Coupled best proximity point theorems for proximally g-Meir-Keeler type mappings in partially ordered metric spaces

*Correspondence: abkar@sci.ikiu.ac.ir Department of Mathematics, Imam Khomeini International University, Qazvin, 34149, Iran Abstract In this paper we first introduce the notion of proximally g-Meir-Keeler type mappings, then we study the existence and uniqueness of coupled best proximity points for these mappings. This generalization is in line with Edelstein’s generalization of Meir-Keeler type mappings, as well as in line with the recent one used in (Eshaghi Gordji in Math. Probl. Eng. 2012:150363, 2012).


Introduction
The Banach contraction principle [] is a classical and powerful tool in nonlinear analysis. This principle has been generalized in different directions by many authors (see, for example, [-] and the references therein). Afterward, Bhaskar and Lakshmikantham [] introduced the notion of coupled fixed points of a given two-variable mapping F. They also established the uniqueness of coupled fixed point for the mapping F, and successfully applied their results to the problem of existence and uniqueness of solution for a periodic boundary value problem. Following their lines of research, some authors have extended these results in several directions (see, for instance, [, ]).
The well-known best approximation theorem, due to Fan [], asserts that if A is a nonempty, compact and convex subset of a normed linear space X, and T is a continuous mapping from A to X, then there exists a point x ∈ A such that the distance of x to Tx is equal to the distance of Tx to A. Such a point x is called a best approximant of T in A. This result was in turn generalized by several authors (see, for example, [-], and the references therein).
In the sequel, X is a nonempty set, and (X, d) is a metric space. In [], Meir and Keeler generalized the well-known Banach contraction principle. In , Meir and Keeler defined weak uniformly strict contraction as follows: given > , there exists δ >  such that ≤ d(x, y) < + δ ⇒ d(Fx, Fy) < for a self-map F on X. They proved that every weakly uniformly strict contraction on a complete metric space (X, d) has a unique fixed point. In this way, they were able to recapture earlier results due to Edelstein, as well as that of Boyd and Wong.
Recently, Eshaghi Gordji et al. [], defined the generalized g-Meir-Keeler type contractions and proved some coupled fixed point theorems under a generalized g-Meir-Keelercontractive condition. In this way, they improved results of Bhaskar and Lakshmikantham []. We shall recall their definitions here.
Definition . [] Let (X, ≤) be a partially ordered set and d be a metric on X. Let F : X × X → X and g : X → X be two given mappings. We say that F is a generalized g-Meir-Keeler type contraction if, for every > , there exists δ( ) >  such that, for all x, y, u, v ∈ X with g(x) ≤ g(u) and g(y) ≥ g(v), They also defined the mixed strict g-monotone property.

Definition . []
Let (X, ≤) be a partially ordered set and F : X × X → X and g : X → X be two given mappings. We say that F has the mixed strict g-monotone property if, for any x, y ∈ X, Since we shall be dealing with coupled best proximity points, the following definitions will be needed.
Definition . Let A, B be nonempty subsets of a metric space (X, d), and T : A → B be a non-self mapping. A point x * ∈ A is called a best proximity point of In this paper we shall use the following symbols: It is well known that A  is contained in the boundary of A (see, for instance, []). For more information on some recent results in this topic, we refer the interested reader to [-]. In [], the authors defined the notion of a coupled best proximity point, as follows.
Definition . Let A, B be nonempty subsets of a metric space (X, d) and F : It is easy to see that if A = B in Definition ., then a coupled best proximity point is a coupled fixed point of F.
[] defined a proximally coupled contraction as follows.
Definition . Let (X, d, ≤) be a partially ordered metric space and A, B be nonempty subsets of X. A mapping F : A × A → B is said to be a proximally coupled contraction if there exists k ∈ (, ) such that whenever Motivated by the results of [] and [], in this paper we first introduce the notions of proximal mixed strict monotone property and proximally Meir-Keeler type functions and prove the existence and uniqueness of coupled best proximity point theorems for these mappings. This will be implemented in Section . In Section , we shall discuss some more generalizations of this notions. An example will be provided to illustrate our result.

Proximally Meir-Keeler type mappings
In this section we will define proximal mixed strict monotone property and proximally Meir-Keeler type mappings, and prove some theorems in this regard. In the next section we deal with some further generalizations of this topics. Therefore, we shall not provide any proof for the statements made in this section, instead we will comment on how these facts can be inferred from their counterparts in Section ; indeed, the next section is devoted to proximally g-Meir-Keeler type mappings, as well as to proximal mixed strict g-monotone property, so that if we put g = identity, we get all the results stated in Section .
Definition . Let (X, d, ≤) be a partially ordered metric space, A, B be nonempty subsets of X, and F : A × A → B be a given mapping. We say that F has the proximal mixed strict monotone property if, for all x, y ∈ A, Definition . Let (X, d, ≤) be a partially ordered metric space, A, B be nonempty subsets of X, and F : A × A → B be a given mapping. We say that F is a proximally Meir-Keeler type function if, for every > , there exist δ( ) >  and k ∈ (, ) such that whenever Proposition . Let (X, d, ≤) be a partially ordered metric space, A, B be nonempty subsets of X, and let F : A × A → B be a given mapping such that the conditions Then F is a proximally Meir-Keeler type function.
From now on, we suppose that (X, d, ≤) is a partially ordered metric space endowed with the following partial ordering: for all (x, y), (u, v) ∈ X × X,

partially ordered metric space, A, B be nonempty subsets of X, and let F : A × A → B be a given mapping. If F is a proximally Meir-Keeler type function, and
Theorem . Let (X, d, ≤) be a partially ordered complete metric space. Let A, B be nonempty closed subsets of the metric space be a given mapping satisfying the following conditions: Theorem . Let (X, d, ≤) be a partially ordered complete metric space. Let A be a nonempty closed subset of the metric space (X, d). Let F : A × A → A be a given mapping satisfying the following conditions: (a) F is continuous; be a given mapping satisfying the following conditions: (a) if {x n } is a nondecreasing sequence in A such that such that x n → x, then x n < x and if {y n } is a nonincreasing sequence in A such that y n → y, then y n ≥ y; Remark . Theorems . and . hold true, if we replace the continuity of F by the following.
If {x n } is a nondecreasing sequence in A such that x n → x, then x n < x and if {y n } is a nonincreasing sequence in A such that y n → y, then y n ≥ y. Now, we consider the product space A × A with the following partial ordering: for all Theorem . Suppose that all the hypotheses of Theorem . hold and, further, for all

Proximally g-Meir-Keeler type mappings
Let X be a nonempty set. We recall that an element (x, y) ∈ X × X is called a coupled coincidence point of two mappings F : X × X → X and g : X → X provided that F(x, y) = g(x) and F(y, x) = g(y) for all x, y ∈ X. Also, we say that F and g are commutative if g(F(x, y)) = F(g(x), g(y)) for all x, y ∈ X.
We now present the following definitions.
Definition . Let (X, d, ≤) be a partially ordered metric space, A, B be nonempty subsets of X, and F : A × A → B and g : A → A be two given mappings. We say that F has the proximal mixed strict g-monotone property provided that for all then g(u  ) < g(u  ), and if Definition . Let (X, d, ≤) be a partially ordered metric space, A, B be nonempty subsets of X, and F : A×A → B and g : A → A be two given mappings. We say that F is a proximally g-Meir-Keeler type function if, for every > , there exist δ( ) >  and k ∈ (, ) such that whenever Proposition . Let (X, d, ≤) be a partially ordered metric space and A, B be nonempty subsets of X, and let F : A × A → B and g : A → A be two given mappings such that the Then F is a proximally g-Meir-Keeler type function.
Proof Suppose () is satisfied. For every > , we set  = / and δ(  ) =  . Now, if Remark . If we set g = I, the identity map, in Proposition ., then Proposition . follows.
From now on, we suppose that (X, d, ≤) is a partially ordered metric space endowed with the following partial ordering: for all (x, y), (u, v) ∈ X × X,

Lemma . Let (X, d, ≤) be a partially ordered metric space and A, B be nonempty subsets of X and let F : A × A → B and g : A → A be two given mappings. If F is a proximally g-Meir-Keeler type function, and
Proof Let x  , x  , y  , y  ∈ A be such that then d(g(x  ), g(x  )) + d(g(y  ), g(y  )) > . Let k  ∈ (, ) be arbitrary. Since F is a proximally g-Meir-Keeler type function, for there exist δ( ) >  and k = k  ∈ (, ) such that for all x  , x  , y  , y  , u  , u  ∈ A for which (g(x  ), g(y  )) ≤ (g(x  ), g(y  )), and This completes the proof.
Remark . If in Lemma ., we set g = I the identity map, Lemma . follows.

Lemma . Let (X, d, ≤) be a partially ordered metric space and A, B be nonempty subsets of X, A  = ∅ and F : A × A → B and g : A → A be two given mappings. If F has the proximal mixed strict g-monotone property, with g(A
Using the fact that F has the proximal mixed strict g-monotone property, together with () and (), we have , g(y  ))) = d (A, B), Also, from the proximal mixed strict g-monotone property of F with (), (), we have , g(y  ))) = d (A, B), Now from (), () we get g(x  ) < g(u), hence the proof is complete.
Lemma . Let (X, d, ≤) be a partially ordered metric space and A, B be nonempty subsets of X, A  = ∅, and F : A × A → B and g : A → A be two given mappings. Let F have the proximal mixed strict g-monotone property, with g( Proof The proof is similar to that of Lemma ., so we omit the details. Theorem . Let (X, d, ≤) be a partially ordered complete metric space. Let A, B be nonempty closed subsets of the metric space (X, d) such that A  = ∅. Let F : A × A → B and g : A → A be two given mappings satisfying the following conditions: (a) F and g are continuous; (b) F has the proximal mixed strict g-monotone property on A such that g( , g(y  ))) = d(A, B) with g(x  ) < g(x  ), and d(g(y  ), F(g(y  ), g(x  ))) = d(A, B) with g(y  ) ≥ g(y  ). Then there exists (x, y) ∈ A × A such that d(g(x), F(g(x), g(y))) = d(A, B) and d(g(y), F(g(y), g(x))) = d (A, B). d(A, B) and d(g(y  ), F(g(y  ), g(x  ))) = d (A, B).
Hence from Lemma . and Lemma . we obtain g(x  ) < g(x  ) and g(y  ) > g(y  ).
Since A is a closed subset of the complete metric space X, there exist x  , y  ∈ A such that lim n→∞ g(x n ) = x  , lim n→∞ g(y n ) = y  .
Note that x n , y n ∈ A  , g(A  ) = A  , so that g(x n ), g(y n ) ∈ A  . Since A  is closed, we conclude that (x  , y  ) ∈ A  × A  , i.e., there exist x, y ∈ A  such that g(x) = x  , g(y) = y  . Therefore Since {g(x n )} is monotone increasing and {g(y n )} is monotone decreasing, we have and d g(y n+ ), F g(y n ), g(x n ) = d (A, B).
Since F is continuous, we have, from (), F g(x n ), g(y n ) → F g(x), g(y) and F g(y n ), g(x n ) → F g(y), g(x) .
Thus, the continuity of the metric d implies that and d g(y n+ ), F g(y n ), g(x n ) → d g(y), F g(y), g(x) .  (b) F has the proximal mixed strict g-monotone property on A such that F(g(A), g(A)) ⊆ g(A); (c) F is a proximally g-Meir-Keeler type function; , and g(y  ) = F(g(y  ), g(x  )) with g(y  ) ≥ g(y  ). Then there exists (x, y) ∈ A × A such that d g(x), F g(x), g(y) = , d g(y), F g(y), g(x) = .
, g(y  ))) = d(A, B) with g(x  ) < g(x  ), and d(g(y  ), F(g(y  ), g(x  ))) = d(A, B) with g(y  ) ≥ g(y  ); (e) if {x n } is a nondecreasing sequence in A such that x n → x, then x n < x and if {y n } is a nonincreasing sequence in A such that y n → y, then y n ≥ y. Then there exists (x, y) ∈ A × A such that d g(x), F g(x), g(y) = d(A, B), d g(y), F g(y), g(x) = d(A, B).
Proof As in the proof of Theorem ., there exist sequences {x n } and {y n } in A  such that d(g(x n+ ), F(g(x n ), g(y n ))) = d(A, B) with g(x n ) < g(x n+ ), ∀n ≥ , and d(g(y n+ ), F(g(y n ), g(x n ))) = d(A, B) with g(y n ) > g(y n+ ), ∀n ≥ .
We shall illustrate the above theorem by the following example. F(x, y) = -xy -  and g(x) = x -.