department of Applied Mathematics National Hsinchu University of Education No 521 Nandah Road Hsinchu City Taiwan2Department of Applied Mathematics Chung Yuan Christian University Taiwan

Correspondence should be addressed to Chao-Hung Chen chenwebmailmlcedutw

Received 12 March 2013 Accepted 20 April 2013

Academic Editor Erdal Karapınar

Copyright copy 2013 Chi-Ming Chen et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We prove two new fixed point theorems in the framework of partially ordered metric spaces Our results generalize and improvemany recent fixed point theorems in the literature

1 Introduction and Preliminaries

Throughout this paper by R+ we denote the set of allnonnegative real numbers while N is the set of all naturalnumbers Let (119883 119889) be a metric space 119863 a subset of 119883 and119891 119863 rarr 119883 a map We say 119891 is contractive if there exists120572 isin [0 1) such that for all 119909 119910 isin 119863

119889 (119891119909 119891119910) le 120572 sdot 119889 (119909 119910) (1)

The well-known Banachrsquos fixed point theorem asserts thatif 119863 = 119883 119891 is contractive and (119883 119889) is complete then119891 has a unique fixed point in 119883 In nonlinear analysis thestudy of fixed points of given mappings satisfying certaincontractive conditions in various abstract spaces has beeninvestigated deeply The Banach contraction principle [1] isone of the initial and crucial results in this direction Alsothis principle has many generalizations For instance Alberand Guerre-Delabriere in [2] suggested a generalization ofthe Banach contractionmapping principle by introducing theconcept of weak contraction in Hilbert spaces In [2] theauthors also proved that the result of Eslamian and Abkar [3]is equivalent to the result of Dutta and Choudhury [4] Laterweakly contractive mappings and mappings satisfying otherweak contractive inequalities have been discussed in severalworks some of which are noted in [4ndash16]

In 2008 Dutta and Choudhury proved the followingtheorem

Theorem 1 (see [4]) Let (119883 119889) be a complete metric spaceand let 119891 119883 rarr 119883 be such that

120595 (119889 (119891119909 119891119910)) le 120595 (119889 (119909 119910)) minus 120601 (119889 (119909 119910))

119891119900119903 119890119886119888ℎ 119909 119910 isin 119883(2)

where 120595 120601 R+ rarr R+ are continuous and nondecreasingand 120595(119905) = 120601(119905) = 0 if and only if 119905 = 0 Then 119891 has a fixedpoint in119883

Recently Eslamian and Abkar [3] proved the followingtheorem

Theorem 2 (see [3]) Let (119883 119889) be a complete metric spaceand let 119891 119883 rarr 119883 be such that

120595 (119889 (119891119909 119891119910)) le 120572 (119889 (119909 119910)) minus 120573 (119889 (119909 119910))

119891119900119903 119890119886119888ℎ 119909 119910 isin 119883(3)

where 120595 120572 120573 R+ rarr R+ are such that 120595 is continuous andnondecreasing 120572 is continuous 120573 is lower semicontinuous and

120595 (119905) minus 120572 (119905) + 120573 (119905) gt 0 forall119905 gt 0

120595 (119905) = 0 119894119891 119886119899119889 119900119899119897119910 119894119891 119905 = 0 120572 (0) = 120573 (0) = 0(4)

Then 119891 has a fixed point in119883

2 Journal of Applied Mathematics

In the recent fixed point theory has developed rapidly inpartially ordered metric spaces (eg [17ndash22])

In 2012 Choudhury andKundu [23] proved the followingfixed point theorem as a generalization of Theorem 2

Theorem 3 (see [23]) Let (119883 ⊑) be a partially ordered set andsuppose that there exists a metric 119889 in 119883 such that (119883 119889) is acomplete metric space and let 119891 119883 rarr 119883 be a nondecreasingmapping such that

120595 (119889 (119891119909 119891119910)) le 120572 (119889 (119909 119910)) minus 120573 (119889 (119909 119910))

119891119900119903 119890119886119888ℎ 119909 119910 isin 119883 119904119906119888ℎ 119905ℎ119886119905 119909 ⊑ 119910(5)

where 120595 120572 120573 R+ rarr R+ are such that 120595 is continuous andnondecreasing 120572 is continuous 120573 is lower semicontinuous and

120595 (119905) minus 120572 (119905) + 120573 (119905) gt 0 forall119905 gt 0

120595 (119905) = 0 119894119891 119886119899119889 119900119899119897119910 119894119891 119905 = 0 120572 (0) = 120573 (0) = 0(6)

Also if any nondecreasing sequence 119909119899 in 119883 converges to ]

then one assumes that

119909119899⊑ ] forall119899 isin N (7)

If there exists 1199090isin 119883 with 119909

0⊑ 1198911199090 then 119891 and 119892 have a

coincidence point in119883

In this paper we prove two new fixed point theorems inthe framework of partially orderedmetric spaces Our resultsgeneralize and improve many recent fixed point theorems inthe literature

2 Fixed Point Results (I)

We start with the following definition

Definition 4 Let (119883 ⊑) be a partially ordered set and 119891

119883 rarr 119883 Then 119891 is said to be monotone nondecreasing iffor 119909 119910 isin 119883

119909 ⊑ 119910 997904rArr 119891119909 ⊑ 119891119910 (8)

Let (119883 ⊑) be a partially ordered set 119909 119910 isin 119883 are said tobe comparable if either 119909 ⊑ 119910 or 119910 ⊑ 119909 holds

In the section we denote by Ψ the class of functions 120595

R+3

rarr R+ satisfying the following conditions

(1205951) 120595 is an increasing continuous function in eachcoordinate

(1205952) for all 119905 isin R+ 120595(119905 119905 119905) le 119905 120595(0 0 119905) le 119905 and120595(119905 0 0) le 119905

Next we denote byΦ the class of functions 120601 R+ rarr R+

satisfying the following conditions

(1206011) 120601 is a continuous nondecreasing function

(1206012) 120601(119905) gt 0 for 119905 gt 0 and 120601(0) = 0

(1206013) 120601 is subadditive that is 120601(119905

1+ 1199052) le 120601(119905

1) + 120601(119905

2) for

all 1199051 1199052gt 0

And we denote the following sets of functions

Θ = 120593 R+

rarr R+ such that 120593 is continuous

Ξ = 120585 R+

rarr R+ such that 120585 is lower continuous

(9)

Let 119883 be a nonempty set and let (119883 ⊑) be a partiallyordered set endowedwith ametric119889Then the triple (119883 ⊑ 119889)is called a partially ordered complete metric space

We now state the main fixed point theorem for(120593 120595 120601 120585)-contractions in partially ordered metric spacesas follows

Theorem 5 Let (119883 ⊑ 119889) be a partially ordered completemetric space Let 119891 119883 rarr 119883 be monotone nondecreasingand

120593 (119889 (119891119909 119891119910)) le 120595 (120601 (119889 (119909 119910)) 120601 (119889 (119909 119891119909)) 120601 (119889 (119910 119891119910)))

minus 120585 (max 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910)) (10)

for all comparable 119909 119910 isin 119883 where 120593 isin Θ 120595 isin Ψ 120601 isin Φ and120585 isin Ξ and

120593 (119905) minus 120601 (119905) + 120585 (119905) gt 0 forall119905 gt 0

120593 (119905) = 0 119894119891 119886119899119889 119900119899119897119910 119894119891 119905 = 0 120601 (0) = 120585 (0) = 0(11)

Suppose that either

(a) 119891 is continuous or

(b) if any nondecreasing sequence 119909119899 in119883 converges to ]

then one assumes that

119909119899⊑ ] forall119899 isin N (12)

If there exists 1199090isin 119883 with 119909

0⊑ 1198911199090 then 119891 has a fixed point

in 119883

Proof Since 119891 is nondecreasing by induction we constructthe sequence 119909

119899 recursively as

119909119899= 119891119899

1199090= 119891119909119899minus1

forall119899 isin N (13)

Thus we also conclude that

1199090⊏ 1199091= 1198911199090⊑ 1199092= 1198911199091⊑ sdot sdot sdot ⊑ 119909

119899= 119891119909119899minus1

⊑ sdot sdot sdot (14)

If any two consecutive terms in (14) are equal then the 119891 hasa fixed point and hence the proof is completed So we mayassume that

119889 (119909119899minus1

119909119899) = 0 forall119899 isin N (15)

Now we claim that 119889(119909119899 119909119899+1

) le 119889(119909119899minus1

119909119899) for all 119899 isin N If

not we assume that 119889(119909119899minus1

119909119899) lt 119889(119909

119899 119909119899+1

) for some 119899 isin N

Journal of Applied Mathematics 3

substituting 119909 = 119909119899and 119910 = 119909

119899+1in (10) and using the

definition of the function 120595 we have

120595 (120601 (119889 (119909119899 119909119899+1

)) 120601 (119889 (119909119899 119891119909119899)) 120601 (119889 (119909

119899+1 119891119909119899+1

)))

= 120595 (120601 (119889 (119909119899 119909119899+1

)) 120601 (119889 (119909119899 119909119899+1

)) 120601 (119889 (119909119899+1

119909119899+2

)))

le 120601 (119889 (119909119899+1

119909119899+2

))

120585 (max 119889 (119909119899 119909119899+1

) 119889 (119909119899 119891119909119899) 119889 (119909

119899+1 119891119909119899+1

))

= 120585 (max 119889 (119909119899 119909119899+1

) 119889 (119909119899 119909119899+1

) 119889 (119909119899+1

119909119899+2

))

= 120585 (119889 (119909119899+1

119909119899+2

))

(16)

and hence

120593 (119889 (119909119899+1

119909119899+2

))

= 120593 (119889 (119891119909119899 119891119909119899+1

))

le 120601 (119889 (119909119899+1

119909119899+2

)) minus 120585 (119889 (119909119899+1

119909119899+2

))

(17)

Since 120593(119905) minus 120601(119905) + 120585(119905) gt 0 for all 119905 gt 0 we have that119889(119909119899+1

119909119899+2

) = 0 which contradicts (15) Therefore weconclude that

119889 (119909119899 119909119899+1

) le 119889 (119909119899minus1

119909119899) forall119899 isin N (18)

From the previous argument we also have that for each 119899 isin N

120593 (119889 (119909119899 119909119899+1

)) le 120601 (119889 (119909119899minus1

119909119899)) minus 120585 (119889 (119909

119899minus1 119909119899)) (19)

It follows in (18) that the sequence 119889(119909119899 119909119899+1

) is monotonedecreasing it must converge to some 120578 ge 0 Taking limit as119899 rarr infin in (19) and using the continuities of 120593 and 120601 and thelower semicontinuous of 120585 we get

120593 (120578) le 120601 (120578) minus 120585 (120578) (20)

which implies that 120578 = 0 So we conclude that

lim119899rarrinfin

119889 (119909119899 119909119899+1

) = 0 (21)

We next claim that 119909119899 is a Cauchy sequence that is for

every 120576 gt 0 there exists 119899 isin N such that if 119901 119902 ge 119899 then119889(119909119901 119909119902) lt 120576

Suppose on the contrary that there exists 120598 gt 0 such thatfor any 119899 isin N there are 119901

119899 119902119899isin Nwith 119901

119899gt 119902119899ge 119899 satisfying

119889 (119909119902119899

119909119901119899

) ge 120598 (22)

Further corresponding to 119902119899ge 119899 we can choose 119901

119899in such

a way that it the smallest integer with 119901119899gt 119902119899ge 119899 and

119889(119909119902119899

119909119901119899

) ge 120598 Therefore 119889(119909119902119899

119909119901119899minus1) lt 120598 Now we have

that for all 119899 isin N

120598 le 119889 (119909119901119899

119909119902119899

)

le 119889 (119909119901119899

119909119901119899minus1) + 119889 (119909

119901119899minus1 119909119902119899

)

lt 119889 (119909119901119899

119909119901119899minus1) + 120598

(23)

By letting 119899 rarr infin we get that

lim119899rarrinfin

119889 (119909119901119899

119909119902119899

) = 120598 (24)

On the other hand we have

119889 (119909119901119899

119909119902119899

)

le 119889 (119909119901119899

119909119901119899minus1) + 119889 (119909

119901119899minus1 119909119902119899minus1) + 119889 (119909

119902119899minus1 119909119902119899

)

119889 (119909119901119899minus1 119909119902119899minus1)

le 119889 (119909119901119899minus1 119909119901119899

) + 119889 (119909119901119899

119909119902119899

) + 119889 (119909119902119899

119909119902119899minus1)

(25)

Letting 119899 rarr infin then we get

lim119899rarrinfin

119889 (119909119901119899minus1 119909119902119899minus1) = 120598 (26)

By (14) we have that the elements119909119901119899

and119909119902119899

are comparableSubstituting 119909 = 119909

119901119899minus1

and 119910 = 119909119902119899minus1

in (10) we have that forall 119899 isin N

120595 (120601 (119889 (119909119901119899minus1 119909119902119899minus1)) 120601 (119889 (119909

119901119899minus1 119891119909119901119899minus1))

120601 (119889 (119909119902119899minus1 119891119909119902119899minus1)))

le 120595 (120601 (119889 (119909119901119899minus1 119909119902119899minus1)) 120601 (119889 (119909

119901119899minus1 119909119901119899

))

120601 (119889 (119909119902119899minus1 119909119902119899

)))

119872 (119909119901119899minus1 119909119902119899minus1)

= max 119889 (119909119901119899minus1 119909119902119899minus1) 119889 (119909

119901119899minus1 119891119909119901119899minus1)

119889 (119909119902119899minus1 119891119909119902119899minus1)

= max 119889 (119909119901119899minus1 119909119902119899minus1) 119889 (119909

119901119899minus1 119909119901119899

)

119889 (119909119902119899minus1 119909119902119899

)

(27)

By the previous argument and using inequality (10) we canconclude that

120593 (120598) le 120595 (120601 (120598) 0 0) minus 120585 (120598)

le 120601 (120598) minus 120585 (120598) (28)

which implies that 120598 = 0 a contradiction Therefore thesequence 119909

119899 is a Cauchy sequence

Since119883 is complete there exists ] isin 119883 such that

lim119899rarrinfin

119909119899= ] (29)

Suppose that (a) holds Then

] = lim119899rarrinfin

119909119899+1

= lim119899rarrinfin

119891119909119899= 119891] (30)

Thus ] is a fixed point in119883

4 Journal of Applied Mathematics

Suppose that (b) holds that is 119909119899⊑ ] for all 119899 isin N

Substituting 119909 = 119909119899and 119910 = ] in (10) we have that

120593 (119889 (119909119899+1

119891]))

= 120593 (119889 (119891119909119899 119891]))

le 120595 (120601 (119889 (119909119899 ])) 120601 (119889 (119909

119899 119891119909119899)) 120601 (119889 (] 119891])))

minus 120585 (max 119889 (119909119899 ]) 119889 (119909

119899 119891119909119899) 119889 (] 119891]))

(31)

Taking limit as 119899 rarr infin in equality (31) we have

120593 (119889 (] 119891])) le 120595 (120601 (0) 120601 (119900) 120601 (119889 (] 119891]))) minus 120585 (119889 (] 119891]))

le 120601 (119889 (] 119891])) minus 120585 (119889 (] 119891]))

(32)

which implies that 119889(] 119891]) = 0 that is ] = 119891] So wecomplete the proof

If we let

120595 (120601 (119889 (119909 119910)) 120601 (119889 (119909 119891119909)) 120601 (119889 (119910 119891119910)))

= max 120601 (119889 (119909 119910)) 120601 (119889 (119909 119891119909)) 120601 (119889 (119910 119891119910)) (33)

it is easy to get the following theorem

Theorem 6 Let (119883 ⊑ 119889) be a partially ordered completemetric space Let 119891 119883 rarr 119883 be monotone nondecreasingand

120593 (119889 (119891119909 119891119910))

le max 120601 (119889 (119909 119910)) 120601 (119889 (119909 119891119909)) 120601 (119889 (119910 119891119910))

minus 120585 (max 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910))

(34)

for all comparable 119909 119910 isin 119883 where 120593 isin Θ 120601 isin Φ and 120585 isin Ξand

120593 (119905) minus 120601 (119905) + 120585 (119905) gt 0 forall119905 gt 0

120593 (119905) = 0 119894119891 119886119899119889 119900119899119897119910 119894119891 119905 = 0 120601 (0) = 120585 (0) = 0(35)

Suppose that either

(a) 119891 is continuous or(b) if any nondecreasing sequence 119909

119899 in119883 converges to ]

then one assumes that

119909119899⊑ ] forall119899 isin N (36)

If there exists 1199090isin 119883 with 119909

0⊑ 1198911199090 then 119891 has a fixed point

in119883

3 Fixed Point Results (II)

In the section we denote by Ψ the class of functions 120595

R+3

rarr R+ satisfying the following conditions

(1205951) 120595 is an increasing and continuous function in eachcoordinate

(1205952) for 119905 isin R+ 120601(119905 119905 119905) le 119905 120601(119905 0 0) le 119905 and 120601(0 0 119905) le119905

Next we denote byΘ the class of functions120593 R+ rarr R+

satisfying the following conditions

(1205931) 120593 is continuous and nondecreasing

(1205932) for 119905 gt 0 120593(119905) gt 0 and 120593(0) = 0

And we denote byΦ the class of functions 120601 R+ rarr R+

satisfying the following conditions

(1206011) 120601 is continuous

(1206012) for 119905 gt 0 120601(119905) gt 0 and 120601(0) = 0

We now state the main fixed point theorem for the(120595 120593 120601)-contractions in partially ordered metric spaces asfollows

Theorem 7 Let (119883 ⊑ 119889) be a partially ordered completemetric space and let 119891 119883 rarr 119883 be monotone nondecreasingand

120593 (119889 (119891119909 119891119910))

le 120595 (120593 (119889 (119909 119910)) 120593 (119889 (119909 119891119909)) 120593 (119889 (119910 119891119910)))

minus 120601 (119872 (119909 119910)) + 119871 sdot 119898 (119909 119910)

(37)

for all comparable 119909 119910 isin 119883 and 120595 isin Ψ 120593 isin Θ 120601 isin Φ where119871 gt 0 and

119872(119909 119910) = max 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910)

119898 (119909 119910)

= min 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910) 119889 (119909 119891119910)

119889 (119910 119891119909)

(38)

Suppose that either

(a) 119891 is continuous or(b) if any nondecreasing sequence 119909

119899 in119883 converges to ]

then one assumes that

119909119899⊑ ] forall119899 isin N (39)

If there exists 1199090isin 119883 with 119909

0⊑ 1198911199090 then 119891 has a fixed point

in 119883

Proof If 1198911199090

= 1199090 then the proof is finished Suppose

that 1199090⊏ 1198911199090 Since 119891 is nondecreasing by induction we

construct the sequence 119909119899 recursively as

119909119899= 119891119899

1199090= 119891119909119899minus1

forall119899 isin N (40)

Thus we also conclude that

1199090⊏ 1199091= 1198911199090⊑ 1199092= 1198911199091⊑ sdot sdot sdot ⊑ 119909

119899= 119891119909119899minus1

⊑ sdot sdot sdot (41)

We now claim that

lim119899rarrinfin

119889 (119909119899 119909119899+1

) = 0 (42)

Journal of Applied Mathematics 5

Put 119909 = 119909119899minus1

and 119910 = 119909119899in (37) Note that

119898(119909119899minus1

119909119899)

= min 119889 (119909119899minus1

119909119899) 119889 (119909

119899minus1 119891119909119899minus1

)

119889 (119909119899 119891119909119899) 119889 (119909

119899minus1 119891119909119899) 119889 (119909

119899 119891119909119899minus1

)

= min 119889 (119909119899minus1

119909119899) 119889 (119909

119899minus1 119909119899) 119889 (119909

119899 119909119899+1

)

119889 (119909119899minus1

119909119899+1

) 119889 (119909119899 119909119899)

= 0

(43)

So we obtain that

120593 (119889 (119909119899 119909119899+1

))

= 120593 (119889 (119891119909119899minus1

119891119909119899))

le 120595 (120593 (119889 (119909119899minus1

119909119899)) 120593 (119889 (119909

119899minus1 119891119909119899minus1

))

120593 (119889 (119909119899 119891119909119899))) minus 120601 (119872 (119909

119899minus1 119909119899))

le 120595 (120593 (119889 (119909119899minus1

119909119899)) 120593 (119889 (119909

119899minus1 119909119899)) 120593 (119889 (119909

119899 119909119899+1

)))

minus 120601 (119872 (119909119899minus1

119909119899))

(44)

where

119872(119909119899minus1

119909119899)

= max 119889 (119909119899minus1

119909119899) 119889 (119909

119899minus1 119891119909119899minus1

) 119889 (119909119899 119891119909119899)

= max 119889 (119909119899minus1

119909119899) 119889 (119909

119899minus1 119909119899) 119889 (119909

119899 119909119899+1

)

(45)

We now claim that

119889 (119909119899 119909119899+1

) lt 119889 (119909119899minus1

119909119899) forall119899 isin N (46)

If not we assume that 119889(119909119899minus1

119909119899) le 119889(119909

119899 119909119899+1

) then120593(119889(119909

119899minus1 119909119899)) le 120593(119889(119909

119899 119909119899+1

)) since 120593 is non-decreasingUsing inequality (44) and the conditions of the function 120595we have that for each 119899 isin N

120593 (119889 (119909119899 119909119899+1

)) le 120593 (119889 (119909119899 119909119899+1

)) minus 120601 (119889 (119909119899 119909119899+1

)) (47)

which implies that120601(119889(119909119899 119909119899+1

))= 0 and hence119889(119909119899 119909119899+1

)=0 This contradicts our initial assumption

From the previous argument we have that for each 119899 isin N

120593 (119889 (119909119899 119909119899+1

)) le 120593 (119889 (119909119899minus1

119909119899)) minus 120601 (119889 (119909

119899minus1 119909119899))

119889 (119909119899 119909119899+1

) lt 119889 (119909119899minus1

119909119899)

(48)

And since the sequence 119889(119909119899 119909119899+1

) is decreasing it mustconverge to some 120578 ge 0 Taking limit as 119899 rarr infin in (48) andby the continuity of 120593 and 120601 we get

120593 (120578) le 120593 (120578) minus 120601 (120578) (49)

and so we conclude that 120601(120578) = 0 and 120578 = 0We next claim that 119909

119899 is Cauchy that is for every 120576 gt 0

there exists 119899 isin N such that if 119901 119902 ge 119899 then 119889(119909119901 119909119902) lt 120576

Suppose on the contrary that there exists 120598 gt 0 such thatfor any 119899 isin N there are 119901

119899 119902119899isin Nwith 119901

119899gt 119902119899ge 119899 satisfying

119889 (119909119902119899

119909119901119899

) ge 120598 (50)

Further corresponding to 119902119899ge 119899 we can choose 119901

119899in such

a way that it the smallest integer with 119901119899gt 119902119899ge 119899 and

119889(119909119902119899

119909119901119899

) ge 120598 Therefore 119889(119909119902119899

119909119901119899minus1) lt 120598 By the rectan-

gular inequality we have

120598 le 119889 (119909119901119899

119909119902119899

)

le 119889 (119909119901119899

119909119901119899minus1) + 119889 (119909

119901119899minus1 119909119902119899

)

lt 119889 (119909119901119899

119909119901119899minus1) + 120598

(51)

Letting 119899 rarr infin then we get

lim119899rarrinfin

119889 (119909119901119899

119909119902119899

) = 120598 (52)

On the other hand we have

119889 (119909119901119899

119909119902119899

)

le 119889 (119909119901119899

119909119901119899minus1) + 119889 (119909

119901119899minus1 119909119902119899minus1) + 119889 (119909

119902119899minus1 119909119902119899

)

119889 (119909119901119899minus1 119909119902119899minus1)

le 119889 (119909119901119899minus1 119909119901119899

) + 119889 (119909119901119899

119909119902119899

) + 119889 (119909119902119899

119909119902119899minus1)

(53)

By letting 119899 rarr infin we get that

lim119899rarrinfin

119889 (119909119901119899minus1 119909119902119899minus1) = 120598 (54)

Using inequalities (37) (52) and (54) and putting 119909 = 119909119901119899minus1

and 119910 = 119909119902119899minus1 we have that

120593 (119889 (119909119901119899

119909119902119899

))

= 120593 (119889 (119891119909119901119899minus1 119891119909119902119899minus1))

le 120595 (120593 (119889 (119909119901119899minus1 119909119902119899minus1)) 120593 (119889 (119909

119901119899minus1 119891119909119901119899minus1))

120593 (119889 (119909119902119899minus1 119891119909119902119899minus1)))

minus 120601 (119872(119909119901119899minus1 119909119902119899minus1)) + 119871 sdot 119898 (119909

119901119899minus1 119909119902119899minus1)

= 120595 (120593 (119889 (119909119901119899minus1 119909119902119899minus1)) 120593 (119889 (119909

119901119899minus1 119909119901119899

))

120593 (119889 (119909119902119899minus1 119909119902119899

)))

minus 120601 (119872(119909119901119899minus1 119909119902119899minus1)) + 119871 sdot 119898 (119909

119901119899minus1 119909119902119899minus1)

(55)

6 Journal of Applied Mathematics

where

119872(119909119901119899minus1 119909119902119899minus1)

= max 119889 (119909119901119899minus1 119909119902119899minus1) 119889 (119909

119901119899minus1 119909119901119899

) 119889 (119909119902119899minus1 119909119902119899

)

119898 (119909119901119899minus1 119909119902119899minus1)

= min 119889 (119909119901119899minus1 119909119902119899minus1) 119889 (119909

119901119899minus1 119909119901119899

)

119889 (119909119902119899minus1 119909119902119899

) 119889 (119909119901119899minus1 119909119902119899

) 119889 (119909119902119899minus1 119909119901119899

)

(56)

Letting 119899 rarr infin then we obtain that

lim119899rarrinfin

119872(119909119901119899minus1 119909119902119899minus1) = 120598

lim119899rarrinfin

119898(119909119901119899minus1 119909119902119899minus1) = 0

120593 (120598) le 120595 (120593 (120598) 0 0) minus 120601 (120598) le 120593 (120598) minus 120601 (120598)

(57)

This implies that 120601(120598) = 0 and hence 120598 = 0 So we get acontraction Therefore 119909

119899 is a Cauchy sequence

Since119883 is complete there exists ] isin 119883 such that

lim119899rarrinfin

119909119899= ] (58)

Suppose that (a) holds Then

] = lim119899rarrinfin

119909119899+1

= lim119899rarrinfin

119891119909119899= 119891] (59)

Thus ] is a fixed point in119883Suppose that (b) holds that is 119909

119899⊑ ] for all 119899 isin N

Substituting 119909 = 119909119899and 119910 = ] in (37) we have that

120593 (119889 (119909119899+1

119891]))

= 120593 (119889 (119891119909119899 119891]))

le 120595 (120593 (119889 (119909119899 ])) 120593 (119889 (119909

119899 119891119909119899)) 120593 (119889 (] 119891])))

minus 120601 (119872 (119909119899 ])) + 119871 sdot 119898 (119909

119899 ])

(60)

where

119872(119909119899 ]) = max 119889 (119909

119899 ]) 119889 (119909

119899 119891119909119899) 119889 (] 119891])

119898 (119909119899 ])

= min 119889 (119909119899 ]) 119889 (119909

119899 119891119909119899) 119889 (] 119891]) 119889 (119909

119899 119891])

119889 (] 119891119909119899)

(61)

Letting 119899 rarr infin then we obtain that

119872(119909119899 ]) 997888rarr 119889 (] 119891]) 119898 (119909

119899 ]) 997888rarr 0

120593 (119889 (] 119891])) le 120595 (120593 (0) 120593 (0) 120593 (119889 (] 119891]))) minus 120601 (119889 (] 119891]))

le 120593 (119889 (] 119891])) minus 120601 (119889 (] 119891]))

(62)

which implies that 119889(] 119891]) = 0 that is ] = 119891] So wecomplete the proof

If we let

120595 (120593 (119889 (119909 119910)) 120593 (119889 (119909 119891119909)) 120593 (119889 (119910 119891119910)))

= max 120593 (119889 (119909 119910)) 120593 (119889 (119909 119891119909)) 120593 (119889 (119910 119891119910)) (63)

it is easy to get the following theorem

Theorem 8 Let (119883 ⊑ 119889) be a partially ordered completemetric space and let 119891 119883 rarr 119883 be monotone nondecreasingand

120593 (119889 (119891119909 119891119910))

le max 120593 (119889 (119909 119910)) 120593 (119889 (119909 119891119909)) 120593 (119889 (119910 119891119910))

minus 120601 (119872 (119909 119910)) + 119871 sdot 119898 (119909 119910)

(64)

for all comparable 119909 119910 isin 119883 and 120593 isin Θ 120601 isin Φ where 119871 gt 0

and

119872(119909 119910) = max 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910)

119898 (119909 119910) = min 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910) 119889 (119909 119891119910)

119889 (119910 119891119909)

(65)

Suppose that either

(a) 119891 is continuous or(b) if any nondecreasing sequence 119909

119899 in119883 converges to ]

then one assumes that

119909119899⊑ ] forall119899 isin N (66)

If there exists 1199090isin 119883 with 119909

0⊑ 1198911199090 then 119891 has a fixed point

in 119883

Acknowledgment

The authors would like to thank the referee(s) for the manyuseful comments and suggestions for the improvement of thepaper

References

[1] S Banach ldquoSur les operations dans les ensembles abstraitset leur application aux equations integeralesrdquo FundamentaMathematicae vol 3 pp 133ndash181 1922

[2] I Ya Alber and S Guerre-Delabriere ldquoPrinciples of weaklycontractive maps in Hilbert spacesrdquo in New Results in OperatorTheory I Gohberg and Yu Lyubich Eds vol 98 of Advancesand Applications pp 7ndash22 Birkhauser Basel Switzerland 1997

[3] M Eslamian and A Abkar ldquoA fixed point theorem for gener-alized weakly contractive mappings in complete metric space rdquoItalian Journal of Pure and Applied Mathematics In press

[4] P N Dutta and B S Choudhury ldquoA generalisation of con-traction principle in metric spacesrdquo Fixed Point Theory andApplications vol 2008 Article ID 406368 8 pages 2008

[5] H Aydi E Karapınar and B Samet ldquoRemarks on some recentfixed point theoremsrdquo Fixed Point Theory and Applications vol2012 article 76 2012

Journal of Applied Mathematics 7

[6] H Aydi E Karapınar and W Shatanawi ldquoCoupled fixed pointresults for (120593 120595)-weakly contractive condition in ordered par-tialmetric spacesrdquoComputers ampMathematics with Applicationsvol 62 no 12 pp 4449ndash4460 2011

[7] W-S Du ldquoOn coincidence point and fixed point theorems fornonlinear multivaluedmapsrdquo Topology and its Applications vol159 no 1 pp 49ndash56 2012

[8] W-S Du ldquoOn generalized weakly directional contractions andapproximate fixed point propertywith applicationsrdquo Fixed PointTheory and Applications vol 2012 article 6 2012

[9] W-S Du and S-X Zheng ldquoNonlinear conditions for coinci-dence point and fixed point theoremsrdquo Taiwanese Journal ofMathematics vol 16 no 3 pp 857ndash868 2012

[10] W-S Du and S-X Zheng ldquoNew nonlinear conditions andinequalities for the existence of coincidence points and fixedpointsrdquo Journal of Applied Mathematics vol 2012 Article ID196759 12 pages 2012

[11] E Karapinar and K Sadarangani ldquoFixed point theory for cyclic(120593 120595)-contractionsrdquo Fixed Point Theory and Applications vol2011 article 69 2011

[12] E Karapınar ldquoFixed point theory for cyclic weak 120601-contractionrdquo Applied Mathematics Letters vol 24 no 6pp 822ndash825 2011

[13] E Karapınar and I S Yuce ldquoFixed point theory for cyclic gen-eralized weak 120601-contraction on partial metric spacesrdquo AbstractandAppliedAnalysis vol 2012 Article ID491542 12 pages 2012

[14] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis vol 47 no 4 pp 2683ndash2693 2001

[15] B Djafari Rouhani and S Moradi ldquoCommon fixed point ofmultivalued generalized 120593-weak contractive mappingsrdquo FixedPoint Theory and Applications Article ID 708984 201013 pages2010

[16] Q Zhang and Y Song ldquoFixed point theory for generalized 120601-weak contractionsrdquo Applied Mathematics Letters vol 22 no 1pp 75ndash78 2009

[17] L B Ciric N Cakic M Rajovic and J S Ume ldquoMonotonegeneralized nonlinear contractions in partially ordered metricspacesrdquo Fixed Point Theory and Applications vol 2008 ArticleID 131294 11 pages 2008

[18] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysis Theory Methods amp Applications A vol 65 no 7 pp1379ndash1393 2006

[19] J Harjani B Lopez and K Sadarangani ldquoFixed point theoremsfor weakly C-contractive mappings in ordered metric spacesrdquoComputers amp Mathematics with Applications vol 61 no 4 pp790ndash796 2011

[20] J Harjani B Lopez and K Sadarangani ldquoA fixed point theoremfor mappings satisfying a contractive condition of rationaltype on a partially ordered metric spacerdquo Abstract and AppliedAnalysis vol 2010 Article ID 190701 8 pages 2010

[21] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[22] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrixequationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004

[23] B S Choudhury and A Kundu ldquo(120595 120572 120573)-weak contractions inpartially ordered metric spacesrdquo Applied Mathematics Lettersvol 25 no 1 pp 6ndash10 2012

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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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

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

In the recent fixed point theory has developed rapidly inpartially ordered metric spaces (eg [17ndash22])

In 2012 Choudhury andKundu [23] proved the followingfixed point theorem as a generalization of Theorem 2

Theorem 3 (see [23]) Let (119883 ⊑) be a partially ordered set andsuppose that there exists a metric 119889 in 119883 such that (119883 119889) is acomplete metric space and let 119891 119883 rarr 119883 be a nondecreasingmapping such that

120595 (119889 (119891119909 119891119910)) le 120572 (119889 (119909 119910)) minus 120573 (119889 (119909 119910))

119891119900119903 119890119886119888ℎ 119909 119910 isin 119883 119904119906119888ℎ 119905ℎ119886119905 119909 ⊑ 119910(5)

where 120595 120572 120573 R+ rarr R+ are such that 120595 is continuous andnondecreasing 120572 is continuous 120573 is lower semicontinuous and

120595 (119905) minus 120572 (119905) + 120573 (119905) gt 0 forall119905 gt 0

120595 (119905) = 0 119894119891 119886119899119889 119900119899119897119910 119894119891 119905 = 0 120572 (0) = 120573 (0) = 0(6)

Also if any nondecreasing sequence 119909119899 in 119883 converges to ]

then one assumes that

119909119899⊑ ] forall119899 isin N (7)

If there exists 1199090isin 119883 with 119909

0⊑ 1198911199090 then 119891 and 119892 have a

coincidence point in119883

In this paper we prove two new fixed point theorems inthe framework of partially orderedmetric spaces Our resultsgeneralize and improve many recent fixed point theorems inthe literature

2 Fixed Point Results (I)

We start with the following definition

Definition 4 Let (119883 ⊑) be a partially ordered set and 119891

119883 rarr 119883 Then 119891 is said to be monotone nondecreasing iffor 119909 119910 isin 119883

119909 ⊑ 119910 997904rArr 119891119909 ⊑ 119891119910 (8)

Let (119883 ⊑) be a partially ordered set 119909 119910 isin 119883 are said tobe comparable if either 119909 ⊑ 119910 or 119910 ⊑ 119909 holds

In the section we denote by Ψ the class of functions 120595

R+3

rarr R+ satisfying the following conditions

(1205951) 120595 is an increasing continuous function in eachcoordinate

(1205952) for all 119905 isin R+ 120595(119905 119905 119905) le 119905 120595(0 0 119905) le 119905 and120595(119905 0 0) le 119905

Next we denote byΦ the class of functions 120601 R+ rarr R+

satisfying the following conditions

(1206011) 120601 is a continuous nondecreasing function

(1206012) 120601(119905) gt 0 for 119905 gt 0 and 120601(0) = 0

(1206013) 120601 is subadditive that is 120601(119905

1+ 1199052) le 120601(119905

1) + 120601(119905

2) for

all 1199051 1199052gt 0

And we denote the following sets of functions

Θ = 120593 R+

rarr R+ such that 120593 is continuous

Ξ = 120585 R+

rarr R+ such that 120585 is lower continuous

(9)

Let 119883 be a nonempty set and let (119883 ⊑) be a partiallyordered set endowedwith ametric119889Then the triple (119883 ⊑ 119889)is called a partially ordered complete metric space

We now state the main fixed point theorem for(120593 120595 120601 120585)-contractions in partially ordered metric spacesas follows

Theorem 5 Let (119883 ⊑ 119889) be a partially ordered completemetric space Let 119891 119883 rarr 119883 be monotone nondecreasingand

120593 (119889 (119891119909 119891119910)) le 120595 (120601 (119889 (119909 119910)) 120601 (119889 (119909 119891119909)) 120601 (119889 (119910 119891119910)))

minus 120585 (max 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910)) (10)

for all comparable 119909 119910 isin 119883 where 120593 isin Θ 120595 isin Ψ 120601 isin Φ and120585 isin Ξ and

120593 (119905) minus 120601 (119905) + 120585 (119905) gt 0 forall119905 gt 0

120593 (119905) = 0 119894119891 119886119899119889 119900119899119897119910 119894119891 119905 = 0 120601 (0) = 120585 (0) = 0(11)

Suppose that either

(a) 119891 is continuous or

(b) if any nondecreasing sequence 119909119899 in119883 converges to ]

then one assumes that

119909119899⊑ ] forall119899 isin N (12)

If there exists 1199090isin 119883 with 119909

0⊑ 1198911199090 then 119891 has a fixed point

in 119883

Proof Since 119891 is nondecreasing by induction we constructthe sequence 119909

119899 recursively as

119909119899= 119891119899

1199090= 119891119909119899minus1

forall119899 isin N (13)

Thus we also conclude that

1199090⊏ 1199091= 1198911199090⊑ 1199092= 1198911199091⊑ sdot sdot sdot ⊑ 119909

119899= 119891119909119899minus1

⊑ sdot sdot sdot (14)

If any two consecutive terms in (14) are equal then the 119891 hasa fixed point and hence the proof is completed So we mayassume that

119889 (119909119899minus1

119909119899) = 0 forall119899 isin N (15)

Now we claim that 119889(119909119899 119909119899+1

) le 119889(119909119899minus1

119909119899) for all 119899 isin N If

not we assume that 119889(119909119899minus1

119909119899) lt 119889(119909

119899 119909119899+1

) for some 119899 isin N

Journal of Applied Mathematics 3

substituting 119909 = 119909119899and 119910 = 119909

119899+1in (10) and using the

definition of the function 120595 we have

120595 (120601 (119889 (119909119899 119909119899+1

)) 120601 (119889 (119909119899 119891119909119899)) 120601 (119889 (119909

119899+1 119891119909119899+1

)))

= 120595 (120601 (119889 (119909119899 119909119899+1

)) 120601 (119889 (119909119899 119909119899+1

)) 120601 (119889 (119909119899+1

119909119899+2

)))

le 120601 (119889 (119909119899+1

119909119899+2

))

120585 (max 119889 (119909119899 119909119899+1

) 119889 (119909119899 119891119909119899) 119889 (119909

119899+1 119891119909119899+1

))

= 120585 (max 119889 (119909119899 119909119899+1

) 119889 (119909119899 119909119899+1

) 119889 (119909119899+1

119909119899+2

))

= 120585 (119889 (119909119899+1

119909119899+2

))

(16)

and hence

120593 (119889 (119909119899+1

119909119899+2

))

= 120593 (119889 (119891119909119899 119891119909119899+1

))

le 120601 (119889 (119909119899+1

119909119899+2

)) minus 120585 (119889 (119909119899+1

119909119899+2

))

(17)

Since 120593(119905) minus 120601(119905) + 120585(119905) gt 0 for all 119905 gt 0 we have that119889(119909119899+1

119909119899+2

) = 0 which contradicts (15) Therefore weconclude that

119889 (119909119899 119909119899+1

) le 119889 (119909119899minus1

119909119899) forall119899 isin N (18)

From the previous argument we also have that for each 119899 isin N

120593 (119889 (119909119899 119909119899+1

)) le 120601 (119889 (119909119899minus1

119909119899)) minus 120585 (119889 (119909

119899minus1 119909119899)) (19)

It follows in (18) that the sequence 119889(119909119899 119909119899+1

) is monotonedecreasing it must converge to some 120578 ge 0 Taking limit as119899 rarr infin in (19) and using the continuities of 120593 and 120601 and thelower semicontinuous of 120585 we get

120593 (120578) le 120601 (120578) minus 120585 (120578) (20)

which implies that 120578 = 0 So we conclude that

lim119899rarrinfin

119889 (119909119899 119909119899+1

) = 0 (21)

We next claim that 119909119899 is a Cauchy sequence that is for

every 120576 gt 0 there exists 119899 isin N such that if 119901 119902 ge 119899 then119889(119909119901 119909119902) lt 120576

Suppose on the contrary that there exists 120598 gt 0 such thatfor any 119899 isin N there are 119901

119899 119902119899isin Nwith 119901

119899gt 119902119899ge 119899 satisfying

119889 (119909119902119899

119909119901119899

) ge 120598 (22)

Further corresponding to 119902119899ge 119899 we can choose 119901

119899in such

a way that it the smallest integer with 119901119899gt 119902119899ge 119899 and

119889(119909119902119899

119909119901119899

) ge 120598 Therefore 119889(119909119902119899

119909119901119899minus1) lt 120598 Now we have

that for all 119899 isin N

120598 le 119889 (119909119901119899

119909119902119899

)

le 119889 (119909119901119899

119909119901119899minus1) + 119889 (119909

119901119899minus1 119909119902119899

)

lt 119889 (119909119901119899

119909119901119899minus1) + 120598

(23)

By letting 119899 rarr infin we get that

lim119899rarrinfin

119889 (119909119901119899

119909119902119899

) = 120598 (24)

On the other hand we have

119889 (119909119901119899

119909119902119899

)

le 119889 (119909119901119899

119909119901119899minus1) + 119889 (119909

119901119899minus1 119909119902119899minus1) + 119889 (119909

119902119899minus1 119909119902119899

)

119889 (119909119901119899minus1 119909119902119899minus1)

le 119889 (119909119901119899minus1 119909119901119899

) + 119889 (119909119901119899

119909119902119899

) + 119889 (119909119902119899

119909119902119899minus1)

(25)

Letting 119899 rarr infin then we get

lim119899rarrinfin

119889 (119909119901119899minus1 119909119902119899minus1) = 120598 (26)

By (14) we have that the elements119909119901119899

and119909119902119899

are comparableSubstituting 119909 = 119909

119901119899minus1

and 119910 = 119909119902119899minus1

in (10) we have that forall 119899 isin N

120595 (120601 (119889 (119909119901119899minus1 119909119902119899minus1)) 120601 (119889 (119909

119901119899minus1 119891119909119901119899minus1))

120601 (119889 (119909119902119899minus1 119891119909119902119899minus1)))

le 120595 (120601 (119889 (119909119901119899minus1 119909119902119899minus1)) 120601 (119889 (119909

119901119899minus1 119909119901119899

))

120601 (119889 (119909119902119899minus1 119909119902119899

)))

119872 (119909119901119899minus1 119909119902119899minus1)

= max 119889 (119909119901119899minus1 119909119902119899minus1) 119889 (119909

119901119899minus1 119891119909119901119899minus1)

119889 (119909119902119899minus1 119891119909119902119899minus1)

= max 119889 (119909119901119899minus1 119909119902119899minus1) 119889 (119909

119901119899minus1 119909119901119899

)

119889 (119909119902119899minus1 119909119902119899

)

(27)

By the previous argument and using inequality (10) we canconclude that

120593 (120598) le 120595 (120601 (120598) 0 0) minus 120585 (120598)

le 120601 (120598) minus 120585 (120598) (28)

which implies that 120598 = 0 a contradiction Therefore thesequence 119909

119899 is a Cauchy sequence

Since119883 is complete there exists ] isin 119883 such that

lim119899rarrinfin

119909119899= ] (29)

Suppose that (a) holds Then

] = lim119899rarrinfin

119909119899+1

= lim119899rarrinfin

119891119909119899= 119891] (30)

Thus ] is a fixed point in119883

4 Journal of Applied Mathematics

Suppose that (b) holds that is 119909119899⊑ ] for all 119899 isin N

Substituting 119909 = 119909119899and 119910 = ] in (10) we have that

120593 (119889 (119909119899+1

119891]))

= 120593 (119889 (119891119909119899 119891]))

le 120595 (120601 (119889 (119909119899 ])) 120601 (119889 (119909

119899 119891119909119899)) 120601 (119889 (] 119891])))

minus 120585 (max 119889 (119909119899 ]) 119889 (119909

119899 119891119909119899) 119889 (] 119891]))

(31)

Taking limit as 119899 rarr infin in equality (31) we have

120593 (119889 (] 119891])) le 120595 (120601 (0) 120601 (119900) 120601 (119889 (] 119891]))) minus 120585 (119889 (] 119891]))

le 120601 (119889 (] 119891])) minus 120585 (119889 (] 119891]))

(32)

which implies that 119889(] 119891]) = 0 that is ] = 119891] So wecomplete the proof

If we let

120595 (120601 (119889 (119909 119910)) 120601 (119889 (119909 119891119909)) 120601 (119889 (119910 119891119910)))

= max 120601 (119889 (119909 119910)) 120601 (119889 (119909 119891119909)) 120601 (119889 (119910 119891119910)) (33)

it is easy to get the following theorem

Theorem 6 Let (119883 ⊑ 119889) be a partially ordered completemetric space Let 119891 119883 rarr 119883 be monotone nondecreasingand

120593 (119889 (119891119909 119891119910))

le max 120601 (119889 (119909 119910)) 120601 (119889 (119909 119891119909)) 120601 (119889 (119910 119891119910))

minus 120585 (max 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910))

(34)

for all comparable 119909 119910 isin 119883 where 120593 isin Θ 120601 isin Φ and 120585 isin Ξand

120593 (119905) minus 120601 (119905) + 120585 (119905) gt 0 forall119905 gt 0

120593 (119905) = 0 119894119891 119886119899119889 119900119899119897119910 119894119891 119905 = 0 120601 (0) = 120585 (0) = 0(35)

Suppose that either

(a) 119891 is continuous or(b) if any nondecreasing sequence 119909

119899 in119883 converges to ]

then one assumes that

119909119899⊑ ] forall119899 isin N (36)

If there exists 1199090isin 119883 with 119909

0⊑ 1198911199090 then 119891 has a fixed point

in119883

3 Fixed Point Results (II)

In the section we denote by Ψ the class of functions 120595

R+3

rarr R+ satisfying the following conditions

(1205951) 120595 is an increasing and continuous function in eachcoordinate

(1205952) for 119905 isin R+ 120601(119905 119905 119905) le 119905 120601(119905 0 0) le 119905 and 120601(0 0 119905) le119905

Next we denote byΘ the class of functions120593 R+ rarr R+

satisfying the following conditions

(1205931) 120593 is continuous and nondecreasing

(1205932) for 119905 gt 0 120593(119905) gt 0 and 120593(0) = 0

And we denote byΦ the class of functions 120601 R+ rarr R+

satisfying the following conditions

(1206011) 120601 is continuous

(1206012) for 119905 gt 0 120601(119905) gt 0 and 120601(0) = 0

We now state the main fixed point theorem for the(120595 120593 120601)-contractions in partially ordered metric spaces asfollows

Theorem 7 Let (119883 ⊑ 119889) be a partially ordered completemetric space and let 119891 119883 rarr 119883 be monotone nondecreasingand

120593 (119889 (119891119909 119891119910))

le 120595 (120593 (119889 (119909 119910)) 120593 (119889 (119909 119891119909)) 120593 (119889 (119910 119891119910)))

minus 120601 (119872 (119909 119910)) + 119871 sdot 119898 (119909 119910)

(37)

for all comparable 119909 119910 isin 119883 and 120595 isin Ψ 120593 isin Θ 120601 isin Φ where119871 gt 0 and

119872(119909 119910) = max 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910)

119898 (119909 119910)

= min 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910) 119889 (119909 119891119910)

119889 (119910 119891119909)

(38)

Suppose that either

(a) 119891 is continuous or(b) if any nondecreasing sequence 119909

119899 in119883 converges to ]

then one assumes that

119909119899⊑ ] forall119899 isin N (39)

If there exists 1199090isin 119883 with 119909

0⊑ 1198911199090 then 119891 has a fixed point

in 119883

Proof If 1198911199090

= 1199090 then the proof is finished Suppose

that 1199090⊏ 1198911199090 Since 119891 is nondecreasing by induction we

construct the sequence 119909119899 recursively as

119909119899= 119891119899

1199090= 119891119909119899minus1

forall119899 isin N (40)

Thus we also conclude that

1199090⊏ 1199091= 1198911199090⊑ 1199092= 1198911199091⊑ sdot sdot sdot ⊑ 119909

119899= 119891119909119899minus1

⊑ sdot sdot sdot (41)

We now claim that

lim119899rarrinfin

119889 (119909119899 119909119899+1

) = 0 (42)

Journal of Applied Mathematics 5

Put 119909 = 119909119899minus1

and 119910 = 119909119899in (37) Note that

119898(119909119899minus1

119909119899)

= min 119889 (119909119899minus1

119909119899) 119889 (119909

119899minus1 119891119909119899minus1

)

119889 (119909119899 119891119909119899) 119889 (119909

119899minus1 119891119909119899) 119889 (119909

119899 119891119909119899minus1

)

= min 119889 (119909119899minus1

119909119899) 119889 (119909

119899minus1 119909119899) 119889 (119909

119899 119909119899+1

)

119889 (119909119899minus1

119909119899+1

) 119889 (119909119899 119909119899)

= 0

(43)

So we obtain that

120593 (119889 (119909119899 119909119899+1

))

= 120593 (119889 (119891119909119899minus1

119891119909119899))

le 120595 (120593 (119889 (119909119899minus1

119909119899)) 120593 (119889 (119909

119899minus1 119891119909119899minus1

))

120593 (119889 (119909119899 119891119909119899))) minus 120601 (119872 (119909

119899minus1 119909119899))

le 120595 (120593 (119889 (119909119899minus1

119909119899)) 120593 (119889 (119909

119899minus1 119909119899)) 120593 (119889 (119909

119899 119909119899+1

)))

minus 120601 (119872 (119909119899minus1

119909119899))

(44)

where

119872(119909119899minus1

119909119899)

= max 119889 (119909119899minus1

119909119899) 119889 (119909

119899minus1 119891119909119899minus1

) 119889 (119909119899 119891119909119899)

= max 119889 (119909119899minus1

119909119899) 119889 (119909

119899minus1 119909119899) 119889 (119909

119899 119909119899+1

)

(45)

We now claim that

119889 (119909119899 119909119899+1

) lt 119889 (119909119899minus1

119909119899) forall119899 isin N (46)

If not we assume that 119889(119909119899minus1

119909119899) le 119889(119909

119899 119909119899+1

) then120593(119889(119909

119899minus1 119909119899)) le 120593(119889(119909

119899 119909119899+1

)) since 120593 is non-decreasingUsing inequality (44) and the conditions of the function 120595we have that for each 119899 isin N

120593 (119889 (119909119899 119909119899+1

)) le 120593 (119889 (119909119899 119909119899+1

)) minus 120601 (119889 (119909119899 119909119899+1

)) (47)

which implies that120601(119889(119909119899 119909119899+1

))= 0 and hence119889(119909119899 119909119899+1

)=0 This contradicts our initial assumption

From the previous argument we have that for each 119899 isin N

120593 (119889 (119909119899 119909119899+1

)) le 120593 (119889 (119909119899minus1

119909119899)) minus 120601 (119889 (119909

119899minus1 119909119899))

119889 (119909119899 119909119899+1

) lt 119889 (119909119899minus1

119909119899)

(48)

And since the sequence 119889(119909119899 119909119899+1

) is decreasing it mustconverge to some 120578 ge 0 Taking limit as 119899 rarr infin in (48) andby the continuity of 120593 and 120601 we get

120593 (120578) le 120593 (120578) minus 120601 (120578) (49)

and so we conclude that 120601(120578) = 0 and 120578 = 0We next claim that 119909

119899 is Cauchy that is for every 120576 gt 0

there exists 119899 isin N such that if 119901 119902 ge 119899 then 119889(119909119901 119909119902) lt 120576

Suppose on the contrary that there exists 120598 gt 0 such thatfor any 119899 isin N there are 119901

119899 119902119899isin Nwith 119901

119899gt 119902119899ge 119899 satisfying

119889 (119909119902119899

119909119901119899

) ge 120598 (50)

Further corresponding to 119902119899ge 119899 we can choose 119901

119899in such

a way that it the smallest integer with 119901119899gt 119902119899ge 119899 and

119889(119909119902119899

119909119901119899

) ge 120598 Therefore 119889(119909119902119899

119909119901119899minus1) lt 120598 By the rectan-

gular inequality we have

120598 le 119889 (119909119901119899

119909119902119899

)

le 119889 (119909119901119899

119909119901119899minus1) + 119889 (119909

119901119899minus1 119909119902119899

)

lt 119889 (119909119901119899

119909119901119899minus1) + 120598

(51)

Letting 119899 rarr infin then we get

lim119899rarrinfin

119889 (119909119901119899

119909119902119899

) = 120598 (52)

On the other hand we have

119889 (119909119901119899

119909119902119899

)

le 119889 (119909119901119899

119909119901119899minus1) + 119889 (119909

119901119899minus1 119909119902119899minus1) + 119889 (119909

119902119899minus1 119909119902119899

)

119889 (119909119901119899minus1 119909119902119899minus1)

le 119889 (119909119901119899minus1 119909119901119899

) + 119889 (119909119901119899

119909119902119899

) + 119889 (119909119902119899

119909119902119899minus1)

(53)

By letting 119899 rarr infin we get that

lim119899rarrinfin

119889 (119909119901119899minus1 119909119902119899minus1) = 120598 (54)

Using inequalities (37) (52) and (54) and putting 119909 = 119909119901119899minus1

and 119910 = 119909119902119899minus1 we have that

120593 (119889 (119909119901119899

119909119902119899

))

= 120593 (119889 (119891119909119901119899minus1 119891119909119902119899minus1))

le 120595 (120593 (119889 (119909119901119899minus1 119909119902119899minus1)) 120593 (119889 (119909

119901119899minus1 119891119909119901119899minus1))

120593 (119889 (119909119902119899minus1 119891119909119902119899minus1)))

minus 120601 (119872(119909119901119899minus1 119909119902119899minus1)) + 119871 sdot 119898 (119909

119901119899minus1 119909119902119899minus1)

= 120595 (120593 (119889 (119909119901119899minus1 119909119902119899minus1)) 120593 (119889 (119909

119901119899minus1 119909119901119899

))

120593 (119889 (119909119902119899minus1 119909119902119899

)))

minus 120601 (119872(119909119901119899minus1 119909119902119899minus1)) + 119871 sdot 119898 (119909

119901119899minus1 119909119902119899minus1)

(55)

6 Journal of Applied Mathematics

where

119872(119909119901119899minus1 119909119902119899minus1)

= max 119889 (119909119901119899minus1 119909119902119899minus1) 119889 (119909

119901119899minus1 119909119901119899

) 119889 (119909119902119899minus1 119909119902119899

)

119898 (119909119901119899minus1 119909119902119899minus1)

= min 119889 (119909119901119899minus1 119909119902119899minus1) 119889 (119909

119901119899minus1 119909119901119899

)

119889 (119909119902119899minus1 119909119902119899

) 119889 (119909119901119899minus1 119909119902119899

) 119889 (119909119902119899minus1 119909119901119899

)

(56)

Letting 119899 rarr infin then we obtain that

lim119899rarrinfin

119872(119909119901119899minus1 119909119902119899minus1) = 120598

lim119899rarrinfin

119898(119909119901119899minus1 119909119902119899minus1) = 0

120593 (120598) le 120595 (120593 (120598) 0 0) minus 120601 (120598) le 120593 (120598) minus 120601 (120598)

(57)

This implies that 120601(120598) = 0 and hence 120598 = 0 So we get acontraction Therefore 119909

119899 is a Cauchy sequence

Since119883 is complete there exists ] isin 119883 such that

lim119899rarrinfin

119909119899= ] (58)

Suppose that (a) holds Then

] = lim119899rarrinfin

119909119899+1

= lim119899rarrinfin

119891119909119899= 119891] (59)

Thus ] is a fixed point in119883Suppose that (b) holds that is 119909

119899⊑ ] for all 119899 isin N

Substituting 119909 = 119909119899and 119910 = ] in (37) we have that

120593 (119889 (119909119899+1

119891]))

= 120593 (119889 (119891119909119899 119891]))

le 120595 (120593 (119889 (119909119899 ])) 120593 (119889 (119909

119899 119891119909119899)) 120593 (119889 (] 119891])))

minus 120601 (119872 (119909119899 ])) + 119871 sdot 119898 (119909

119899 ])

(60)

where

119872(119909119899 ]) = max 119889 (119909

119899 ]) 119889 (119909

119899 119891119909119899) 119889 (] 119891])

119898 (119909119899 ])

= min 119889 (119909119899 ]) 119889 (119909

119899 119891119909119899) 119889 (] 119891]) 119889 (119909

119899 119891])

119889 (] 119891119909119899)

(61)

Letting 119899 rarr infin then we obtain that

119872(119909119899 ]) 997888rarr 119889 (] 119891]) 119898 (119909

119899 ]) 997888rarr 0

120593 (119889 (] 119891])) le 120595 (120593 (0) 120593 (0) 120593 (119889 (] 119891]))) minus 120601 (119889 (] 119891]))

le 120593 (119889 (] 119891])) minus 120601 (119889 (] 119891]))

(62)

which implies that 119889(] 119891]) = 0 that is ] = 119891] So wecomplete the proof

If we let

120595 (120593 (119889 (119909 119910)) 120593 (119889 (119909 119891119909)) 120593 (119889 (119910 119891119910)))

= max 120593 (119889 (119909 119910)) 120593 (119889 (119909 119891119909)) 120593 (119889 (119910 119891119910)) (63)

it is easy to get the following theorem

Theorem 8 Let (119883 ⊑ 119889) be a partially ordered completemetric space and let 119891 119883 rarr 119883 be monotone nondecreasingand

120593 (119889 (119891119909 119891119910))

le max 120593 (119889 (119909 119910)) 120593 (119889 (119909 119891119909)) 120593 (119889 (119910 119891119910))

minus 120601 (119872 (119909 119910)) + 119871 sdot 119898 (119909 119910)

(64)

for all comparable 119909 119910 isin 119883 and 120593 isin Θ 120601 isin Φ where 119871 gt 0

and

119872(119909 119910) = max 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910)

119898 (119909 119910) = min 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910) 119889 (119909 119891119910)

119889 (119910 119891119909)

(65)

Suppose that either

(a) 119891 is continuous or(b) if any nondecreasing sequence 119909

119899 in119883 converges to ]

then one assumes that

119909119899⊑ ] forall119899 isin N (66)

If there exists 1199090isin 119883 with 119909

0⊑ 1198911199090 then 119891 has a fixed point

in 119883

Acknowledgment

The authors would like to thank the referee(s) for the manyuseful comments and suggestions for the improvement of thepaper

References

[1] S Banach ldquoSur les operations dans les ensembles abstraitset leur application aux equations integeralesrdquo FundamentaMathematicae vol 3 pp 133ndash181 1922

[2] I Ya Alber and S Guerre-Delabriere ldquoPrinciples of weaklycontractive maps in Hilbert spacesrdquo in New Results in OperatorTheory I Gohberg and Yu Lyubich Eds vol 98 of Advancesand Applications pp 7ndash22 Birkhauser Basel Switzerland 1997

[3] M Eslamian and A Abkar ldquoA fixed point theorem for gener-alized weakly contractive mappings in complete metric space rdquoItalian Journal of Pure and Applied Mathematics In press

[4] P N Dutta and B S Choudhury ldquoA generalisation of con-traction principle in metric spacesrdquo Fixed Point Theory andApplications vol 2008 Article ID 406368 8 pages 2008

[5] H Aydi E Karapınar and B Samet ldquoRemarks on some recentfixed point theoremsrdquo Fixed Point Theory and Applications vol2012 article 76 2012

Journal of Applied Mathematics 7

[6] H Aydi E Karapınar and W Shatanawi ldquoCoupled fixed pointresults for (120593 120595)-weakly contractive condition in ordered par-tialmetric spacesrdquoComputers ampMathematics with Applicationsvol 62 no 12 pp 4449ndash4460 2011

[7] W-S Du ldquoOn coincidence point and fixed point theorems fornonlinear multivaluedmapsrdquo Topology and its Applications vol159 no 1 pp 49ndash56 2012

[8] W-S Du ldquoOn generalized weakly directional contractions andapproximate fixed point propertywith applicationsrdquo Fixed PointTheory and Applications vol 2012 article 6 2012

[9] W-S Du and S-X Zheng ldquoNonlinear conditions for coinci-dence point and fixed point theoremsrdquo Taiwanese Journal ofMathematics vol 16 no 3 pp 857ndash868 2012

[10] W-S Du and S-X Zheng ldquoNew nonlinear conditions andinequalities for the existence of coincidence points and fixedpointsrdquo Journal of Applied Mathematics vol 2012 Article ID196759 12 pages 2012

[11] E Karapinar and K Sadarangani ldquoFixed point theory for cyclic(120593 120595)-contractionsrdquo Fixed Point Theory and Applications vol2011 article 69 2011

[12] E Karapınar ldquoFixed point theory for cyclic weak 120601-contractionrdquo Applied Mathematics Letters vol 24 no 6pp 822ndash825 2011

[13] E Karapınar and I S Yuce ldquoFixed point theory for cyclic gen-eralized weak 120601-contraction on partial metric spacesrdquo AbstractandAppliedAnalysis vol 2012 Article ID491542 12 pages 2012

[14] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis vol 47 no 4 pp 2683ndash2693 2001

[15] B Djafari Rouhani and S Moradi ldquoCommon fixed point ofmultivalued generalized 120593-weak contractive mappingsrdquo FixedPoint Theory and Applications Article ID 708984 201013 pages2010

[16] Q Zhang and Y Song ldquoFixed point theory for generalized 120601-weak contractionsrdquo Applied Mathematics Letters vol 22 no 1pp 75ndash78 2009

[17] L B Ciric N Cakic M Rajovic and J S Ume ldquoMonotonegeneralized nonlinear contractions in partially ordered metricspacesrdquo Fixed Point Theory and Applications vol 2008 ArticleID 131294 11 pages 2008

[18] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysis Theory Methods amp Applications A vol 65 no 7 pp1379ndash1393 2006

[19] J Harjani B Lopez and K Sadarangani ldquoFixed point theoremsfor weakly C-contractive mappings in ordered metric spacesrdquoComputers amp Mathematics with Applications vol 61 no 4 pp790ndash796 2011

[20] J Harjani B Lopez and K Sadarangani ldquoA fixed point theoremfor mappings satisfying a contractive condition of rationaltype on a partially ordered metric spacerdquo Abstract and AppliedAnalysis vol 2010 Article ID 190701 8 pages 2010

[21] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[22] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrixequationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004

[23] B S Choudhury and A Kundu ldquo(120595 120572 120573)-weak contractions inpartially ordered metric spacesrdquo Applied Mathematics Lettersvol 25 no 1 pp 6ndash10 2012

Submit your manuscripts athttpwwwhindawicom

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

substituting 119909 = 119909119899and 119910 = 119909

119899+1in (10) and using the

definition of the function 120595 we have

120595 (120601 (119889 (119909119899 119909119899+1

)) 120601 (119889 (119909119899 119891119909119899)) 120601 (119889 (119909

119899+1 119891119909119899+1

)))

= 120595 (120601 (119889 (119909119899 119909119899+1

)) 120601 (119889 (119909119899 119909119899+1

)) 120601 (119889 (119909119899+1

119909119899+2

)))

le 120601 (119889 (119909119899+1

119909119899+2

))

120585 (max 119889 (119909119899 119909119899+1

) 119889 (119909119899 119891119909119899) 119889 (119909

119899+1 119891119909119899+1

))

= 120585 (max 119889 (119909119899 119909119899+1

) 119889 (119909119899 119909119899+1

) 119889 (119909119899+1

119909119899+2

))

= 120585 (119889 (119909119899+1

119909119899+2

))

(16)

and hence

120593 (119889 (119909119899+1

119909119899+2

))

= 120593 (119889 (119891119909119899 119891119909119899+1

))

le 120601 (119889 (119909119899+1

119909119899+2

)) minus 120585 (119889 (119909119899+1

119909119899+2

))

(17)

Since 120593(119905) minus 120601(119905) + 120585(119905) gt 0 for all 119905 gt 0 we have that119889(119909119899+1

119909119899+2

) = 0 which contradicts (15) Therefore weconclude that

119889 (119909119899 119909119899+1

) le 119889 (119909119899minus1

119909119899) forall119899 isin N (18)

From the previous argument we also have that for each 119899 isin N

120593 (119889 (119909119899 119909119899+1

)) le 120601 (119889 (119909119899minus1

119909119899)) minus 120585 (119889 (119909

119899minus1 119909119899)) (19)

It follows in (18) that the sequence 119889(119909119899 119909119899+1

) is monotonedecreasing it must converge to some 120578 ge 0 Taking limit as119899 rarr infin in (19) and using the continuities of 120593 and 120601 and thelower semicontinuous of 120585 we get

120593 (120578) le 120601 (120578) minus 120585 (120578) (20)

which implies that 120578 = 0 So we conclude that

lim119899rarrinfin

119889 (119909119899 119909119899+1

) = 0 (21)

We next claim that 119909119899 is a Cauchy sequence that is for

every 120576 gt 0 there exists 119899 isin N such that if 119901 119902 ge 119899 then119889(119909119901 119909119902) lt 120576

Suppose on the contrary that there exists 120598 gt 0 such thatfor any 119899 isin N there are 119901

119899 119902119899isin Nwith 119901

119899gt 119902119899ge 119899 satisfying

119889 (119909119902119899

119909119901119899

) ge 120598 (22)

Further corresponding to 119902119899ge 119899 we can choose 119901

119899in such

a way that it the smallest integer with 119901119899gt 119902119899ge 119899 and

119889(119909119902119899

119909119901119899

) ge 120598 Therefore 119889(119909119902119899

119909119901119899minus1) lt 120598 Now we have

that for all 119899 isin N

120598 le 119889 (119909119901119899

119909119902119899

)

le 119889 (119909119901119899

119909119901119899minus1) + 119889 (119909

119901119899minus1 119909119902119899

)

lt 119889 (119909119901119899

119909119901119899minus1) + 120598

(23)

By letting 119899 rarr infin we get that

lim119899rarrinfin

119889 (119909119901119899

119909119902119899

) = 120598 (24)

On the other hand we have

119889 (119909119901119899

119909119902119899

)

le 119889 (119909119901119899

119909119901119899minus1) + 119889 (119909

119901119899minus1 119909119902119899minus1) + 119889 (119909

119902119899minus1 119909119902119899

)

119889 (119909119901119899minus1 119909119902119899minus1)

le 119889 (119909119901119899minus1 119909119901119899

) + 119889 (119909119901119899

119909119902119899

) + 119889 (119909119902119899

119909119902119899minus1)

(25)

Letting 119899 rarr infin then we get

lim119899rarrinfin

119889 (119909119901119899minus1 119909119902119899minus1) = 120598 (26)

By (14) we have that the elements119909119901119899

and119909119902119899

are comparableSubstituting 119909 = 119909

119901119899minus1

and 119910 = 119909119902119899minus1

in (10) we have that forall 119899 isin N

120595 (120601 (119889 (119909119901119899minus1 119909119902119899minus1)) 120601 (119889 (119909

119901119899minus1 119891119909119901119899minus1))

120601 (119889 (119909119902119899minus1 119891119909119902119899minus1)))

le 120595 (120601 (119889 (119909119901119899minus1 119909119902119899minus1)) 120601 (119889 (119909

119901119899minus1 119909119901119899

))

120601 (119889 (119909119902119899minus1 119909119902119899

)))

119872 (119909119901119899minus1 119909119902119899minus1)

= max 119889 (119909119901119899minus1 119909119902119899minus1) 119889 (119909

119901119899minus1 119891119909119901119899minus1)

119889 (119909119902119899minus1 119891119909119902119899minus1)

= max 119889 (119909119901119899minus1 119909119902119899minus1) 119889 (119909

119901119899minus1 119909119901119899

)

119889 (119909119902119899minus1 119909119902119899

)

(27)

By the previous argument and using inequality (10) we canconclude that

120593 (120598) le 120595 (120601 (120598) 0 0) minus 120585 (120598)

le 120601 (120598) minus 120585 (120598) (28)

which implies that 120598 = 0 a contradiction Therefore thesequence 119909

119899 is a Cauchy sequence

Since119883 is complete there exists ] isin 119883 such that

lim119899rarrinfin

119909119899= ] (29)

Suppose that (a) holds Then

] = lim119899rarrinfin

119909119899+1

= lim119899rarrinfin

119891119909119899= 119891] (30)

Thus ] is a fixed point in119883

4 Journal of Applied Mathematics

Suppose that (b) holds that is 119909119899⊑ ] for all 119899 isin N

Substituting 119909 = 119909119899and 119910 = ] in (10) we have that

120593 (119889 (119909119899+1

119891]))

= 120593 (119889 (119891119909119899 119891]))

le 120595 (120601 (119889 (119909119899 ])) 120601 (119889 (119909

119899 119891119909119899)) 120601 (119889 (] 119891])))

minus 120585 (max 119889 (119909119899 ]) 119889 (119909

119899 119891119909119899) 119889 (] 119891]))

(31)

Taking limit as 119899 rarr infin in equality (31) we have

120593 (119889 (] 119891])) le 120595 (120601 (0) 120601 (119900) 120601 (119889 (] 119891]))) minus 120585 (119889 (] 119891]))

le 120601 (119889 (] 119891])) minus 120585 (119889 (] 119891]))

(32)

which implies that 119889(] 119891]) = 0 that is ] = 119891] So wecomplete the proof

If we let

120595 (120601 (119889 (119909 119910)) 120601 (119889 (119909 119891119909)) 120601 (119889 (119910 119891119910)))

= max 120601 (119889 (119909 119910)) 120601 (119889 (119909 119891119909)) 120601 (119889 (119910 119891119910)) (33)

it is easy to get the following theorem

Theorem 6 Let (119883 ⊑ 119889) be a partially ordered completemetric space Let 119891 119883 rarr 119883 be monotone nondecreasingand

120593 (119889 (119891119909 119891119910))

le max 120601 (119889 (119909 119910)) 120601 (119889 (119909 119891119909)) 120601 (119889 (119910 119891119910))

minus 120585 (max 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910))

(34)

for all comparable 119909 119910 isin 119883 where 120593 isin Θ 120601 isin Φ and 120585 isin Ξand

120593 (119905) minus 120601 (119905) + 120585 (119905) gt 0 forall119905 gt 0

120593 (119905) = 0 119894119891 119886119899119889 119900119899119897119910 119894119891 119905 = 0 120601 (0) = 120585 (0) = 0(35)

Suppose that either

(a) 119891 is continuous or(b) if any nondecreasing sequence 119909

119899 in119883 converges to ]

then one assumes that

119909119899⊑ ] forall119899 isin N (36)

If there exists 1199090isin 119883 with 119909

0⊑ 1198911199090 then 119891 has a fixed point

in119883

3 Fixed Point Results (II)

In the section we denote by Ψ the class of functions 120595

R+3

rarr R+ satisfying the following conditions

(1205951) 120595 is an increasing and continuous function in eachcoordinate

(1205952) for 119905 isin R+ 120601(119905 119905 119905) le 119905 120601(119905 0 0) le 119905 and 120601(0 0 119905) le119905

Next we denote byΘ the class of functions120593 R+ rarr R+

satisfying the following conditions

(1205931) 120593 is continuous and nondecreasing

(1205932) for 119905 gt 0 120593(119905) gt 0 and 120593(0) = 0

And we denote byΦ the class of functions 120601 R+ rarr R+

satisfying the following conditions

(1206011) 120601 is continuous

(1206012) for 119905 gt 0 120601(119905) gt 0 and 120601(0) = 0

We now state the main fixed point theorem for the(120595 120593 120601)-contractions in partially ordered metric spaces asfollows

Theorem 7 Let (119883 ⊑ 119889) be a partially ordered completemetric space and let 119891 119883 rarr 119883 be monotone nondecreasingand

120593 (119889 (119891119909 119891119910))

le 120595 (120593 (119889 (119909 119910)) 120593 (119889 (119909 119891119909)) 120593 (119889 (119910 119891119910)))

minus 120601 (119872 (119909 119910)) + 119871 sdot 119898 (119909 119910)

(37)

for all comparable 119909 119910 isin 119883 and 120595 isin Ψ 120593 isin Θ 120601 isin Φ where119871 gt 0 and

119872(119909 119910) = max 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910)

119898 (119909 119910)

= min 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910) 119889 (119909 119891119910)

119889 (119910 119891119909)

(38)

Suppose that either

(a) 119891 is continuous or(b) if any nondecreasing sequence 119909

119899 in119883 converges to ]

then one assumes that

119909119899⊑ ] forall119899 isin N (39)

If there exists 1199090isin 119883 with 119909

0⊑ 1198911199090 then 119891 has a fixed point

in 119883

Proof If 1198911199090

= 1199090 then the proof is finished Suppose

that 1199090⊏ 1198911199090 Since 119891 is nondecreasing by induction we

construct the sequence 119909119899 recursively as

119909119899= 119891119899

1199090= 119891119909119899minus1

forall119899 isin N (40)

Thus we also conclude that

1199090⊏ 1199091= 1198911199090⊑ 1199092= 1198911199091⊑ sdot sdot sdot ⊑ 119909

119899= 119891119909119899minus1

⊑ sdot sdot sdot (41)

We now claim that

lim119899rarrinfin

119889 (119909119899 119909119899+1

) = 0 (42)

Journal of Applied Mathematics 5

Put 119909 = 119909119899minus1

and 119910 = 119909119899in (37) Note that

119898(119909119899minus1

119909119899)

= min 119889 (119909119899minus1

119909119899) 119889 (119909

119899minus1 119891119909119899minus1

)

119889 (119909119899 119891119909119899) 119889 (119909

119899minus1 119891119909119899) 119889 (119909

119899 119891119909119899minus1

)

= min 119889 (119909119899minus1

119909119899) 119889 (119909

119899minus1 119909119899) 119889 (119909

119899 119909119899+1

)

119889 (119909119899minus1

119909119899+1

) 119889 (119909119899 119909119899)

= 0

(43)

So we obtain that

120593 (119889 (119909119899 119909119899+1

))

= 120593 (119889 (119891119909119899minus1

119891119909119899))

le 120595 (120593 (119889 (119909119899minus1

119909119899)) 120593 (119889 (119909

119899minus1 119891119909119899minus1

))

120593 (119889 (119909119899 119891119909119899))) minus 120601 (119872 (119909

119899minus1 119909119899))

le 120595 (120593 (119889 (119909119899minus1

119909119899)) 120593 (119889 (119909

119899minus1 119909119899)) 120593 (119889 (119909

119899 119909119899+1

)))

minus 120601 (119872 (119909119899minus1

119909119899))

(44)

where

119872(119909119899minus1

119909119899)

= max 119889 (119909119899minus1

119909119899) 119889 (119909

119899minus1 119891119909119899minus1

) 119889 (119909119899 119891119909119899)

= max 119889 (119909119899minus1

119909119899) 119889 (119909

119899minus1 119909119899) 119889 (119909

119899 119909119899+1

)

(45)

We now claim that

119889 (119909119899 119909119899+1

) lt 119889 (119909119899minus1

119909119899) forall119899 isin N (46)

If not we assume that 119889(119909119899minus1

119909119899) le 119889(119909

119899 119909119899+1

) then120593(119889(119909

119899minus1 119909119899)) le 120593(119889(119909

119899 119909119899+1

)) since 120593 is non-decreasingUsing inequality (44) and the conditions of the function 120595we have that for each 119899 isin N

120593 (119889 (119909119899 119909119899+1

)) le 120593 (119889 (119909119899 119909119899+1

)) minus 120601 (119889 (119909119899 119909119899+1

)) (47)

which implies that120601(119889(119909119899 119909119899+1

))= 0 and hence119889(119909119899 119909119899+1

)=0 This contradicts our initial assumption

From the previous argument we have that for each 119899 isin N

120593 (119889 (119909119899 119909119899+1

)) le 120593 (119889 (119909119899minus1

119909119899)) minus 120601 (119889 (119909

119899minus1 119909119899))

119889 (119909119899 119909119899+1

) lt 119889 (119909119899minus1

119909119899)

(48)

And since the sequence 119889(119909119899 119909119899+1

) is decreasing it mustconverge to some 120578 ge 0 Taking limit as 119899 rarr infin in (48) andby the continuity of 120593 and 120601 we get

120593 (120578) le 120593 (120578) minus 120601 (120578) (49)

and so we conclude that 120601(120578) = 0 and 120578 = 0We next claim that 119909

119899 is Cauchy that is for every 120576 gt 0

there exists 119899 isin N such that if 119901 119902 ge 119899 then 119889(119909119901 119909119902) lt 120576

Suppose on the contrary that there exists 120598 gt 0 such thatfor any 119899 isin N there are 119901

119899 119902119899isin Nwith 119901

119899gt 119902119899ge 119899 satisfying

119889 (119909119902119899

119909119901119899

) ge 120598 (50)

Further corresponding to 119902119899ge 119899 we can choose 119901

119899in such

a way that it the smallest integer with 119901119899gt 119902119899ge 119899 and

119889(119909119902119899

119909119901119899

) ge 120598 Therefore 119889(119909119902119899

119909119901119899minus1) lt 120598 By the rectan-

gular inequality we have

120598 le 119889 (119909119901119899

119909119902119899

)

le 119889 (119909119901119899

119909119901119899minus1) + 119889 (119909

119901119899minus1 119909119902119899

)

lt 119889 (119909119901119899

119909119901119899minus1) + 120598

(51)

Letting 119899 rarr infin then we get

lim119899rarrinfin

119889 (119909119901119899

119909119902119899

) = 120598 (52)

On the other hand we have

119889 (119909119901119899

119909119902119899

)

le 119889 (119909119901119899

119909119901119899minus1) + 119889 (119909

119901119899minus1 119909119902119899minus1) + 119889 (119909

119902119899minus1 119909119902119899

)

119889 (119909119901119899minus1 119909119902119899minus1)

le 119889 (119909119901119899minus1 119909119901119899

) + 119889 (119909119901119899

119909119902119899

) + 119889 (119909119902119899

119909119902119899minus1)

(53)

By letting 119899 rarr infin we get that

lim119899rarrinfin

119889 (119909119901119899minus1 119909119902119899minus1) = 120598 (54)

Using inequalities (37) (52) and (54) and putting 119909 = 119909119901119899minus1

and 119910 = 119909119902119899minus1 we have that

120593 (119889 (119909119901119899

119909119902119899

))

= 120593 (119889 (119891119909119901119899minus1 119891119909119902119899minus1))

le 120595 (120593 (119889 (119909119901119899minus1 119909119902119899minus1)) 120593 (119889 (119909

119901119899minus1 119891119909119901119899minus1))

120593 (119889 (119909119902119899minus1 119891119909119902119899minus1)))

minus 120601 (119872(119909119901119899minus1 119909119902119899minus1)) + 119871 sdot 119898 (119909

119901119899minus1 119909119902119899minus1)

= 120595 (120593 (119889 (119909119901119899minus1 119909119902119899minus1)) 120593 (119889 (119909

119901119899minus1 119909119901119899

))

120593 (119889 (119909119902119899minus1 119909119902119899

)))

minus 120601 (119872(119909119901119899minus1 119909119902119899minus1)) + 119871 sdot 119898 (119909

119901119899minus1 119909119902119899minus1)

(55)

6 Journal of Applied Mathematics

where

119872(119909119901119899minus1 119909119902119899minus1)

= max 119889 (119909119901119899minus1 119909119902119899minus1) 119889 (119909

119901119899minus1 119909119901119899

) 119889 (119909119902119899minus1 119909119902119899

)

119898 (119909119901119899minus1 119909119902119899minus1)

= min 119889 (119909119901119899minus1 119909119902119899minus1) 119889 (119909

119901119899minus1 119909119901119899

)

119889 (119909119902119899minus1 119909119902119899

) 119889 (119909119901119899minus1 119909119902119899

) 119889 (119909119902119899minus1 119909119901119899

)

(56)

Letting 119899 rarr infin then we obtain that

lim119899rarrinfin

119872(119909119901119899minus1 119909119902119899minus1) = 120598

lim119899rarrinfin

119898(119909119901119899minus1 119909119902119899minus1) = 0

120593 (120598) le 120595 (120593 (120598) 0 0) minus 120601 (120598) le 120593 (120598) minus 120601 (120598)

(57)

This implies that 120601(120598) = 0 and hence 120598 = 0 So we get acontraction Therefore 119909

119899 is a Cauchy sequence

Since119883 is complete there exists ] isin 119883 such that

lim119899rarrinfin

119909119899= ] (58)

Suppose that (a) holds Then

] = lim119899rarrinfin

119909119899+1

= lim119899rarrinfin

119891119909119899= 119891] (59)

Thus ] is a fixed point in119883Suppose that (b) holds that is 119909

119899⊑ ] for all 119899 isin N

Substituting 119909 = 119909119899and 119910 = ] in (37) we have that

120593 (119889 (119909119899+1

119891]))

= 120593 (119889 (119891119909119899 119891]))

le 120595 (120593 (119889 (119909119899 ])) 120593 (119889 (119909

119899 119891119909119899)) 120593 (119889 (] 119891])))

minus 120601 (119872 (119909119899 ])) + 119871 sdot 119898 (119909

119899 ])

(60)

where

119872(119909119899 ]) = max 119889 (119909

119899 ]) 119889 (119909

119899 119891119909119899) 119889 (] 119891])

119898 (119909119899 ])

= min 119889 (119909119899 ]) 119889 (119909

119899 119891119909119899) 119889 (] 119891]) 119889 (119909

119899 119891])

119889 (] 119891119909119899)

(61)

Letting 119899 rarr infin then we obtain that

119872(119909119899 ]) 997888rarr 119889 (] 119891]) 119898 (119909

119899 ]) 997888rarr 0

120593 (119889 (] 119891])) le 120595 (120593 (0) 120593 (0) 120593 (119889 (] 119891]))) minus 120601 (119889 (] 119891]))

le 120593 (119889 (] 119891])) minus 120601 (119889 (] 119891]))

(62)

which implies that 119889(] 119891]) = 0 that is ] = 119891] So wecomplete the proof

If we let

120595 (120593 (119889 (119909 119910)) 120593 (119889 (119909 119891119909)) 120593 (119889 (119910 119891119910)))

= max 120593 (119889 (119909 119910)) 120593 (119889 (119909 119891119909)) 120593 (119889 (119910 119891119910)) (63)

it is easy to get the following theorem

Theorem 8 Let (119883 ⊑ 119889) be a partially ordered completemetric space and let 119891 119883 rarr 119883 be monotone nondecreasingand

120593 (119889 (119891119909 119891119910))

le max 120593 (119889 (119909 119910)) 120593 (119889 (119909 119891119909)) 120593 (119889 (119910 119891119910))

minus 120601 (119872 (119909 119910)) + 119871 sdot 119898 (119909 119910)

(64)

for all comparable 119909 119910 isin 119883 and 120593 isin Θ 120601 isin Φ where 119871 gt 0

and

119872(119909 119910) = max 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910)

119898 (119909 119910) = min 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910) 119889 (119909 119891119910)

119889 (119910 119891119909)

(65)

Suppose that either

(a) 119891 is continuous or(b) if any nondecreasing sequence 119909

119899 in119883 converges to ]

then one assumes that

119909119899⊑ ] forall119899 isin N (66)

If there exists 1199090isin 119883 with 119909

0⊑ 1198911199090 then 119891 has a fixed point

in 119883

Acknowledgment

The authors would like to thank the referee(s) for the manyuseful comments and suggestions for the improvement of thepaper

References

[1] S Banach ldquoSur les operations dans les ensembles abstraitset leur application aux equations integeralesrdquo FundamentaMathematicae vol 3 pp 133ndash181 1922

[2] I Ya Alber and S Guerre-Delabriere ldquoPrinciples of weaklycontractive maps in Hilbert spacesrdquo in New Results in OperatorTheory I Gohberg and Yu Lyubich Eds vol 98 of Advancesand Applications pp 7ndash22 Birkhauser Basel Switzerland 1997

[3] M Eslamian and A Abkar ldquoA fixed point theorem for gener-alized weakly contractive mappings in complete metric space rdquoItalian Journal of Pure and Applied Mathematics In press

[4] P N Dutta and B S Choudhury ldquoA generalisation of con-traction principle in metric spacesrdquo Fixed Point Theory andApplications vol 2008 Article ID 406368 8 pages 2008

[5] H Aydi E Karapınar and B Samet ldquoRemarks on some recentfixed point theoremsrdquo Fixed Point Theory and Applications vol2012 article 76 2012

Journal of Applied Mathematics 7

[6] H Aydi E Karapınar and W Shatanawi ldquoCoupled fixed pointresults for (120593 120595)-weakly contractive condition in ordered par-tialmetric spacesrdquoComputers ampMathematics with Applicationsvol 62 no 12 pp 4449ndash4460 2011

[7] W-S Du ldquoOn coincidence point and fixed point theorems fornonlinear multivaluedmapsrdquo Topology and its Applications vol159 no 1 pp 49ndash56 2012

[8] W-S Du ldquoOn generalized weakly directional contractions andapproximate fixed point propertywith applicationsrdquo Fixed PointTheory and Applications vol 2012 article 6 2012

[9] W-S Du and S-X Zheng ldquoNonlinear conditions for coinci-dence point and fixed point theoremsrdquo Taiwanese Journal ofMathematics vol 16 no 3 pp 857ndash868 2012

[10] W-S Du and S-X Zheng ldquoNew nonlinear conditions andinequalities for the existence of coincidence points and fixedpointsrdquo Journal of Applied Mathematics vol 2012 Article ID196759 12 pages 2012

[11] E Karapinar and K Sadarangani ldquoFixed point theory for cyclic(120593 120595)-contractionsrdquo Fixed Point Theory and Applications vol2011 article 69 2011

[12] E Karapınar ldquoFixed point theory for cyclic weak 120601-contractionrdquo Applied Mathematics Letters vol 24 no 6pp 822ndash825 2011

[13] E Karapınar and I S Yuce ldquoFixed point theory for cyclic gen-eralized weak 120601-contraction on partial metric spacesrdquo AbstractandAppliedAnalysis vol 2012 Article ID491542 12 pages 2012

[14] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis vol 47 no 4 pp 2683ndash2693 2001

[15] B Djafari Rouhani and S Moradi ldquoCommon fixed point ofmultivalued generalized 120593-weak contractive mappingsrdquo FixedPoint Theory and Applications Article ID 708984 201013 pages2010

[16] Q Zhang and Y Song ldquoFixed point theory for generalized 120601-weak contractionsrdquo Applied Mathematics Letters vol 22 no 1pp 75ndash78 2009

[17] L B Ciric N Cakic M Rajovic and J S Ume ldquoMonotonegeneralized nonlinear contractions in partially ordered metricspacesrdquo Fixed Point Theory and Applications vol 2008 ArticleID 131294 11 pages 2008

[18] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysis Theory Methods amp Applications A vol 65 no 7 pp1379ndash1393 2006

[19] J Harjani B Lopez and K Sadarangani ldquoFixed point theoremsfor weakly C-contractive mappings in ordered metric spacesrdquoComputers amp Mathematics with Applications vol 61 no 4 pp790ndash796 2011

[20] J Harjani B Lopez and K Sadarangani ldquoA fixed point theoremfor mappings satisfying a contractive condition of rationaltype on a partially ordered metric spacerdquo Abstract and AppliedAnalysis vol 2010 Article ID 190701 8 pages 2010

[21] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[22] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrixequationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004

[23] B S Choudhury and A Kundu ldquo(120595 120572 120573)-weak contractions inpartially ordered metric spacesrdquo Applied Mathematics Lettersvol 25 no 1 pp 6ndash10 2012

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

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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

4 Journal of Applied Mathematics

Suppose that (b) holds that is 119909119899⊑ ] for all 119899 isin N

Substituting 119909 = 119909119899and 119910 = ] in (10) we have that

120593 (119889 (119909119899+1

119891]))

= 120593 (119889 (119891119909119899 119891]))

le 120595 (120601 (119889 (119909119899 ])) 120601 (119889 (119909

119899 119891119909119899)) 120601 (119889 (] 119891])))

minus 120585 (max 119889 (119909119899 ]) 119889 (119909

119899 119891119909119899) 119889 (] 119891]))

(31)

Taking limit as 119899 rarr infin in equality (31) we have

120593 (119889 (] 119891])) le 120595 (120601 (0) 120601 (119900) 120601 (119889 (] 119891]))) minus 120585 (119889 (] 119891]))

le 120601 (119889 (] 119891])) minus 120585 (119889 (] 119891]))

(32)

which implies that 119889(] 119891]) = 0 that is ] = 119891] So wecomplete the proof

If we let

120595 (120601 (119889 (119909 119910)) 120601 (119889 (119909 119891119909)) 120601 (119889 (119910 119891119910)))

= max 120601 (119889 (119909 119910)) 120601 (119889 (119909 119891119909)) 120601 (119889 (119910 119891119910)) (33)

it is easy to get the following theorem

Theorem 6 Let (119883 ⊑ 119889) be a partially ordered completemetric space Let 119891 119883 rarr 119883 be monotone nondecreasingand

120593 (119889 (119891119909 119891119910))

le max 120601 (119889 (119909 119910)) 120601 (119889 (119909 119891119909)) 120601 (119889 (119910 119891119910))

minus 120585 (max 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910))

(34)

for all comparable 119909 119910 isin 119883 where 120593 isin Θ 120601 isin Φ and 120585 isin Ξand

120593 (119905) minus 120601 (119905) + 120585 (119905) gt 0 forall119905 gt 0

120593 (119905) = 0 119894119891 119886119899119889 119900119899119897119910 119894119891 119905 = 0 120601 (0) = 120585 (0) = 0(35)

Suppose that either

(a) 119891 is continuous or(b) if any nondecreasing sequence 119909

119899 in119883 converges to ]

then one assumes that

119909119899⊑ ] forall119899 isin N (36)

If there exists 1199090isin 119883 with 119909

0⊑ 1198911199090 then 119891 has a fixed point

in119883

3 Fixed Point Results (II)

In the section we denote by Ψ the class of functions 120595

R+3

rarr R+ satisfying the following conditions

(1205951) 120595 is an increasing and continuous function in eachcoordinate

(1205952) for 119905 isin R+ 120601(119905 119905 119905) le 119905 120601(119905 0 0) le 119905 and 120601(0 0 119905) le119905

Next we denote byΘ the class of functions120593 R+ rarr R+

satisfying the following conditions

(1205931) 120593 is continuous and nondecreasing

(1205932) for 119905 gt 0 120593(119905) gt 0 and 120593(0) = 0

And we denote byΦ the class of functions 120601 R+ rarr R+

satisfying the following conditions

(1206011) 120601 is continuous

(1206012) for 119905 gt 0 120601(119905) gt 0 and 120601(0) = 0

We now state the main fixed point theorem for the(120595 120593 120601)-contractions in partially ordered metric spaces asfollows

Theorem 7 Let (119883 ⊑ 119889) be a partially ordered completemetric space and let 119891 119883 rarr 119883 be monotone nondecreasingand

120593 (119889 (119891119909 119891119910))

le 120595 (120593 (119889 (119909 119910)) 120593 (119889 (119909 119891119909)) 120593 (119889 (119910 119891119910)))

minus 120601 (119872 (119909 119910)) + 119871 sdot 119898 (119909 119910)

(37)

for all comparable 119909 119910 isin 119883 and 120595 isin Ψ 120593 isin Θ 120601 isin Φ where119871 gt 0 and

119872(119909 119910) = max 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910)

119898 (119909 119910)

= min 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910) 119889 (119909 119891119910)

119889 (119910 119891119909)

(38)

Suppose that either

(a) 119891 is continuous or(b) if any nondecreasing sequence 119909

119899 in119883 converges to ]

then one assumes that

119909119899⊑ ] forall119899 isin N (39)

If there exists 1199090isin 119883 with 119909

0⊑ 1198911199090 then 119891 has a fixed point

in 119883

Proof If 1198911199090

= 1199090 then the proof is finished Suppose

that 1199090⊏ 1198911199090 Since 119891 is nondecreasing by induction we

construct the sequence 119909119899 recursively as

119909119899= 119891119899

1199090= 119891119909119899minus1

forall119899 isin N (40)

Thus we also conclude that

1199090⊏ 1199091= 1198911199090⊑ 1199092= 1198911199091⊑ sdot sdot sdot ⊑ 119909

119899= 119891119909119899minus1

⊑ sdot sdot sdot (41)

We now claim that

lim119899rarrinfin

119889 (119909119899 119909119899+1

) = 0 (42)

Journal of Applied Mathematics 5

Put 119909 = 119909119899minus1

and 119910 = 119909119899in (37) Note that

119898(119909119899minus1

119909119899)

= min 119889 (119909119899minus1

119909119899) 119889 (119909

119899minus1 119891119909119899minus1

)

119889 (119909119899 119891119909119899) 119889 (119909

119899minus1 119891119909119899) 119889 (119909

119899 119891119909119899minus1

)

= min 119889 (119909119899minus1

119909119899) 119889 (119909

119899minus1 119909119899) 119889 (119909

119899 119909119899+1

)

119889 (119909119899minus1

119909119899+1

) 119889 (119909119899 119909119899)

= 0

(43)

So we obtain that

120593 (119889 (119909119899 119909119899+1

))

= 120593 (119889 (119891119909119899minus1

119891119909119899))

le 120595 (120593 (119889 (119909119899minus1

119909119899)) 120593 (119889 (119909

119899minus1 119891119909119899minus1

))

120593 (119889 (119909119899 119891119909119899))) minus 120601 (119872 (119909

119899minus1 119909119899))

le 120595 (120593 (119889 (119909119899minus1

119909119899)) 120593 (119889 (119909

119899minus1 119909119899)) 120593 (119889 (119909

119899 119909119899+1

)))

minus 120601 (119872 (119909119899minus1

119909119899))

(44)

where

119872(119909119899minus1

119909119899)

= max 119889 (119909119899minus1

119909119899) 119889 (119909

119899minus1 119891119909119899minus1

) 119889 (119909119899 119891119909119899)

= max 119889 (119909119899minus1

119909119899) 119889 (119909

119899minus1 119909119899) 119889 (119909

119899 119909119899+1

)

(45)

We now claim that

119889 (119909119899 119909119899+1

) lt 119889 (119909119899minus1

119909119899) forall119899 isin N (46)

If not we assume that 119889(119909119899minus1

119909119899) le 119889(119909

119899 119909119899+1

) then120593(119889(119909

119899minus1 119909119899)) le 120593(119889(119909

119899 119909119899+1

)) since 120593 is non-decreasingUsing inequality (44) and the conditions of the function 120595we have that for each 119899 isin N

120593 (119889 (119909119899 119909119899+1

)) le 120593 (119889 (119909119899 119909119899+1

)) minus 120601 (119889 (119909119899 119909119899+1

)) (47)

which implies that120601(119889(119909119899 119909119899+1

))= 0 and hence119889(119909119899 119909119899+1

)=0 This contradicts our initial assumption

From the previous argument we have that for each 119899 isin N

120593 (119889 (119909119899 119909119899+1

)) le 120593 (119889 (119909119899minus1

119909119899)) minus 120601 (119889 (119909

119899minus1 119909119899))

119889 (119909119899 119909119899+1

) lt 119889 (119909119899minus1

119909119899)

(48)

And since the sequence 119889(119909119899 119909119899+1

) is decreasing it mustconverge to some 120578 ge 0 Taking limit as 119899 rarr infin in (48) andby the continuity of 120593 and 120601 we get

120593 (120578) le 120593 (120578) minus 120601 (120578) (49)

and so we conclude that 120601(120578) = 0 and 120578 = 0We next claim that 119909

119899 is Cauchy that is for every 120576 gt 0

there exists 119899 isin N such that if 119901 119902 ge 119899 then 119889(119909119901 119909119902) lt 120576

Suppose on the contrary that there exists 120598 gt 0 such thatfor any 119899 isin N there are 119901

119899 119902119899isin Nwith 119901

119899gt 119902119899ge 119899 satisfying

119889 (119909119902119899

119909119901119899

) ge 120598 (50)

Further corresponding to 119902119899ge 119899 we can choose 119901

119899in such

a way that it the smallest integer with 119901119899gt 119902119899ge 119899 and

119889(119909119902119899

119909119901119899

) ge 120598 Therefore 119889(119909119902119899

119909119901119899minus1) lt 120598 By the rectan-

gular inequality we have

120598 le 119889 (119909119901119899

119909119902119899

)

le 119889 (119909119901119899

119909119901119899minus1) + 119889 (119909

119901119899minus1 119909119902119899

)

lt 119889 (119909119901119899

119909119901119899minus1) + 120598

(51)

Letting 119899 rarr infin then we get

lim119899rarrinfin

119889 (119909119901119899

119909119902119899

) = 120598 (52)

On the other hand we have

119889 (119909119901119899

119909119902119899

)

le 119889 (119909119901119899

119909119901119899minus1) + 119889 (119909

119901119899minus1 119909119902119899minus1) + 119889 (119909

119902119899minus1 119909119902119899

)

119889 (119909119901119899minus1 119909119902119899minus1)

le 119889 (119909119901119899minus1 119909119901119899

) + 119889 (119909119901119899

119909119902119899

) + 119889 (119909119902119899

119909119902119899minus1)

(53)

By letting 119899 rarr infin we get that

lim119899rarrinfin

119889 (119909119901119899minus1 119909119902119899minus1) = 120598 (54)

Using inequalities (37) (52) and (54) and putting 119909 = 119909119901119899minus1

and 119910 = 119909119902119899minus1 we have that

120593 (119889 (119909119901119899

119909119902119899

))

= 120593 (119889 (119891119909119901119899minus1 119891119909119902119899minus1))

le 120595 (120593 (119889 (119909119901119899minus1 119909119902119899minus1)) 120593 (119889 (119909

119901119899minus1 119891119909119901119899minus1))

120593 (119889 (119909119902119899minus1 119891119909119902119899minus1)))

minus 120601 (119872(119909119901119899minus1 119909119902119899minus1)) + 119871 sdot 119898 (119909

119901119899minus1 119909119902119899minus1)

= 120595 (120593 (119889 (119909119901119899minus1 119909119902119899minus1)) 120593 (119889 (119909

119901119899minus1 119909119901119899

))

120593 (119889 (119909119902119899minus1 119909119902119899

)))

minus 120601 (119872(119909119901119899minus1 119909119902119899minus1)) + 119871 sdot 119898 (119909

119901119899minus1 119909119902119899minus1)

(55)

6 Journal of Applied Mathematics

where

119872(119909119901119899minus1 119909119902119899minus1)

= max 119889 (119909119901119899minus1 119909119902119899minus1) 119889 (119909

119901119899minus1 119909119901119899

) 119889 (119909119902119899minus1 119909119902119899

)

119898 (119909119901119899minus1 119909119902119899minus1)

= min 119889 (119909119901119899minus1 119909119902119899minus1) 119889 (119909

119901119899minus1 119909119901119899

)

119889 (119909119902119899minus1 119909119902119899

) 119889 (119909119901119899minus1 119909119902119899

) 119889 (119909119902119899minus1 119909119901119899

)

(56)

Letting 119899 rarr infin then we obtain that

lim119899rarrinfin

119872(119909119901119899minus1 119909119902119899minus1) = 120598

lim119899rarrinfin

119898(119909119901119899minus1 119909119902119899minus1) = 0

120593 (120598) le 120595 (120593 (120598) 0 0) minus 120601 (120598) le 120593 (120598) minus 120601 (120598)

(57)

This implies that 120601(120598) = 0 and hence 120598 = 0 So we get acontraction Therefore 119909

119899 is a Cauchy sequence

Since119883 is complete there exists ] isin 119883 such that

lim119899rarrinfin

119909119899= ] (58)

Suppose that (a) holds Then

] = lim119899rarrinfin

119909119899+1

= lim119899rarrinfin

119891119909119899= 119891] (59)

Thus ] is a fixed point in119883Suppose that (b) holds that is 119909

119899⊑ ] for all 119899 isin N

Substituting 119909 = 119909119899and 119910 = ] in (37) we have that

120593 (119889 (119909119899+1

119891]))

= 120593 (119889 (119891119909119899 119891]))

le 120595 (120593 (119889 (119909119899 ])) 120593 (119889 (119909

119899 119891119909119899)) 120593 (119889 (] 119891])))

minus 120601 (119872 (119909119899 ])) + 119871 sdot 119898 (119909

119899 ])

(60)

where

119872(119909119899 ]) = max 119889 (119909

119899 ]) 119889 (119909

119899 119891119909119899) 119889 (] 119891])

119898 (119909119899 ])

= min 119889 (119909119899 ]) 119889 (119909

119899 119891119909119899) 119889 (] 119891]) 119889 (119909

119899 119891])

119889 (] 119891119909119899)

(61)

Letting 119899 rarr infin then we obtain that

119872(119909119899 ]) 997888rarr 119889 (] 119891]) 119898 (119909

119899 ]) 997888rarr 0

120593 (119889 (] 119891])) le 120595 (120593 (0) 120593 (0) 120593 (119889 (] 119891]))) minus 120601 (119889 (] 119891]))

le 120593 (119889 (] 119891])) minus 120601 (119889 (] 119891]))

(62)

which implies that 119889(] 119891]) = 0 that is ] = 119891] So wecomplete the proof

If we let

120595 (120593 (119889 (119909 119910)) 120593 (119889 (119909 119891119909)) 120593 (119889 (119910 119891119910)))

= max 120593 (119889 (119909 119910)) 120593 (119889 (119909 119891119909)) 120593 (119889 (119910 119891119910)) (63)

it is easy to get the following theorem

Theorem 8 Let (119883 ⊑ 119889) be a partially ordered completemetric space and let 119891 119883 rarr 119883 be monotone nondecreasingand

120593 (119889 (119891119909 119891119910))

le max 120593 (119889 (119909 119910)) 120593 (119889 (119909 119891119909)) 120593 (119889 (119910 119891119910))

minus 120601 (119872 (119909 119910)) + 119871 sdot 119898 (119909 119910)

(64)

for all comparable 119909 119910 isin 119883 and 120593 isin Θ 120601 isin Φ where 119871 gt 0

and

119872(119909 119910) = max 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910)

119898 (119909 119910) = min 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910) 119889 (119909 119891119910)

119889 (119910 119891119909)

(65)

Suppose that either

(a) 119891 is continuous or(b) if any nondecreasing sequence 119909

119899 in119883 converges to ]

then one assumes that

119909119899⊑ ] forall119899 isin N (66)

If there exists 1199090isin 119883 with 119909

0⊑ 1198911199090 then 119891 has a fixed point

in 119883

Acknowledgment

The authors would like to thank the referee(s) for the manyuseful comments and suggestions for the improvement of thepaper

References

[1] S Banach ldquoSur les operations dans les ensembles abstraitset leur application aux equations integeralesrdquo FundamentaMathematicae vol 3 pp 133ndash181 1922

[2] I Ya Alber and S Guerre-Delabriere ldquoPrinciples of weaklycontractive maps in Hilbert spacesrdquo in New Results in OperatorTheory I Gohberg and Yu Lyubich Eds vol 98 of Advancesand Applications pp 7ndash22 Birkhauser Basel Switzerland 1997

[3] M Eslamian and A Abkar ldquoA fixed point theorem for gener-alized weakly contractive mappings in complete metric space rdquoItalian Journal of Pure and Applied Mathematics In press

[4] P N Dutta and B S Choudhury ldquoA generalisation of con-traction principle in metric spacesrdquo Fixed Point Theory andApplications vol 2008 Article ID 406368 8 pages 2008

[5] H Aydi E Karapınar and B Samet ldquoRemarks on some recentfixed point theoremsrdquo Fixed Point Theory and Applications vol2012 article 76 2012

Journal of Applied Mathematics 7

[6] H Aydi E Karapınar and W Shatanawi ldquoCoupled fixed pointresults for (120593 120595)-weakly contractive condition in ordered par-tialmetric spacesrdquoComputers ampMathematics with Applicationsvol 62 no 12 pp 4449ndash4460 2011

[7] W-S Du ldquoOn coincidence point and fixed point theorems fornonlinear multivaluedmapsrdquo Topology and its Applications vol159 no 1 pp 49ndash56 2012

[8] W-S Du ldquoOn generalized weakly directional contractions andapproximate fixed point propertywith applicationsrdquo Fixed PointTheory and Applications vol 2012 article 6 2012

[9] W-S Du and S-X Zheng ldquoNonlinear conditions for coinci-dence point and fixed point theoremsrdquo Taiwanese Journal ofMathematics vol 16 no 3 pp 857ndash868 2012

[10] W-S Du and S-X Zheng ldquoNew nonlinear conditions andinequalities for the existence of coincidence points and fixedpointsrdquo Journal of Applied Mathematics vol 2012 Article ID196759 12 pages 2012

[11] E Karapinar and K Sadarangani ldquoFixed point theory for cyclic(120593 120595)-contractionsrdquo Fixed Point Theory and Applications vol2011 article 69 2011

[12] E Karapınar ldquoFixed point theory for cyclic weak 120601-contractionrdquo Applied Mathematics Letters vol 24 no 6pp 822ndash825 2011

[13] E Karapınar and I S Yuce ldquoFixed point theory for cyclic gen-eralized weak 120601-contraction on partial metric spacesrdquo AbstractandAppliedAnalysis vol 2012 Article ID491542 12 pages 2012

[14] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis vol 47 no 4 pp 2683ndash2693 2001

[15] B Djafari Rouhani and S Moradi ldquoCommon fixed point ofmultivalued generalized 120593-weak contractive mappingsrdquo FixedPoint Theory and Applications Article ID 708984 201013 pages2010

[16] Q Zhang and Y Song ldquoFixed point theory for generalized 120601-weak contractionsrdquo Applied Mathematics Letters vol 22 no 1pp 75ndash78 2009

[17] L B Ciric N Cakic M Rajovic and J S Ume ldquoMonotonegeneralized nonlinear contractions in partially ordered metricspacesrdquo Fixed Point Theory and Applications vol 2008 ArticleID 131294 11 pages 2008

[18] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysis Theory Methods amp Applications A vol 65 no 7 pp1379ndash1393 2006

[19] J Harjani B Lopez and K Sadarangani ldquoFixed point theoremsfor weakly C-contractive mappings in ordered metric spacesrdquoComputers amp Mathematics with Applications vol 61 no 4 pp790ndash796 2011

[20] J Harjani B Lopez and K Sadarangani ldquoA fixed point theoremfor mappings satisfying a contractive condition of rationaltype on a partially ordered metric spacerdquo Abstract and AppliedAnalysis vol 2010 Article ID 190701 8 pages 2010

[21] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[22] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrixequationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004

[23] B S Choudhury and A Kundu ldquo(120595 120572 120573)-weak contractions inpartially ordered metric spacesrdquo Applied Mathematics Lettersvol 25 no 1 pp 6ndash10 2012

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 Applied Mathematics 5

Put 119909 = 119909119899minus1

and 119910 = 119909119899in (37) Note that

119898(119909119899minus1

119909119899)

= min 119889 (119909119899minus1

119909119899) 119889 (119909

119899minus1 119891119909119899minus1

)

119889 (119909119899 119891119909119899) 119889 (119909

119899minus1 119891119909119899) 119889 (119909

119899 119891119909119899minus1

)

= min 119889 (119909119899minus1

119909119899) 119889 (119909

119899minus1 119909119899) 119889 (119909

119899 119909119899+1

)

119889 (119909119899minus1

119909119899+1

) 119889 (119909119899 119909119899)

= 0

(43)

So we obtain that

120593 (119889 (119909119899 119909119899+1

))

= 120593 (119889 (119891119909119899minus1

119891119909119899))

le 120595 (120593 (119889 (119909119899minus1

119909119899)) 120593 (119889 (119909

119899minus1 119891119909119899minus1

))

120593 (119889 (119909119899 119891119909119899))) minus 120601 (119872 (119909

119899minus1 119909119899))

le 120595 (120593 (119889 (119909119899minus1

119909119899)) 120593 (119889 (119909

119899minus1 119909119899)) 120593 (119889 (119909

119899 119909119899+1

)))

minus 120601 (119872 (119909119899minus1

119909119899))

(44)

where

119872(119909119899minus1

119909119899)

= max 119889 (119909119899minus1

119909119899) 119889 (119909

119899minus1 119891119909119899minus1

) 119889 (119909119899 119891119909119899)

= max 119889 (119909119899minus1

119909119899) 119889 (119909

119899minus1 119909119899) 119889 (119909

119899 119909119899+1

)

(45)

We now claim that

119889 (119909119899 119909119899+1

) lt 119889 (119909119899minus1

119909119899) forall119899 isin N (46)

If not we assume that 119889(119909119899minus1

119909119899) le 119889(119909

119899 119909119899+1

) then120593(119889(119909

119899minus1 119909119899)) le 120593(119889(119909

119899 119909119899+1

)) since 120593 is non-decreasingUsing inequality (44) and the conditions of the function 120595we have that for each 119899 isin N

120593 (119889 (119909119899 119909119899+1

)) le 120593 (119889 (119909119899 119909119899+1

)) minus 120601 (119889 (119909119899 119909119899+1

)) (47)

which implies that120601(119889(119909119899 119909119899+1

))= 0 and hence119889(119909119899 119909119899+1

)=0 This contradicts our initial assumption

From the previous argument we have that for each 119899 isin N

120593 (119889 (119909119899 119909119899+1

)) le 120593 (119889 (119909119899minus1

119909119899)) minus 120601 (119889 (119909

119899minus1 119909119899))

119889 (119909119899 119909119899+1

) lt 119889 (119909119899minus1

119909119899)

(48)

And since the sequence 119889(119909119899 119909119899+1

) is decreasing it mustconverge to some 120578 ge 0 Taking limit as 119899 rarr infin in (48) andby the continuity of 120593 and 120601 we get

120593 (120578) le 120593 (120578) minus 120601 (120578) (49)

and so we conclude that 120601(120578) = 0 and 120578 = 0We next claim that 119909

119899 is Cauchy that is for every 120576 gt 0

there exists 119899 isin N such that if 119901 119902 ge 119899 then 119889(119909119901 119909119902) lt 120576

Suppose on the contrary that there exists 120598 gt 0 such thatfor any 119899 isin N there are 119901

119899 119902119899isin Nwith 119901

119899gt 119902119899ge 119899 satisfying

119889 (119909119902119899

119909119901119899

) ge 120598 (50)

Further corresponding to 119902119899ge 119899 we can choose 119901

119899in such

a way that it the smallest integer with 119901119899gt 119902119899ge 119899 and

119889(119909119902119899

119909119901119899

) ge 120598 Therefore 119889(119909119902119899

119909119901119899minus1) lt 120598 By the rectan-

gular inequality we have

120598 le 119889 (119909119901119899

119909119902119899

)

le 119889 (119909119901119899

119909119901119899minus1) + 119889 (119909

119901119899minus1 119909119902119899

)

lt 119889 (119909119901119899

119909119901119899minus1) + 120598

(51)

Letting 119899 rarr infin then we get

lim119899rarrinfin

119889 (119909119901119899

119909119902119899

) = 120598 (52)

On the other hand we have

119889 (119909119901119899

119909119902119899

)

le 119889 (119909119901119899

119909119901119899minus1) + 119889 (119909

119901119899minus1 119909119902119899minus1) + 119889 (119909

119902119899minus1 119909119902119899

)

119889 (119909119901119899minus1 119909119902119899minus1)

le 119889 (119909119901119899minus1 119909119901119899

) + 119889 (119909119901119899

119909119902119899

) + 119889 (119909119902119899

119909119902119899minus1)

(53)

By letting 119899 rarr infin we get that

lim119899rarrinfin

119889 (119909119901119899minus1 119909119902119899minus1) = 120598 (54)

Using inequalities (37) (52) and (54) and putting 119909 = 119909119901119899minus1

and 119910 = 119909119902119899minus1 we have that

120593 (119889 (119909119901119899

119909119902119899

))

= 120593 (119889 (119891119909119901119899minus1 119891119909119902119899minus1))

le 120595 (120593 (119889 (119909119901119899minus1 119909119902119899minus1)) 120593 (119889 (119909

119901119899minus1 119891119909119901119899minus1))

120593 (119889 (119909119902119899minus1 119891119909119902119899minus1)))

minus 120601 (119872(119909119901119899minus1 119909119902119899minus1)) + 119871 sdot 119898 (119909

119901119899minus1 119909119902119899minus1)

= 120595 (120593 (119889 (119909119901119899minus1 119909119902119899minus1)) 120593 (119889 (119909

119901119899minus1 119909119901119899

))

120593 (119889 (119909119902119899minus1 119909119902119899

)))

minus 120601 (119872(119909119901119899minus1 119909119902119899minus1)) + 119871 sdot 119898 (119909

119901119899minus1 119909119902119899minus1)

(55)

6 Journal of Applied Mathematics

where

119872(119909119901119899minus1 119909119902119899minus1)

= max 119889 (119909119901119899minus1 119909119902119899minus1) 119889 (119909

119901119899minus1 119909119901119899

) 119889 (119909119902119899minus1 119909119902119899

)

119898 (119909119901119899minus1 119909119902119899minus1)

= min 119889 (119909119901119899minus1 119909119902119899minus1) 119889 (119909

119901119899minus1 119909119901119899

)

119889 (119909119902119899minus1 119909119902119899

) 119889 (119909119901119899minus1 119909119902119899

) 119889 (119909119902119899minus1 119909119901119899

)

(56)

Letting 119899 rarr infin then we obtain that

lim119899rarrinfin

119872(119909119901119899minus1 119909119902119899minus1) = 120598

lim119899rarrinfin

119898(119909119901119899minus1 119909119902119899minus1) = 0

120593 (120598) le 120595 (120593 (120598) 0 0) minus 120601 (120598) le 120593 (120598) minus 120601 (120598)

(57)

This implies that 120601(120598) = 0 and hence 120598 = 0 So we get acontraction Therefore 119909

119899 is a Cauchy sequence

Since119883 is complete there exists ] isin 119883 such that

lim119899rarrinfin

119909119899= ] (58)

Suppose that (a) holds Then

] = lim119899rarrinfin

119909119899+1

= lim119899rarrinfin

119891119909119899= 119891] (59)

Thus ] is a fixed point in119883Suppose that (b) holds that is 119909

119899⊑ ] for all 119899 isin N

Substituting 119909 = 119909119899and 119910 = ] in (37) we have that

120593 (119889 (119909119899+1

119891]))

= 120593 (119889 (119891119909119899 119891]))

le 120595 (120593 (119889 (119909119899 ])) 120593 (119889 (119909

119899 119891119909119899)) 120593 (119889 (] 119891])))

minus 120601 (119872 (119909119899 ])) + 119871 sdot 119898 (119909

119899 ])

(60)

where

119872(119909119899 ]) = max 119889 (119909

119899 ]) 119889 (119909

119899 119891119909119899) 119889 (] 119891])

119898 (119909119899 ])

= min 119889 (119909119899 ]) 119889 (119909

119899 119891119909119899) 119889 (] 119891]) 119889 (119909

119899 119891])

119889 (] 119891119909119899)

(61)

Letting 119899 rarr infin then we obtain that

119872(119909119899 ]) 997888rarr 119889 (] 119891]) 119898 (119909

119899 ]) 997888rarr 0

120593 (119889 (] 119891])) le 120595 (120593 (0) 120593 (0) 120593 (119889 (] 119891]))) minus 120601 (119889 (] 119891]))

le 120593 (119889 (] 119891])) minus 120601 (119889 (] 119891]))

(62)

which implies that 119889(] 119891]) = 0 that is ] = 119891] So wecomplete the proof

If we let

120595 (120593 (119889 (119909 119910)) 120593 (119889 (119909 119891119909)) 120593 (119889 (119910 119891119910)))

= max 120593 (119889 (119909 119910)) 120593 (119889 (119909 119891119909)) 120593 (119889 (119910 119891119910)) (63)

it is easy to get the following theorem

Theorem 8 Let (119883 ⊑ 119889) be a partially ordered completemetric space and let 119891 119883 rarr 119883 be monotone nondecreasingand

120593 (119889 (119891119909 119891119910))

le max 120593 (119889 (119909 119910)) 120593 (119889 (119909 119891119909)) 120593 (119889 (119910 119891119910))

minus 120601 (119872 (119909 119910)) + 119871 sdot 119898 (119909 119910)

(64)

for all comparable 119909 119910 isin 119883 and 120593 isin Θ 120601 isin Φ where 119871 gt 0

and

119872(119909 119910) = max 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910)

119898 (119909 119910) = min 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910) 119889 (119909 119891119910)

119889 (119910 119891119909)

(65)

Suppose that either

(a) 119891 is continuous or(b) if any nondecreasing sequence 119909

119899 in119883 converges to ]

then one assumes that

119909119899⊑ ] forall119899 isin N (66)

If there exists 1199090isin 119883 with 119909

0⊑ 1198911199090 then 119891 has a fixed point

in 119883

Acknowledgment

The authors would like to thank the referee(s) for the manyuseful comments and suggestions for the improvement of thepaper

References

[1] S Banach ldquoSur les operations dans les ensembles abstraitset leur application aux equations integeralesrdquo FundamentaMathematicae vol 3 pp 133ndash181 1922

[2] I Ya Alber and S Guerre-Delabriere ldquoPrinciples of weaklycontractive maps in Hilbert spacesrdquo in New Results in OperatorTheory I Gohberg and Yu Lyubich Eds vol 98 of Advancesand Applications pp 7ndash22 Birkhauser Basel Switzerland 1997

[3] M Eslamian and A Abkar ldquoA fixed point theorem for gener-alized weakly contractive mappings in complete metric space rdquoItalian Journal of Pure and Applied Mathematics In press

[4] P N Dutta and B S Choudhury ldquoA generalisation of con-traction principle in metric spacesrdquo Fixed Point Theory andApplications vol 2008 Article ID 406368 8 pages 2008

[5] H Aydi E Karapınar and B Samet ldquoRemarks on some recentfixed point theoremsrdquo Fixed Point Theory and Applications vol2012 article 76 2012

Journal of Applied Mathematics 7

[6] H Aydi E Karapınar and W Shatanawi ldquoCoupled fixed pointresults for (120593 120595)-weakly contractive condition in ordered par-tialmetric spacesrdquoComputers ampMathematics with Applicationsvol 62 no 12 pp 4449ndash4460 2011

[7] W-S Du ldquoOn coincidence point and fixed point theorems fornonlinear multivaluedmapsrdquo Topology and its Applications vol159 no 1 pp 49ndash56 2012

[8] W-S Du ldquoOn generalized weakly directional contractions andapproximate fixed point propertywith applicationsrdquo Fixed PointTheory and Applications vol 2012 article 6 2012

[9] W-S Du and S-X Zheng ldquoNonlinear conditions for coinci-dence point and fixed point theoremsrdquo Taiwanese Journal ofMathematics vol 16 no 3 pp 857ndash868 2012

[10] W-S Du and S-X Zheng ldquoNew nonlinear conditions andinequalities for the existence of coincidence points and fixedpointsrdquo Journal of Applied Mathematics vol 2012 Article ID196759 12 pages 2012

[11] E Karapinar and K Sadarangani ldquoFixed point theory for cyclic(120593 120595)-contractionsrdquo Fixed Point Theory and Applications vol2011 article 69 2011

[12] E Karapınar ldquoFixed point theory for cyclic weak 120601-contractionrdquo Applied Mathematics Letters vol 24 no 6pp 822ndash825 2011

[13] E Karapınar and I S Yuce ldquoFixed point theory for cyclic gen-eralized weak 120601-contraction on partial metric spacesrdquo AbstractandAppliedAnalysis vol 2012 Article ID491542 12 pages 2012

[14] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis vol 47 no 4 pp 2683ndash2693 2001

[15] B Djafari Rouhani and S Moradi ldquoCommon fixed point ofmultivalued generalized 120593-weak contractive mappingsrdquo FixedPoint Theory and Applications Article ID 708984 201013 pages2010

[16] Q Zhang and Y Song ldquoFixed point theory for generalized 120601-weak contractionsrdquo Applied Mathematics Letters vol 22 no 1pp 75ndash78 2009

[17] L B Ciric N Cakic M Rajovic and J S Ume ldquoMonotonegeneralized nonlinear contractions in partially ordered metricspacesrdquo Fixed Point Theory and Applications vol 2008 ArticleID 131294 11 pages 2008

[18] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysis Theory Methods amp Applications A vol 65 no 7 pp1379ndash1393 2006

[19] J Harjani B Lopez and K Sadarangani ldquoFixed point theoremsfor weakly C-contractive mappings in ordered metric spacesrdquoComputers amp Mathematics with Applications vol 61 no 4 pp790ndash796 2011

[20] J Harjani B Lopez and K Sadarangani ldquoA fixed point theoremfor mappings satisfying a contractive condition of rationaltype on a partially ordered metric spacerdquo Abstract and AppliedAnalysis vol 2010 Article ID 190701 8 pages 2010

[21] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[22] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrixequationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004

[23] B S Choudhury and A Kundu ldquo(120595 120572 120573)-weak contractions inpartially ordered metric spacesrdquo Applied Mathematics Lettersvol 25 no 1 pp 6ndash10 2012

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 Applied Mathematics

where

119872(119909119901119899minus1 119909119902119899minus1)

= max 119889 (119909119901119899minus1 119909119902119899minus1) 119889 (119909

119901119899minus1 119909119901119899

) 119889 (119909119902119899minus1 119909119902119899

)

119898 (119909119901119899minus1 119909119902119899minus1)

= min 119889 (119909119901119899minus1 119909119902119899minus1) 119889 (119909

119901119899minus1 119909119901119899

)

119889 (119909119902119899minus1 119909119902119899

) 119889 (119909119901119899minus1 119909119902119899

) 119889 (119909119902119899minus1 119909119901119899

)

(56)

Letting 119899 rarr infin then we obtain that

lim119899rarrinfin

119872(119909119901119899minus1 119909119902119899minus1) = 120598

lim119899rarrinfin

119898(119909119901119899minus1 119909119902119899minus1) = 0

120593 (120598) le 120595 (120593 (120598) 0 0) minus 120601 (120598) le 120593 (120598) minus 120601 (120598)

(57)

This implies that 120601(120598) = 0 and hence 120598 = 0 So we get acontraction Therefore 119909

119899 is a Cauchy sequence

Since119883 is complete there exists ] isin 119883 such that

lim119899rarrinfin

119909119899= ] (58)

Suppose that (a) holds Then

] = lim119899rarrinfin

119909119899+1

= lim119899rarrinfin

119891119909119899= 119891] (59)

Thus ] is a fixed point in119883Suppose that (b) holds that is 119909

119899⊑ ] for all 119899 isin N

Substituting 119909 = 119909119899and 119910 = ] in (37) we have that

120593 (119889 (119909119899+1

119891]))

= 120593 (119889 (119891119909119899 119891]))

le 120595 (120593 (119889 (119909119899 ])) 120593 (119889 (119909

119899 119891119909119899)) 120593 (119889 (] 119891])))

minus 120601 (119872 (119909119899 ])) + 119871 sdot 119898 (119909

119899 ])

(60)

where

119872(119909119899 ]) = max 119889 (119909

119899 ]) 119889 (119909

119899 119891119909119899) 119889 (] 119891])

119898 (119909119899 ])

= min 119889 (119909119899 ]) 119889 (119909

119899 119891119909119899) 119889 (] 119891]) 119889 (119909

119899 119891])

119889 (] 119891119909119899)

(61)

Letting 119899 rarr infin then we obtain that

119872(119909119899 ]) 997888rarr 119889 (] 119891]) 119898 (119909

119899 ]) 997888rarr 0

120593 (119889 (] 119891])) le 120595 (120593 (0) 120593 (0) 120593 (119889 (] 119891]))) minus 120601 (119889 (] 119891]))

le 120593 (119889 (] 119891])) minus 120601 (119889 (] 119891]))

(62)

which implies that 119889(] 119891]) = 0 that is ] = 119891] So wecomplete the proof

If we let

120595 (120593 (119889 (119909 119910)) 120593 (119889 (119909 119891119909)) 120593 (119889 (119910 119891119910)))

= max 120593 (119889 (119909 119910)) 120593 (119889 (119909 119891119909)) 120593 (119889 (119910 119891119910)) (63)

it is easy to get the following theorem

Theorem 8 Let (119883 ⊑ 119889) be a partially ordered completemetric space and let 119891 119883 rarr 119883 be monotone nondecreasingand

120593 (119889 (119891119909 119891119910))

le max 120593 (119889 (119909 119910)) 120593 (119889 (119909 119891119909)) 120593 (119889 (119910 119891119910))

minus 120601 (119872 (119909 119910)) + 119871 sdot 119898 (119909 119910)

(64)

for all comparable 119909 119910 isin 119883 and 120593 isin Θ 120601 isin Φ where 119871 gt 0

and

119872(119909 119910) = max 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910)

119898 (119909 119910) = min 119889 (119909 119910) 119889 (119909 119891119909) 119889 (119910 119891119910) 119889 (119909 119891119910)

119889 (119910 119891119909)

(65)

Suppose that either

(a) 119891 is continuous or(b) if any nondecreasing sequence 119909

119899 in119883 converges to ]

then one assumes that

119909119899⊑ ] forall119899 isin N (66)

If there exists 1199090isin 119883 with 119909

0⊑ 1198911199090 then 119891 has a fixed point

in 119883

Acknowledgment

The authors would like to thank the referee(s) for the manyuseful comments and suggestions for the improvement of thepaper

References

[1] S Banach ldquoSur les operations dans les ensembles abstraitset leur application aux equations integeralesrdquo FundamentaMathematicae vol 3 pp 133ndash181 1922

[2] I Ya Alber and S Guerre-Delabriere ldquoPrinciples of weaklycontractive maps in Hilbert spacesrdquo in New Results in OperatorTheory I Gohberg and Yu Lyubich Eds vol 98 of Advancesand Applications pp 7ndash22 Birkhauser Basel Switzerland 1997

[3] M Eslamian and A Abkar ldquoA fixed point theorem for gener-alized weakly contractive mappings in complete metric space rdquoItalian Journal of Pure and Applied Mathematics In press

[4] P N Dutta and B S Choudhury ldquoA generalisation of con-traction principle in metric spacesrdquo Fixed Point Theory andApplications vol 2008 Article ID 406368 8 pages 2008

[5] H Aydi E Karapınar and B Samet ldquoRemarks on some recentfixed point theoremsrdquo Fixed Point Theory and Applications vol2012 article 76 2012

Journal of Applied Mathematics 7

[6] H Aydi E Karapınar and W Shatanawi ldquoCoupled fixed pointresults for (120593 120595)-weakly contractive condition in ordered par-tialmetric spacesrdquoComputers ampMathematics with Applicationsvol 62 no 12 pp 4449ndash4460 2011

[7] W-S Du ldquoOn coincidence point and fixed point theorems fornonlinear multivaluedmapsrdquo Topology and its Applications vol159 no 1 pp 49ndash56 2012

[8] W-S Du ldquoOn generalized weakly directional contractions andapproximate fixed point propertywith applicationsrdquo Fixed PointTheory and Applications vol 2012 article 6 2012

[9] W-S Du and S-X Zheng ldquoNonlinear conditions for coinci-dence point and fixed point theoremsrdquo Taiwanese Journal ofMathematics vol 16 no 3 pp 857ndash868 2012

[10] W-S Du and S-X Zheng ldquoNew nonlinear conditions andinequalities for the existence of coincidence points and fixedpointsrdquo Journal of Applied Mathematics vol 2012 Article ID196759 12 pages 2012

[11] E Karapinar and K Sadarangani ldquoFixed point theory for cyclic(120593 120595)-contractionsrdquo Fixed Point Theory and Applications vol2011 article 69 2011

[12] E Karapınar ldquoFixed point theory for cyclic weak 120601-contractionrdquo Applied Mathematics Letters vol 24 no 6pp 822ndash825 2011

[13] E Karapınar and I S Yuce ldquoFixed point theory for cyclic gen-eralized weak 120601-contraction on partial metric spacesrdquo AbstractandAppliedAnalysis vol 2012 Article ID491542 12 pages 2012

[14] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis vol 47 no 4 pp 2683ndash2693 2001

[15] B Djafari Rouhani and S Moradi ldquoCommon fixed point ofmultivalued generalized 120593-weak contractive mappingsrdquo FixedPoint Theory and Applications Article ID 708984 201013 pages2010

[16] Q Zhang and Y Song ldquoFixed point theory for generalized 120601-weak contractionsrdquo Applied Mathematics Letters vol 22 no 1pp 75ndash78 2009

[17] L B Ciric N Cakic M Rajovic and J S Ume ldquoMonotonegeneralized nonlinear contractions in partially ordered metricspacesrdquo Fixed Point Theory and Applications vol 2008 ArticleID 131294 11 pages 2008

[18] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysis Theory Methods amp Applications A vol 65 no 7 pp1379ndash1393 2006

[19] J Harjani B Lopez and K Sadarangani ldquoFixed point theoremsfor weakly C-contractive mappings in ordered metric spacesrdquoComputers amp Mathematics with Applications vol 61 no 4 pp790ndash796 2011

[20] J Harjani B Lopez and K Sadarangani ldquoA fixed point theoremfor mappings satisfying a contractive condition of rationaltype on a partially ordered metric spacerdquo Abstract and AppliedAnalysis vol 2010 Article ID 190701 8 pages 2010

[21] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[22] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrixequationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004

[23] B S Choudhury and A Kundu ldquo(120595 120572 120573)-weak contractions inpartially ordered metric spacesrdquo Applied Mathematics Lettersvol 25 no 1 pp 6ndash10 2012

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 Applied Mathematics 7

[6] H Aydi E Karapınar and W Shatanawi ldquoCoupled fixed pointresults for (120593 120595)-weakly contractive condition in ordered par-tialmetric spacesrdquoComputers ampMathematics with Applicationsvol 62 no 12 pp 4449ndash4460 2011

[7] W-S Du ldquoOn coincidence point and fixed point theorems fornonlinear multivaluedmapsrdquo Topology and its Applications vol159 no 1 pp 49ndash56 2012

[8] W-S Du ldquoOn generalized weakly directional contractions andapproximate fixed point propertywith applicationsrdquo Fixed PointTheory and Applications vol 2012 article 6 2012

[9] W-S Du and S-X Zheng ldquoNonlinear conditions for coinci-dence point and fixed point theoremsrdquo Taiwanese Journal ofMathematics vol 16 no 3 pp 857ndash868 2012

[10] W-S Du and S-X Zheng ldquoNew nonlinear conditions andinequalities for the existence of coincidence points and fixedpointsrdquo Journal of Applied Mathematics vol 2012 Article ID196759 12 pages 2012

[11] E Karapinar and K Sadarangani ldquoFixed point theory for cyclic(120593 120595)-contractionsrdquo Fixed Point Theory and Applications vol2011 article 69 2011

[12] E Karapınar ldquoFixed point theory for cyclic weak 120601-contractionrdquo Applied Mathematics Letters vol 24 no 6pp 822ndash825 2011

[13] E Karapınar and I S Yuce ldquoFixed point theory for cyclic gen-eralized weak 120601-contraction on partial metric spacesrdquo AbstractandAppliedAnalysis vol 2012 Article ID491542 12 pages 2012

[14] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis vol 47 no 4 pp 2683ndash2693 2001

[15] B Djafari Rouhani and S Moradi ldquoCommon fixed point ofmultivalued generalized 120593-weak contractive mappingsrdquo FixedPoint Theory and Applications Article ID 708984 201013 pages2010

[16] Q Zhang and Y Song ldquoFixed point theory for generalized 120601-weak contractionsrdquo Applied Mathematics Letters vol 22 no 1pp 75ndash78 2009

[17] L B Ciric N Cakic M Rajovic and J S Ume ldquoMonotonegeneralized nonlinear contractions in partially ordered metricspacesrdquo Fixed Point Theory and Applications vol 2008 ArticleID 131294 11 pages 2008

[18] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysis Theory Methods amp Applications A vol 65 no 7 pp1379ndash1393 2006

[19] J Harjani B Lopez and K Sadarangani ldquoFixed point theoremsfor weakly C-contractive mappings in ordered metric spacesrdquoComputers amp Mathematics with Applications vol 61 no 4 pp790ndash796 2011

[20] J Harjani B Lopez and K Sadarangani ldquoA fixed point theoremfor mappings satisfying a contractive condition of rationaltype on a partially ordered metric spacerdquo Abstract and AppliedAnalysis vol 2010 Article ID 190701 8 pages 2010

[21] J J Nieto and R Rodrıguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[22] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrixequationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004

[23] B S Choudhury and A Kundu ldquo(120595 120572 120573)-weak contractions inpartially ordered metric spacesrdquo Applied Mathematics Lettersvol 25 no 1 pp 6ndash10 2012

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