A New Coupled Fixed Point Result in Extended Metric Spaces with an Application to Study the Stability of Set-Valued Functional Equations

Abstract

In this paper, we introduce a new coupled fixed point theorem in a generalized metric space and utilize the same to study the stability for a system of set-valued functional equations.

1. Introduction and Preliminaries

Discussing the stability of functional equations, we pursue the inquiry put forward in 1940 by Ulam [1] which states that the solution of an equation varying marginally from a given solution, should of need be near the solution for the given equation. Three popular techniques to establish the stability from a perspective Hyers–Ulam of functional equations are the direct technique [2], the technique of invariant means [3, 4] and the fixed point technique (see [5]). In the last technique, different definitely known fixed point theorems are utilized, just as some new have been demonstrated and utilized in a specific circumstance. In 1991, Baker [6] studied the stability of functional equations via the banach fixed point theorem. Since fixed point technique of Baker, Radu [7] gave the stability of an equation of functional by the option of fixed point technique which was presented by Diaz and Margolis [8]. The fixed point technique has given a great deal of impact in the advancement of the stability of functional equations. We allude to numerous papers of stability of equations of functional utilizing the fixed point technique in references on the stability of equations of functional (see [8–10]).

In 2008, Park and An [11] used the fixed point technique to study the stability of functional equations due to Cauchy–Jensen. In 2009, Gao et al. [12] defined the generalized Cauchy–Jensen equation as follows:

Let be an abelian group and -divisible, where , the set of all natural numbers, and be a normed space with the norm . For any function , the equation

for each and is said to be the generalized equation of Cauchy–Jensen. In special case, when , the equation is called the Cauchy–Jensen equation.

Recently, in 2018, Kaskasem et al. [13] introduced the stability by Hyers–Ulam–Rassias of the generalized set-valued functional equations of Cauchy-Jensen given by

for each

The objective of our paper is basically two fold. The first goal is introduce a new fixed point technique dealing with coupled fixed point results for nonlinear contractive mappings on the generalized metric space due to Diaz et al. [8]. The second goal is to apply our new coupled fixed point results to study the stability for the following coupled system of the generalized set-valued functional equations of Cauchy–Jensen:

for all and

Next, we recall some preliminaries that will be used in the main results of this paper.

Let be a Banach space. We defined the following:(i) = the set of power sets of ;(ii) = all bounded and closed subsets of ;(iii) = all convex and closed subsets of ;(iv) = all convex closed and bounded subsets of .

Definition 1 [12]. On , they consider the addition and the scalar multiplication as follows:where and , the set of all real numbers. Also, we define the following:ThenAlso, when is convex, we obtainfor all . For any set , the distance function and the support function are defined byFor all sets , the Hausdorff distance between and is defined bywhere is the closed unit ball in .

Proposition 1 [13]. For any and , the following properties hold:(1);(2).

Definition 2 [14]. Let be a set. A distance mapping is said to be a generalized metric on if the following conditions are hold:  for all if and only if ;  for all ;  for all ;  every -Cauchy sequence in is convergent, i.e., for a sequence , implies the existence of an element with , ( is unique by and ).By the fact that not every two points in have necessarily a finite distance. One might call such a space a generalized complete metric space.

Example 1. Let Define on as follows:Then is a generalized metric space.

Definition 3 [15]. Let be a partially ordered space and let . The function is said to have the mixed monotone property if is nondecreasing monotone in and is nonincreasing monotone in , that is, for each ,

Definition 4 [15]. A pair is called a coupled fixed point of the function if and .

2. Main Coupled Fixed Point Results

Theorem 1. Suppose that is a complete generalized metric space and the function be a continuous mapping having the mixed monotone property on . Assume that there exists a such that for , the following holds:for all and . If there exist such that and . Then the following alternative holds: either.
(i)for all , we have(ii) has a coupled fixed point in , that is, there exist such that and

Proof. By the given assumptions, there exists such that and . Then, we can define such that and , then and . Also there exists such that and . Since has the mixed monotone property, we have,Continuing in this way, we construct two sequences and in such thatfor all .
There are two mutually exclusive possibilities: either
(a) for every integer one haswhich is exactly the alternative of the conclusion of the theorem, or else(b)some integer one hasNow, we need to show that (b) implies alternative of the conclusion of the theorem.
If case (b) holds, let denote a particular one. For definiteness, one could choose the smallest of all integer , such thatThen, by (15), since and we getAlso,However at this point, the triangle property (12 in Definition 2 infers that, at whatever point , one has for each , thatSince , then the sequence and similarly the sequence are Cauchy sequences and by (15) in Definition 2 they are convergent. In other words, there exist such thatAt last, we guarantee and , since is continuous at then we have

Remark 1. Let be a mapping from into itself. If we put and in Theorem 1, then one can deduce the following theorem.

Theorem 2. Suppose that is a partially ordered complete generalized metric space and the function be a continuous strictly contractive mapping, that is, there exists a number such thatIf there exists with
Then the following alternative holds: either
(I)for all , we have(II) has a coupled fixed point in , that is, there exist such that

Proof 2. By the given assumptions, there exists such that Then, we can define such that then Also there exists such that Since has the mixed monotone property, we have,Continuing in this way, we construct two sequences in such thatfor all .
There are two mutually exclusive possibilities: either
(A)for every integer one has which is exactly the alternative of the conclusion of the theorem, or else(B)some integer one hasNow, we need to show that (B) implies alternative (II) of the conclusion of the theorem.
If case (B) holds, let denote a particular one. For definiteness, one could choose the smallest of all integer , such thatThen, by (30), since , we getHowever at this point, the triangle property (12) in Definition 3 infers that, at whatever point , one has for each , thatSince , then the sequence is a Cauchy sequence and by (15) in Definition 2 it is convergent. In other words, there exists a point such thatAt last, we guarantee since is continuous at then we have

Remark 2. We note that the contractive condition in [8, Theorem 1.6] is slightly stronger than the condition (30) of Theorem 2.

3. Stability of the Cauchy–Jensen Functional Equations

Let be a real normed space and be a real banach space.

Definition 5. Let be two set-valued mappings.
(1) The coupled generalized Cauchy-Jensen set-valued functional equation is defined byfor all and (2)Every solution of the generalized Cauchy-Jensen set-valued functional equation is called a Cauchy–Jensen set-valued mapping.

Theorem 3. Let be two set-valued mappings defined on into such that there exists a function
satisfyingfor all and . If there exists such thatfor all , then there exists unique generalized Cauchy–Jensen set-valued mappings such thatfor all . Moreover, if there exist positive real number and with such that diam for all , then are singleton sets.

Proof 3. First, we consider the set and introduce the generalized metric on X as follows:where and . Then is a complete generalized metric space (see[[16], Theorem (3)]). Now, we consider a linear mapping such thatwhere .
Next, we show that is a strictly contractive mapping with Lipschitz constant . Let with and for some . It follows from (48) thatfor all From Proposition 1, (44), (45) and (52) we obtain thatfor all . Hence, , that is, . Therefore, we suppose that and in (43) since is convex, we havefor all . Then, we havefor all Thus, by (2), we havefor all
Similarly, one can deduce thatand sofor all .
By Theorem 1, there exist two mappings such that the following conditions hold;
(a) is a coupled fixed point of , that is, and for all . Then we have(b) The sequences and converge to respectively. This implies the following equality:(c) We obtain that and which implies to the following inequality:Thus the inequalities (46) hold.
It follows from (41) and (42) thatfor all . Thus, we haveSo, we haveand similarly, one can get thatfor all . Moreover, let and be positive real numbers with and for all . Then, we havefor all . Since we have
This implies that is a singleton set. This completes the proof.

Open Problem 13. Can our results in this paper be extended in generalized metric spaces as in Aydi and Czerwik [17] and Karapinar et al. [18].

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

The authors read and approved the final manuscript.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through General Research Project under grant number G.R.P-57-40.