Research ArticleThe Existence and Uniqueness of Coupled BestProximity Point for Proximally Coupled Contraction ina Complete Ordered Metric Space

V Pragadeeswarar1 M Marudai1 P Kumam2 and K Sitthithakerngkiet3

1 Department of Mathematics Bharathidasan University Tiruchirappalli Tamil Nadu 620 024 India2Department of Mathematics Faculty of Science King Mongkutrsquos University of Technology Thonburi (KMUTT) Bang ModThung Khru Bangkok 10140 Thailand

3Department of Mathematics Faculty of Applied Science King Mongkutrsquos University of North Bangkok (KMUTNB) WongsawangBangsue Bangkok 10800 Thailand

Correspondence should be addressed to P Kumam poomkumkmuttacth and K Sitthithakerngkiet kanokwanskmutnbacth

Received 2 February 2014 Accepted 18 April 2014 Published 14 May 2014

Academic Editor Salvador Romaguera

Copyright copy 2014 V Pragadeeswarar et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We prove the existence and uniqueness of coupled best proximity point for mappings satisfying the proximally coupled contractionin a complete ordered metric space Further our result provides an extension of a result due to Bhaskar and Lakshmikantham

1 Introduction

One of the most useful tools in the study of nonlinearfunctional equation is to describe many problems in physicschemistry and engineering It can be formulated in terms offinding the fixed points of a nonlinear self-mapping Fixedpoint theory investigates the techniques for determining asolution of the pattern 119879119909 = 119909 where 119879 is a self-mappingdefined on a subset119860 of ametric space119883 Noteworthy a fixedpoint 119909 of 119879 on can be written by 119889(119909 119879119909) = 0

A well-known principle that guarantees a unique fixedpoint solution is the Banach contraction principle [1] whichstates on a complete metric space 119883 for a contraction self-mapping (ie 119889(119879119909 119879119910) le 120572119889(119909 119910) for all 119909 119910 isin 119883 where120572 is a nonnegative number such that 120572 lt 1) Over theyears this principle has been generalized in many wayssee also [2ndash6] Recently an interesting way is to study theextension of Banach contraction principle to the case of non-self-mappings Certainly a contraction non-self-mapping 119879

119860 rarr 119861 does not necessarily have a fixed point where 119860 and119861 are nonempty subsets of a metric space119883

Ultimately one proceeds to find an approximate solution119909 isin 119860 which is closest to 119879119909 in the sense that distance

119889(119909 119879119909) is minimumwhich implies that 119909 and119879119909 are in closeproximity to each other Indeed the best approximation the-orems and the best proximity point theorem investigate theexistence of an approximate solution to fixed point problemsfor the non-self-mappings In 1969 Fan [7] guarantees at leastone solution to the minimization problem min

119909isin119860119909 minus 119879119909

where 119860 is a nonempty compact convex subset of a normedlinear space119883 and119879 119860 rarr 119883 is a continuous function Suchan element 119909 isin 119860 satisfying the above condition is called abest approximant of119879 in119860 Note that if119909 isin 119860 is a best approx-imant then 119909 minus 119879119909 need not be the optimum As a matterof fact the best proximity point theorems have been exploredto find sufficient conditions for the existence of an element119909 such that the error 119889(119909 119879119909) is minimum

To have a concrete lower bound let us consider twononempty subsets 119860 119861 of a metric space 119883 and a mapping119879 119860 rarr 119861 The natural question is whether one can findan element 119909lowast isin 119860 such that 119889(119909lowast 119879119909lowast) = 119889(119860 119861) where119889(119860 119861) = min119889(119909 119879119909) 119909 isin 119860 Since 119889(119909 119879119909) ge 119889(119860 119861)

for all 119909 isin 119860 the optimal solution to the problem ofminimiz-ing the real valued function 119909 rarr 119889(119909 119879119909) over the domain119860 of the mapping 119879 will be the one for which the value119889(119860 119861) is attained A point that satisfies the condition

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 274062 7 pageshttpdxdoiorg1011552014274062

2 Abstract and Applied Analysis

119889(119909 119879119909) = 119889(119860 119861) is called a best proximity point of 119879Furthermore the best proximity theorems also regress to anextension of fixed point theorem that is a best proximitypoint becomes a fixed point if either 119889(119860 119861) = 0 or themapping under consideration is a self-mapping

The existence and convergence of best proximity pointsare an interesting topic of optimization theory For moredetails on this approach we refer the reader to De la Senand Agarwal [8] Kumam et al [9ndash11] di Bari et al [12]Eldred and Veeramani [13] Al-Thagafi and Shahzad [14]Sadiq Basha and Veeramani [15] Kim and Lee [16] Kirk et al[17] Sankar Raj [18] Karapınar et al [19] and Jleli and Samet[20] The study of best proximity point in the setting of par-tially ordered metric space attracted recently the attention ofmany authors see [21ndash33]

Nowwe recall the definition of coupled fixed point Let119883be a nonempty set and 119865 119883 times119883 rarr 119883 a given mapping Anelement (119909 119910) isin 119883 times 119883 is called a coupled fixed point of themapping 119865 if 119865(119909 119910) = 119909 and 119865(119910 119909) = 119910 In 2006 Bhaskarand Lakshmikantham [34] proved some coupled fixed pointtheorems for mappings satisfying the mixed monotonemapping Indeed let (119883 le) be a partially ordered setthe mapping 119865 is said to have the mixed monotone propertyif

1199091 1199092isin 119883 119909

1le 1199092997904rArr 119865 (119909

1 119910) le 119865 (119909

2 119910)

forall119910 isin 119883

1199101 1199102isin 119883 119910

1le 1199102997904rArr 119865 (119909 119910

1) ge 119865 (119909 119910

2)

forall119909 isin 119883

(1)

Their results investigate a large class of problems and showthe existence and uniqueness of a solution for a periodicboundary value problem For more details on this conceptone may go through the references [35ndash37]

Motivated by the above theorems we introduce theconcept of proximally mixed monotone property and proxi-mally coupled contraction We also explore the existence anduniqueness of coupled best proximity points in the setting ofpartially ordered metric spaces thereby producing optimalapproximate solutions for that function with respect to bothcoordinates Further we attempt to give the generalization ofthe results in [34]

2 Preliminaries

Let 119883 be a nonempty set such that (119883 119889) is a metric spaceUnless otherwise specified it is assumed throughout thissection that119860 and119861 are nonempty subsets of themetric space(119883 119889) the following notions are used subsequently

119889 (119860 119861) = inf 119889 (119909 119910) 119909 isin 119860 119910 isin 119861

1198600= 119909 isin 119860 119889 (119909 119910) = 119889 (119860 119861) for some 119910 isin 119861

1198610= 119910 isin 119861 119889 (119909 119910) = 119889 (119860 119861) for some 119909 isin 119860

(2)

In [17] the authors discussed sufficient conditions whichguarantee the nonemptiness of 119860

0and 119861

0 Also in [15]

the authors proved that 1198600is contained in the boundary of

119860 Moreover the authors proved that 1198600is contained in the

boundary of 119860 in the setting of normed linear spaces

Definition 1 Let (119883 119889 le) be a partially ordered metric spaceand 119860 119861 nonempty subsets of 119883 A mapping 119865 119860 times

119860 rarr 119861 is said to have proximal mixed monotone propertyif 119865(119909 119910) is proximally nondecreasing in 119909 and is proximallynonincreasing in 119910 that is for all 119909 119910 isin 119860

1199091le 1199092

119889 (1199061 119865 (1199091 119910)) = 119889 (119860 119861)

119889 (1199062 119865 (1199092 119910)) = 119889 (119860 119861)

997904rArr 1199061le 1199062

1199101le 1199102

119889 (1199063 119865 (119909 119910

1)) = 119889 (119860 119861)

119889 (1199064 119865 (119909 119910

2)) = 119889 (119860 119861)

997904rArr 1199064le 1199063

(3)

where 1199091 1199092 1199101 1199102 1199061 1199062 1199063 1199064isin 119860

One can see that if 119860 = 119861 in the above definition thenotion of proximal mixedmonotone property reduces to thatof mixed monotone property

Lemma2 Let (119883 119889 le) be a partially orderedmetric space and119860 119861 nonempty subsets of 119883 Assume that 119860

0is nonempty A

mapping 119865 119860 times 119860 rarr 119861 has proximal mixed monotoneproperty with 119865(119860

0times 1198600) sube 119861

0 then for any 119909

0 1199091 1199092 1199100

1199101in 1198600

1199090le 1199091 119910

0ge 1199101

119889 (1199091 119865 (1199090 1199100)) = 119889 (119860 119861)

119889 (1199092 119865 (1199091 1199101)) = 119889 (119860 119861)

997904rArr 1199091le 1199092 (4)

Proof By hypothesis119865(1198600times1198600) sube 1198610 119865(1199091 1199100) isin 1198610 Hence

there exists 119909lowast1isin 119860 such that

119889 (119909lowast

1 119865 (1199091 1199100)) = 119889 (119860 119861) (5)

Using 119865 is proximal mixed monotone (in particular 119865 isproximally nondecreasing in 119909) to (4) and (5) we get

1199090le 1199091

119889 (1199091 119865 (1199090 1199100)) = 119889 (119860 119861)

119889 (119909lowast

1 119865 (1199091 1199100)) = 119889 (119860 119861)

997904rArr 1199091le 119909lowast

1 (6)

Analogously using 119865 is proximal mixed monotone (in par-ticular 119865 is proximally nonincreasing in 119910) to (4) and (5) weget

1199101le 1199100

119889 (1199092 119865 (1199091 1199101)) = 119889 (119860 119861)

119889 (119909lowast

1 119865 (1199091 1199100)) = 119889 (119860 119861)

997904rArr 119909lowast

1le 1199092 (7)

From (6) and (7) one can conclude that 1199091le 1199092 Hence the

proof is completed

Lemma3 Let (119883 119889 le) be a partially orderedmetric space and119860 119861 nonempty subsets of 119883 Assume that 119860

0is nonempty A

mapping 119865 119860 times 119860 rarr 119861 has proximal mixed monotone

Abstract and Applied Analysis 3

property with 119865(1198600times 1198600) sube 119861

0 then for any 119909

0 1199091 1199100 1199101

1199102in 1198600

1199090le 1199091 119910

0ge 1199101

119889 (1199101 119865 (1199100 1199090)) = 119889 (119860 119861)

119889 (1199102 119865 (1199101 1199091)) = 119889 (119860 119861)

997904rArr 1199101ge 1199102 (8)

Proof The proof is same as Lemma 2

Definition 4 Let (119883 119889 le) be a partially ordered metric spaceand 119860 119861 nonempty subsets of 119883 A mapping 119865 119860 times 119860 rarr

119861 is said to be proximally coupled contraction if there exists119896 isin (0 1) such that whenever

1199091le 1199092 119910

1ge 1199102

119889 (1199061 119865 (1199091 1199101)) = 119889 (119860 119861)

119889 (1199062 119865 (1199092 1199102)) = 119889 (119860 119861)

997904rArr 119889 (1199061 1199062)

le119896

2[119889 (1199091 1199092) + 119889 (119910

1 1199102)]

(9)

where 1199091 1199092 1199101 1199102 1199061 1199062isin 119860

One can see that if 119860 = 119861 in the above definition thenotion of proximally coupled contraction reduces to that ofcoupled contraction

3 Coupled Best Proximity Point Theorems

Let (119883 119889 le) be a partially ordered complete metric spaceendowed with the product space 119883 times 119883 with the followingpartial order

for (119909 119910) (119906 V) isin 119883 times 119883

(119906 V) le (119909 119910) lArrrArr 119909 ge 119906 119910 le V(10)

Theorem 5 Let (119883 le 119889) be a partially ordered completemetric space Let 119860 and 119861 be nonempty closed subsets of themetric space (119883 119889) such that 119860

0= 0 Let 119865 119860 times 119860 rarr 119861

satisfy the following conditions

(i) 119865 is a continuous proximally coupled contraction hav-ing the proximal mixed monotone property on 119860 suchthat 119865(119860

0times 1198600) sube 1198610

(ii) There exist elements (1199090 1199100) and (119909

1 1199101) in 119860

0times 1198600

such that

119889 (1199091 119865 (1199090 1199100)) = 119889 (119860 119861) 119908119894119905ℎ 119909

0le 1199091

119889 (1199101 119865 (1199100 1199090)) = 119889 (119860 119861) 119908119894119905ℎ 119910

0ge 1199101

(11)

Then there exist (119909 119910) isin 119860 times 119860 such that 119889(119909 119865(119909 119910)) =

119889(119860 119861) and 119889(119910 119865(119910 119909)) = 119889(119860 119861)

Proof By hypothesis there exist elements (1199090 1199100) and (119909

1 1199101)

in 1198600times 1198600such that

119889 (1199091 119865 (1199090 1199100)) = 119889 (119860 119861) with 119909

0le 1199091

119889 (1199101 119865 (1199100 1199090)) = 119889 (119860 119861) with 119910

0ge 1199101

(12)

Because of the fact that 119865(1198600times 1198600) sube 119861

0 there exists an

element (1199092 1199102) in 119860

0times 1198600such that

119889 (1199092 119865 (1199091 1199101)) = 119889 (119860 119861)

119889 (1199102 119865 (1199101 1199091)) = 119889 (119860 119861)

(13)

Hence from Lemmas 2 and 3 we obtain 1199091le 1199092and 119910

1ge 1199102

Continuing this process we can construct the sequences(119909119899) and (119910

119899) in 119860

0such that

119889 (119909119899+1

119865 (119909119899 119910119899)) = 119889 (119860 119861) forall119899 isin N (14)

with 1199090le 1199091le 1199092le sdot sdot sdot 119909

119899le 119909119899+1

sdot sdot sdot and

119889 (119910119899+1

119865 (119910119899 119909119899)) = 119889 (119860 119861) forall119899 isin N (15)

with 1199100ge 1199101ge 1199102ge sdot sdot sdot 119910

119899ge 119910119899+1

sdot sdot sdot Then119889(119909

119899 119865(119909119899minus1

119910119899minus1

)) = 119889(119860 119861) 119889(119909119899+1

119865(119909119899 119910119899)) =

119889(119860 119861) and also we have 119909119899minus1

le 119909119899 119910119899minus1

ge 119910119899 forall119899 isin N Now

using that 119865 is proximally coupled contraction on 119860 we get

119889 (119909119899 119909119899+1

) le119896

2[119889 (119909119899minus1

119909119899) + 119889 (119910

119899minus1 119910119899)] forall119899 isin N

(16)

Similarly

119889 (119910119899 119910119899+1

) le119896

2[119889 (119910119899minus1

119910119899) + 119889 (119909

119899minus1 119909119899)] forall119899 isin N

(17)

Adding (16) and (17) we get

119889 (119909119899 119909119899+1

) + 119889 (119910119899 119910119899+1

)

le 119896 [119889 (119909119899minus1

119909119899) + 119889 (119910

119899minus1 119910119899)]

le 119896 [119896 [119889 (119909119899minus2

119909119899minus1

) + 119889 (119910119899minus2

119910119899minus1

)]]

le sdot sdot sdot

le 119896119899

[119889 (1199090 1199091) + 119889 (119910

0 1199101)]

(18)

Finally we get

119889 (119909119899 119909119899+1

) + 119889 (119910119899 119910119899+1

) le 119896119899

[119889 (1199090 1199091) + 119889 (119910

0 1199101)]

(19)

Now we prove that (119909119899) and (119910

119899) are Cauchy sequences

For 119899 gt 119898 regarding triangle inequality and (19) one canobserve that

119889 (119909119899 119909119898) + 119889 (119910

119899 119910119898)

le 119889 (119909119899 119909119899minus1

) + 119889 (119910119899 119910119899minus1

) + sdot sdot sdot + 119889 (119909119898 119909119898+1

)

+ 119889 (119910119898 119910119898+1

)

le (119896119899minus1

+ sdot sdot sdot + 119896119898

) [119889 (1199090 1199091) + 119889 (119910

0 1199101)]

le119896119898

1 minus 119896[119889 (1199090 1199091) + 119889 (119910

0 1199101)]

(20)

4 Abstract and Applied Analysis

Let 120598 gt 0 be given Choose a natural number 119872 such that(119896119898

(1 minus 119896))[119889(1199090 1199091) + 119889(119910

0 1199101)] lt 120598 for all 119898 gt 119872 Thus

119889(119909119899 119909119898)+119889(119910

119899 119910119898) lt 120598 for 119899 gt 119898 Therefore the sequences

(119909119899) and (119910

119899) are Cauchy

Since 119860 is closed subset of a complete metric space 119883these sequences have limits Thus there exists 119909 119910 isin 119860 suchthat119909

119899rarr 119909 and119910

119899rarr 119910Therefore (119909

119899 119910119899) rarr (119909 119910) in119860times

119860 Since 119865 is continuous we have 119865(119909119899 119910119899) rarr 119865(119909 119910) and

119865(119910119899 119909119899) rarr 119865(119910 119909)

Hence the continuity of the metric function 119889

implies that 119889(119909119899+1

119865(119909119899 119910119899)) rarr 119889(119909 119865(119909 119910)) and

119889(119910119899+1

119865(119910119899 119909119899)) rarr 119889(119910 119865(119910 119909)) But from (14)

and (15) we get that the sequences 119889(119909119899+1

119865(119909119899 119910119899))

and 119889(119910119899+1

119865(119910119899 119909119899)) are constant sequences with the

value 119889(119860 119861) Therefore 119889(119909 119865(119909 119910)) = 119889(119860 119861) and119889(119910 119865(119910 119909)) = 119889(119860 119861) This completes the proof of thetheorem

Corollary 6 Let (119883 le 119889) be a partially ordered completemetric space Let 119860 be nonempty closed subset of the metricspace (119883 119889) Let 119865 119860 times 119860 rarr 119860 satisfy the followingconditions

(i) 119865 is continuous having the proximal mixed monotoneproperty and proximally coupled contraction on 119860

(ii) There exist (1199090 1199100) and (119909

1 1199101) in119860times119860 such that 119909

1=

119865(1199090 1199100) with 119909

0le 1199091and 119910

1= 119865(119910

0 1199090) with 119910

0ge

1199101

Then there exist (119909 119910) isin 119860times119860 such that 119889(119909 119865(119909 119910)) = 0 and119889(119910 119865(119910 119909)) = 0

In what follows we prove that Theorem 5 is still validfor 119865 not necessarily continuous assuming the followinghypothesis in 119860

(119909119899) is a nondecreasing sequence in 119860

such that 119909119899997888rarr 119909 then 119909

119899le 119909

(119910119899) is a nonincreasing sequence in 119860

such that 119910119899997888rarr 119910 then 119910 le 119910

119899

(21)

Theorem 7 Assume the conditions (21) and 1198600is closed in119883

instead of continuity of 119865 in Theorem 5 then the conclusion ofTheorem 5 holds

Proof Following the proof of Theorem 5 there existsequences (119909

119899) and (119910

119899) in 119860 satisfying the following

conditions

119889 (119909119899+1

119865 (119909119899 119910119899)) = 119889 (119860 119861) with 119909

119899le 119909119899+1

forall119899 isin N

(22)

119889 (119910119899+1

119865 (119910119899 119909119899)) = 119889 (119860 119861) with 119910

119899ge 119910119899+1

forall119899 isin N

(23)

Moreover (119909119899) converges to 119909 and 119910

119899converges to 119910 in 119860

From (21) we get 119909119899

le 119909 and 119910119899

ge 119910 Note that thesequences (119909

119899) and (119910

119899) are in119860

0and119860

0is closedTherefore

(119909 119910) isin 1198600times 1198600 Since 119865(119860

0times 1198600) sube 119861

0 there exists

119865(119909 119910) and119865(119910 119909) are in1198610Therefore there exists (119909lowast 119910lowast) isin

1198600times 1198600such that

119889 (119909lowast

119865 (119909 119910)) = 119889 (119860 119861) (24)

119889 (119910lowast

119865 (119910 119909)) = 119889 (119860 119861) (25)

Since 119909119899le 119909 and 119910

119899ge 119910 By using 119865 is proximally coupled

contraction for (22) and (24) also for (25) and (23) we get

119889 (119909119899+1

119909lowast

) le119896

2[119889 (119909119899 119909) + 119889 (119910

119899 119910)] forall119899

119889 (119910lowast

119910119899+1

) le119896

2[119889 (119910 119910

119899) + 119889 (119909 119909

119899)] forall119899

(26)

Since 119909119899rarr 119909 and 119910

119899rarr 119910 by taking limit on the above two

inequality we get 119909 = 119909lowast and119910 = 119910

lowast Consequently the resultfollows

Corollary 8 Assume the conditions (21) instead of continuityof 119865 in Corollary 6 then the conclusion of Corollary 6 holds

Now we present an example where it can be appreciatedthat hypotheses in Theorems 5 and 7 do not guaranteeuniqueness of the coupled best proximity point

Example 9 Let 119883 = (0 1) (1 0) (minus1 0) (0 minus1) sub R2 andconsider the usual order (119909 119910) ⪯ (119911 119905) hArr 119909 le 119911 and 119910 le 119905

Thus (119883 ⪯) is a partially ordered set Besides (119883 1198892) is

a complete metric space considering 1198892the euclidean metric

Let 119860 = (0 1) (1 0) and 119861 = (0 minus1) (minus1 0) be closedsubsets of119883 Then 119889(119860 119861) = radic2119860 = 119860

0and 119861 = 119861

0 Let 119865

119860 times 119860 rarr 119861 be defined as 119865((1199091 1199092) (1199101 1199102)) = (minus119909

2 minus1199091)

Then it can be seen that 119865 is continuous such that 119865(1198600times

1198600) sube 1198610 The only comparable pairs of points in119860 are 119909 ⪯ 119909

for 119909 isin 119860 hence proximal mixed monotone property andproximally coupled contraction on 119860 are satisfied trivially

It can be shown that the other hypotheses of the theoremare also satisfiedHowever119865has three coupled best proximitypoints ((0 1) (0 1)) ((0 1) (1 0)) and ((1 0) (1 0))

One can prove that the coupled best proximity point is infact unique provided that the product space 119860 times 119860 endowedwith the partial order mentioned earlier has the followingproperty

Every pair of elements has either a lower bound

or an upper bound(27)

It is known that this condition is equivalent to the followingFor every pair of (119909 119910) (119909lowast 119910lowast) isin 119860 times 119860 there exists

(1199111 1199112) in 119860 times 119860

that is comparable to (119909 119910) (119909lowast

119910lowast

) (28)

Abstract and Applied Analysis 5

Theorem 10 In addition to the hypothesis ofTheorem 5 (respTheorem 7) suppose that for any two elements (119909 119910) and(119909lowast

119910lowast

) in 1198600times 1198600

there exists (1199111 1199112) isin 119860

0times 1198600

such that (1199111 1199112) is comparable to (119909 119910) (119909lowast 119910lowast)

(29)

then 119865 has a unique coupled best proximity point

Proof FromTheorem 5 (respTheorem 7) the set of coupledbest proximity points of 119865 is nonempty Suppose that thereexist (119909 119910) and (119909

lowast

119910lowast

) in 119860 times 119860 which are coupled bestproximity points That is

119889 (119909 119865 (119909 119910)) = 119889 (119860 119861) 119889 (119910 119865 (119910 119909)) = 119889 (119860 119861)

119889 (119909lowast

119865 (119909lowast

119910lowast

)) = 119889 (119860 119861)

119889 (119910lowast

119865 (119910lowast

119909lowast

)) = 119889 (119860 119861)

(30)

We distinguish two cases

Case 1 Suppose that (119909 119910) is comparable Let (119909 119910) becomparable to (119909lowast 119910lowast) with respect to the ordering in119860times119860Apply 119865 as proximally coupled contraction to 119889(119909 119865(119909 119910)) =119889(119860 119861) and 119889(119909lowast 119865(119909lowast 119910lowast)) = 119889(119860 119861) there exists 119896 isin (0 1)

such that

119889 (119909 119909lowast

) le119896

2[119889 (119909 119909

lowast

) + 119889 (119910 119910lowast

)] (31)

Similarly one can prove that

119889 (119910 119910lowast

) le119896

2[119889 (119910 119910

lowast

) + 119889 (119909 119909lowast

)] (32)

Adding (31) and (32) we get

119889 (119909 119909lowast

) + 119889 (119910 119910lowast

) le 119896 [119889 (119909 119909lowast

) + 119889 (119910 119910lowast

)] (33)

This implies that 119889(119909 119909lowast) + 119889(119910 119910lowast

) = 0 hence 119909 = 119909lowast and

119910 = 119910lowast

Case 2 Suppose that (119909 119910) is not comparable Let (119909 119910) benoncomparable to (119909

lowast

119910lowast

) then there exists (1199061 V1) isin 119860

0times

1198600which is comparable to (119909 119910) and (119909

lowast

119910lowast

)Since119865(119860

0times1198600) sube 1198610 there exists (119906

2 V2) isin 1198600times1198600such

that 119889(1199062 119865(1199061 V1)) = 119889(119860 119861) and 119889(V

2 119865(V1 1199061)) = 119889(119860 119861)

Without loss of generality assume that (1199061 V1) le (119909 119910) (ie

119909 ge 1199061and 119910 le V

1) Note that (119906

1 V1) le (119909 119910) implies that

(119910 119909) le (V1 1199061) From Lemmas 2 and 3 we get

1199061le 119909 V

1ge 119910

119889 (1199062 119865 (1199061 V1)) = 119889 (119860 119861)

119889 (119909 119865 (119909 119910)) = 119889 (119860 119861)

997904rArr 1199062le 119909

1199061le 119909 V

1ge 119910

119889 (V2 119865 (V1 1199061)) = 119889 (119860 119861)

119889 (119910 119865 (119910 119909)) = 119889 (119860 119861)

997904rArr V2ge 119910

(34)

From the above two inequalities we obtain (1199062 V2) le (119909 119910)

Continuing this process we get sequences 119906119899 and V

119899 such

that 119889(119906119899+1

119865(119906119899 V119899)) = 119889(119860 119861) and 119889(V

119899+1 119865(V119899 119906119899)) =

119889(119860 119861) with (119906119899 V119899) le (119909 119910) forall119899 isin N By using that 119865 is a

proximally coupled contraction we get

119906119899le 119909 V

119899ge 119910

119889 (119906119899+1

119865 (119906119899 V119899)) = 119889 (119860 119861)

119889 (119909 119865 (119909 119910)) = 119889 (119860 119861)

dArr

119889 (119906119899+1

119909) le119896

2[119889 (119906119899 119909) + 119889 (V

119899 119910)] forall119899 isin N

(35)

Similarly we can prove that

119910 le V119899 119909 ge 119906

119899

119889 (119910 119865 (119910 119909)) = 119889 (119860 119861)

119889 (V119899+1

119865 (V119899 119906119899)) = 119889 (119860 119861)

dArr

119889 (119910 V119899+1

) le119896

2[119889 (119910 V

119899) + 119889 (119909 119906

119899)] forall119899 isin N

(36)

Adding (35) and (36) we obtain

119889 (119906119899+1

119909) + 119889 (119910 V119899+1

) le 119896 [119889 (119906119899 119909) + 119889 (V

119899 119910)]

119889 (119906119899+1

119909) + 119889 (119910 V119899+1

)

le 119896 [119889 (119906119899 119909) + 119889 (V

119899 119910)]

le 1198962

[119889 (119906119899minus1

119909) + 119889 (V119899minus1

119910)]

sdot sdot sdot

le 119896119899+1

[119889 (1199060 119909) + 119889 (V

0 119910)]

(37)

As 119899 rarr infin we get 119889(119906119899+1

119909) + 119889(119910 V119899+1

) rarr 0 so that119906119899rarr 119909 and V

119899rarr 119910

Analogously one can prove that 119906119899rarr 119909lowast and V

119899rarr 119910lowast

Therefore 119909 = 119909lowast and 119910 = 119910

lowast Hence the proof is completed

Example 11 Let 119883 = R be endowed with usual metric andwith the usual order in R

Suppose that 119860 = [1 2] and 119861 = [minus2 minus1] Then 119860 and 119861

are nonempty closed subsets of 119883 and 1198600= 1 and 119861

0= minus1

Also note that 119889(119860 119861) = 2Now consider the function 119865 119860 times 119860 rarr 119861 defined as

119865 (119909 119910) =minus119909 minus 119910 minus 2

4 (38)

Then it can be seen that 119865 is continuous and 119865(1 1) = minus1Hence119865(119860

0times1198600) sube 1198610 It is easy to see that other hypotheses

of the Theorem 10 are also satisfied Further it is easy to seethat (1 1) is the unique element satisfying the conclusion ofTheorem 10

The following result due to Fan [7] is a corollaryTheorem 10 by taking 119860 = 119861

6 Abstract and Applied Analysis

Corollary 12 In addition to the hypothesis of Corollary 6(resp Corollary 8) suppose that for any two elements (119909 119910)and (119909lowast 119910lowast) in 119860 times 119860

119905ℎ119890119903119890 119890119909119894119904119905119904 (1199111 1199112) isin 119860 times 119860

119904119906119888ℎ 119905ℎ119886119905 (1199111 1199112) 119894119904 119888119900119898119901119886119903119886119887119897119890 119905119900 (119909 119910) (119909

lowast

119910lowast

)

(39)

then 119865 has a unique coupled fixed point

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research Dr K Sitthithakerngkiet was funded by KingMongkutrsquos University of Technology North Bangkok (Con-tract no KMUTNB-GEN-57-19) Moreover the third authorwas supported by the Higher Education Research Promotionand National Research University Project of Thailand Officeof the Higher Education Commission (NRU2557)

References

[1] S Banach ldquoSur les operations dans les ensembles abstraits etleurs applications auxequations integralesrdquo Fundamenta Math-ematicae vol 3 pp 133ndash181 1922

[2] A D Arvanitakis ldquoA proof of the generalized Banach con-traction conjecturerdquo Proceedings of the American MathematicalSociety vol 131 no 12 pp 3647ndash3656 2003

[3] B S Choudhury and K Das ldquoA new contraction principle inmenger spacesrdquo Acta Mathematica Sinica vol 24 no 8 pp1379ndash1386 2008

[4] T Suzuki ldquoA generalized banach contraction principle thatcharacterizesmetric completenessrdquo Proceedings of the AmericanMathematical Society vol 136 no 5 pp 1861ndash1869 2008

[5] H Alikhani D Gopal M A Miandaragh Sh Rezapour andN Shahzad ldquoSome endpoint results for 120573-generalized weakcontractive multifunctionsrdquo The Scientific World Journal vol2013 Article ID 948472 7 pages 2013

[6] P Kumam F Rouzkard M Imdad and D Gopal ldquoFixedpoint theorems on ordered metric spaces through a rationalcontractionrdquoAbstract and Applied Analysis vol 2013 Article ID206515 9 pages 2013

[7] K Fan ldquoExtensions of two fixed point theorems of F EBrowderrdquoMathematische Zeitschrift vol 112 no 3 pp 234ndash2401969

[8] M De la Sen and R P Agarwal ldquoSome fixed point-type resultsfor a class of extended cyclic self-mappings with a more generalcontractive conditionrdquo Fixed PointTheory and Applications vol2011 article 59 2011

[9] P Kumam P Salimi and C Vetro ldquoBest proximity point resultsfor modified -proximal C-contraction mappingsrdquo Fixed PointTheory and Applications vol 2014 article 99 2014

[10] P Kumam H Aydi E Karapinar and W Sintunavarat ldquoBestproximity points and extension of Mizoguchi-Takahashirsquos fixedpoint theoremsrdquo Fixed Point Theory and Applications vol 2013article 242 2013

[11] P Kumam and A Roldn-Lpez-de-Hierro ldquoOn existence anduniqueness of g-best proximity points under (120593 120579 120572 119892)-con-tractivity conditions and consequencesrdquo Abstract and AppliedAnalysis vol 2014 Article ID 234027 14 pages 2014

[12] C di Bari T Suzuki and C Vetro ldquoBest proximity points forcyclic Meir-Keeler contractionsrdquo Nonlinear Analysis TheoryMethods and Applications vol 69 no 11 pp 3790ndash3794 2008

[13] A A Eldred and P Veeramani ldquoExistence and convergence ofbest proximity pointsrdquo Journal of Mathematical Analysis andApplications vol 323 no 2 pp 1001ndash1006 2006

[14] M A Al-Thagafi and N Shahzad ldquoConvergence and existenceresults for best proximity pointsrdquo Nonlinear Analysis TheoryMethods and Applications vol 70 no 10 pp 3665ndash3671 2009

[15] S Sadiq Basha and P Veeramani ldquoBest proximity pair theoremsfor multifunctions with open fibresrdquo Journal of ApproximationTheory vol 103 no 1 pp 119ndash129 2000

[16] W K Kim and K H Lee ldquoExistence of best proximity pairsand equilibrium pairsrdquo Journal of Mathematical Analysis andApplications vol 316 no 2 pp 433ndash446 2006

[17] W A Kirk S Reich and P Veeramani ldquoProximinal retracts andbest proximity pair theoremsrdquo Numerical Functional Analysisand Optimization vol 24 no 7-8 pp 851ndash862 2003

[18] V Sankar Raj ldquoA best proximity point theorem for weakly con-tractive non-self-mappingsrdquo Nonlinear Analysis Theory Meth-ods and Applications vol 74 no 14 pp 4804ndash4808 2011

[19] E Karapınar V Pragadeeswarar and M Marudai ldquoBest prox-imity point for generalized proximal weak contractions incomplete metric spacerdquo Journal of Applied Mathematics vol2014 Article ID 150941 6 pages 2014

[20] M Jleli and B Samet ldquoRemarks on the paper best proximitypoint theorems An exploration of a common solution toapproximation and optimization problemsrdquoAppliedMathemat-ics and Computation vol 228 pp 366ndash370 2014

[21] S S Basha ldquoDiscrete optimization in partially ordered setsrdquoJournal of Global Optimization vol 54 no 3 pp 511ndash517 2012

[22] A Abkar and M Gabeleh ldquoBest proximity points for cyclicmappings in ordered metric spacesrdquo Journal of OptimizationTheory and Applications vol 150 no 1 pp 188ndash193 2011

[23] A Abkar and M Gabeleh ldquoGeneralized cyclic contractions inpartially orderedmetric spacesrdquoOptimization Letters vol 6 no8 pp 1819ndash1830 2012

[24] W Sintunavarat and P Kumam ldquoCoupled best proximity pointtheorem in metric spacesrdquo Fixed PointTheory and Applicationsvol 2012 article 93 2012

[25] V Pragadeeswarar and M Marudai ldquoBest proximity pointsapproximation and optimization in partially ordered metricspacesrdquo Optimization Letters vol 7 pp 1883ndash1892 2013

[26] V Pragadeeswarar and M Marudai ldquoBest proximity points forgeneralized proximal weak contractions in partially orderedmetric spacesrdquo Optimization Letters 2013

[27] C Mongkolkeha and P Kumam ldquoBest proximity point The-orems for generalized cyclic contractions in ordered metricSpacesrdquo Journal of Optimization Theory and Applications vol155 pp 215ndash226 2012

[28] W Sanhan C Mongkolkeha and P Kumam ldquoGeneralizedproximal -contraction mappings and Best proximity pointsrdquoAbstract and Applied Analysis vol 2012 article 42 2012

[29] C Mongkolkeha and P Kumam ldquoBest proximity points forasymptotic proximal pointwise weaker Meir-Keeler-type -contraction mappingsrdquo Journal of the Egyptian MathematicalSociety vol 21 no 2 pp 87ndash90 2013

Abstract and Applied Analysis 7

[30] CMongkolkeha andPKumam ldquoSome commonbest proximitypoints for proximity commuting mappingsrdquo Optimization Let-ters vol 7 no 8 pp 1825ndash1836 2013

[31] C Mongkolkeha Y J Cho and P Kumam ldquoBest proximitypoints for generalized proxinal C-Contraction mappings inmetric spaces with partial ordersrdquo Journal of Inequalities andApplications vol 2013 article 94 2013

[32] C Mongkolkeha Y J Cho and P Kumam ldquoBest proximitypoints for Geraghtyrsquos proximal contraction mappingsrdquo FixedPoint Theory and Applications vol 2013 article 180 2013

[33] C Mongkolkeha C Kongban and P Kumam ldquoThe existenceand uniqueness of best proximity point theorems for gener-alized almost contractionrdquo Abstract and Applied Analysis vol2014 Article ID 813614 11 pages 2014

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

[35] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 70 no 12 pp 4341ndash4349 2009

[36] N V Luong and N X Thuan ldquoCoupled fixed point theoremsfor mixed monotone mappings and an application to integralequationsrdquo Computers and Mathematics with Applications vol62 no 11 pp 4238ndash4248 2011

[37] W Shatanawi ldquoPartially ordered cone metric spaces and cou-pled fixed point resultsrdquo Computers and Mathematics withApplications vol 60 no 8 pp 2508ndash2515 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

119889(119909 119879119909) = 119889(119860 119861) is called a best proximity point of 119879Furthermore the best proximity theorems also regress to anextension of fixed point theorem that is a best proximitypoint becomes a fixed point if either 119889(119860 119861) = 0 or themapping under consideration is a self-mapping

The existence and convergence of best proximity pointsare an interesting topic of optimization theory For moredetails on this approach we refer the reader to De la Senand Agarwal [8] Kumam et al [9ndash11] di Bari et al [12]Eldred and Veeramani [13] Al-Thagafi and Shahzad [14]Sadiq Basha and Veeramani [15] Kim and Lee [16] Kirk et al[17] Sankar Raj [18] Karapınar et al [19] and Jleli and Samet[20] The study of best proximity point in the setting of par-tially ordered metric space attracted recently the attention ofmany authors see [21ndash33]

Nowwe recall the definition of coupled fixed point Let119883be a nonempty set and 119865 119883 times119883 rarr 119883 a given mapping Anelement (119909 119910) isin 119883 times 119883 is called a coupled fixed point of themapping 119865 if 119865(119909 119910) = 119909 and 119865(119910 119909) = 119910 In 2006 Bhaskarand Lakshmikantham [34] proved some coupled fixed pointtheorems for mappings satisfying the mixed monotonemapping Indeed let (119883 le) be a partially ordered setthe mapping 119865 is said to have the mixed monotone propertyif

1199091 1199092isin 119883 119909

1le 1199092997904rArr 119865 (119909

1 119910) le 119865 (119909

2 119910)

forall119910 isin 119883

1199101 1199102isin 119883 119910

1le 1199102997904rArr 119865 (119909 119910

1) ge 119865 (119909 119910

2)

forall119909 isin 119883

(1)

Their results investigate a large class of problems and showthe existence and uniqueness of a solution for a periodicboundary value problem For more details on this conceptone may go through the references [35ndash37]

Motivated by the above theorems we introduce theconcept of proximally mixed monotone property and proxi-mally coupled contraction We also explore the existence anduniqueness of coupled best proximity points in the setting ofpartially ordered metric spaces thereby producing optimalapproximate solutions for that function with respect to bothcoordinates Further we attempt to give the generalization ofthe results in [34]

2 Preliminaries

Let 119883 be a nonempty set such that (119883 119889) is a metric spaceUnless otherwise specified it is assumed throughout thissection that119860 and119861 are nonempty subsets of themetric space(119883 119889) the following notions are used subsequently

119889 (119860 119861) = inf 119889 (119909 119910) 119909 isin 119860 119910 isin 119861

1198600= 119909 isin 119860 119889 (119909 119910) = 119889 (119860 119861) for some 119910 isin 119861

1198610= 119910 isin 119861 119889 (119909 119910) = 119889 (119860 119861) for some 119909 isin 119860

(2)

In [17] the authors discussed sufficient conditions whichguarantee the nonemptiness of 119860

0and 119861

0 Also in [15]

the authors proved that 1198600is contained in the boundary of

119860 Moreover the authors proved that 1198600is contained in the

boundary of 119860 in the setting of normed linear spaces

Definition 1 Let (119883 119889 le) be a partially ordered metric spaceand 119860 119861 nonempty subsets of 119883 A mapping 119865 119860 times

119860 rarr 119861 is said to have proximal mixed monotone propertyif 119865(119909 119910) is proximally nondecreasing in 119909 and is proximallynonincreasing in 119910 that is for all 119909 119910 isin 119860

1199091le 1199092

119889 (1199061 119865 (1199091 119910)) = 119889 (119860 119861)

119889 (1199062 119865 (1199092 119910)) = 119889 (119860 119861)

997904rArr 1199061le 1199062

1199101le 1199102

119889 (1199063 119865 (119909 119910

1)) = 119889 (119860 119861)

119889 (1199064 119865 (119909 119910

2)) = 119889 (119860 119861)

997904rArr 1199064le 1199063

(3)

where 1199091 1199092 1199101 1199102 1199061 1199062 1199063 1199064isin 119860

One can see that if 119860 = 119861 in the above definition thenotion of proximal mixedmonotone property reduces to thatof mixed monotone property

Lemma2 Let (119883 119889 le) be a partially orderedmetric space and119860 119861 nonempty subsets of 119883 Assume that 119860

0is nonempty A

mapping 119865 119860 times 119860 rarr 119861 has proximal mixed monotoneproperty with 119865(119860

0times 1198600) sube 119861

0 then for any 119909

0 1199091 1199092 1199100

1199101in 1198600

1199090le 1199091 119910

0ge 1199101

119889 (1199091 119865 (1199090 1199100)) = 119889 (119860 119861)

119889 (1199092 119865 (1199091 1199101)) = 119889 (119860 119861)

997904rArr 1199091le 1199092 (4)

Proof By hypothesis119865(1198600times1198600) sube 1198610 119865(1199091 1199100) isin 1198610 Hence

there exists 119909lowast1isin 119860 such that

119889 (119909lowast

1 119865 (1199091 1199100)) = 119889 (119860 119861) (5)

Using 119865 is proximal mixed monotone (in particular 119865 isproximally nondecreasing in 119909) to (4) and (5) we get

1199090le 1199091

119889 (1199091 119865 (1199090 1199100)) = 119889 (119860 119861)

119889 (119909lowast

1 119865 (1199091 1199100)) = 119889 (119860 119861)

997904rArr 1199091le 119909lowast

1 (6)

Analogously using 119865 is proximal mixed monotone (in par-ticular 119865 is proximally nonincreasing in 119910) to (4) and (5) weget

1199101le 1199100

119889 (1199092 119865 (1199091 1199101)) = 119889 (119860 119861)

119889 (119909lowast

1 119865 (1199091 1199100)) = 119889 (119860 119861)

997904rArr 119909lowast

1le 1199092 (7)

From (6) and (7) one can conclude that 1199091le 1199092 Hence the

proof is completed

Lemma3 Let (119883 119889 le) be a partially orderedmetric space and119860 119861 nonempty subsets of 119883 Assume that 119860

0is nonempty A

mapping 119865 119860 times 119860 rarr 119861 has proximal mixed monotone

Abstract and Applied Analysis 3

property with 119865(1198600times 1198600) sube 119861

0 then for any 119909

0 1199091 1199100 1199101

1199102in 1198600

1199090le 1199091 119910

0ge 1199101

119889 (1199101 119865 (1199100 1199090)) = 119889 (119860 119861)

119889 (1199102 119865 (1199101 1199091)) = 119889 (119860 119861)

997904rArr 1199101ge 1199102 (8)

Proof The proof is same as Lemma 2

Definition 4 Let (119883 119889 le) be a partially ordered metric spaceand 119860 119861 nonempty subsets of 119883 A mapping 119865 119860 times 119860 rarr

119861 is said to be proximally coupled contraction if there exists119896 isin (0 1) such that whenever

1199091le 1199092 119910

1ge 1199102

119889 (1199061 119865 (1199091 1199101)) = 119889 (119860 119861)

119889 (1199062 119865 (1199092 1199102)) = 119889 (119860 119861)

997904rArr 119889 (1199061 1199062)

le119896

2[119889 (1199091 1199092) + 119889 (119910

1 1199102)]

(9)

where 1199091 1199092 1199101 1199102 1199061 1199062isin 119860

One can see that if 119860 = 119861 in the above definition thenotion of proximally coupled contraction reduces to that ofcoupled contraction

3 Coupled Best Proximity Point Theorems

Let (119883 119889 le) be a partially ordered complete metric spaceendowed with the product space 119883 times 119883 with the followingpartial order

for (119909 119910) (119906 V) isin 119883 times 119883

(119906 V) le (119909 119910) lArrrArr 119909 ge 119906 119910 le V(10)

Theorem 5 Let (119883 le 119889) be a partially ordered completemetric space Let 119860 and 119861 be nonempty closed subsets of themetric space (119883 119889) such that 119860

0= 0 Let 119865 119860 times 119860 rarr 119861

satisfy the following conditions

(i) 119865 is a continuous proximally coupled contraction hav-ing the proximal mixed monotone property on 119860 suchthat 119865(119860

0times 1198600) sube 1198610

(ii) There exist elements (1199090 1199100) and (119909

1 1199101) in 119860

0times 1198600

such that

119889 (1199091 119865 (1199090 1199100)) = 119889 (119860 119861) 119908119894119905ℎ 119909

0le 1199091

119889 (1199101 119865 (1199100 1199090)) = 119889 (119860 119861) 119908119894119905ℎ 119910

0ge 1199101

(11)

Then there exist (119909 119910) isin 119860 times 119860 such that 119889(119909 119865(119909 119910)) =

119889(119860 119861) and 119889(119910 119865(119910 119909)) = 119889(119860 119861)

Proof By hypothesis there exist elements (1199090 1199100) and (119909

1 1199101)

in 1198600times 1198600such that

119889 (1199091 119865 (1199090 1199100)) = 119889 (119860 119861) with 119909

0le 1199091

119889 (1199101 119865 (1199100 1199090)) = 119889 (119860 119861) with 119910

0ge 1199101

(12)

Because of the fact that 119865(1198600times 1198600) sube 119861

0 there exists an

element (1199092 1199102) in 119860

0times 1198600such that

119889 (1199092 119865 (1199091 1199101)) = 119889 (119860 119861)

119889 (1199102 119865 (1199101 1199091)) = 119889 (119860 119861)

(13)

Hence from Lemmas 2 and 3 we obtain 1199091le 1199092and 119910

1ge 1199102

Continuing this process we can construct the sequences(119909119899) and (119910

119899) in 119860

0such that

119889 (119909119899+1

119865 (119909119899 119910119899)) = 119889 (119860 119861) forall119899 isin N (14)

with 1199090le 1199091le 1199092le sdot sdot sdot 119909

119899le 119909119899+1

sdot sdot sdot and

119889 (119910119899+1

119865 (119910119899 119909119899)) = 119889 (119860 119861) forall119899 isin N (15)

with 1199100ge 1199101ge 1199102ge sdot sdot sdot 119910

119899ge 119910119899+1

sdot sdot sdot Then119889(119909

119899 119865(119909119899minus1

119910119899minus1

)) = 119889(119860 119861) 119889(119909119899+1

119865(119909119899 119910119899)) =

119889(119860 119861) and also we have 119909119899minus1

le 119909119899 119910119899minus1

ge 119910119899 forall119899 isin N Now

using that 119865 is proximally coupled contraction on 119860 we get

119889 (119909119899 119909119899+1

) le119896

2[119889 (119909119899minus1

119909119899) + 119889 (119910

119899minus1 119910119899)] forall119899 isin N

(16)

Similarly

119889 (119910119899 119910119899+1

) le119896

2[119889 (119910119899minus1

119910119899) + 119889 (119909

119899minus1 119909119899)] forall119899 isin N

(17)

Adding (16) and (17) we get

119889 (119909119899 119909119899+1

) + 119889 (119910119899 119910119899+1

)

le 119896 [119889 (119909119899minus1

119909119899) + 119889 (119910

119899minus1 119910119899)]

le 119896 [119896 [119889 (119909119899minus2

119909119899minus1

) + 119889 (119910119899minus2

119910119899minus1

)]]

le sdot sdot sdot

le 119896119899

[119889 (1199090 1199091) + 119889 (119910

0 1199101)]

(18)

Finally we get

119889 (119909119899 119909119899+1

) + 119889 (119910119899 119910119899+1

) le 119896119899

[119889 (1199090 1199091) + 119889 (119910

0 1199101)]

(19)

Now we prove that (119909119899) and (119910

119899) are Cauchy sequences

For 119899 gt 119898 regarding triangle inequality and (19) one canobserve that

119889 (119909119899 119909119898) + 119889 (119910

119899 119910119898)

le 119889 (119909119899 119909119899minus1

) + 119889 (119910119899 119910119899minus1

) + sdot sdot sdot + 119889 (119909119898 119909119898+1

)

+ 119889 (119910119898 119910119898+1

)

le (119896119899minus1

+ sdot sdot sdot + 119896119898

) [119889 (1199090 1199091) + 119889 (119910

0 1199101)]

le119896119898

1 minus 119896[119889 (1199090 1199091) + 119889 (119910

0 1199101)]

(20)

4 Abstract and Applied Analysis

Let 120598 gt 0 be given Choose a natural number 119872 such that(119896119898

(1 minus 119896))[119889(1199090 1199091) + 119889(119910

0 1199101)] lt 120598 for all 119898 gt 119872 Thus

119889(119909119899 119909119898)+119889(119910

119899 119910119898) lt 120598 for 119899 gt 119898 Therefore the sequences

(119909119899) and (119910

119899) are Cauchy

Since 119860 is closed subset of a complete metric space 119883these sequences have limits Thus there exists 119909 119910 isin 119860 suchthat119909

119899rarr 119909 and119910

119899rarr 119910Therefore (119909

119899 119910119899) rarr (119909 119910) in119860times

119860 Since 119865 is continuous we have 119865(119909119899 119910119899) rarr 119865(119909 119910) and

119865(119910119899 119909119899) rarr 119865(119910 119909)

Hence the continuity of the metric function 119889

implies that 119889(119909119899+1

119865(119909119899 119910119899)) rarr 119889(119909 119865(119909 119910)) and

119889(119910119899+1

119865(119910119899 119909119899)) rarr 119889(119910 119865(119910 119909)) But from (14)

and (15) we get that the sequences 119889(119909119899+1

119865(119909119899 119910119899))

and 119889(119910119899+1

119865(119910119899 119909119899)) are constant sequences with the

value 119889(119860 119861) Therefore 119889(119909 119865(119909 119910)) = 119889(119860 119861) and119889(119910 119865(119910 119909)) = 119889(119860 119861) This completes the proof of thetheorem

Corollary 6 Let (119883 le 119889) be a partially ordered completemetric space Let 119860 be nonempty closed subset of the metricspace (119883 119889) Let 119865 119860 times 119860 rarr 119860 satisfy the followingconditions

(i) 119865 is continuous having the proximal mixed monotoneproperty and proximally coupled contraction on 119860

(ii) There exist (1199090 1199100) and (119909

1 1199101) in119860times119860 such that 119909

1=

119865(1199090 1199100) with 119909

0le 1199091and 119910

1= 119865(119910

0 1199090) with 119910

0ge

1199101

Then there exist (119909 119910) isin 119860times119860 such that 119889(119909 119865(119909 119910)) = 0 and119889(119910 119865(119910 119909)) = 0

In what follows we prove that Theorem 5 is still validfor 119865 not necessarily continuous assuming the followinghypothesis in 119860

(119909119899) is a nondecreasing sequence in 119860

such that 119909119899997888rarr 119909 then 119909

119899le 119909

(119910119899) is a nonincreasing sequence in 119860

such that 119910119899997888rarr 119910 then 119910 le 119910

119899

(21)

Theorem 7 Assume the conditions (21) and 1198600is closed in119883

instead of continuity of 119865 in Theorem 5 then the conclusion ofTheorem 5 holds

Proof Following the proof of Theorem 5 there existsequences (119909

119899) and (119910

119899) in 119860 satisfying the following

conditions

119889 (119909119899+1

119865 (119909119899 119910119899)) = 119889 (119860 119861) with 119909

119899le 119909119899+1

forall119899 isin N

(22)

119889 (119910119899+1

119865 (119910119899 119909119899)) = 119889 (119860 119861) with 119910

119899ge 119910119899+1

forall119899 isin N

(23)

Moreover (119909119899) converges to 119909 and 119910

119899converges to 119910 in 119860

From (21) we get 119909119899

le 119909 and 119910119899

ge 119910 Note that thesequences (119909

119899) and (119910

119899) are in119860

0and119860

0is closedTherefore

(119909 119910) isin 1198600times 1198600 Since 119865(119860

0times 1198600) sube 119861

0 there exists

119865(119909 119910) and119865(119910 119909) are in1198610Therefore there exists (119909lowast 119910lowast) isin

1198600times 1198600such that

119889 (119909lowast

119865 (119909 119910)) = 119889 (119860 119861) (24)

119889 (119910lowast

119865 (119910 119909)) = 119889 (119860 119861) (25)

Since 119909119899le 119909 and 119910

119899ge 119910 By using 119865 is proximally coupled

contraction for (22) and (24) also for (25) and (23) we get

119889 (119909119899+1

119909lowast

) le119896

2[119889 (119909119899 119909) + 119889 (119910

119899 119910)] forall119899

119889 (119910lowast

119910119899+1

) le119896

2[119889 (119910 119910

119899) + 119889 (119909 119909

119899)] forall119899

(26)

Since 119909119899rarr 119909 and 119910

119899rarr 119910 by taking limit on the above two

inequality we get 119909 = 119909lowast and119910 = 119910

lowast Consequently the resultfollows

Corollary 8 Assume the conditions (21) instead of continuityof 119865 in Corollary 6 then the conclusion of Corollary 6 holds

Now we present an example where it can be appreciatedthat hypotheses in Theorems 5 and 7 do not guaranteeuniqueness of the coupled best proximity point

Example 9 Let 119883 = (0 1) (1 0) (minus1 0) (0 minus1) sub R2 andconsider the usual order (119909 119910) ⪯ (119911 119905) hArr 119909 le 119911 and 119910 le 119905

Thus (119883 ⪯) is a partially ordered set Besides (119883 1198892) is

a complete metric space considering 1198892the euclidean metric

Let 119860 = (0 1) (1 0) and 119861 = (0 minus1) (minus1 0) be closedsubsets of119883 Then 119889(119860 119861) = radic2119860 = 119860

0and 119861 = 119861

0 Let 119865

119860 times 119860 rarr 119861 be defined as 119865((1199091 1199092) (1199101 1199102)) = (minus119909

2 minus1199091)

Then it can be seen that 119865 is continuous such that 119865(1198600times

1198600) sube 1198610 The only comparable pairs of points in119860 are 119909 ⪯ 119909

for 119909 isin 119860 hence proximal mixed monotone property andproximally coupled contraction on 119860 are satisfied trivially

It can be shown that the other hypotheses of the theoremare also satisfiedHowever119865has three coupled best proximitypoints ((0 1) (0 1)) ((0 1) (1 0)) and ((1 0) (1 0))

One can prove that the coupled best proximity point is infact unique provided that the product space 119860 times 119860 endowedwith the partial order mentioned earlier has the followingproperty

Every pair of elements has either a lower bound

or an upper bound(27)

It is known that this condition is equivalent to the followingFor every pair of (119909 119910) (119909lowast 119910lowast) isin 119860 times 119860 there exists

(1199111 1199112) in 119860 times 119860

that is comparable to (119909 119910) (119909lowast

119910lowast

) (28)

Abstract and Applied Analysis 5

Theorem 10 In addition to the hypothesis ofTheorem 5 (respTheorem 7) suppose that for any two elements (119909 119910) and(119909lowast

119910lowast

) in 1198600times 1198600

there exists (1199111 1199112) isin 119860

0times 1198600

such that (1199111 1199112) is comparable to (119909 119910) (119909lowast 119910lowast)

(29)

then 119865 has a unique coupled best proximity point

Proof FromTheorem 5 (respTheorem 7) the set of coupledbest proximity points of 119865 is nonempty Suppose that thereexist (119909 119910) and (119909

lowast

119910lowast

) in 119860 times 119860 which are coupled bestproximity points That is

119889 (119909 119865 (119909 119910)) = 119889 (119860 119861) 119889 (119910 119865 (119910 119909)) = 119889 (119860 119861)

119889 (119909lowast

119865 (119909lowast

119910lowast

)) = 119889 (119860 119861)

119889 (119910lowast

119865 (119910lowast

119909lowast

)) = 119889 (119860 119861)

(30)

We distinguish two cases

Case 1 Suppose that (119909 119910) is comparable Let (119909 119910) becomparable to (119909lowast 119910lowast) with respect to the ordering in119860times119860Apply 119865 as proximally coupled contraction to 119889(119909 119865(119909 119910)) =119889(119860 119861) and 119889(119909lowast 119865(119909lowast 119910lowast)) = 119889(119860 119861) there exists 119896 isin (0 1)

such that

119889 (119909 119909lowast

) le119896

2[119889 (119909 119909

lowast

) + 119889 (119910 119910lowast

)] (31)

Similarly one can prove that

119889 (119910 119910lowast

) le119896

2[119889 (119910 119910

lowast

) + 119889 (119909 119909lowast

)] (32)

Adding (31) and (32) we get

119889 (119909 119909lowast

) + 119889 (119910 119910lowast

) le 119896 [119889 (119909 119909lowast

) + 119889 (119910 119910lowast

)] (33)

This implies that 119889(119909 119909lowast) + 119889(119910 119910lowast

) = 0 hence 119909 = 119909lowast and

119910 = 119910lowast

Case 2 Suppose that (119909 119910) is not comparable Let (119909 119910) benoncomparable to (119909

lowast

119910lowast

) then there exists (1199061 V1) isin 119860

0times

1198600which is comparable to (119909 119910) and (119909

lowast

119910lowast

)Since119865(119860

0times1198600) sube 1198610 there exists (119906

2 V2) isin 1198600times1198600such

that 119889(1199062 119865(1199061 V1)) = 119889(119860 119861) and 119889(V

2 119865(V1 1199061)) = 119889(119860 119861)

Without loss of generality assume that (1199061 V1) le (119909 119910) (ie

119909 ge 1199061and 119910 le V

1) Note that (119906

1 V1) le (119909 119910) implies that

(119910 119909) le (V1 1199061) From Lemmas 2 and 3 we get

1199061le 119909 V

1ge 119910

119889 (1199062 119865 (1199061 V1)) = 119889 (119860 119861)

119889 (119909 119865 (119909 119910)) = 119889 (119860 119861)

997904rArr 1199062le 119909

1199061le 119909 V

1ge 119910

119889 (V2 119865 (V1 1199061)) = 119889 (119860 119861)

119889 (119910 119865 (119910 119909)) = 119889 (119860 119861)

997904rArr V2ge 119910

(34)

From the above two inequalities we obtain (1199062 V2) le (119909 119910)

Continuing this process we get sequences 119906119899 and V

119899 such

that 119889(119906119899+1

119865(119906119899 V119899)) = 119889(119860 119861) and 119889(V

119899+1 119865(V119899 119906119899)) =

119889(119860 119861) with (119906119899 V119899) le (119909 119910) forall119899 isin N By using that 119865 is a

proximally coupled contraction we get

119906119899le 119909 V

119899ge 119910

119889 (119906119899+1

119865 (119906119899 V119899)) = 119889 (119860 119861)

119889 (119909 119865 (119909 119910)) = 119889 (119860 119861)

dArr

119889 (119906119899+1

119909) le119896

2[119889 (119906119899 119909) + 119889 (V

119899 119910)] forall119899 isin N

(35)

Similarly we can prove that

119910 le V119899 119909 ge 119906

119899

119889 (119910 119865 (119910 119909)) = 119889 (119860 119861)

119889 (V119899+1

119865 (V119899 119906119899)) = 119889 (119860 119861)

dArr

119889 (119910 V119899+1

) le119896

2[119889 (119910 V

119899) + 119889 (119909 119906

119899)] forall119899 isin N

(36)

Adding (35) and (36) we obtain

119889 (119906119899+1

119909) + 119889 (119910 V119899+1

) le 119896 [119889 (119906119899 119909) + 119889 (V

119899 119910)]

119889 (119906119899+1

119909) + 119889 (119910 V119899+1

)

le 119896 [119889 (119906119899 119909) + 119889 (V

119899 119910)]

le 1198962

[119889 (119906119899minus1

119909) + 119889 (V119899minus1

119910)]

sdot sdot sdot

le 119896119899+1

[119889 (1199060 119909) + 119889 (V

0 119910)]

(37)

As 119899 rarr infin we get 119889(119906119899+1

119909) + 119889(119910 V119899+1

) rarr 0 so that119906119899rarr 119909 and V

119899rarr 119910

Analogously one can prove that 119906119899rarr 119909lowast and V

119899rarr 119910lowast

Therefore 119909 = 119909lowast and 119910 = 119910

lowast Hence the proof is completed

Example 11 Let 119883 = R be endowed with usual metric andwith the usual order in R

Suppose that 119860 = [1 2] and 119861 = [minus2 minus1] Then 119860 and 119861

are nonempty closed subsets of 119883 and 1198600= 1 and 119861

0= minus1

Also note that 119889(119860 119861) = 2Now consider the function 119865 119860 times 119860 rarr 119861 defined as

119865 (119909 119910) =minus119909 minus 119910 minus 2

4 (38)

Then it can be seen that 119865 is continuous and 119865(1 1) = minus1Hence119865(119860

0times1198600) sube 1198610 It is easy to see that other hypotheses

of the Theorem 10 are also satisfied Further it is easy to seethat (1 1) is the unique element satisfying the conclusion ofTheorem 10

The following result due to Fan [7] is a corollaryTheorem 10 by taking 119860 = 119861

6 Abstract and Applied Analysis

Corollary 12 In addition to the hypothesis of Corollary 6(resp Corollary 8) suppose that for any two elements (119909 119910)and (119909lowast 119910lowast) in 119860 times 119860

119905ℎ119890119903119890 119890119909119894119904119905119904 (1199111 1199112) isin 119860 times 119860

119904119906119888ℎ 119905ℎ119886119905 (1199111 1199112) 119894119904 119888119900119898119901119886119903119886119887119897119890 119905119900 (119909 119910) (119909

lowast

119910lowast

)

(39)

then 119865 has a unique coupled fixed point

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research Dr K Sitthithakerngkiet was funded by KingMongkutrsquos University of Technology North Bangkok (Con-tract no KMUTNB-GEN-57-19) Moreover the third authorwas supported by the Higher Education Research Promotionand National Research University Project of Thailand Officeof the Higher Education Commission (NRU2557)

References

[1] S Banach ldquoSur les operations dans les ensembles abstraits etleurs applications auxequations integralesrdquo Fundamenta Math-ematicae vol 3 pp 133ndash181 1922

[2] A D Arvanitakis ldquoA proof of the generalized Banach con-traction conjecturerdquo Proceedings of the American MathematicalSociety vol 131 no 12 pp 3647ndash3656 2003

[3] B S Choudhury and K Das ldquoA new contraction principle inmenger spacesrdquo Acta Mathematica Sinica vol 24 no 8 pp1379ndash1386 2008

[4] T Suzuki ldquoA generalized banach contraction principle thatcharacterizesmetric completenessrdquo Proceedings of the AmericanMathematical Society vol 136 no 5 pp 1861ndash1869 2008

[5] H Alikhani D Gopal M A Miandaragh Sh Rezapour andN Shahzad ldquoSome endpoint results for 120573-generalized weakcontractive multifunctionsrdquo The Scientific World Journal vol2013 Article ID 948472 7 pages 2013

[6] P Kumam F Rouzkard M Imdad and D Gopal ldquoFixedpoint theorems on ordered metric spaces through a rationalcontractionrdquoAbstract and Applied Analysis vol 2013 Article ID206515 9 pages 2013

[7] K Fan ldquoExtensions of two fixed point theorems of F EBrowderrdquoMathematische Zeitschrift vol 112 no 3 pp 234ndash2401969

[8] M De la Sen and R P Agarwal ldquoSome fixed point-type resultsfor a class of extended cyclic self-mappings with a more generalcontractive conditionrdquo Fixed PointTheory and Applications vol2011 article 59 2011

[9] P Kumam P Salimi and C Vetro ldquoBest proximity point resultsfor modified -proximal C-contraction mappingsrdquo Fixed PointTheory and Applications vol 2014 article 99 2014

[10] P Kumam H Aydi E Karapinar and W Sintunavarat ldquoBestproximity points and extension of Mizoguchi-Takahashirsquos fixedpoint theoremsrdquo Fixed Point Theory and Applications vol 2013article 242 2013

[11] P Kumam and A Roldn-Lpez-de-Hierro ldquoOn existence anduniqueness of g-best proximity points under (120593 120579 120572 119892)-con-tractivity conditions and consequencesrdquo Abstract and AppliedAnalysis vol 2014 Article ID 234027 14 pages 2014

[12] C di Bari T Suzuki and C Vetro ldquoBest proximity points forcyclic Meir-Keeler contractionsrdquo Nonlinear Analysis TheoryMethods and Applications vol 69 no 11 pp 3790ndash3794 2008

[13] A A Eldred and P Veeramani ldquoExistence and convergence ofbest proximity pointsrdquo Journal of Mathematical Analysis andApplications vol 323 no 2 pp 1001ndash1006 2006

[14] M A Al-Thagafi and N Shahzad ldquoConvergence and existenceresults for best proximity pointsrdquo Nonlinear Analysis TheoryMethods and Applications vol 70 no 10 pp 3665ndash3671 2009

[15] S Sadiq Basha and P Veeramani ldquoBest proximity pair theoremsfor multifunctions with open fibresrdquo Journal of ApproximationTheory vol 103 no 1 pp 119ndash129 2000

[16] W K Kim and K H Lee ldquoExistence of best proximity pairsand equilibrium pairsrdquo Journal of Mathematical Analysis andApplications vol 316 no 2 pp 433ndash446 2006

[17] W A Kirk S Reich and P Veeramani ldquoProximinal retracts andbest proximity pair theoremsrdquo Numerical Functional Analysisand Optimization vol 24 no 7-8 pp 851ndash862 2003

[18] V Sankar Raj ldquoA best proximity point theorem for weakly con-tractive non-self-mappingsrdquo Nonlinear Analysis Theory Meth-ods and Applications vol 74 no 14 pp 4804ndash4808 2011

[19] E Karapınar V Pragadeeswarar and M Marudai ldquoBest prox-imity point for generalized proximal weak contractions incomplete metric spacerdquo Journal of Applied Mathematics vol2014 Article ID 150941 6 pages 2014

[20] M Jleli and B Samet ldquoRemarks on the paper best proximitypoint theorems An exploration of a common solution toapproximation and optimization problemsrdquoAppliedMathemat-ics and Computation vol 228 pp 366ndash370 2014

[21] S S Basha ldquoDiscrete optimization in partially ordered setsrdquoJournal of Global Optimization vol 54 no 3 pp 511ndash517 2012

[22] A Abkar and M Gabeleh ldquoBest proximity points for cyclicmappings in ordered metric spacesrdquo Journal of OptimizationTheory and Applications vol 150 no 1 pp 188ndash193 2011

[23] A Abkar and M Gabeleh ldquoGeneralized cyclic contractions inpartially orderedmetric spacesrdquoOptimization Letters vol 6 no8 pp 1819ndash1830 2012

[24] W Sintunavarat and P Kumam ldquoCoupled best proximity pointtheorem in metric spacesrdquo Fixed PointTheory and Applicationsvol 2012 article 93 2012

[25] V Pragadeeswarar and M Marudai ldquoBest proximity pointsapproximation and optimization in partially ordered metricspacesrdquo Optimization Letters vol 7 pp 1883ndash1892 2013

[26] V Pragadeeswarar and M Marudai ldquoBest proximity points forgeneralized proximal weak contractions in partially orderedmetric spacesrdquo Optimization Letters 2013

[27] C Mongkolkeha and P Kumam ldquoBest proximity point The-orems for generalized cyclic contractions in ordered metricSpacesrdquo Journal of Optimization Theory and Applications vol155 pp 215ndash226 2012

[28] W Sanhan C Mongkolkeha and P Kumam ldquoGeneralizedproximal -contraction mappings and Best proximity pointsrdquoAbstract and Applied Analysis vol 2012 article 42 2012

[29] C Mongkolkeha and P Kumam ldquoBest proximity points forasymptotic proximal pointwise weaker Meir-Keeler-type -contraction mappingsrdquo Journal of the Egyptian MathematicalSociety vol 21 no 2 pp 87ndash90 2013

Abstract and Applied Analysis 7

[30] CMongkolkeha andPKumam ldquoSome commonbest proximitypoints for proximity commuting mappingsrdquo Optimization Let-ters vol 7 no 8 pp 1825ndash1836 2013

[31] C Mongkolkeha Y J Cho and P Kumam ldquoBest proximitypoints for generalized proxinal C-Contraction mappings inmetric spaces with partial ordersrdquo Journal of Inequalities andApplications vol 2013 article 94 2013

[32] C Mongkolkeha Y J Cho and P Kumam ldquoBest proximitypoints for Geraghtyrsquos proximal contraction mappingsrdquo FixedPoint Theory and Applications vol 2013 article 180 2013

[33] C Mongkolkeha C Kongban and P Kumam ldquoThe existenceand uniqueness of best proximity point theorems for gener-alized almost contractionrdquo Abstract and Applied Analysis vol2014 Article ID 813614 11 pages 2014

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

[35] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 70 no 12 pp 4341ndash4349 2009

[36] N V Luong and N X Thuan ldquoCoupled fixed point theoremsfor mixed monotone mappings and an application to integralequationsrdquo Computers and Mathematics with Applications vol62 no 11 pp 4238ndash4248 2011

[37] W Shatanawi ldquoPartially ordered cone metric spaces and cou-pled fixed point resultsrdquo Computers and Mathematics withApplications vol 60 no 8 pp 2508ndash2515 2010

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

Abstract and Applied Analysis 3

property with 119865(1198600times 1198600) sube 119861

0 then for any 119909

0 1199091 1199100 1199101

1199102in 1198600

1199090le 1199091 119910

0ge 1199101

119889 (1199101 119865 (1199100 1199090)) = 119889 (119860 119861)

119889 (1199102 119865 (1199101 1199091)) = 119889 (119860 119861)

997904rArr 1199101ge 1199102 (8)

Proof The proof is same as Lemma 2

Definition 4 Let (119883 119889 le) be a partially ordered metric spaceand 119860 119861 nonempty subsets of 119883 A mapping 119865 119860 times 119860 rarr

119861 is said to be proximally coupled contraction if there exists119896 isin (0 1) such that whenever

1199091le 1199092 119910

1ge 1199102

119889 (1199061 119865 (1199091 1199101)) = 119889 (119860 119861)

119889 (1199062 119865 (1199092 1199102)) = 119889 (119860 119861)

997904rArr 119889 (1199061 1199062)

le119896

2[119889 (1199091 1199092) + 119889 (119910

1 1199102)]

(9)

where 1199091 1199092 1199101 1199102 1199061 1199062isin 119860

One can see that if 119860 = 119861 in the above definition thenotion of proximally coupled contraction reduces to that ofcoupled contraction

3 Coupled Best Proximity Point Theorems

Let (119883 119889 le) be a partially ordered complete metric spaceendowed with the product space 119883 times 119883 with the followingpartial order

for (119909 119910) (119906 V) isin 119883 times 119883

(119906 V) le (119909 119910) lArrrArr 119909 ge 119906 119910 le V(10)

Theorem 5 Let (119883 le 119889) be a partially ordered completemetric space Let 119860 and 119861 be nonempty closed subsets of themetric space (119883 119889) such that 119860

0= 0 Let 119865 119860 times 119860 rarr 119861

satisfy the following conditions

(i) 119865 is a continuous proximally coupled contraction hav-ing the proximal mixed monotone property on 119860 suchthat 119865(119860

0times 1198600) sube 1198610

(ii) There exist elements (1199090 1199100) and (119909

1 1199101) in 119860

0times 1198600

such that

119889 (1199091 119865 (1199090 1199100)) = 119889 (119860 119861) 119908119894119905ℎ 119909

0le 1199091

119889 (1199101 119865 (1199100 1199090)) = 119889 (119860 119861) 119908119894119905ℎ 119910

0ge 1199101

(11)

Then there exist (119909 119910) isin 119860 times 119860 such that 119889(119909 119865(119909 119910)) =

119889(119860 119861) and 119889(119910 119865(119910 119909)) = 119889(119860 119861)

Proof By hypothesis there exist elements (1199090 1199100) and (119909

1 1199101)

in 1198600times 1198600such that

119889 (1199091 119865 (1199090 1199100)) = 119889 (119860 119861) with 119909

0le 1199091

119889 (1199101 119865 (1199100 1199090)) = 119889 (119860 119861) with 119910

0ge 1199101

(12)

Because of the fact that 119865(1198600times 1198600) sube 119861

0 there exists an

element (1199092 1199102) in 119860

0times 1198600such that

119889 (1199092 119865 (1199091 1199101)) = 119889 (119860 119861)

119889 (1199102 119865 (1199101 1199091)) = 119889 (119860 119861)

(13)

Hence from Lemmas 2 and 3 we obtain 1199091le 1199092and 119910

1ge 1199102

Continuing this process we can construct the sequences(119909119899) and (119910

119899) in 119860

0such that

119889 (119909119899+1

119865 (119909119899 119910119899)) = 119889 (119860 119861) forall119899 isin N (14)

with 1199090le 1199091le 1199092le sdot sdot sdot 119909

119899le 119909119899+1

sdot sdot sdot and

119889 (119910119899+1

119865 (119910119899 119909119899)) = 119889 (119860 119861) forall119899 isin N (15)

with 1199100ge 1199101ge 1199102ge sdot sdot sdot 119910

119899ge 119910119899+1

sdot sdot sdot Then119889(119909

119899 119865(119909119899minus1

119910119899minus1

)) = 119889(119860 119861) 119889(119909119899+1

119865(119909119899 119910119899)) =

119889(119860 119861) and also we have 119909119899minus1

le 119909119899 119910119899minus1

ge 119910119899 forall119899 isin N Now

using that 119865 is proximally coupled contraction on 119860 we get

119889 (119909119899 119909119899+1

) le119896

2[119889 (119909119899minus1

119909119899) + 119889 (119910

119899minus1 119910119899)] forall119899 isin N

(16)

Similarly

119889 (119910119899 119910119899+1

) le119896

2[119889 (119910119899minus1

119910119899) + 119889 (119909

119899minus1 119909119899)] forall119899 isin N

(17)

Adding (16) and (17) we get

119889 (119909119899 119909119899+1

) + 119889 (119910119899 119910119899+1

)

le 119896 [119889 (119909119899minus1

119909119899) + 119889 (119910

119899minus1 119910119899)]

le 119896 [119896 [119889 (119909119899minus2

119909119899minus1

) + 119889 (119910119899minus2

119910119899minus1

)]]

le sdot sdot sdot

le 119896119899

[119889 (1199090 1199091) + 119889 (119910

0 1199101)]

(18)

Finally we get

119889 (119909119899 119909119899+1

) + 119889 (119910119899 119910119899+1

) le 119896119899

[119889 (1199090 1199091) + 119889 (119910

0 1199101)]

(19)

Now we prove that (119909119899) and (119910

119899) are Cauchy sequences

For 119899 gt 119898 regarding triangle inequality and (19) one canobserve that

119889 (119909119899 119909119898) + 119889 (119910

119899 119910119898)

le 119889 (119909119899 119909119899minus1

) + 119889 (119910119899 119910119899minus1

) + sdot sdot sdot + 119889 (119909119898 119909119898+1

)

+ 119889 (119910119898 119910119898+1

)

le (119896119899minus1

+ sdot sdot sdot + 119896119898

) [119889 (1199090 1199091) + 119889 (119910

0 1199101)]

le119896119898

1 minus 119896[119889 (1199090 1199091) + 119889 (119910

0 1199101)]

(20)

4 Abstract and Applied Analysis

Let 120598 gt 0 be given Choose a natural number 119872 such that(119896119898

(1 minus 119896))[119889(1199090 1199091) + 119889(119910

0 1199101)] lt 120598 for all 119898 gt 119872 Thus

119889(119909119899 119909119898)+119889(119910

119899 119910119898) lt 120598 for 119899 gt 119898 Therefore the sequences

(119909119899) and (119910

119899) are Cauchy

Since 119860 is closed subset of a complete metric space 119883these sequences have limits Thus there exists 119909 119910 isin 119860 suchthat119909

119899rarr 119909 and119910

119899rarr 119910Therefore (119909

119899 119910119899) rarr (119909 119910) in119860times

119860 Since 119865 is continuous we have 119865(119909119899 119910119899) rarr 119865(119909 119910) and

119865(119910119899 119909119899) rarr 119865(119910 119909)

Hence the continuity of the metric function 119889

implies that 119889(119909119899+1

119865(119909119899 119910119899)) rarr 119889(119909 119865(119909 119910)) and

119889(119910119899+1

119865(119910119899 119909119899)) rarr 119889(119910 119865(119910 119909)) But from (14)

and (15) we get that the sequences 119889(119909119899+1

119865(119909119899 119910119899))

and 119889(119910119899+1

119865(119910119899 119909119899)) are constant sequences with the

value 119889(119860 119861) Therefore 119889(119909 119865(119909 119910)) = 119889(119860 119861) and119889(119910 119865(119910 119909)) = 119889(119860 119861) This completes the proof of thetheorem

Corollary 6 Let (119883 le 119889) be a partially ordered completemetric space Let 119860 be nonempty closed subset of the metricspace (119883 119889) Let 119865 119860 times 119860 rarr 119860 satisfy the followingconditions

(i) 119865 is continuous having the proximal mixed monotoneproperty and proximally coupled contraction on 119860

(ii) There exist (1199090 1199100) and (119909

1 1199101) in119860times119860 such that 119909

1=

119865(1199090 1199100) with 119909

0le 1199091and 119910

1= 119865(119910

0 1199090) with 119910

0ge

1199101

Then there exist (119909 119910) isin 119860times119860 such that 119889(119909 119865(119909 119910)) = 0 and119889(119910 119865(119910 119909)) = 0

In what follows we prove that Theorem 5 is still validfor 119865 not necessarily continuous assuming the followinghypothesis in 119860

(119909119899) is a nondecreasing sequence in 119860

such that 119909119899997888rarr 119909 then 119909

119899le 119909

(119910119899) is a nonincreasing sequence in 119860

such that 119910119899997888rarr 119910 then 119910 le 119910

119899

(21)

Theorem 7 Assume the conditions (21) and 1198600is closed in119883

instead of continuity of 119865 in Theorem 5 then the conclusion ofTheorem 5 holds

Proof Following the proof of Theorem 5 there existsequences (119909

119899) and (119910

119899) in 119860 satisfying the following

conditions

119889 (119909119899+1

119865 (119909119899 119910119899)) = 119889 (119860 119861) with 119909

119899le 119909119899+1

forall119899 isin N

(22)

119889 (119910119899+1

119865 (119910119899 119909119899)) = 119889 (119860 119861) with 119910

119899ge 119910119899+1

forall119899 isin N

(23)

Moreover (119909119899) converges to 119909 and 119910

119899converges to 119910 in 119860

From (21) we get 119909119899

le 119909 and 119910119899

ge 119910 Note that thesequences (119909

119899) and (119910

119899) are in119860

0and119860

0is closedTherefore

(119909 119910) isin 1198600times 1198600 Since 119865(119860

0times 1198600) sube 119861

0 there exists

119865(119909 119910) and119865(119910 119909) are in1198610Therefore there exists (119909lowast 119910lowast) isin

1198600times 1198600such that

119889 (119909lowast

119865 (119909 119910)) = 119889 (119860 119861) (24)

119889 (119910lowast

119865 (119910 119909)) = 119889 (119860 119861) (25)

Since 119909119899le 119909 and 119910

119899ge 119910 By using 119865 is proximally coupled

contraction for (22) and (24) also for (25) and (23) we get

119889 (119909119899+1

119909lowast

) le119896

2[119889 (119909119899 119909) + 119889 (119910

119899 119910)] forall119899

119889 (119910lowast

119910119899+1

) le119896

2[119889 (119910 119910

119899) + 119889 (119909 119909

119899)] forall119899

(26)

Since 119909119899rarr 119909 and 119910

119899rarr 119910 by taking limit on the above two

inequality we get 119909 = 119909lowast and119910 = 119910

lowast Consequently the resultfollows

Corollary 8 Assume the conditions (21) instead of continuityof 119865 in Corollary 6 then the conclusion of Corollary 6 holds

Now we present an example where it can be appreciatedthat hypotheses in Theorems 5 and 7 do not guaranteeuniqueness of the coupled best proximity point

Example 9 Let 119883 = (0 1) (1 0) (minus1 0) (0 minus1) sub R2 andconsider the usual order (119909 119910) ⪯ (119911 119905) hArr 119909 le 119911 and 119910 le 119905

Thus (119883 ⪯) is a partially ordered set Besides (119883 1198892) is

a complete metric space considering 1198892the euclidean metric

Let 119860 = (0 1) (1 0) and 119861 = (0 minus1) (minus1 0) be closedsubsets of119883 Then 119889(119860 119861) = radic2119860 = 119860

0and 119861 = 119861

0 Let 119865

119860 times 119860 rarr 119861 be defined as 119865((1199091 1199092) (1199101 1199102)) = (minus119909

2 minus1199091)

Then it can be seen that 119865 is continuous such that 119865(1198600times

1198600) sube 1198610 The only comparable pairs of points in119860 are 119909 ⪯ 119909

for 119909 isin 119860 hence proximal mixed monotone property andproximally coupled contraction on 119860 are satisfied trivially

It can be shown that the other hypotheses of the theoremare also satisfiedHowever119865has three coupled best proximitypoints ((0 1) (0 1)) ((0 1) (1 0)) and ((1 0) (1 0))

One can prove that the coupled best proximity point is infact unique provided that the product space 119860 times 119860 endowedwith the partial order mentioned earlier has the followingproperty

Every pair of elements has either a lower bound

or an upper bound(27)

It is known that this condition is equivalent to the followingFor every pair of (119909 119910) (119909lowast 119910lowast) isin 119860 times 119860 there exists

(1199111 1199112) in 119860 times 119860

that is comparable to (119909 119910) (119909lowast

119910lowast

) (28)

Abstract and Applied Analysis 5

Theorem 10 In addition to the hypothesis ofTheorem 5 (respTheorem 7) suppose that for any two elements (119909 119910) and(119909lowast

119910lowast

) in 1198600times 1198600

there exists (1199111 1199112) isin 119860

0times 1198600

such that (1199111 1199112) is comparable to (119909 119910) (119909lowast 119910lowast)

(29)

then 119865 has a unique coupled best proximity point

Proof FromTheorem 5 (respTheorem 7) the set of coupledbest proximity points of 119865 is nonempty Suppose that thereexist (119909 119910) and (119909

lowast

119910lowast

) in 119860 times 119860 which are coupled bestproximity points That is

119889 (119909 119865 (119909 119910)) = 119889 (119860 119861) 119889 (119910 119865 (119910 119909)) = 119889 (119860 119861)

119889 (119909lowast

119865 (119909lowast

119910lowast

)) = 119889 (119860 119861)

119889 (119910lowast

119865 (119910lowast

119909lowast

)) = 119889 (119860 119861)

(30)

We distinguish two cases

Case 1 Suppose that (119909 119910) is comparable Let (119909 119910) becomparable to (119909lowast 119910lowast) with respect to the ordering in119860times119860Apply 119865 as proximally coupled contraction to 119889(119909 119865(119909 119910)) =119889(119860 119861) and 119889(119909lowast 119865(119909lowast 119910lowast)) = 119889(119860 119861) there exists 119896 isin (0 1)

such that

119889 (119909 119909lowast

) le119896

2[119889 (119909 119909

lowast

) + 119889 (119910 119910lowast

)] (31)

Similarly one can prove that

119889 (119910 119910lowast

) le119896

2[119889 (119910 119910

lowast

) + 119889 (119909 119909lowast

)] (32)

Adding (31) and (32) we get

119889 (119909 119909lowast

) + 119889 (119910 119910lowast

) le 119896 [119889 (119909 119909lowast

) + 119889 (119910 119910lowast

)] (33)

This implies that 119889(119909 119909lowast) + 119889(119910 119910lowast

) = 0 hence 119909 = 119909lowast and

119910 = 119910lowast

Case 2 Suppose that (119909 119910) is not comparable Let (119909 119910) benoncomparable to (119909

lowast

119910lowast

) then there exists (1199061 V1) isin 119860

0times

1198600which is comparable to (119909 119910) and (119909

lowast

119910lowast

)Since119865(119860

0times1198600) sube 1198610 there exists (119906

2 V2) isin 1198600times1198600such

that 119889(1199062 119865(1199061 V1)) = 119889(119860 119861) and 119889(V

2 119865(V1 1199061)) = 119889(119860 119861)

Without loss of generality assume that (1199061 V1) le (119909 119910) (ie

119909 ge 1199061and 119910 le V

1) Note that (119906

1 V1) le (119909 119910) implies that

(119910 119909) le (V1 1199061) From Lemmas 2 and 3 we get

1199061le 119909 V

1ge 119910

119889 (1199062 119865 (1199061 V1)) = 119889 (119860 119861)

119889 (119909 119865 (119909 119910)) = 119889 (119860 119861)

997904rArr 1199062le 119909

1199061le 119909 V

1ge 119910

119889 (V2 119865 (V1 1199061)) = 119889 (119860 119861)

119889 (119910 119865 (119910 119909)) = 119889 (119860 119861)

997904rArr V2ge 119910

(34)

From the above two inequalities we obtain (1199062 V2) le (119909 119910)

Continuing this process we get sequences 119906119899 and V

119899 such

that 119889(119906119899+1

119865(119906119899 V119899)) = 119889(119860 119861) and 119889(V

119899+1 119865(V119899 119906119899)) =

119889(119860 119861) with (119906119899 V119899) le (119909 119910) forall119899 isin N By using that 119865 is a

proximally coupled contraction we get

119906119899le 119909 V

119899ge 119910

119889 (119906119899+1

119865 (119906119899 V119899)) = 119889 (119860 119861)

119889 (119909 119865 (119909 119910)) = 119889 (119860 119861)

dArr

119889 (119906119899+1

119909) le119896

2[119889 (119906119899 119909) + 119889 (V

119899 119910)] forall119899 isin N

(35)

Similarly we can prove that

119910 le V119899 119909 ge 119906

119899

119889 (119910 119865 (119910 119909)) = 119889 (119860 119861)

119889 (V119899+1

119865 (V119899 119906119899)) = 119889 (119860 119861)

dArr

119889 (119910 V119899+1

) le119896

2[119889 (119910 V

119899) + 119889 (119909 119906

119899)] forall119899 isin N

(36)

Adding (35) and (36) we obtain

119889 (119906119899+1

119909) + 119889 (119910 V119899+1

) le 119896 [119889 (119906119899 119909) + 119889 (V

119899 119910)]

119889 (119906119899+1

119909) + 119889 (119910 V119899+1

)

le 119896 [119889 (119906119899 119909) + 119889 (V

119899 119910)]

le 1198962

[119889 (119906119899minus1

119909) + 119889 (V119899minus1

119910)]

sdot sdot sdot

le 119896119899+1

[119889 (1199060 119909) + 119889 (V

0 119910)]

(37)

As 119899 rarr infin we get 119889(119906119899+1

119909) + 119889(119910 V119899+1

) rarr 0 so that119906119899rarr 119909 and V

119899rarr 119910

Analogously one can prove that 119906119899rarr 119909lowast and V

119899rarr 119910lowast

Therefore 119909 = 119909lowast and 119910 = 119910

lowast Hence the proof is completed

Example 11 Let 119883 = R be endowed with usual metric andwith the usual order in R

Suppose that 119860 = [1 2] and 119861 = [minus2 minus1] Then 119860 and 119861

are nonempty closed subsets of 119883 and 1198600= 1 and 119861

0= minus1

Also note that 119889(119860 119861) = 2Now consider the function 119865 119860 times 119860 rarr 119861 defined as

119865 (119909 119910) =minus119909 minus 119910 minus 2

4 (38)

Then it can be seen that 119865 is continuous and 119865(1 1) = minus1Hence119865(119860

0times1198600) sube 1198610 It is easy to see that other hypotheses

of the Theorem 10 are also satisfied Further it is easy to seethat (1 1) is the unique element satisfying the conclusion ofTheorem 10

The following result due to Fan [7] is a corollaryTheorem 10 by taking 119860 = 119861

6 Abstract and Applied Analysis

Corollary 12 In addition to the hypothesis of Corollary 6(resp Corollary 8) suppose that for any two elements (119909 119910)and (119909lowast 119910lowast) in 119860 times 119860

119905ℎ119890119903119890 119890119909119894119904119905119904 (1199111 1199112) isin 119860 times 119860

119904119906119888ℎ 119905ℎ119886119905 (1199111 1199112) 119894119904 119888119900119898119901119886119903119886119887119897119890 119905119900 (119909 119910) (119909

lowast

119910lowast

)

(39)

then 119865 has a unique coupled fixed point

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research Dr K Sitthithakerngkiet was funded by KingMongkutrsquos University of Technology North Bangkok (Con-tract no KMUTNB-GEN-57-19) Moreover the third authorwas supported by the Higher Education Research Promotionand National Research University Project of Thailand Officeof the Higher Education Commission (NRU2557)

References

[1] S Banach ldquoSur les operations dans les ensembles abstraits etleurs applications auxequations integralesrdquo Fundamenta Math-ematicae vol 3 pp 133ndash181 1922

[2] A D Arvanitakis ldquoA proof of the generalized Banach con-traction conjecturerdquo Proceedings of the American MathematicalSociety vol 131 no 12 pp 3647ndash3656 2003

[3] B S Choudhury and K Das ldquoA new contraction principle inmenger spacesrdquo Acta Mathematica Sinica vol 24 no 8 pp1379ndash1386 2008

[4] T Suzuki ldquoA generalized banach contraction principle thatcharacterizesmetric completenessrdquo Proceedings of the AmericanMathematical Society vol 136 no 5 pp 1861ndash1869 2008

[5] H Alikhani D Gopal M A Miandaragh Sh Rezapour andN Shahzad ldquoSome endpoint results for 120573-generalized weakcontractive multifunctionsrdquo The Scientific World Journal vol2013 Article ID 948472 7 pages 2013

[6] P Kumam F Rouzkard M Imdad and D Gopal ldquoFixedpoint theorems on ordered metric spaces through a rationalcontractionrdquoAbstract and Applied Analysis vol 2013 Article ID206515 9 pages 2013

[7] K Fan ldquoExtensions of two fixed point theorems of F EBrowderrdquoMathematische Zeitschrift vol 112 no 3 pp 234ndash2401969

[8] M De la Sen and R P Agarwal ldquoSome fixed point-type resultsfor a class of extended cyclic self-mappings with a more generalcontractive conditionrdquo Fixed PointTheory and Applications vol2011 article 59 2011

[9] P Kumam P Salimi and C Vetro ldquoBest proximity point resultsfor modified -proximal C-contraction mappingsrdquo Fixed PointTheory and Applications vol 2014 article 99 2014

[10] P Kumam H Aydi E Karapinar and W Sintunavarat ldquoBestproximity points and extension of Mizoguchi-Takahashirsquos fixedpoint theoremsrdquo Fixed Point Theory and Applications vol 2013article 242 2013

[11] P Kumam and A Roldn-Lpez-de-Hierro ldquoOn existence anduniqueness of g-best proximity points under (120593 120579 120572 119892)-con-tractivity conditions and consequencesrdquo Abstract and AppliedAnalysis vol 2014 Article ID 234027 14 pages 2014

[12] C di Bari T Suzuki and C Vetro ldquoBest proximity points forcyclic Meir-Keeler contractionsrdquo Nonlinear Analysis TheoryMethods and Applications vol 69 no 11 pp 3790ndash3794 2008

[13] A A Eldred and P Veeramani ldquoExistence and convergence ofbest proximity pointsrdquo Journal of Mathematical Analysis andApplications vol 323 no 2 pp 1001ndash1006 2006

[14] M A Al-Thagafi and N Shahzad ldquoConvergence and existenceresults for best proximity pointsrdquo Nonlinear Analysis TheoryMethods and Applications vol 70 no 10 pp 3665ndash3671 2009

[15] S Sadiq Basha and P Veeramani ldquoBest proximity pair theoremsfor multifunctions with open fibresrdquo Journal of ApproximationTheory vol 103 no 1 pp 119ndash129 2000

[16] W K Kim and K H Lee ldquoExistence of best proximity pairsand equilibrium pairsrdquo Journal of Mathematical Analysis andApplications vol 316 no 2 pp 433ndash446 2006

[17] W A Kirk S Reich and P Veeramani ldquoProximinal retracts andbest proximity pair theoremsrdquo Numerical Functional Analysisand Optimization vol 24 no 7-8 pp 851ndash862 2003

[18] V Sankar Raj ldquoA best proximity point theorem for weakly con-tractive non-self-mappingsrdquo Nonlinear Analysis Theory Meth-ods and Applications vol 74 no 14 pp 4804ndash4808 2011

[19] E Karapınar V Pragadeeswarar and M Marudai ldquoBest prox-imity point for generalized proximal weak contractions incomplete metric spacerdquo Journal of Applied Mathematics vol2014 Article ID 150941 6 pages 2014

[20] M Jleli and B Samet ldquoRemarks on the paper best proximitypoint theorems An exploration of a common solution toapproximation and optimization problemsrdquoAppliedMathemat-ics and Computation vol 228 pp 366ndash370 2014

[21] S S Basha ldquoDiscrete optimization in partially ordered setsrdquoJournal of Global Optimization vol 54 no 3 pp 511ndash517 2012

[22] A Abkar and M Gabeleh ldquoBest proximity points for cyclicmappings in ordered metric spacesrdquo Journal of OptimizationTheory and Applications vol 150 no 1 pp 188ndash193 2011

[23] A Abkar and M Gabeleh ldquoGeneralized cyclic contractions inpartially orderedmetric spacesrdquoOptimization Letters vol 6 no8 pp 1819ndash1830 2012

[24] W Sintunavarat and P Kumam ldquoCoupled best proximity pointtheorem in metric spacesrdquo Fixed PointTheory and Applicationsvol 2012 article 93 2012

[25] V Pragadeeswarar and M Marudai ldquoBest proximity pointsapproximation and optimization in partially ordered metricspacesrdquo Optimization Letters vol 7 pp 1883ndash1892 2013

[26] V Pragadeeswarar and M Marudai ldquoBest proximity points forgeneralized proximal weak contractions in partially orderedmetric spacesrdquo Optimization Letters 2013

[27] C Mongkolkeha and P Kumam ldquoBest proximity point The-orems for generalized cyclic contractions in ordered metricSpacesrdquo Journal of Optimization Theory and Applications vol155 pp 215ndash226 2012

[28] W Sanhan C Mongkolkeha and P Kumam ldquoGeneralizedproximal -contraction mappings and Best proximity pointsrdquoAbstract and Applied Analysis vol 2012 article 42 2012

[29] C Mongkolkeha and P Kumam ldquoBest proximity points forasymptotic proximal pointwise weaker Meir-Keeler-type -contraction mappingsrdquo Journal of the Egyptian MathematicalSociety vol 21 no 2 pp 87ndash90 2013

Abstract and Applied Analysis 7

[30] CMongkolkeha andPKumam ldquoSome commonbest proximitypoints for proximity commuting mappingsrdquo Optimization Let-ters vol 7 no 8 pp 1825ndash1836 2013

[31] C Mongkolkeha Y J Cho and P Kumam ldquoBest proximitypoints for generalized proxinal C-Contraction mappings inmetric spaces with partial ordersrdquo Journal of Inequalities andApplications vol 2013 article 94 2013

[32] C Mongkolkeha Y J Cho and P Kumam ldquoBest proximitypoints for Geraghtyrsquos proximal contraction mappingsrdquo FixedPoint Theory and Applications vol 2013 article 180 2013

[33] C Mongkolkeha C Kongban and P Kumam ldquoThe existenceand uniqueness of best proximity point theorems for gener-alized almost contractionrdquo Abstract and Applied Analysis vol2014 Article ID 813614 11 pages 2014

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

[35] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 70 no 12 pp 4341ndash4349 2009

[36] N V Luong and N X Thuan ldquoCoupled fixed point theoremsfor mixed monotone mappings and an application to integralequationsrdquo Computers and Mathematics with Applications vol62 no 11 pp 4238ndash4248 2011

[37] W Shatanawi ldquoPartially ordered cone metric spaces and cou-pled fixed point resultsrdquo Computers and Mathematics withApplications vol 60 no 8 pp 2508ndash2515 2010

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

4 Abstract and Applied Analysis

Let 120598 gt 0 be given Choose a natural number 119872 such that(119896119898

(1 minus 119896))[119889(1199090 1199091) + 119889(119910

0 1199101)] lt 120598 for all 119898 gt 119872 Thus

119889(119909119899 119909119898)+119889(119910

119899 119910119898) lt 120598 for 119899 gt 119898 Therefore the sequences

(119909119899) and (119910

119899) are Cauchy

Since 119860 is closed subset of a complete metric space 119883these sequences have limits Thus there exists 119909 119910 isin 119860 suchthat119909

119899rarr 119909 and119910

119899rarr 119910Therefore (119909

119899 119910119899) rarr (119909 119910) in119860times

119860 Since 119865 is continuous we have 119865(119909119899 119910119899) rarr 119865(119909 119910) and

119865(119910119899 119909119899) rarr 119865(119910 119909)

Hence the continuity of the metric function 119889

implies that 119889(119909119899+1

119865(119909119899 119910119899)) rarr 119889(119909 119865(119909 119910)) and

119889(119910119899+1

119865(119910119899 119909119899)) rarr 119889(119910 119865(119910 119909)) But from (14)

and (15) we get that the sequences 119889(119909119899+1

119865(119909119899 119910119899))

and 119889(119910119899+1

119865(119910119899 119909119899)) are constant sequences with the

value 119889(119860 119861) Therefore 119889(119909 119865(119909 119910)) = 119889(119860 119861) and119889(119910 119865(119910 119909)) = 119889(119860 119861) This completes the proof of thetheorem

Corollary 6 Let (119883 le 119889) be a partially ordered completemetric space Let 119860 be nonempty closed subset of the metricspace (119883 119889) Let 119865 119860 times 119860 rarr 119860 satisfy the followingconditions

(i) 119865 is continuous having the proximal mixed monotoneproperty and proximally coupled contraction on 119860

(ii) There exist (1199090 1199100) and (119909

1 1199101) in119860times119860 such that 119909

1=

119865(1199090 1199100) with 119909

0le 1199091and 119910

1= 119865(119910

0 1199090) with 119910

0ge

1199101

Then there exist (119909 119910) isin 119860times119860 such that 119889(119909 119865(119909 119910)) = 0 and119889(119910 119865(119910 119909)) = 0

In what follows we prove that Theorem 5 is still validfor 119865 not necessarily continuous assuming the followinghypothesis in 119860

(119909119899) is a nondecreasing sequence in 119860

such that 119909119899997888rarr 119909 then 119909

119899le 119909

(119910119899) is a nonincreasing sequence in 119860

such that 119910119899997888rarr 119910 then 119910 le 119910

119899

(21)

Theorem 7 Assume the conditions (21) and 1198600is closed in119883

instead of continuity of 119865 in Theorem 5 then the conclusion ofTheorem 5 holds

Proof Following the proof of Theorem 5 there existsequences (119909

119899) and (119910

119899) in 119860 satisfying the following

conditions

119889 (119909119899+1

119865 (119909119899 119910119899)) = 119889 (119860 119861) with 119909

119899le 119909119899+1

forall119899 isin N

(22)

119889 (119910119899+1

119865 (119910119899 119909119899)) = 119889 (119860 119861) with 119910

119899ge 119910119899+1

forall119899 isin N

(23)

Moreover (119909119899) converges to 119909 and 119910

119899converges to 119910 in 119860

From (21) we get 119909119899

le 119909 and 119910119899

ge 119910 Note that thesequences (119909

119899) and (119910

119899) are in119860

0and119860

0is closedTherefore

(119909 119910) isin 1198600times 1198600 Since 119865(119860

0times 1198600) sube 119861

0 there exists

119865(119909 119910) and119865(119910 119909) are in1198610Therefore there exists (119909lowast 119910lowast) isin

1198600times 1198600such that

119889 (119909lowast

119865 (119909 119910)) = 119889 (119860 119861) (24)

119889 (119910lowast

119865 (119910 119909)) = 119889 (119860 119861) (25)

Since 119909119899le 119909 and 119910

119899ge 119910 By using 119865 is proximally coupled

contraction for (22) and (24) also for (25) and (23) we get

119889 (119909119899+1

119909lowast

) le119896

2[119889 (119909119899 119909) + 119889 (119910

119899 119910)] forall119899

119889 (119910lowast

119910119899+1

) le119896

2[119889 (119910 119910

119899) + 119889 (119909 119909

119899)] forall119899

(26)

Since 119909119899rarr 119909 and 119910

119899rarr 119910 by taking limit on the above two

inequality we get 119909 = 119909lowast and119910 = 119910

lowast Consequently the resultfollows

Corollary 8 Assume the conditions (21) instead of continuityof 119865 in Corollary 6 then the conclusion of Corollary 6 holds

Now we present an example where it can be appreciatedthat hypotheses in Theorems 5 and 7 do not guaranteeuniqueness of the coupled best proximity point

Example 9 Let 119883 = (0 1) (1 0) (minus1 0) (0 minus1) sub R2 andconsider the usual order (119909 119910) ⪯ (119911 119905) hArr 119909 le 119911 and 119910 le 119905

Thus (119883 ⪯) is a partially ordered set Besides (119883 1198892) is

a complete metric space considering 1198892the euclidean metric

Let 119860 = (0 1) (1 0) and 119861 = (0 minus1) (minus1 0) be closedsubsets of119883 Then 119889(119860 119861) = radic2119860 = 119860

0and 119861 = 119861

0 Let 119865

119860 times 119860 rarr 119861 be defined as 119865((1199091 1199092) (1199101 1199102)) = (minus119909

2 minus1199091)

Then it can be seen that 119865 is continuous such that 119865(1198600times

1198600) sube 1198610 The only comparable pairs of points in119860 are 119909 ⪯ 119909

for 119909 isin 119860 hence proximal mixed monotone property andproximally coupled contraction on 119860 are satisfied trivially

It can be shown that the other hypotheses of the theoremare also satisfiedHowever119865has three coupled best proximitypoints ((0 1) (0 1)) ((0 1) (1 0)) and ((1 0) (1 0))

One can prove that the coupled best proximity point is infact unique provided that the product space 119860 times 119860 endowedwith the partial order mentioned earlier has the followingproperty

Every pair of elements has either a lower bound

or an upper bound(27)

It is known that this condition is equivalent to the followingFor every pair of (119909 119910) (119909lowast 119910lowast) isin 119860 times 119860 there exists

(1199111 1199112) in 119860 times 119860

that is comparable to (119909 119910) (119909lowast

119910lowast

) (28)

Abstract and Applied Analysis 5

Theorem 10 In addition to the hypothesis ofTheorem 5 (respTheorem 7) suppose that for any two elements (119909 119910) and(119909lowast

119910lowast

) in 1198600times 1198600

there exists (1199111 1199112) isin 119860

0times 1198600

such that (1199111 1199112) is comparable to (119909 119910) (119909lowast 119910lowast)

(29)

then 119865 has a unique coupled best proximity point

Proof FromTheorem 5 (respTheorem 7) the set of coupledbest proximity points of 119865 is nonempty Suppose that thereexist (119909 119910) and (119909

lowast

119910lowast

) in 119860 times 119860 which are coupled bestproximity points That is

119889 (119909 119865 (119909 119910)) = 119889 (119860 119861) 119889 (119910 119865 (119910 119909)) = 119889 (119860 119861)

119889 (119909lowast

119865 (119909lowast

119910lowast

)) = 119889 (119860 119861)

119889 (119910lowast

119865 (119910lowast

119909lowast

)) = 119889 (119860 119861)

(30)

We distinguish two cases

Case 1 Suppose that (119909 119910) is comparable Let (119909 119910) becomparable to (119909lowast 119910lowast) with respect to the ordering in119860times119860Apply 119865 as proximally coupled contraction to 119889(119909 119865(119909 119910)) =119889(119860 119861) and 119889(119909lowast 119865(119909lowast 119910lowast)) = 119889(119860 119861) there exists 119896 isin (0 1)

such that

119889 (119909 119909lowast

) le119896

2[119889 (119909 119909

lowast

) + 119889 (119910 119910lowast

)] (31)

Similarly one can prove that

119889 (119910 119910lowast

) le119896

2[119889 (119910 119910

lowast

) + 119889 (119909 119909lowast

)] (32)

Adding (31) and (32) we get

119889 (119909 119909lowast

) + 119889 (119910 119910lowast

) le 119896 [119889 (119909 119909lowast

) + 119889 (119910 119910lowast

)] (33)

This implies that 119889(119909 119909lowast) + 119889(119910 119910lowast

) = 0 hence 119909 = 119909lowast and

119910 = 119910lowast

Case 2 Suppose that (119909 119910) is not comparable Let (119909 119910) benoncomparable to (119909

lowast

119910lowast

) then there exists (1199061 V1) isin 119860

0times

1198600which is comparable to (119909 119910) and (119909

lowast

119910lowast

)Since119865(119860

0times1198600) sube 1198610 there exists (119906

2 V2) isin 1198600times1198600such

that 119889(1199062 119865(1199061 V1)) = 119889(119860 119861) and 119889(V

2 119865(V1 1199061)) = 119889(119860 119861)

Without loss of generality assume that (1199061 V1) le (119909 119910) (ie

119909 ge 1199061and 119910 le V

1) Note that (119906

1 V1) le (119909 119910) implies that

(119910 119909) le (V1 1199061) From Lemmas 2 and 3 we get

1199061le 119909 V

1ge 119910

119889 (1199062 119865 (1199061 V1)) = 119889 (119860 119861)

119889 (119909 119865 (119909 119910)) = 119889 (119860 119861)

997904rArr 1199062le 119909

1199061le 119909 V

1ge 119910

119889 (V2 119865 (V1 1199061)) = 119889 (119860 119861)

119889 (119910 119865 (119910 119909)) = 119889 (119860 119861)

997904rArr V2ge 119910

(34)

From the above two inequalities we obtain (1199062 V2) le (119909 119910)

Continuing this process we get sequences 119906119899 and V

119899 such

that 119889(119906119899+1

119865(119906119899 V119899)) = 119889(119860 119861) and 119889(V

119899+1 119865(V119899 119906119899)) =

119889(119860 119861) with (119906119899 V119899) le (119909 119910) forall119899 isin N By using that 119865 is a

proximally coupled contraction we get

119906119899le 119909 V

119899ge 119910

119889 (119906119899+1

119865 (119906119899 V119899)) = 119889 (119860 119861)

119889 (119909 119865 (119909 119910)) = 119889 (119860 119861)

dArr

119889 (119906119899+1

119909) le119896

2[119889 (119906119899 119909) + 119889 (V

119899 119910)] forall119899 isin N

(35)

Similarly we can prove that

119910 le V119899 119909 ge 119906

119899

119889 (119910 119865 (119910 119909)) = 119889 (119860 119861)

119889 (V119899+1

119865 (V119899 119906119899)) = 119889 (119860 119861)

dArr

119889 (119910 V119899+1

) le119896

2[119889 (119910 V

119899) + 119889 (119909 119906

119899)] forall119899 isin N

(36)

Adding (35) and (36) we obtain

119889 (119906119899+1

119909) + 119889 (119910 V119899+1

) le 119896 [119889 (119906119899 119909) + 119889 (V

119899 119910)]

119889 (119906119899+1

119909) + 119889 (119910 V119899+1

)

le 119896 [119889 (119906119899 119909) + 119889 (V

119899 119910)]

le 1198962

[119889 (119906119899minus1

119909) + 119889 (V119899minus1

119910)]

sdot sdot sdot

le 119896119899+1

[119889 (1199060 119909) + 119889 (V

0 119910)]

(37)

As 119899 rarr infin we get 119889(119906119899+1

119909) + 119889(119910 V119899+1

) rarr 0 so that119906119899rarr 119909 and V

119899rarr 119910

Analogously one can prove that 119906119899rarr 119909lowast and V

119899rarr 119910lowast

Therefore 119909 = 119909lowast and 119910 = 119910

lowast Hence the proof is completed

Example 11 Let 119883 = R be endowed with usual metric andwith the usual order in R

Suppose that 119860 = [1 2] and 119861 = [minus2 minus1] Then 119860 and 119861

are nonempty closed subsets of 119883 and 1198600= 1 and 119861

0= minus1

Also note that 119889(119860 119861) = 2Now consider the function 119865 119860 times 119860 rarr 119861 defined as

119865 (119909 119910) =minus119909 minus 119910 minus 2

4 (38)

Then it can be seen that 119865 is continuous and 119865(1 1) = minus1Hence119865(119860

0times1198600) sube 1198610 It is easy to see that other hypotheses

of the Theorem 10 are also satisfied Further it is easy to seethat (1 1) is the unique element satisfying the conclusion ofTheorem 10

The following result due to Fan [7] is a corollaryTheorem 10 by taking 119860 = 119861

6 Abstract and Applied Analysis

Corollary 12 In addition to the hypothesis of Corollary 6(resp Corollary 8) suppose that for any two elements (119909 119910)and (119909lowast 119910lowast) in 119860 times 119860

119905ℎ119890119903119890 119890119909119894119904119905119904 (1199111 1199112) isin 119860 times 119860

119904119906119888ℎ 119905ℎ119886119905 (1199111 1199112) 119894119904 119888119900119898119901119886119903119886119887119897119890 119905119900 (119909 119910) (119909

lowast

119910lowast

)

(39)

then 119865 has a unique coupled fixed point

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research Dr K Sitthithakerngkiet was funded by KingMongkutrsquos University of Technology North Bangkok (Con-tract no KMUTNB-GEN-57-19) Moreover the third authorwas supported by the Higher Education Research Promotionand National Research University Project of Thailand Officeof the Higher Education Commission (NRU2557)

References

[1] S Banach ldquoSur les operations dans les ensembles abstraits etleurs applications auxequations integralesrdquo Fundamenta Math-ematicae vol 3 pp 133ndash181 1922

[2] A D Arvanitakis ldquoA proof of the generalized Banach con-traction conjecturerdquo Proceedings of the American MathematicalSociety vol 131 no 12 pp 3647ndash3656 2003

[3] B S Choudhury and K Das ldquoA new contraction principle inmenger spacesrdquo Acta Mathematica Sinica vol 24 no 8 pp1379ndash1386 2008

[4] T Suzuki ldquoA generalized banach contraction principle thatcharacterizesmetric completenessrdquo Proceedings of the AmericanMathematical Society vol 136 no 5 pp 1861ndash1869 2008

[5] H Alikhani D Gopal M A Miandaragh Sh Rezapour andN Shahzad ldquoSome endpoint results for 120573-generalized weakcontractive multifunctionsrdquo The Scientific World Journal vol2013 Article ID 948472 7 pages 2013

[6] P Kumam F Rouzkard M Imdad and D Gopal ldquoFixedpoint theorems on ordered metric spaces through a rationalcontractionrdquoAbstract and Applied Analysis vol 2013 Article ID206515 9 pages 2013

[7] K Fan ldquoExtensions of two fixed point theorems of F EBrowderrdquoMathematische Zeitschrift vol 112 no 3 pp 234ndash2401969

[8] M De la Sen and R P Agarwal ldquoSome fixed point-type resultsfor a class of extended cyclic self-mappings with a more generalcontractive conditionrdquo Fixed PointTheory and Applications vol2011 article 59 2011

[9] P Kumam P Salimi and C Vetro ldquoBest proximity point resultsfor modified -proximal C-contraction mappingsrdquo Fixed PointTheory and Applications vol 2014 article 99 2014

[10] P Kumam H Aydi E Karapinar and W Sintunavarat ldquoBestproximity points and extension of Mizoguchi-Takahashirsquos fixedpoint theoremsrdquo Fixed Point Theory and Applications vol 2013article 242 2013

[11] P Kumam and A Roldn-Lpez-de-Hierro ldquoOn existence anduniqueness of g-best proximity points under (120593 120579 120572 119892)-con-tractivity conditions and consequencesrdquo Abstract and AppliedAnalysis vol 2014 Article ID 234027 14 pages 2014

[12] C di Bari T Suzuki and C Vetro ldquoBest proximity points forcyclic Meir-Keeler contractionsrdquo Nonlinear Analysis TheoryMethods and Applications vol 69 no 11 pp 3790ndash3794 2008

[13] A A Eldred and P Veeramani ldquoExistence and convergence ofbest proximity pointsrdquo Journal of Mathematical Analysis andApplications vol 323 no 2 pp 1001ndash1006 2006

[14] M A Al-Thagafi and N Shahzad ldquoConvergence and existenceresults for best proximity pointsrdquo Nonlinear Analysis TheoryMethods and Applications vol 70 no 10 pp 3665ndash3671 2009

[15] S Sadiq Basha and P Veeramani ldquoBest proximity pair theoremsfor multifunctions with open fibresrdquo Journal of ApproximationTheory vol 103 no 1 pp 119ndash129 2000

[16] W K Kim and K H Lee ldquoExistence of best proximity pairsand equilibrium pairsrdquo Journal of Mathematical Analysis andApplications vol 316 no 2 pp 433ndash446 2006

[17] W A Kirk S Reich and P Veeramani ldquoProximinal retracts andbest proximity pair theoremsrdquo Numerical Functional Analysisand Optimization vol 24 no 7-8 pp 851ndash862 2003

[18] V Sankar Raj ldquoA best proximity point theorem for weakly con-tractive non-self-mappingsrdquo Nonlinear Analysis Theory Meth-ods and Applications vol 74 no 14 pp 4804ndash4808 2011

[19] E Karapınar V Pragadeeswarar and M Marudai ldquoBest prox-imity point for generalized proximal weak contractions incomplete metric spacerdquo Journal of Applied Mathematics vol2014 Article ID 150941 6 pages 2014

[20] M Jleli and B Samet ldquoRemarks on the paper best proximitypoint theorems An exploration of a common solution toapproximation and optimization problemsrdquoAppliedMathemat-ics and Computation vol 228 pp 366ndash370 2014

[21] S S Basha ldquoDiscrete optimization in partially ordered setsrdquoJournal of Global Optimization vol 54 no 3 pp 511ndash517 2012

[22] A Abkar and M Gabeleh ldquoBest proximity points for cyclicmappings in ordered metric spacesrdquo Journal of OptimizationTheory and Applications vol 150 no 1 pp 188ndash193 2011

[23] A Abkar and M Gabeleh ldquoGeneralized cyclic contractions inpartially orderedmetric spacesrdquoOptimization Letters vol 6 no8 pp 1819ndash1830 2012

[24] W Sintunavarat and P Kumam ldquoCoupled best proximity pointtheorem in metric spacesrdquo Fixed PointTheory and Applicationsvol 2012 article 93 2012

[25] V Pragadeeswarar and M Marudai ldquoBest proximity pointsapproximation and optimization in partially ordered metricspacesrdquo Optimization Letters vol 7 pp 1883ndash1892 2013

[26] V Pragadeeswarar and M Marudai ldquoBest proximity points forgeneralized proximal weak contractions in partially orderedmetric spacesrdquo Optimization Letters 2013

[27] C Mongkolkeha and P Kumam ldquoBest proximity point The-orems for generalized cyclic contractions in ordered metricSpacesrdquo Journal of Optimization Theory and Applications vol155 pp 215ndash226 2012

[28] W Sanhan C Mongkolkeha and P Kumam ldquoGeneralizedproximal -contraction mappings and Best proximity pointsrdquoAbstract and Applied Analysis vol 2012 article 42 2012

[29] C Mongkolkeha and P Kumam ldquoBest proximity points forasymptotic proximal pointwise weaker Meir-Keeler-type -contraction mappingsrdquo Journal of the Egyptian MathematicalSociety vol 21 no 2 pp 87ndash90 2013

Abstract and Applied Analysis 7

[30] CMongkolkeha andPKumam ldquoSome commonbest proximitypoints for proximity commuting mappingsrdquo Optimization Let-ters vol 7 no 8 pp 1825ndash1836 2013

[31] C Mongkolkeha Y J Cho and P Kumam ldquoBest proximitypoints for generalized proxinal C-Contraction mappings inmetric spaces with partial ordersrdquo Journal of Inequalities andApplications vol 2013 article 94 2013

[32] C Mongkolkeha Y J Cho and P Kumam ldquoBest proximitypoints for Geraghtyrsquos proximal contraction mappingsrdquo FixedPoint Theory and Applications vol 2013 article 180 2013

[33] C Mongkolkeha C Kongban and P Kumam ldquoThe existenceand uniqueness of best proximity point theorems for gener-alized almost contractionrdquo Abstract and Applied Analysis vol2014 Article ID 813614 11 pages 2014

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

[35] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 70 no 12 pp 4341ndash4349 2009

[36] N V Luong and N X Thuan ldquoCoupled fixed point theoremsfor mixed monotone mappings and an application to integralequationsrdquo Computers and Mathematics with Applications vol62 no 11 pp 4238ndash4248 2011

[37] W Shatanawi ldquoPartially ordered cone metric spaces and cou-pled fixed point resultsrdquo Computers and Mathematics withApplications vol 60 no 8 pp 2508ndash2515 2010

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

Abstract and Applied Analysis 5

Theorem 10 In addition to the hypothesis ofTheorem 5 (respTheorem 7) suppose that for any two elements (119909 119910) and(119909lowast

119910lowast

) in 1198600times 1198600

there exists (1199111 1199112) isin 119860

0times 1198600

such that (1199111 1199112) is comparable to (119909 119910) (119909lowast 119910lowast)

(29)

then 119865 has a unique coupled best proximity point

Proof FromTheorem 5 (respTheorem 7) the set of coupledbest proximity points of 119865 is nonempty Suppose that thereexist (119909 119910) and (119909

lowast

119910lowast

) in 119860 times 119860 which are coupled bestproximity points That is

119889 (119909 119865 (119909 119910)) = 119889 (119860 119861) 119889 (119910 119865 (119910 119909)) = 119889 (119860 119861)

119889 (119909lowast

119865 (119909lowast

119910lowast

)) = 119889 (119860 119861)

119889 (119910lowast

119865 (119910lowast

119909lowast

)) = 119889 (119860 119861)

(30)

We distinguish two cases

Case 1 Suppose that (119909 119910) is comparable Let (119909 119910) becomparable to (119909lowast 119910lowast) with respect to the ordering in119860times119860Apply 119865 as proximally coupled contraction to 119889(119909 119865(119909 119910)) =119889(119860 119861) and 119889(119909lowast 119865(119909lowast 119910lowast)) = 119889(119860 119861) there exists 119896 isin (0 1)

such that

119889 (119909 119909lowast

) le119896

2[119889 (119909 119909

lowast

) + 119889 (119910 119910lowast

)] (31)

Similarly one can prove that

119889 (119910 119910lowast

) le119896

2[119889 (119910 119910

lowast

) + 119889 (119909 119909lowast

)] (32)

Adding (31) and (32) we get

119889 (119909 119909lowast

) + 119889 (119910 119910lowast

) le 119896 [119889 (119909 119909lowast

) + 119889 (119910 119910lowast

)] (33)

This implies that 119889(119909 119909lowast) + 119889(119910 119910lowast

) = 0 hence 119909 = 119909lowast and

119910 = 119910lowast

Case 2 Suppose that (119909 119910) is not comparable Let (119909 119910) benoncomparable to (119909

lowast

119910lowast

) then there exists (1199061 V1) isin 119860

0times

1198600which is comparable to (119909 119910) and (119909

lowast

119910lowast

)Since119865(119860

0times1198600) sube 1198610 there exists (119906

2 V2) isin 1198600times1198600such

that 119889(1199062 119865(1199061 V1)) = 119889(119860 119861) and 119889(V

2 119865(V1 1199061)) = 119889(119860 119861)

Without loss of generality assume that (1199061 V1) le (119909 119910) (ie

119909 ge 1199061and 119910 le V

1) Note that (119906

1 V1) le (119909 119910) implies that

(119910 119909) le (V1 1199061) From Lemmas 2 and 3 we get

1199061le 119909 V

1ge 119910

119889 (1199062 119865 (1199061 V1)) = 119889 (119860 119861)

119889 (119909 119865 (119909 119910)) = 119889 (119860 119861)

997904rArr 1199062le 119909

1199061le 119909 V

1ge 119910

119889 (V2 119865 (V1 1199061)) = 119889 (119860 119861)

119889 (119910 119865 (119910 119909)) = 119889 (119860 119861)

997904rArr V2ge 119910

(34)

From the above two inequalities we obtain (1199062 V2) le (119909 119910)

Continuing this process we get sequences 119906119899 and V

119899 such

that 119889(119906119899+1

119865(119906119899 V119899)) = 119889(119860 119861) and 119889(V

119899+1 119865(V119899 119906119899)) =

119889(119860 119861) with (119906119899 V119899) le (119909 119910) forall119899 isin N By using that 119865 is a

proximally coupled contraction we get

119906119899le 119909 V

119899ge 119910

119889 (119906119899+1

119865 (119906119899 V119899)) = 119889 (119860 119861)

119889 (119909 119865 (119909 119910)) = 119889 (119860 119861)

dArr

119889 (119906119899+1

119909) le119896

2[119889 (119906119899 119909) + 119889 (V

119899 119910)] forall119899 isin N

(35)

Similarly we can prove that

119910 le V119899 119909 ge 119906

119899

119889 (119910 119865 (119910 119909)) = 119889 (119860 119861)

119889 (V119899+1

119865 (V119899 119906119899)) = 119889 (119860 119861)

dArr

119889 (119910 V119899+1

) le119896

2[119889 (119910 V

119899) + 119889 (119909 119906

119899)] forall119899 isin N

(36)

Adding (35) and (36) we obtain

119889 (119906119899+1

119909) + 119889 (119910 V119899+1

) le 119896 [119889 (119906119899 119909) + 119889 (V

119899 119910)]

119889 (119906119899+1

119909) + 119889 (119910 V119899+1

)

le 119896 [119889 (119906119899 119909) + 119889 (V

119899 119910)]

le 1198962

[119889 (119906119899minus1

119909) + 119889 (V119899minus1

119910)]

sdot sdot sdot

le 119896119899+1

[119889 (1199060 119909) + 119889 (V

0 119910)]

(37)

As 119899 rarr infin we get 119889(119906119899+1

119909) + 119889(119910 V119899+1

) rarr 0 so that119906119899rarr 119909 and V

119899rarr 119910

Analogously one can prove that 119906119899rarr 119909lowast and V

119899rarr 119910lowast

Therefore 119909 = 119909lowast and 119910 = 119910

lowast Hence the proof is completed

Example 11 Let 119883 = R be endowed with usual metric andwith the usual order in R

Suppose that 119860 = [1 2] and 119861 = [minus2 minus1] Then 119860 and 119861

are nonempty closed subsets of 119883 and 1198600= 1 and 119861

0= minus1

Also note that 119889(119860 119861) = 2Now consider the function 119865 119860 times 119860 rarr 119861 defined as

119865 (119909 119910) =minus119909 minus 119910 minus 2

4 (38)

Then it can be seen that 119865 is continuous and 119865(1 1) = minus1Hence119865(119860

0times1198600) sube 1198610 It is easy to see that other hypotheses

of the Theorem 10 are also satisfied Further it is easy to seethat (1 1) is the unique element satisfying the conclusion ofTheorem 10

The following result due to Fan [7] is a corollaryTheorem 10 by taking 119860 = 119861

6 Abstract and Applied Analysis

Corollary 12 In addition to the hypothesis of Corollary 6(resp Corollary 8) suppose that for any two elements (119909 119910)and (119909lowast 119910lowast) in 119860 times 119860

119905ℎ119890119903119890 119890119909119894119904119905119904 (1199111 1199112) isin 119860 times 119860

119904119906119888ℎ 119905ℎ119886119905 (1199111 1199112) 119894119904 119888119900119898119901119886119903119886119887119897119890 119905119900 (119909 119910) (119909

lowast

119910lowast

)

(39)

then 119865 has a unique coupled fixed point

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research Dr K Sitthithakerngkiet was funded by KingMongkutrsquos University of Technology North Bangkok (Con-tract no KMUTNB-GEN-57-19) Moreover the third authorwas supported by the Higher Education Research Promotionand National Research University Project of Thailand Officeof the Higher Education Commission (NRU2557)

References

[1] S Banach ldquoSur les operations dans les ensembles abstraits etleurs applications auxequations integralesrdquo Fundamenta Math-ematicae vol 3 pp 133ndash181 1922

[2] A D Arvanitakis ldquoA proof of the generalized Banach con-traction conjecturerdquo Proceedings of the American MathematicalSociety vol 131 no 12 pp 3647ndash3656 2003

[3] B S Choudhury and K Das ldquoA new contraction principle inmenger spacesrdquo Acta Mathematica Sinica vol 24 no 8 pp1379ndash1386 2008

[4] T Suzuki ldquoA generalized banach contraction principle thatcharacterizesmetric completenessrdquo Proceedings of the AmericanMathematical Society vol 136 no 5 pp 1861ndash1869 2008

[5] H Alikhani D Gopal M A Miandaragh Sh Rezapour andN Shahzad ldquoSome endpoint results for 120573-generalized weakcontractive multifunctionsrdquo The Scientific World Journal vol2013 Article ID 948472 7 pages 2013

[6] P Kumam F Rouzkard M Imdad and D Gopal ldquoFixedpoint theorems on ordered metric spaces through a rationalcontractionrdquoAbstract and Applied Analysis vol 2013 Article ID206515 9 pages 2013

[7] K Fan ldquoExtensions of two fixed point theorems of F EBrowderrdquoMathematische Zeitschrift vol 112 no 3 pp 234ndash2401969

[8] M De la Sen and R P Agarwal ldquoSome fixed point-type resultsfor a class of extended cyclic self-mappings with a more generalcontractive conditionrdquo Fixed PointTheory and Applications vol2011 article 59 2011

[9] P Kumam P Salimi and C Vetro ldquoBest proximity point resultsfor modified -proximal C-contraction mappingsrdquo Fixed PointTheory and Applications vol 2014 article 99 2014

[10] P Kumam H Aydi E Karapinar and W Sintunavarat ldquoBestproximity points and extension of Mizoguchi-Takahashirsquos fixedpoint theoremsrdquo Fixed Point Theory and Applications vol 2013article 242 2013

[11] P Kumam and A Roldn-Lpez-de-Hierro ldquoOn existence anduniqueness of g-best proximity points under (120593 120579 120572 119892)-con-tractivity conditions and consequencesrdquo Abstract and AppliedAnalysis vol 2014 Article ID 234027 14 pages 2014

[12] C di Bari T Suzuki and C Vetro ldquoBest proximity points forcyclic Meir-Keeler contractionsrdquo Nonlinear Analysis TheoryMethods and Applications vol 69 no 11 pp 3790ndash3794 2008

[13] A A Eldred and P Veeramani ldquoExistence and convergence ofbest proximity pointsrdquo Journal of Mathematical Analysis andApplications vol 323 no 2 pp 1001ndash1006 2006

[14] M A Al-Thagafi and N Shahzad ldquoConvergence and existenceresults for best proximity pointsrdquo Nonlinear Analysis TheoryMethods and Applications vol 70 no 10 pp 3665ndash3671 2009

[15] S Sadiq Basha and P Veeramani ldquoBest proximity pair theoremsfor multifunctions with open fibresrdquo Journal of ApproximationTheory vol 103 no 1 pp 119ndash129 2000

[16] W K Kim and K H Lee ldquoExistence of best proximity pairsand equilibrium pairsrdquo Journal of Mathematical Analysis andApplications vol 316 no 2 pp 433ndash446 2006

[17] W A Kirk S Reich and P Veeramani ldquoProximinal retracts andbest proximity pair theoremsrdquo Numerical Functional Analysisand Optimization vol 24 no 7-8 pp 851ndash862 2003

[18] V Sankar Raj ldquoA best proximity point theorem for weakly con-tractive non-self-mappingsrdquo Nonlinear Analysis Theory Meth-ods and Applications vol 74 no 14 pp 4804ndash4808 2011

[19] E Karapınar V Pragadeeswarar and M Marudai ldquoBest prox-imity point for generalized proximal weak contractions incomplete metric spacerdquo Journal of Applied Mathematics vol2014 Article ID 150941 6 pages 2014

[20] M Jleli and B Samet ldquoRemarks on the paper best proximitypoint theorems An exploration of a common solution toapproximation and optimization problemsrdquoAppliedMathemat-ics and Computation vol 228 pp 366ndash370 2014

[21] S S Basha ldquoDiscrete optimization in partially ordered setsrdquoJournal of Global Optimization vol 54 no 3 pp 511ndash517 2012

[22] A Abkar and M Gabeleh ldquoBest proximity points for cyclicmappings in ordered metric spacesrdquo Journal of OptimizationTheory and Applications vol 150 no 1 pp 188ndash193 2011

[23] A Abkar and M Gabeleh ldquoGeneralized cyclic contractions inpartially orderedmetric spacesrdquoOptimization Letters vol 6 no8 pp 1819ndash1830 2012

[24] W Sintunavarat and P Kumam ldquoCoupled best proximity pointtheorem in metric spacesrdquo Fixed PointTheory and Applicationsvol 2012 article 93 2012

[25] V Pragadeeswarar and M Marudai ldquoBest proximity pointsapproximation and optimization in partially ordered metricspacesrdquo Optimization Letters vol 7 pp 1883ndash1892 2013

[26] V Pragadeeswarar and M Marudai ldquoBest proximity points forgeneralized proximal weak contractions in partially orderedmetric spacesrdquo Optimization Letters 2013

[27] C Mongkolkeha and P Kumam ldquoBest proximity point The-orems for generalized cyclic contractions in ordered metricSpacesrdquo Journal of Optimization Theory and Applications vol155 pp 215ndash226 2012

[28] W Sanhan C Mongkolkeha and P Kumam ldquoGeneralizedproximal -contraction mappings and Best proximity pointsrdquoAbstract and Applied Analysis vol 2012 article 42 2012

[29] C Mongkolkeha and P Kumam ldquoBest proximity points forasymptotic proximal pointwise weaker Meir-Keeler-type -contraction mappingsrdquo Journal of the Egyptian MathematicalSociety vol 21 no 2 pp 87ndash90 2013

Abstract and Applied Analysis 7

[30] CMongkolkeha andPKumam ldquoSome commonbest proximitypoints for proximity commuting mappingsrdquo Optimization Let-ters vol 7 no 8 pp 1825ndash1836 2013

[31] C Mongkolkeha Y J Cho and P Kumam ldquoBest proximitypoints for generalized proxinal C-Contraction mappings inmetric spaces with partial ordersrdquo Journal of Inequalities andApplications vol 2013 article 94 2013

[32] C Mongkolkeha Y J Cho and P Kumam ldquoBest proximitypoints for Geraghtyrsquos proximal contraction mappingsrdquo FixedPoint Theory and Applications vol 2013 article 180 2013

[33] C Mongkolkeha C Kongban and P Kumam ldquoThe existenceand uniqueness of best proximity point theorems for gener-alized almost contractionrdquo Abstract and Applied Analysis vol2014 Article ID 813614 11 pages 2014

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

[35] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 70 no 12 pp 4341ndash4349 2009

[36] N V Luong and N X Thuan ldquoCoupled fixed point theoremsfor mixed monotone mappings and an application to integralequationsrdquo Computers and Mathematics with Applications vol62 no 11 pp 4238ndash4248 2011

[37] W Shatanawi ldquoPartially ordered cone metric spaces and cou-pled fixed point resultsrdquo Computers and Mathematics withApplications vol 60 no 8 pp 2508ndash2515 2010

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 Abstract and Applied Analysis

Corollary 12 In addition to the hypothesis of Corollary 6(resp Corollary 8) suppose that for any two elements (119909 119910)and (119909lowast 119910lowast) in 119860 times 119860

119905ℎ119890119903119890 119890119909119894119904119905119904 (1199111 1199112) isin 119860 times 119860

119904119906119888ℎ 119905ℎ119886119905 (1199111 1199112) 119894119904 119888119900119898119901119886119903119886119887119897119890 119905119900 (119909 119910) (119909

lowast

119910lowast

)

(39)

then 119865 has a unique coupled fixed point

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research Dr K Sitthithakerngkiet was funded by KingMongkutrsquos University of Technology North Bangkok (Con-tract no KMUTNB-GEN-57-19) Moreover the third authorwas supported by the Higher Education Research Promotionand National Research University Project of Thailand Officeof the Higher Education Commission (NRU2557)

References

[1] S Banach ldquoSur les operations dans les ensembles abstraits etleurs applications auxequations integralesrdquo Fundamenta Math-ematicae vol 3 pp 133ndash181 1922

[2] A D Arvanitakis ldquoA proof of the generalized Banach con-traction conjecturerdquo Proceedings of the American MathematicalSociety vol 131 no 12 pp 3647ndash3656 2003

[3] B S Choudhury and K Das ldquoA new contraction principle inmenger spacesrdquo Acta Mathematica Sinica vol 24 no 8 pp1379ndash1386 2008

[4] T Suzuki ldquoA generalized banach contraction principle thatcharacterizesmetric completenessrdquo Proceedings of the AmericanMathematical Society vol 136 no 5 pp 1861ndash1869 2008

[5] H Alikhani D Gopal M A Miandaragh Sh Rezapour andN Shahzad ldquoSome endpoint results for 120573-generalized weakcontractive multifunctionsrdquo The Scientific World Journal vol2013 Article ID 948472 7 pages 2013

[6] P Kumam F Rouzkard M Imdad and D Gopal ldquoFixedpoint theorems on ordered metric spaces through a rationalcontractionrdquoAbstract and Applied Analysis vol 2013 Article ID206515 9 pages 2013

[7] K Fan ldquoExtensions of two fixed point theorems of F EBrowderrdquoMathematische Zeitschrift vol 112 no 3 pp 234ndash2401969

[8] M De la Sen and R P Agarwal ldquoSome fixed point-type resultsfor a class of extended cyclic self-mappings with a more generalcontractive conditionrdquo Fixed PointTheory and Applications vol2011 article 59 2011

[9] P Kumam P Salimi and C Vetro ldquoBest proximity point resultsfor modified -proximal C-contraction mappingsrdquo Fixed PointTheory and Applications vol 2014 article 99 2014

[10] P Kumam H Aydi E Karapinar and W Sintunavarat ldquoBestproximity points and extension of Mizoguchi-Takahashirsquos fixedpoint theoremsrdquo Fixed Point Theory and Applications vol 2013article 242 2013

[11] P Kumam and A Roldn-Lpez-de-Hierro ldquoOn existence anduniqueness of g-best proximity points under (120593 120579 120572 119892)-con-tractivity conditions and consequencesrdquo Abstract and AppliedAnalysis vol 2014 Article ID 234027 14 pages 2014

[12] C di Bari T Suzuki and C Vetro ldquoBest proximity points forcyclic Meir-Keeler contractionsrdquo Nonlinear Analysis TheoryMethods and Applications vol 69 no 11 pp 3790ndash3794 2008

[13] A A Eldred and P Veeramani ldquoExistence and convergence ofbest proximity pointsrdquo Journal of Mathematical Analysis andApplications vol 323 no 2 pp 1001ndash1006 2006

[14] M A Al-Thagafi and N Shahzad ldquoConvergence and existenceresults for best proximity pointsrdquo Nonlinear Analysis TheoryMethods and Applications vol 70 no 10 pp 3665ndash3671 2009

[15] S Sadiq Basha and P Veeramani ldquoBest proximity pair theoremsfor multifunctions with open fibresrdquo Journal of ApproximationTheory vol 103 no 1 pp 119ndash129 2000

[16] W K Kim and K H Lee ldquoExistence of best proximity pairsand equilibrium pairsrdquo Journal of Mathematical Analysis andApplications vol 316 no 2 pp 433ndash446 2006

[17] W A Kirk S Reich and P Veeramani ldquoProximinal retracts andbest proximity pair theoremsrdquo Numerical Functional Analysisand Optimization vol 24 no 7-8 pp 851ndash862 2003

[18] V Sankar Raj ldquoA best proximity point theorem for weakly con-tractive non-self-mappingsrdquo Nonlinear Analysis Theory Meth-ods and Applications vol 74 no 14 pp 4804ndash4808 2011

[19] E Karapınar V Pragadeeswarar and M Marudai ldquoBest prox-imity point for generalized proximal weak contractions incomplete metric spacerdquo Journal of Applied Mathematics vol2014 Article ID 150941 6 pages 2014

[20] M Jleli and B Samet ldquoRemarks on the paper best proximitypoint theorems An exploration of a common solution toapproximation and optimization problemsrdquoAppliedMathemat-ics and Computation vol 228 pp 366ndash370 2014

[21] S S Basha ldquoDiscrete optimization in partially ordered setsrdquoJournal of Global Optimization vol 54 no 3 pp 511ndash517 2012

[22] A Abkar and M Gabeleh ldquoBest proximity points for cyclicmappings in ordered metric spacesrdquo Journal of OptimizationTheory and Applications vol 150 no 1 pp 188ndash193 2011

[23] A Abkar and M Gabeleh ldquoGeneralized cyclic contractions inpartially orderedmetric spacesrdquoOptimization Letters vol 6 no8 pp 1819ndash1830 2012

[24] W Sintunavarat and P Kumam ldquoCoupled best proximity pointtheorem in metric spacesrdquo Fixed PointTheory and Applicationsvol 2012 article 93 2012

[25] V Pragadeeswarar and M Marudai ldquoBest proximity pointsapproximation and optimization in partially ordered metricspacesrdquo Optimization Letters vol 7 pp 1883ndash1892 2013

[26] V Pragadeeswarar and M Marudai ldquoBest proximity points forgeneralized proximal weak contractions in partially orderedmetric spacesrdquo Optimization Letters 2013

[27] C Mongkolkeha and P Kumam ldquoBest proximity point The-orems for generalized cyclic contractions in ordered metricSpacesrdquo Journal of Optimization Theory and Applications vol155 pp 215ndash226 2012

[28] W Sanhan C Mongkolkeha and P Kumam ldquoGeneralizedproximal -contraction mappings and Best proximity pointsrdquoAbstract and Applied Analysis vol 2012 article 42 2012

[29] C Mongkolkeha and P Kumam ldquoBest proximity points forasymptotic proximal pointwise weaker Meir-Keeler-type -contraction mappingsrdquo Journal of the Egyptian MathematicalSociety vol 21 no 2 pp 87ndash90 2013

Abstract and Applied Analysis 7

[30] CMongkolkeha andPKumam ldquoSome commonbest proximitypoints for proximity commuting mappingsrdquo Optimization Let-ters vol 7 no 8 pp 1825ndash1836 2013

[31] C Mongkolkeha Y J Cho and P Kumam ldquoBest proximitypoints for generalized proxinal C-Contraction mappings inmetric spaces with partial ordersrdquo Journal of Inequalities andApplications vol 2013 article 94 2013

[32] C Mongkolkeha Y J Cho and P Kumam ldquoBest proximitypoints for Geraghtyrsquos proximal contraction mappingsrdquo FixedPoint Theory and Applications vol 2013 article 180 2013

[33] C Mongkolkeha C Kongban and P Kumam ldquoThe existenceand uniqueness of best proximity point theorems for gener-alized almost contractionrdquo Abstract and Applied Analysis vol2014 Article ID 813614 11 pages 2014

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

[35] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 70 no 12 pp 4341ndash4349 2009

[36] N V Luong and N X Thuan ldquoCoupled fixed point theoremsfor mixed monotone mappings and an application to integralequationsrdquo Computers and Mathematics with Applications vol62 no 11 pp 4238ndash4248 2011

[37] W Shatanawi ldquoPartially ordered cone metric spaces and cou-pled fixed point resultsrdquo Computers and Mathematics withApplications vol 60 no 8 pp 2508ndash2515 2010

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

Abstract and Applied Analysis 7

[30] CMongkolkeha andPKumam ldquoSome commonbest proximitypoints for proximity commuting mappingsrdquo Optimization Let-ters vol 7 no 8 pp 1825ndash1836 2013

[31] C Mongkolkeha Y J Cho and P Kumam ldquoBest proximitypoints for generalized proxinal C-Contraction mappings inmetric spaces with partial ordersrdquo Journal of Inequalities andApplications vol 2013 article 94 2013

[32] C Mongkolkeha Y J Cho and P Kumam ldquoBest proximitypoints for Geraghtyrsquos proximal contraction mappingsrdquo FixedPoint Theory and Applications vol 2013 article 180 2013

[33] C Mongkolkeha C Kongban and P Kumam ldquoThe existenceand uniqueness of best proximity point theorems for gener-alized almost contractionrdquo Abstract and Applied Analysis vol2014 Article ID 813614 11 pages 2014

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

[35] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 70 no 12 pp 4341ndash4349 2009

[36] N V Luong and N X Thuan ldquoCoupled fixed point theoremsfor mixed monotone mappings and an application to integralequationsrdquo Computers and Mathematics with Applications vol62 no 11 pp 4238ndash4248 2011

[37] W Shatanawi ldquoPartially ordered cone metric spaces and cou-pled fixed point resultsrdquo Computers and Mathematics withApplications vol 60 no 8 pp 2508ndash2515 2010

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