Coupled best proximity point theorems for proximally g-meir-keelertypemappings in partially ordered metric spaces

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1 Abkar et al. Fixe Point Theory an Applications 2015) 2015:107 DOI /s R E S E A R C H Open Access Couple best proximity point theorems for proximally g-meir-keelertypemappings in partially orere metric spaces Ali Abkar *, Samaneh Ghos an Azizollah Azizi * Corresponence: Department of Mathematics, Imam Khomeini International University, Qazvin, 34149, Iran Abstract In this paper we first introuce the notion of proximally g-meir-keeler type mappings, then we stuy the existence an uniqueness of couple best proximity points for these mappings. This generalization is in line with Eelstein s generalization of Meir-Keeler type mappings, as well as in line with the recent one use in Eshaghi Gorji in Math. Probl. Eng. 2012:150363, 2012). MSC: 47H10; 54H25; 47J25 Keywors: couple best proximity point; proximally g-meir-keeler type mapping; proximalmixestrictg-monotone property 1 Introuction The Banach contraction principle [1] is a classical an powerful tool in nonlinear analysis. This principle has been generalize in ifferent irections by many authors see, for example, [2 6] an the references therein). Afterwar, Bhaskar an Lakshmikantham [7] introuce the notion of couple fixe points of a given two-variable mapping F. They also establishe the uniqueness of couplefixepoint for the mapping F, an successfully applie their results to the problem of existence an uniqueness of solution for a perioic bounary value problem. Following their lines of research, some authors have extene these results in several irections see, for instance, [8, 9]). The well-known best approximation theorem, ue to Fan [10], asserts that if A is a nonempty, compact an convex subset of a norme linear space X, ant is a continuous mapping from A to X, then there exists a point x A such that the istance of x to Tx is equal to the istance of Tx to A. Suchapointx is calle a best approximant of T in A. This result was in turn generalize by several authors see, for example, [11 13], an the references therein). In the sequel, X is a nonempty set, an X, ) is a metric space. In [2], Meir an Keeler generalize the well-known Banach contraction principle. In 1969, Meir an Keeler efine weak uniformly strict contraction as follows: given ɛ >0,thereexistsδ >0suchthat ɛ x, y)<ɛ + δ Fx, Fy)<ɛ 2015 Abkar et al. This article is istribute uner the terms of the Creative Commons Attribution 4.0 International License which permits unrestricte use, istribution, an reprouction in any meium, provie you give appropriate creit to the original authors) an the source, provie a link to the Creative Commons license, an inicate if changes were mae.

2 Abkar et al. Fixe Point Theory an Applications 2015) 2015:107 Page 2 of 16 for a self-map F on X. They prove that every weakly uniformly strict contraction on a complete metric space X, ) hasauniquefixepoint.inthisway,theywereableto recapture earlier results ue to Eelstein, as well as that of Boy an Wong. Recently, Eshaghi Gorji et al. [14], efine the generalize g-meir-keeler type contractions an prove some couple fixe point theorems uner a generalize g-meir-keelercontractive conition. In this way, they improve results of Bhaskar an Lakshmikantham [7]. We shall recall their efinitions here. Definition 1.1 [14] LetX, ) be a partially orere set an be a metric on X. LetF : X X X an g : X X be two given mappings. We say that F is a generalize g-meir- Keeler type contraction if, for every ɛ >0,thereexistsδɛ)>0suchthat,forallx, y, u, v X with gx) gu)angy) gv), ɛ 1 2[ gx), gu) ) + gy), gv) )] < ɛ + δɛ) Fx, y), Fu, v) ) < ɛ. They also efine the mixe strict g-monotone property. Definition 1.2 [14] LetX, ) be a partially orere set an F : X X X an g : X X be two given mappings. We say that F hasthe mixe strict g-monotone property if, for any x, y X, x 1, x 2 X, gx 1 )<gx 2 ) Fx 1, y)<fx 2, y), y 1, y 2 X, gy 1 )<gy 2 ) Fx, y 1 )>Fx, y 2 ). Since we shall be ealing with couple best proximity points, the following efinitions will be neee. Definition 1.3 Let A, B be nonempty subsets of a metric space X, ), an T : A B be a non-self mapping. A point x A is calle a best proximity point of T if x, Tx )) = A, B), where A, B):=inf { x, y):x A, y B }. In this paper we shall use the following symbols: A 0 = { x A : x, y)=a, B)forsomey B }, B 0 = { y B : x, y)=a, B)forsomex A }. It is well known that A 0 is containe in the bounary of A see, for instance, [15]). For more information on some recent results in this topic, we refer the intereste reaer to [16 20]. In [21], the authors efine the notion of a couple best proximity point, as follows. Definition 1.4 Let A, B be nonempty subsets of a metric space X, ) anf : A A B be a mapping. A point x 1, x 2 ) A A is calle a couple best proximity point of F if x 1, Fx 1, x 2 ) ) = x 2, Fx 2, x 1 ) ) = A, B).

3 Abkar et al. Fixe Point Theory an Applications 2015) 2015:107 Page 3 of 16 It is easy to see that if A = B in Definition 1.4, thenacouplebestproximitypointisa couplefixepointoff. Pragaeeswarar et al. [22] efine a proximally couple contraction as follows. Definition 1.5 Let X,, ) be a partially orere metric space an A, B be nonempty subsets of X. A mapping F : A A B is sai to be a proximally couple contraction if there exists k 0, 1) such that whenever x 1 x 2, y 1 y 2, u 1, Fx 1, y 1 )) = A, B), u 2, Fx 2, y 2 )) = A, B), it follows that u 1, u 2 ) k 2[ x1, x 2 )+y 1, y 2 ) ], where x 1, x 2, y 1, y 2, u 1, u 2 A. Motivate by the results of [14]an[22], in this paper we first introuce the notions of proximal mixe strict monotone property an proximally Meir-Keeler type functions an prove the existence an uniqueness of couple best proximity point theorems for these mappings. This will be implemente in Section 2. InSection3, weshalliscusssome more generalizations of this notions. An example will be provie to illustrate our result. 2 Proximally Meir-Keeler type mappings In this section we will efine proximal mixe strict monotone property an proximally Meir-Keeler type mappings, an prove some theorems in this regar. In the next section we eal with some further generalizations of this topics. Therefore, we shall not provie any proof for the statements mae in this section, instea we will comment on how these facts can be inferre from their counterparts in Section 3; inee, the next section is evote to proximally g-meir-keeler type mappings, as well as to proximal mixe strict g-monotone property, so that if we put g = ientity, we get all the results state in Section 2. Definition 2.1 Let X,, ) be a partiallyorere metric space, A, B be nonempty subsets of X,anF : A A B be a given mapping. We say that F has the proximal mixe strict monotone property if, for all x, y A, x 1 < x 2, u 1, Fx 1, y)) = A, B), u 2, Fx 2, y)) = A, B), then u 1 < u 2 an if y 1 y 2, u 3, Fx, y 1 )) = A, B), u 4, Fx, y 2 )) = A, B), then u 4 u 3,wherex 1, x 2, y 1, y 2, u 1, u 2, u 3, u 4 A.

4 Abkar et al. Fixe Point Theory an Applications 2015) 2015:107 Page 4 of 16 Definition 2.2 Let X,, ) be a partiallyorere metric space, A, B be nonempty subsets of X, anf : A A B be a given mapping. We say that F is a proximally Meir-Keeler type function if, for every ɛ >0,thereexistδɛ)>0ank 0, 1) such that whenever x 1 x 2, y 1 y 2, u 1, Fx 1, y 1 )) = A, B), u 2, Fx 2, y 2 )) = A, B), then ɛ k 2[ x1, x 2 )+y 1, y 2 ) ] < ɛ + δɛ) u 1, u 2 )<ɛ, where x 1, x 2, y 1, y 2, u 1, u 2 A. Proposition 2.3 Let X,, ) be a partially orere metric space, A, B be nonempty subsets of X, an let F : A A B be a given mapping such that the conitions x 1 x 2, y 1 y 2, u 1, Fx 1, y 1 )) = A, B), u 2, Fx 2, y 2 )) = A, B), imply k 0, 1); u 1, u 2 ) k 2[ x1, x 2 )+y 1, y 2 ) ], where x 1, x 2, y 1, y 2, u 1, u 2 A. Then F is a proximally Meir-Keeler type function. From now on, we suppose that X,, ) is a partially orere metric space enowe with the following partial orering: for all x, y), u, v) X X, u, v) x, y) u < x, v y. Lemma 2.4 Let X,, ) be a partially orere metric space, A, B be nonempty subsets of X, an let F : A A B be a given mapping. If F is a proximally Meir-Keeler type function, an x 1, y 1 ) x 2, y 2 ), u 1, Fx 1, y 1 )) = A, B), u 2, Fx 2, y 2 )) = A, B), then k 0, 1); u 1, u 2 ) k 2[ x1, x 2 )+y 1, y 2 ) ], where x 1, x 2, y 1, y 2, u 1, u 2 A. Theorem 2.5 Let X,, ) be a partially orere complete metric space. Let A, Bbe nonempty close subsets of the metric space X, ) such that A 0. Let F : A A B

5 Abkar et al. Fixe Point Theory an Applications 2015) 2015:107 Page 5 of 16 be a given mapping satisfying the following conitions: a) F is continuous; b) F has the proximal mixe strict monotone property on A such that FA 0, A 0 ) B 0 ; c) F is a proximally Meir-Keeler type function; ) there exist elements x 0, y 0 ), x 1, y 1 ) A 0 A 0 such that x 1, Fx 0, y 0 )) = A, B) with x 0 < x 1, an y 1, Fy 0, x 0 )) = A, B) with y 0 y 1. Then there exists x, y) A Asuchthatx, Fx, y)) = A, B) an y, Fy, x)) = A, B). Theorem 2.6 Let X,, ) be a partially orere complete metric space. Let A be a nonempty close subset of the metric space X, ). Let F : A A A be a given mapping satisfying the following conitions: a) F is continuous; b) F has the proximal mixe strict monotone property on A; c) F is a proximally Meir-Keeler type function; ) there exist elements x 0, y 0 ), x 1, y 1 ) A A such that x 1 = Fx 0, y 0 ) with x 0 < x 1, an y 1 = Fy 0, x 0 ) with y 0 y 1. Then there exists x, y) A Asuchthatx, Fx, y)) = 0 an y, Fy, x)) = 0. Theorem 2.7 Let X,, ) be a partially orere complete metric space. Let A, Bbe nonempty close subsets of the metric space X, ) such that A 0. Let F : A A B be a given mapping satisfying the following conitions: a) if {x n } is a nonecreasing sequence in A such that such that x n x, then x n < x an if {y n } is a nonincreasing sequence in A such that y n y, then y n y; b) F has the proximal mixe strict monotone property on A such that FA 0, A 0 ) B 0 ; c) F is a proximally Meir-Keeler type function; ) there exist elements x 0, y 0 ), x 1, y 1 ) A 0 A 0 such that x 1, Fx 0, y 0 )) = A, B) with x 0 < x 1, an y 1, Fy 0, x 0 )) = A, B) with y 0 y 1. Then there exists x, y) A Asuchthatx, Fx, y)) = A, B) an y, Fy, x)) = A, B). Remark 2.8 Theorems 2.5 an 2.6 hol true, if we replace the continuity of F by the following. If {x n } is a nonecreasing sequence in A such that x n x, thenx n < x an if {y n } is a nonincreasing sequence in A such that y n y,theny n y. Now, we consier the prouct space A A with the following partial orering: for all x, y), u, v) A A, u, v) x, y) u < x, v y. Theorem 2.9 Suppose that all the hypotheses of Theorem 2.5 hol an, further, for all x, y), x, y ) A 0 A 0, there exists u, v) A 0 A 0 such that u, v) is comparable with x, y), x, y ). Then there exists a unique x, y) A Asuchthatx, Fx, y)) = A, B) an y, Fy, x)) = A, B). Example 2.10 Let X be the real numbers enowe with usual metric an consier the usual orering x, y) u, v) x u, y v.supposethata =[3,+ )anb =, 3].

6 Abkar et al. Fixe Point Theory an Applications 2015) 2015:107 Page 6 of 16 Then A, B are nonempty close subsets of X. Also,wehaveA, B) =6,A 0 = {3} an B 0 = { 3}. Define a mapping F : A A B as follows: Fx, y)= x y. 2 Then F is continuous an F3, 3) = 3, i.e., FA 0, A 0 ) B 0. Note also that the other hypotheses of Theorem 2.9 are satisfie, then there exists a unique point 3, 3) A A such that 3, F3, 3)) = 6 = A, B). 3 Proximally g-meir-keeler type mappings Let X be a nonempty set. We recall that an element x, y) X X is calle a couple coincience point of two mappings F : X X X an g : X X provie that Fx, y)=gx)anfy, x)=gy) for all x, y X.Also,wesaythatF an g are commutative if gfx, y)) = Fgx), gy)) for all x, y X. We now present the following efinitions. Definition 3.1 Let X,, ) be a partiallyoreremetric space, A, B be nonempty subsets of X, anf : A A B an g : A A be two given mappings. We say that F has the proximal mixe strict g-monotone property provie that for all x, y A, if gx 1 )<gx 2 ), gu 1 ), Fgx 1 ), gy))) = A, B), gu 2 ), Fgx 2 ), gy))) = A, B), then gu 1 )<gu 2 ), an if gy 1 ) gy 2 ), gu 3 ), Fgx), gy 1 ))) = A, B), gu 4 ), Fgx), gy 2 ))) = A, B), then gu 4 ) gu 3 ), where x 1, x 2, y 1, y 2, u 1, u 2, u 3, u 4 A. Definition 3.2 Let X,, ) be a partiallyoreremetric space, A, B be nonempty subsets of X,anF : A A B an g : A A be two given mappings. We say that F is a proximally g-meir-keeler type function if, for every ɛ >0,thereexistδɛ)>0ank 0, 1) such that whenever gx 1 ) gx 2 ), gy 1 ) gy 2 ), gu 1 ), Fgx 1 ), gy 1 ))) = A, B), gu 2 ), Fgx 2 ), gy 2 ))) = A, B), then ɛ k 2[ gx1 ), gx 2 ) ) + gy 1 ), gy 2 ) )] < ɛ + δɛ) gu 1 ), gu 2 ) ) < ɛ, where x 1, x 2, y 1, y 2, u 1, u 2 A. Proposition 3.3 Let X,, ) be a partially orere metric space an A, B be nonempty subsets of X, an let F : A A Bang: A A be two given mappings such that the

7 Abkar et al. Fixe Point Theory an Applications 2015) 2015:107 Page 7 of 16 conitions gx 1 ) gx 2 ), gy 1 ) gy 2 ), gu 1 ), Fgx 1 ), gy 1 ))) = A, B), gu 2 ), Fgx 2 ), gy 2 ))) = A, B), imply k 0, 1); gu 1 ), gu 2 ) ) k 2[ gx1 ), gx 2 ) ) + gy 1 ), gy 2 ) )], 1) where x 1, x 2, y 1, y 2, u 1, u 2 A. Then F is a proximally g-meir-keeler type function. Proof Suppose 1) issatisfie.foreveryɛ >0,wesetɛ 0 = ɛ/2 an δɛ 0 )=ɛ 0.Now,if ɛ 0 k 2 [gx 1), gx 2 )) + gy 1 ), gy 2 ))] < ɛ + δɛ), it follows from 1) thatgu 1 ), gu 2 )) k 2 [gx 1), gx 2 )) + gy 1 ), gy 2 ))] < ɛ 0 + δɛ 0 )=ɛ, i.e., gu 1 ), gu 2 )) < ɛ. Remark 3.4 If we set g = I, the ientity map, in Proposition 3.3, thenproposition2.3 follows. From now on, we suppose that X,, ) is a partially orere metric space enowe with the following partial orering: for all x, y), u, v) X X, u, v) x, y) u < x, v y. Lemma 3.5 Let X,, ) be a partially orere metric space an A, B be nonempty subsets of X an let F : A A Bang: A A be two given mappings. If F is a proximally g-meir-keeler type function, an gx 1 ), gy 1 )) gx 2 ), gy 2 )), gu 1 ), Fgx 1 ), gy 1 ))) = A, B), gu 2 ), Fgx 2 ), gy 2 ))) = A, B), then k 0, 1); gu 1 ), gu 2 ) ) k 2[ gx1 ), gx 2 ) ) + gy 1 ), gy 2 ) )], where x 1, x 2, y 1, y 2, u 1, u 2 A. Proof Let x 1, x 2, y 1, y 2 A be such that gx 1 ), gy 1 )) gx 2 ), gy 2 )), gu 1 ), Fgx 1 ), gy 1 ))) = A, B), gu 2 ), Fgx 2 ), gy 2 ))) = A, B), then gx 1 ), gx 2 )) + gy 1 ), gy 2 )) > 0. Let k 0 0, 1) be arbitrary. Since F is a proximally g-meir-keeler type function, for ɛ = k 0 [ gx1 ), gx 2 ) ) + gy 1 ), gy 2 ) )], 2

8 Abkar et al. Fixe Point Theory an Applications 2015) 2015:107 Page 8 of 16 there exist δɛ) >0ank = k 0 0, 1) such that for all x 1, x 2, y 1, y 2, u 1, u 2 A for which gx 1 ), gy 1 )) gx 2 ), gy 2 )), an { gu 1 ), Fgx 1 ), gy 1 ))) = A, B), gu 2 ), Fgx 2 ), gy 2 ))) = A, B), we have ɛ k 2[ gx1 ), gx 2 ) ) + gy 1 ), gy 2 ) )] < ɛ + δɛ) gu 1 ), gu 2 ) ) < ɛ, therefore, gu 1 ), gu 2 ) ) < k 0 [ gx1 ), gx 2 ) ) + gy 1 ), gy 2 ) )] 2 = k 2[ gx1 ), gx 2 ) ) + gy 1 ), gy 2 ) )]. This completes the proof. Remark 3.6 If in Lemma 3.5,wesetg = I the ientity map, Lemma 2.4 follows. Lemma 3.7 Let X,, ) be a partially orere metric space an A, B be nonempty subsets of X, A 0 an F : A A Bang: A A be two given mappings. If F has the proximal mixe strict g-monotone property, with ga 0 )=A 0, FA 0, A 0 ) B 0, an if gx 1 ), gy 1 )) gx 2 ), gy 2 )), gx 2 ), F gx 1 ), gy 1 ) )) = A, B), 2) gu), F gx 2 ), gy 2 ) )) = A, B), 3) where x 1, x 2, y 1, y 2, u A 0, then gx 2 )<gu). Proof Since ga 0 )=A 0, FA 0, A 0 ) B 0, it follows that Fgx 2 ), gy 1 )) B 0. Hence there exists gu 1 ) A 0 such that g u 1), F gx2 ), gy 1 ) )) = A, B). 4) Using the fact that F has the proximal mixe strict g-monotone property, together with 2)an4), we have gx 1 )<gx 2 ), gx 2 ), Fgx 1 ), gy 1 ))) = A, B), gu 1 ), Fgx 2), gy 1 ))) = A, B), then gx 2 )<g u 1). 5)

9 Abkar et al. Fixe Point Theory an Applications 2015) 2015:107 Page 9 of 16 Also, from the proximal mixe strict g-monotone propertyof F with 3), 5), we have gy 2 ) gy 1 ), gu), Fgx 2 ), gy 2 ))) = A, B), gu 1 ), Fgx 2), gy 1 ))) = A, B), then g u 1) gu). 6) Now from 5), 6)wegetgx 2 )<gu), hence the proof is complete. Lemma 3.8 Let X,, ) be a partially orere metric space an A, B be nonempty subsets of X, A 0, an F : A A Bang: A A be two given mappings. Let F have the proximal mixe strict g-monotone property, with ga 0 )=A 0, FA 0, A 0 ) B 0. If gx 1 ), gy 1 )) gx 2 ), gy 2 )), gy 2 ), Fgy 1 ), gx 1 ))) = A, B), gu), Fgy 2 ), gx 2 ))) = A, B), where x 1, x 2, y 1, y 2, u A 0, then gy 2 )>gu). Proof TheproofissimilartothatofLemma3.7, so we omit the etails. Theorem 3.9 Let X,, ) be a partially orere complete metric space. Let A, Bbe nonempty close subsets of the metric space X, ) such that A 0. Let F : A A B an g : A Abetwogivenmappingssatisfyingthefollowingconitions: a) F an g are continuous; b) F has the proximal mixe strict g-monotone property on A such that ga 0 )=A 0, FA 0, A 0 ) B 0 ; c) F is a proximally g-meir-keeler type function; ) there exist elements x 0, y 0 ), x 1, y 1 ) A 0 A 0 such that gx 1 ), Fgx 0 ), gy 0 ))) = A, B) with gx 0 )<gx 1 ), an gy 1 ), Fgy 0 ), gx 0 ))) = A, B) with gy 0 ) gy 1 ). Then there exists x, y) A Asuchthatgx), Fgx), gy))) = A, B) an gy), Fgy), gx))) = A, B). Proof Let x 0, y 0 ), x 1, y 1 ) A 0 A 0 be such that gx 1 ), Fgx 0 ), gy 0 ))) = A, B) with gx 0 )<gx 1 ), an gy 1 ), Fgy 0 ), gx 0 ))) = A, B)withgy 0 ) gy 1 ). Since FA 0, A 0 ) B 0, ga 0 )=A 0, there exists an element x 2, y 2 ) A 0 A 0 such that gx 2 ), Fgx 1 ), gy 1 ))) = A, B)angy 2 ), Fgy 1 ), gx 1 ))) = A, B). Hence from Lemma 3.7 an Lemma 3.8 we obtain gx 1 )<gx 2 )angy 1 )>gy 2 ). Continuing this process, we construct the sequences {x n } an {y n } in A 0 such that with gx n+1 ), F gx n ), gy n ) )) = A, B), n 0, gx 0 )<gx 1 )< < gx n )<gx n+1 )< 7)

10 Abkar et al. Fixe Point Theory an Applications 2015) 2015:107 Page 10 of 16 an gy n+1 ), F gy n ), gx n ) )) = A, B), n 0, with gy 0 )>gy 1 )> > gy n )>gy n+1 )>. 8) Define δ n := gx n ), gx n+1 ) ) + gy n ), gy n+1 ) ). 9) Since F is a proximally g-meir-keeler type function, gx n ), Fgx n 1 ), gy n 1 ))) = A, B), an gx n+1 ), F gx n ), gy n ) )) = A, B), gx n 1 ) gx n ), gy n 1 ) gy n ), it follows from Lemma 3.5 that gx n ), gx n+1 ) ) k 2[ gxn 1 ), gx n ) ) + gy n 1 ), gy n ) )]. 10) We also have gy n ), Fgy n 1 ), gx n 1 ))) = A, B), an gy n+1 ), F gy n ), gx n ) )) = A, B), gx n 1 )<gx n ), gy n 1 )>gy n ), hence using the fact that F is a proximally g-meir-keeler type function, it follows from Lemma 3.5 that gy n ), gy n+1 ) ) k 2[ gyn 1 ), gy n ) ) + gx n 1 ), gx n ) )]. 11) It now follows from 9), 10), an 11)that δ n kδ n 1 ) k 2 δ n 2 ) k n δ 0 ) or δ n k n δ 0 ). 12) Now, we prove that {gx n )} an {gy n )} are Cauchy sequences. For n > m, it follows from the triangle inequality an 12)that gx n ), gx m ) ) + gy n ), gy m ) ) gx n ), gx n 1 ) ) + gy n ), gy n 1 ) ) + + gx m+1 ), gx m ) ) + gy m+1 ), gy m ) ) k n k m) δ 0 ) km 1 k δ 0).

11 Abkar et al. Fixe Point Theory an Applications 2015) 2015:107 Page 11 of 16 Let ɛ >0begiven.ChooseanaturalnumberN such that km 1 k δ 0)<ɛ,form > N.Thus, gx n ), gx m ) ) + gy n ), gy m ) ) < ɛ, n > m. Therefore, {gx n )} an {gy n )} are Cauchy sequences. Since A is a close subset of the complete metric space X,thereexistx 1, y 1 A such that lim n gx n)=x 1, lim gy n)=y 1. n Note that x n, y n A 0, ga 0 )=A 0,sothatgx n ), gy n ) A 0.SinceA 0 is close, we conclue that x 1, y 1 ) A 0 A 0, i.e.,thereexistx, y A 0 such that gx)=x 1, gy)=y 1. Therefore gx n ) gx), gy n ) gy). 13) Since {gx n )} is monotone increasing an {gy n )} is monotone ecreasing, we have gx n )<gx), gy n )>gy). 14) From 7), 8), gx n+1 ), F gx n ), gy n ) )) = A, B) 15) an gy n+1 ), F gy n ), gx n ) )) = A, B). 16) Since F is continuous, we have, from 13), F gx n ), gy n ) ) F gx), gy) ) an F gy n ), gx n ) ) F gy), gx) ). Thus, the continuity of the metric implies that gx n+1 ), F gx n ), gy n ) )) gx), F gx), gy) )) 17) an gy n+1 ), F gy n ), gx n ) )) gy), F gy), gx) )). 18) Therefore from 15), 16), 17), 18), gx), F gx), gy) )) = A, B), gy), F gy), gx) )) = A, B). Remark 3.10 If we set g = I in Theorem 3.9,Theorem2.5 follows.

12 Abkar et al. Fixe Point Theory an Applications 2015) 2015:107 Page 12 of 16 Corollary 3.11 Let X,, ) be a partially orere complete metric space. Let A be a nonempty close subsets of the metric space X, ). Let F : A A Aang: A Abe two given mappings satisfying the following conitions: a) F an g are continuous; b) F has the proximal mixe strict g-monotone property on A such that FgA), ga)) ga); c) F is a proximally g-meir-keeler type function; ) there exist elements x 0, y 0 ), x 1, y 1 ) A A such that gx 1 )=Fgx 0 ), gy 0 )) with gx 0 )<gx 1 ), an gy 1 )=Fgy 0 ), gx 0 )) with gy 0 ) gy 1 ). Then there exists x, y) A Asuchthat gx), F gx), gy) )) =0, gy), F gy), gx) )) =0. Theorem 3.12 Let X,, ) be a partially orere complete metric space. Let A, Bbe nonempty close subsets of the metric space X, ) such that A 0. Let F : A A B an g : A Abetwogivenmappingssatisfyingthefollowingconitions: a) g is continuous; b) F has the proximal mixe strict g-monotone property on A such that FA 0, A 0 ) B 0, ga 0 )=A 0 ; c) F is a proximally g-meir-keeler type function; ) there exist elements x 0, y 0 ), x 1, y 1 ) A 0 A 0 such that gx 1 ), Fgx 0 ), gy 0 ))) = A, B) with gx 0 )<gx 1 ), an gy 1 ), Fgy 0 ), gx 0 ))) = A, B) with gy 0 ) gy 1 ); e) if {x n } is a nonecreasing sequence in A such that x n x, then x n < x an if {y n } is a nonincreasing sequence in A such that y n y, then y n y. Then there exists x, y) A Asuchthat gx), F gx), gy) )) = A, B), gy), F gy), gx) )) = A, B). Proof As in the proof of Theorem 3.9,thereexistsequences{x n } an {y n } in A 0 such that gx n+1 ), Fgx n ), gy n ))) = A, B)with gx n )<gx n+1 ), n 0, 19) an gy n+1 ), Fgy n ), gx n ))) = A, B)with gy n )>gy n+1 ), n 0. 20) Also, gx n ) gx)angy n ) gy). From e), we get gx n )<gx), an gy n ) gy). Since FA 0, A 0 ) B 0, it follows that Fgx), gy)) an Fgy), gx)) are in B 0. Therefore, there exists x 1, y 1 ) A 0 A 0 such that x 1, Fgx), gy))) = A, B)any 1, Fgy), gx))) = A, B). Since ga 0 )=A 0,thereexistx, y A 0 such that gx )=x 1, gy )=y 1.Hence, g x ), F gx), gy) )) = A, B) 21)

13 Abkar et al. Fixe Point Theory an Applications 2015) 2015:107 Page 13 of 16 an g y ), F gy), gx) )) = A, B). 22) Since gx n )<gx), gy n ) gy)anf is a proximally g-meir-keeler type function, it follows from Lemma 3.5,19), an 21)that gx n+1 ), g x )) k 2[ gxn ), gx) ) + gy n ), gy) )]. Similarly, from Lemma 3.5 an 20)an22)weobtain gy n+1 ), g y )) k 2[ gyn ), gy) ) + gx n ), gx) )]. By taking the limit of the above two inequalities, we get gx)=gx )angy)=gy ). Remark 3.13 Corollary 3.11 holstrueifwereplacethecontinuityoff by the following statement. If {x n } is a nonecreasing sequence in A such that x n x, thenx n < x an if {y n } is a nonincreasing sequence in A such that y n y,theny n y. We now consier the prouct space A A with the following partial orering: for all x, y), u, v) A A, u, v) x, y) u < x, v y. Theorem 3.14 Suppose that all the hypotheses of Theorem 3.9 hol an, further, for all x, y), x, y ) A 0 A 0, there exists u, v) A 0 A 0 such that u, v) is comparable to x, y), x, y )with respect to the orering in A A). Then there exists a unique x, y) A A such that gx), Fgx), gy))) = A, B) an gy), Fgy), gx))) = A, B). Proof By Theorem 3.9, there exists an element x, y) A A such that gx), F gx), gy) )) = A, B) 23) an gy), F gy), gx) )) = A, B). 24) Now, suppose that there exists an element x, y ) A A such that gx ), Fgx ), gy ))) = A, B)angy ), Fgy ), gx ))) = A, B). First, let gx), gy)) be comparable to gx ), gy )) with respect to the orering in A A. Since gx), Fgx), gy))) = A, B), an gx ), Fgx ), gy ))) = A, B), it follows from Lemma 3.5 that k 0, 1); gx), g x )) k 2[ gx), g x )) + gy), g y ))]. 25)

14 Abkar et al. Fixe Point Theory an Applications 2015) 2015:107 Page 14 of 16 Similarly, from gy), Fgy), gx))) = A, B), gy ), Fgy ), gx ))) = A, B) anlemma 3.5,wehave gy), g y )) k 2[ gx), g x )) + gy), g y ))]. 26) Aing 25)an26), weget gx), g x )) + gy), g y )) k [ gx), g x )) + gy), g y ))]. 27) It follows that gx), gx )) + gy), gy )) = 0 an so gx)=gx )angy)=gy ). Secon, let gx), gy)) is not comparable to gx ), gy )), then there exists gu 1 ), gv 1 )) in A 0 A 0 which is comparable to gx), gy)) an gx ), gy )). Since FA 0, A 0 ) B 0 an ga 0 )=A 0,thereexistsgu 2 ), gv 2 )) A 0 A 0 such that gu 2 ), Fgu 1 ), gv 1 ))) = A, B) an gv 2 ), Fgv 1 ), gu 1 ))) = A, B). Without loss of generality assume that gu 1 ), gv 1 )) gx), gy)), i.e., gu 1 )<gx) an gv 1 ) gy). Therefore gy), gx)) gv 1 ), gu 1 )). Now, from Lemmas 3.7 an 3.8,wehave gu 1 ), gv 1 )) gx), gy)), gu 2 ), Fgu 1 ), gv 1 ))) = A, B), gx), Fgx), gy))) = A, B) =A, B), then gu 2 )<gx)aninthesameway gu 1 ), gv 1 )) gx), gy)), gv 2 ), Fgv 1 ), gu 1 ))) = A, B), gy), Fgy), gx))) = A, B) =A, B), then gv 2 )>gy). Continuing this process, we obtain sequences {x n } an {y n } such that gu n )<gx), gv n )> gy), gu n+1 ), F gu n ), gv n ) )) = A, B), 28) gv n+1 ), F gv n ), gu n ) )) = A, B). 29) Since F is a proximally g-meir-keeler type function, from Lemma 3.5,23), 28), k 0, 1); gu n+1 ), gx) ) k 2[ gun ), gx) ) + gv n ), gy) )], n 0. 30) Similarly from Lemma 3.5,24), 29), we have gv n+1 ), gy) ) k 2[ gvn ), gy) ) + gu n ), gx) )], n 0. 31) Aing 30)an31), we have gu n+1 ), gx) ) + gv n+1 ), gy) ) k [ gu n ), gx) ) + gv n ), gy) )]

15 Abkar et al. Fixe Point Theory an Applications 2015) 2015:107 Page 15 of 16 k 2[ gu n 1 ), gx) ) + gv n 1 ), gy) )] k n+1[ gu 0 ), gx) ) + gv 0 ), gy) )]. As n,wegetgu n+1 ), gx)) + gv n+1 ), gy)) 0, i.e., gu n ) gx) angv n ) gy). Similarly, we can prove gu n ) gx )angv n ) gy ). Hence, gx) =gx )an gy)=gy ) an the proof is complete. We shall illustrate the above theorem by the following example. Example 3.15 Let X be the real numbers enowe with usual metric an consier the usual orering x, y) u, v) x u, y v.supposethata =[2,+ )anb =, 2]. Then A, B are nonempty close subsets of X. Also,wehaveA, B)=4anA 0 = {2} an B 0 = { 2}. Define the mappings F : A A B an g : A A as follows: Fx, y)= x y 4 4 an gx)=2x 2. Then F an g are continuous an F2, 2) = 2 an g2) = 2, i.e., FA 0, A 0 ) B 0 an ga 0 )=A 0. Note also that the other hypotheses of Theorem 3.14 are satisfie, then there exists a unique point 2, 2) A A such that g2), Fg2), g2))) = 4 = A, B). Competing interests The authors eclare that they have no competing interests. Authors contributions All authors contribute equally an significantly in writing this paper. All authors rea an approve the final manuscript. Acknowlegements This research was one while the first author was on a sabbatical leave at State University of New York at Albany SUNYA); he wishes to thank SUNYA for its hospitality an Imam Khomeini International University for the financial support. Receive: 25 March 2015 Accepte: 17 June 2015 References 1. Banach, S: Sur les operations ans les ensembles abstraits et leur application aux equations integrales. Funam. Math. 3, ) 2. Meir,A,Keeler,E:Atheoremoncontractionmappings.J.Math.Anal.Appl.28, ) 3. Nieto, JJ, Roriguez-Lopez, R: Contractive mapping theorems in partially orere sets an applications to orinary ifferential equations. Orer 22, ) 4. Suzuki, T: Meir-Keeler contractions of integral type are still Meir-Keeler contractions. Int. J. Math. Math. Sci. 2007, Article ID ). oi: /2007/ Suzuki, T: A generalize Banach contraction principle which characterizes metric completeness. Proc. Am. Math. Soc. 136, ) 6. Suzuki, T: Fixe point theorems an convergence theorems for some generalize nonexpansive mappings. J. Math. Anal. Appl. 340, ) 7. Bhaskar, TG, Lakshmikantham, V: Fixe point theorems in partially orere metric spaces an applications. Nonlinear Anal. 65, ) 8. Lakshmikantham, V, Ćirić, LB: Couple fixe point theorems for nonlinear contractions in partially orere metric spaces. Nonlinear Anal. 70, ) 9. Samet, B: Couple fixe point theorems for a generalize Meir-Keeler contraction in partially orere metric spaces. Nonlinear Anal. 72, ) 10. Fan, K: Extensions of two fixe point theorems of F.E. Brower. Math. Z. 122, ) 11. Samet, B: Some results on best proximity points. J. Optim. Theory Appl. 159, ) 12. Abkar, A, Gabeleh, M: Global optimal solutions of noncyclic mappings in metric spaces. J. Optim. Theory Appl. 153, ) 13. Abkar, A, Gabeleh, M: Generalize cyclic contractions in partially orere metric spaces. Optim. Lett. 68), )

16 Abkar et al. Fixe Point Theory an Applications 2015) 2015:107 Page 16 of Eshaghi Gorji, M, Cho, YJ, Ghos, S, Ghos, M, Haian Dehkori, M: Couple fixe-point theorems for contractions in partially orere metric spaces an applications. Math. Probl. Eng. 2012,ArticleID ) 15. Saiq Basha, S, Veeramani, P: Best proximity pair theorems for multifunctions with open fibers. J. Approx. Theory 103, ) 16. Abkar, A, Gabeleh, M: A note on some best proximity point theorems prove uner P-property. Abstr. Appl. Anal. 2013, Article ID ) 17. Suzuki, T, Kikkawa, M, Vetro, C: The existence of best proximity points in metric spaces with the property UC. Nonlinear Anal. 71, ) 18. Gabeleh, M: Proximal weakly contractive an proximal nonexpansive non-self mappings in metric an Banach spaces. J. Optim. Theory Appl. 158, ) 19. Gabeleh, M: Best proximity point theorems via proximal non-self mappings. J. Optim. Theory Appl. 164, ) 20. Karapinar, E, Agarwal, RP: A note on couple fixe point theorems for α-ψ-contractive-type mappings in partially orere metric spaces. Fixe Point Theory Appl. 2013, ) 21. Sintunavarat, W, Kumam, P: Couple best proximity point theorem in metric spaces. Fixe Point Theory Appl. 2012, ) 22. Pragaeeswarar, V, Maruai, M, Kumam, P, Sitthithakerngkiet, K: The existence an uniqueness of couple best proximity point for proximally couple contraction in a complete orere metric space. Abstr. Appl. Anal. 2014, Article ID )

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