COUPLED BEST PROXIMITY POINT THEOREMS FOR MIXED g-MONOTONE MAPPINGS IN PARTIALLY ORDERED METRIC SPACES

1Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology (VIT) University, Vellore 632 014, T.N., India 2Department of Mathematics, Govt. Girls Polytechnic College Raipur, Chhattishgarh, India 3Department of Mathematics, Faculty of Science, Indira Gandhi National Tribal University, Lalpur, Amarkantak, Anuppur, Madhya Pradesh 484 887, India 4Department of Mathematics Education, Faculty of Tarbiya, Institute for Islamic Studies Ma’arif NU (IAIMNU) Metro Lampung, Metro-34111, Indonesia


INTRODUCTION AND PRELIMINARIES
The classical contraction mapping principal of Banach is one of the most useful and fundamental results in fixed point theory. Several authors studied and extended it in many directions. Here, if we take A = B, then this definition reduced to Definition 1.2. Definition 1.6. [2] Let (X, ≤) be a partially ordered set and T : X × X → X. We say that T has the mixed monotone property if T(x,y) is monotone nondecreasing in x and is monotone nonincreasing in y, that is, for any x, y ∈ X x 1 , x 2 ∈ X x 1 ≤ x 2 ⇒ T (x 1 , y) ≤ T (x 2 , y) y 1 , y 2 ∈ X y 1 ≤ y 2 ⇒ T (x, y 1 ) ≥ T (x, y 2 ). Definition 1.7. [20] A mapping T : A × A → B is said to be the proximal mixed monotone property if T(x, y) is proximally nondecreasing in x and is proximally nonincreasing in y, that is If we take A = B in the above definition, then proximal mixed monotone property reduces to mixed monotone property.  for all r > 0 and lim t→0 + ψ(t) = 0.
In [18] Luong and Thuan obtained a result of coupled fixed. Following is the main theoretical results of Luong and Thuan.
Theorem 1.10. [18] Let (X, ≤) be a partially ordered set and suppose there is a metric d on X such that (X,d) is a complete metric space. Let T : X × X → X be mapping having the mixed monotone property on X such that for all x, y, u, v ∈ X with x ≥ u and y ≤ v, where ψ ∈ Ψ and φ ∈ Φ. If there exist x 0 , y 0 ∈ X such that x 0 ≤ T (x 0 , y 0 ) and y 0 ≥ T (y 0 , x 0 ). Suppose either (a) T is continuous or (b) X has the following property: (i) if a non-decreasing sequence {x n } → x, then x n ≤ x for all n, (ii) if a non-increasing sequence {y n } → y, then y ≥ y n for all n.
Then there exist x, y ∈ X such that T (x, y) = x and T (y, x) = y.
Recently, Kumam et al. [20] extended the results of Luong and Thuan [18]. They also introduced the concept of the proximal mixed monotone property and established coupled best proximity point theorem. Following is the main results of Kumam et al. [20]. (i) T is a continuous proximally coupled weak (ψ, φ ) contraction on A having the proximal mixed monotone property on A such that T (A 0 , A 0 ) ⊆ B 0 ;.
Motivated by the results of [18] and [20], we present the coupled best proximity point and coupled fixed point, and by defining the concept of proximal mixed g-monotone mapping and proximally coupled weak (ψ, φ ) contraction on A. The existence and uniqueness of coupled best proximity points are obtained in partially ordered metric spaces. We also provide an example to support of our results.

MAIN RESULTS
In this section we first present following definitions.
Definition 2.1. Let (X, d, ≤) be a partially ordered metric space. Let A, B be nonempty subsets of X, and T : A × A → B and g : A → A be two given mappings. We say that T has the proximal mixed g-monotone property provided that for all x, y ∈ A, if Definition 2.2. Let (X, d, ≤) be a partially ordered metric space and A, B are nonempty subsets of X. Let T : A × A → B and g : A → A be two given mappings. T is said to be proximally coupled weak (ψ, φ ) contraction on A, whenever Lemma 2.3. Let (X, d, ≤) be a partially ordered metric space and A, B be nonempty subsets of X, A 0 = / 0 and T : A × A → B and g : A → A be two given mappings. If T has the proximal mixed Using the fact that T has the proximal mixed g-monotone property, together with 2.2 and 2.3, we get Also, from the proximal mixed g-monotone property of T with 2.2 and 2.4, we get .
Proof. The proof is the same as Lemma 2.3, so we omit the details.
Following is the main result of this paper. (b) T has the proximal mixed g-monotone property on A such that g( Then there exists (x, y) ∈ A × A such that d(g(x), T (g(x), g(y))) = d(A, B) and d(g(y), T (g(y), g(x))) = d(A, B).
(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.
Proof. As in the proof of Theorem 2.5, there exist sequences {x n } and {y n } in A 0 such that ) f or all n ≥ 0, and (2.32) d g(y n+1 ), T g(y n ), g(x n ) = d(A, B) with g(y n ) ≥ g(y n+1 ) f or all n ≥ 0.
By taking the limit of the above two inequalities, we get g(x) = g(x * ) and g(y) = g(y * ). Hence, from (2.33) and (2.34), we get d g(x), T (g(x), g(y)) = d(A, B) and d(g(y), T g(y), g(x)) = d (A, B). The only comparable pairs of points in A are gx ≤ gx for x ∈ A, hence proximal mixed gmonotone property and proximally coupled weak (ψ, φ ) contraction on A are satisfied trivially.
One can prove that the coupled best proximity point is in fact unique, provided that the product space A × A endowed with the partial order mentioned earlier has the following property: Every pair o f elements has either a lower bound or an upper bound.
It is known that this condition is equivalent to the following. For every pair of (x, y), (z,t) ∈ A × A, there exists (u, v) ∈ A × A that is comparable to (x, y) and (z,t).
Theorem 2.11. Suppose that all the hypotheses of Theorem 2.5 hold and further, for all Case 1: Let (g(x), g(y)) be comparable to (g(z), g(t)) with respect to the ordering in A × A.
Therefore d g(u n ), g(x) + d g(v n ), g(y) is a decreasing sequence. Hence there exists r ≥ 0 such that lim n→∞ d g(u n ), g(x) + d g(v n ), g(y) = r.
We shall show that r = 0. Suppose, to the contrary, that r > 0. On taking the limit as n → ∞ in which is a contradiction. Hence, r = 0, that is, so that g(u n ) → g(x) and g(v n ) → g(y). Analogously, one can prove that g(u n ) → g(z) and g(v n ) → g(t). Therefore, g(x) = g(z) and g(y) = g(t). Hence the proof is complete.
Considering g is assumed to be the identity mappings in Theorem 2.11 then we obtained the following result.  (with respect to the ordering in A × A). Then there exists a unique (x, y) ∈ A × A such that d g(x), T g(x), g(y)) = 0 and d g(y), T g(y), g(x)) = 0.
We shall illustrate our results by the following example.
Example 2.14. Let X = R and d(x, y) =| x − y | be the usual metric on X and let the usual
Then there exists (x, y) ∈ A × A such that d(g(x), T (g(x), g(y))) = 0 and d(g(y), T (g(y), g(x))) = 0. Remark 3.6. This results also true if we replace the continuity of T by the condition (e) of Theorem 2.8.

DATA AVAILABILITY
No data were used to support the study.