Best proximity points of α − β − ψ − proximal contractive mappings in partially ordered complete metric spaces

In this paper, we define α−β−ψ−proximal contractive mappings in partially ordered metric spaces and prove the existence of best proximity points of these maps in partially ordered complete metric spaces. These results extend/generalize the results of Asgari and Badehian, J. Nonl. Sci. and Appl., 2015. We provide illustrative examples in support of our theorems.


Introduction
In recent research in the field of nonlinear functional analysis, many researchers are interested in dealing with non-selfmaps to determine the distance between two nonempty subsets of a metric space (X, d).Let A and B be two nonempty subsets of a metric space (X, d) and T : A → B be a nonselfmapping.Then d(x, T x) ≥ d(A, B), for all x ∈ A, where d(A, B) = inf{d(x, y) : x ∈ A, y ∈ B}.In general, for a non-selfmapping T : A → B, the fixed point equation T x = x may not have a solution.In such cases, one intend to find an approximate solution x ∈ A such that d(x, T x) = d(A, B).Best approximation theorems and best proximity point theorems are relevant in this regard.For instance, let us consider the well known classical best approximation theorem by Ky Fan [12].
Theorem 1.1 ( [12]).Let A be a non-empty compact convex subset of a normed linear space X and T : A → X be a continuous function.Then there exists x ∈ A such that x − T x = d(T x, A) = inf{ T x − a : a ∈ A}.
On the other hand, though best approximation theorems only ensure the existence of approximate solutions, in this case, such results need not yield optimal solutions.But, best proximity point theorems provide sufficient conditions that assure the existence of approximate solutions which are optimal.In this regard, the best proximity point evolves as a generalization of the best approximation.The authors Basha [5], Choudhury, Maity and Konar, [9,10] and Kutbi, Chandok and Sintunavarat [16] tried to reduce the problem of finding approximate solutions to that of finding optimal approximate solutions.
In recent years, the existence of best proximity points is an interesting aspect of optimization theory which attracted the attention of many researchers.For example, Abkar and Gableh [2], Basha [6], Caballaro, Harjani and Sadarangani [7], Eldred [11], Gabeleh [13] and Karapinar [15] and the related references cited in these papers, worked in this area.
A best proximity point becomes a fixed point if the underlying mapping is a selfmapping.Therefore, it is concluded that best proximity point theorems generalize fixed point theorems in a natural way.For more works on the existence of best proximity points, we refer [1,3,8,14,17] and references therein.
Our purpose here is to establish best proximity point theorems in partially ordered metric spaces.
We recall the following notation and definitions.Let (X, d, ) be a partially ordered metric space and let A and B be nonempty subsets of X.

Definition 1.2 ([4]
).Let (X, ) be a partially ordered space with metric d.We say that f : X → X is an α−β−ψ−contractive mapping if there exist three functions α, β : for all x, y ∈ X with x y.
We say that f is an α−β−admissible mapping, if for all x, y ∈ X with x y hold In 2015, Asgari and Badehian [4], proved fixed point theorems for α−β− ψ−contractive mappings in partially ordered space with complete metric.

Theorem 1.2 ([4]
).Let (X, ) be a partially ordered space with complete metric d.Let f : X → X be a nondecreasing, α−β−ψ−contractive mapping satisfying the following conditions: In Section 2, we introduce a notion of α−β−proximal admissible mappings and α−β−ψ−proximal contractive mappings that we consider to prove our main results in Section 3. In Section 4, we draw some corollaries and provide examples in support of our main results.

Preliminaries
Definition 2.1.Let (X, d, ) be a partially ordered metric space, A, B be two nonempty subsets of X, α, β : A × A → [0, ∞) be functions, C α > 0, C β ≥ 0 be two constants and T : A → B be a non-selfmapping.We say that T is an α−β−proximal admissible, if for all x, y, u, v ∈ A, with x y We define a partial order on X by (x, y) (u, v) if and only if x ≤ u and y ≤ v, for all (x, y), (u, v) ∈ X, where is the usual order on [0, ∞).We define T : A → B by We also define functions α, β : Definition 2.2.Let (X, d, ) be a partially ordered metric space and A and B be nonempty closed subsets of X.We say that T : A → B is an α−β−ψ−proximal contractive mapping if there exist functions α, β : A × A → [0, ∞), ψ ∈ Ψ such that for all x, y, u, v ∈ A with x y holds: If α(x, y) = 1 = β(x, y), for all x, y ∈ A in (1) then we say that T is a ψ-proximal contractive mappings.

Remark 2.2. Here we observe that if
) and X = A∪B, with the Euclidean metric d.We define a partial order on X by (x, y) (u, v) if and only if x ≤ u and y ≤ v, for all (x, y), (u, v) ∈ X.
Clearly, holds d(A, B) = 1.We define T : A → B by 8 t, for all t ≥ 0. We will show that T is an α−β−ψ−proximal contractive mapping.
In the following, we prove our main results.

Main Results
Theorem 3.1.Let (X, d, ) be a partially ordered complete metric space.Let A, B be non-empty closed subsets of X with A 0 is nonempty and closed.Let T : A → B be a proximally increasing non-selfmapping such that the following conditions hold: (i) T is continuous, (ii) T is an α−β−ψ−proximal contractive mapping, (iii) T is an α−β−proximal admissible, (iv) T (A 0 ) ⊆ B 0 , (v) there exist elements x 0 , x 1 ∈ A 0 such that x 0 x 1 and d(x 20 Best proximity points of α−β−ψ−proximal contractive mappings. . .Then T has a best proximity point in A 0 . Proof.By condition (v), there exist x 0 , x 1 ∈ A 0 such that x 0 x 1 and ( 4) Since T (A 0 ) ⊆ B 0 , we have T x 1 ∈ B 0 and hence there exists an element Since T is proximally increasing on A, from ( 4) and ( 5), we have x 1 x 2 .On continuing this process, we get a sequence {x n } in A 0 such that for some n 0 ∈ N, then x n 0 is the best proximity point of T and hence the conclusion of the theorem follows.Now, we assume with out loss of generality that any two consecutive elements of {x n } are distinct.
From condition (iii), condition (vi) and ( 6), the following holds: (7) Since T is an α−β−ψ−proximal contractive mapping and by considering (7), we have Again by condition (iii), condition (vi), ( 6) and ( 7), we have ( 9) Therefore, by considering (9) and by the fact that T is an α−β−ψ−proximal contractive mapping, we have and hence On continuing this process, we obtain for n = 1, 2, 3, ... and Since ψ ∈ Ψ, we have ψ n d(x 0 , x 1 ) → 0 as n → ∞.Now, we show that {x n } is a Cauchy sequence.We fix > 0 and choose Therefore by applying triangle inequality, we have Hence {x n } is a Cauchy sequence.Since A 0 is a closed subset of a complete metric space and hence it is complete, there exists x ∈ A 0 such that x n → x.Since T is continuous, by letting n → ∞ in (6), we obtain d(x, T x) = d(A, B).Hence x is a best proximity point of T .
If we drop the continuity assumption from Theorem 3.1, we obtain the following result.Theorem 3.2.Let (X, d, ) be a partially ordered complete metric space.Let A, B be non-empty closed subsets of X with A 0 is nonempty and closed.Let T : A → B be proximally increasing non-selfmapping such that the following conditions hold: (i) T is an α−β−ψ−proximal contractive mapping and T is an α−β− proximal admissible; (ii) T (A 0 ) ⊆ B 0 ; (iii) there exist elements x 0 , x 1 ∈ A 0 such that x 0 x 1 and d(x Then T has a best proximity point in A 0 . Proof.From the proof of Theorem 3.1, we have the sequence {x n } is Cauchy and x n → x ∈ A 0 .Since T (A 0 ) ⊆ B 0 , then T (x) ∈ B 0 and hence there exists In the proof of Theorem 3.1, we obtained that {x n } is a nondecreasing sequence satisfying α(x n , x n+1 ) ≥ C α and β(x n , x n+1 ) ≤ C β .Therefore, by condition (v), it follows that α(x n , x) ≥ C α and β(x n , x) ≤ C β , and condition (vi), we have x n x for n ∈ N.
We now claim that z = x.Since d(x n+1 , T x n ) = d(A, B), by combining this equation with (10) and by the fact that T is an α−β−ψ−proximal contractive mapping, we have and therefore If n → ∞, we obtain x = z.Hence x is a best proximity point of T .
Lemma 3.1.In addition to the hypotheses of Theorem 3.1 (Theorem 3.2), if x is a best proximity point of T with x u, α(x, u) ≥ C α and β(x, u) ≤ C β for some u ∈ A 0 , then there exists a sequence {u n } ⊆ A 0 such that d(u n , T u n−1 ) = d(A, B), x u n , for n = 1, 2, 3, . . .and u n → x as n → ∞.
Proof.Let x be a best proximity point of T , i.e., (11) d(x, T x) = d(A, B).
Let u ∈ A 0 such that x u.We set u 0 = u.Since T (A 0 ) ⊆ B 0 and u = u 0 ∈ A 0 , we have T u 0 ∈ B 0 .Hence there exists u 1 ∈ A such that (12) d(u 1 , T u 0 ) = d(A, B).
From (12), by the definition of A 0 and B 0 , we have u 1 ∈ A 0 .Since T is proximally increasing on A 0 , from x u = u 0 , ( 11) and ( 12), we have x u 1 .
On continuing this process we can construct a sequence {u n } in A 0 such that satisfying (14) x u n , for n = 1, 2, 3, . . .
Then T has a unique best proximity point in A 0 .
Proof.By the proof of Theorem 3.1 (Theorem 3.2), the set of best proximity points of T is nonempty.Let x, y be two best proximity points of T in A 0 .By our assumption, we have there exists u ∈ A 0 such that x u, y u, α(x, u) ≥ C α and β(x, u) ≤ C β and α(y, u) ≥ C α and β(y, u) ≤ C β .Now by applying Lemma 3.1, it follows that there exists a sequence {u n } ⊆ A 0 such that u n → x and u n → y as n → ∞.Hence by the uniqueness of limits we have x = y.

Corollaries and Examples
Corollary 4.1.Let (X, d, ) be a partially ordered complete metric space.Let A, B be non-empty closed subsets of X with A 0 is nonempty and closed.Let T : A → B be proximally increasing non-selfmapping such that for all x, y, u, v ∈ A with x y hold: (i) d(u, T x) = d(v, T y) = d(A, B) implies that d(u, v) ≤ kd(x, y), for some k ∈ [0, 1), (ii) T is continuous and T (A 0 ) ⊆ B 0 , (iii) T is ψ− proximal contractive mapping and T is proximal admissible, (iv) there exist elements x 0 , x 1 ∈ A 0 such that x 0 x 1 and d(x 1 , T x 0 ) = d(A, B), Then T has a best proximity point in A 0 .