Coupled Fixed Point for ( α , ψ )-Contractive in Partially Ordered Metric Spaces Using Compatible Mappings

In this paper, first we introduce notions of (α, ψ)-contractive and (α)-admissible for a pair of map and prove a coupled coincidence point theorem for compatible mappings using these notions. Our work extends and generalizes the results of Mursaleen et al. [1]. At the end, we will provide an example in support of our result.


Introduction
Fixed point theorems give the conditions under which maps have solutions.
Fixed point theory is a beautiful mixture of Analysis, Topology and Geometry.Fixed points Theory has been playing a vital role in the study of nonlinear phenomena.In particular, fixed point techniques have been applied in diverse fields as Biology, Chemistry, and Economics, Engineering, Game theory and Physics.The usefulness of the concrete applications has increased enormously due to the development of accurate techniques for computing fixed points.
The fixed point theory has many important applications in numerical methods like Newton-Raphson Method and establishing Picard's Existence Theorem regarding existence and uniqueness of solution of first order differential equation, existence of solution of integral equations and a system of linear equations.The credit of making the concept of fixed point theory useful and popular goes to polish mathematician Stefan Banach.In 1922, Banachproved a fixed point theorem, which ensures the existence and uniqueness of a fixed point under appropriate conditions.This result of Banach is known as Banach fixed point theoremor contraction mapping prin-ciple, "Let x be any non empty set and ( ) , X d be a completemetric space If T is mapping of X into itself satis- fying ( ) ( ) , , d Tx Ty kd x y ≤ for each , x y X ∈ where 0 1 k ≤ < , then T has a unique fixed point in X".This principle provides a technique for solving a variety of applied problems in Mathematical sciences and Engineering and guarantees the existence and uniqueness of fixed points of certain self maps of metric spaces and provides a constructive method to find out fixed points.Now the question arise what type of problems have the fixed point.The fixed point problems can be elaborated in the following manner: 1) What functions/maps have a fixed point?
2) How do we determine the fixed point?
3) Is the fixed point unique?Currently, fixed point theory has been receiving much attention on in partially ordered metric spaces; that is, metric spaces endowed with a partial ordering.Turinici [2] extending the Banach contraction principle in the setting of partially ordered sets and laid the foundation a new trend in fixed point theory.Ran and Reurings [3] developed some applications of Turinici's theorem to matrix equations and established some results in this direction.The results were further extended by Nieto and Rodŕguez-Ĺpez [4] [5] for non-decreasing mappings.Bhaskar and Lakshmikantham [6] [7] introduced the new notion of coupled fixed points for the mappings satisfying the mixed monotone property in partially ordered spaces and discussed the existence and uniqueness of a solution for a periodic boundary value problem.Later on, Lakshmikantham and Ciríc [8] proved coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces.
Choudhury and Kundu [9], proved the coupled coincidence result for compatible mappings in the settings of partially ordered metric space.Recently, Samet et al. [10] [11] have introduced the notion of α-ψ-contractive and α-admissible mapping and proved fixed point theorems for such mappings in complete metric spaces.For more results regarding coupled fixed points in various metric spaces one can refer to [12]- [23].

Mathematical Preliminaries
In order to obtain our results we need to consider the followings.Definition 2.1.[6].Let ( ) , X ≤ be a partially ordered set and X X X × → be a mapping.Then a map F is said to have the mixed monotone property if ( ) , F x y is monotone non-decreasing in x and is monotone non-increasing in y; that is, for any , x y X ∈ , , , F x y F x y ≤ and , , F x y F x y ≥ .
Definition 2.2.[6].An element ( ) ( ) Definition 2.3.[8].Let ( ) , X d be a partially ordered set and : F X X X × → and : g X X → be two mappings.We say F has the mixed g-monotone property if F is monotone g-non-decreasing in its first argument and is monotone g-non-increasing in its second argument, that is, for any , Definition 2.4.[8].An element ( ) ( ) ( ) , F y x g y = .Choudhury et al. [9] introduced the notion of compatible maps in partially ordered metric spaces as follows: Definition 2.5.[9].The mappings F and g where : In order to obtain our results we need to consider the followings.Definition 2.6.[1].Denote by Ψ the family of non-decreasing functions 2) ψ ∞ → ∞ is non-decreasing and right continuous, then ( ) Definition 2.8.[1].Let ( ) , X d be a partially ordered metric space and : → +∞ be two mappings.Then F is said to be (α)-admissible if x y u v X ∈ .Now, we will introduce our notions: Definition 2.10.Let ( ) , X d be a partially ordered metric space and : F X X X × → and : g X X → be two mappings.Then the maps F and g are said to be (α, ψ)-contractive if there exist two functions and ( ) ( ) : 0, X X α × → +∞ be mappings.Then F and g are said to be (α)-admissible if , , , x y u v X ∈ .

Main Results
Recently, Mursaleen et al. [1] proved the following coupled fixed point theorem with α-ψ-contractive conditions in partial ordered metric spaces: X ≤ be a partially ordered set and there exists a metric d on X such that ( ) be mapping and suppose F has mixed monotone property.Suppose there exists ψ ∈ Ψ and : 0, X X α × → +∞ Such that for , , , x y u v X ∈ , the following holds: , with x u ≥ and y v ≤ .
2) There exists , , X ≤ be a partially ordered set and there exists a metric d on X such that ( ) , X d is a complete metric space.Let : F X X X × → be mapping and : g X X → be another mapping.Suppose F has g-mixed monotone property and there exists ψ ∈ Ψ and [ ] For all , , , x y u v X ∈ with ( ) ( ) Suppose also that 1) F and g are (α)-admissible.
2) There exists , , g is continuous and F and g are compatible in X. , , , F y x g y = .
Continuing this process, we can construct two sequences { } Then, since ( ) ( ) and ( ) ( ) . Therefore, by g-mixed monotone property of F, we have From above, we conclude that ( ) ( ) ( ) ( ) and n n n n g x g x g y g y Thus, by mathematical induction, we conclude that (3.4) holds for all 0 n ≥ .If following holds for some n N ∈ , ( ) ( ) ( ) ( ) ( ) ( ) F x y F y x F x y F y x g x g y g x g y α α = ≥ .
Thus by mathematical induction, we have , , , 1 Similarly, we have ≥ for all, n N ∈ . (3.6)   From (3.3) and conditions 1) and 2) of hypothesis, we get Similarly, we have On adding (3.7) and (3.8), we get m n n ε > > , then by using the triangle inequality, we have Now suppose that (3.4) holds for some fixed 0 n ≥ . 2 coincidence point of g and F in X.