Stability Results for Enriched Contraction Mappings in Convex Metric Spaces

Abstract

In this paper, we obtain some stability results of fixed point sets for a sequence of enriched contraction mappings in the setting of convex metric spaces. In particular, two types of convergence of mappings, namely, -convergence and -convergence are considered. We also illustrate our results by an application to an initial value problem for an ordinary differential equation.

1. Introduction

Stability is a term that refers to the limiting characteristics of a system. The stability of fixed points is the study of the relationship between the convergence of a sequence of mappings on a metric space or a Banach space and the sequence of its fixed points. If the fixed point sets of a sequence of mappings converge to the set of fixed points of their limit mappings, the sequence is said to be stable. We recommend the references [1–8] for some interesting results on the stability of fixed point sets of families of mappings in various settings. In this connection, see also the paper by Reich and Shafrir [9]. Pointwise and uniform convergence play a significant role in the study of fixed point stability. The preceding concepts, however, do not hold true when the domain of definition of all mappings in the study is neither the same space nor a unique nonempty subset of it. Barbet and Nachi [10] overcame this problem by introducing two new ideas of convergence called -convergence and -convergence, which they used to derive stability results in metric spaces. These results generalize the corresponding results of Bonsall [2], Fraser and Nadler [4], and Nadler [6]. The Barbet-Nachi work is unique in the sense that it redefines pointwise and uniform convergence for operators defined on the subsets of a space rather than the entire metric space. -convergence generalized pointwise convergence, while -convergence extended uniform convergence. Afterwards, many authors generalized these results in different settings for various type of mappings (see, for example, [11–15]).

On the other hand, Takahashi [16] considered the concept of convex metric space and looked at a few fixed point theorems for nonexpansive mappings in this space. He observed that the concept of fixed point theorems in Banach spaces can be extended to convex metric spaces. Subsequently, Ciric [17], Naimpally et al. [18], and many others studied fixed point theorems for different classes of mappings in convex metric spaces as well as a subclass of it (compare [19, 20]). In addition, Berinde and Păcurar [21, 22] obtained some fixed point theorems for a class of enriched contraction mappings.

In this paper, we study some convergence results in the sense of Barbet and Nachi [10] for a sequence of enriched contraction mappings and their fixed points in convex metric spaces as well as a subclass of it.

2. Preliminaries

We present some definitions, notions, and facts from the literature in this section. Throughout this paper, we use the notation to denote the set of real numbers and to denote

Definition 1 (see [16]). Let be a metric space. A continuous function is said to be a convex structure on if, for all and any , for any

A metric space endowed with a convex structure is called a convex metric space and is usually denoted by .

Obviously, any linear normed space and each of its convex subsets are convex metric spaces, with the natural convex structure

for all and . However, the converse is not true: there are a number of examples of convex metric spaces that cannot be nested in any Banach space [16]. Let be a convex metric space and be a self mapping. denotes the set of all fixed points of , that is,

Let be a metric space and . A mapping is called as geodesic path from to if

for every The image of forms a geodesic joining and . It is noted that the geodesic segment joining and is not necessarily unique. In some results of this paper, we shall deal with a subclass of convex metric spaces called -hyperbolic metric space (or -hyperbolic space) in the sense of Kohlenbach [23].

Definition 2 (see [23]). The triplet is called a hyperbolic metric space if is a metric space and the function satisfies the following conditions for all and :
(W1)
(W2)
(W3)
(W4)

Remark 3. If for all then it can be seen that all normed linear spaces are included in these spaces.

Remark 4. If conditions (W1)–(W3) are satisfied, then is the hyperbolic type space considered by Goebel and Kirk [24].

We shall write

to indicate the point in a given hyperbolic metric space.

3. -Enriched Contraction Mapping

First, we recall the following definitions and results from Berinde and Păcurar [21] which play an important role in this paper.

Definition 5. Let be a Banach space. A mapping is said to be a -enriched contraction mapping if there exist and such that for all ,

Remark 6. It is pointed out in [21] that every contraction mapping is a -enriched mapping. The class of quasicontraction mapping and the class of -enriched contraction mappings are independent in nature. We present some examples to support our claim.

Example 1 (see [21]). Let with the usual metric and be a mapping defined as

Then, . is a -enriched contraction mapping for any At and ,

for any Thus, is not a quasicontraction mapping.

Example 2. Let with the usual metric and be a mapping defined as

Then, . Now, we show that is a quasicontraction mapping. If and , then

for any . On the other hand, at and ,

is not a -enriched contraction mapping for any

Now, we consider Definition 5 in convex spaces:

Definition 7 (see [6]). Let be a convex metric space. A mapping is said to be an enriched contraction mapping if there exist and such that

Define the mapping by . Then, is a Banach contraction, and hence, it has a fixed point in complete metric spaces.

4. Barbet-Nachi-Type Convergence

Following the conditions developed by Barbet and Nachi [10], we define the following conceptions of convergence for enriched contraction mappings in convex metric spaces, which generalize the notions of pointwise and uniform convergence.

Let be a family of nonempty subsets of the convex metric space and a family of enriched contraction mappings [10]. Then, is said to be a -limit of the sequence , or equivalently, satisfies the property if the following condition holds:

: for every , there exists a sequence in such that

where stands for the graph of .

Further, is a -limit of the sequence of enriched contraction mappings , or equivalently, satisfies the property if the following condition holds [10]:

For all sequences in , there exists a sequence in such that

Lemma 8. Let be a convex metric space, a family of nonempty subsets of , and a sequence of enriched contraction mappings. Define the mapping by where . Then, for any , .

Proof. Let . That is, . That is, .
Conversely, assume . This means that , and hence, we have But , which implies .
That is, . Hence .

5. Stability Results for -Convergence

The following stability result improves and generalizes the results of Fraser and Nadler [4] and Nachi [25].

Theorem 9. Let be a convex metric space and a family of nonempty subsets of . is a family of enriched contraction mappings satisfying the property . If for all , is a fixed point of , then the sequence converges to

Proof. Since satisfies property , , there exists a sequence in such that and . Thus, But . Therefore, Taking the limit as , we obtain Since , we get .

Corollary 10. Let be a convex metric space and a family of enriched contraction mappings satisfying the property . If for all , is a fixed point of , then the sequence converges to

Proof. By taking for in Theorem 9, we find the desired conclusion easily.

The following theorem proves the existence of a fixed point for a -limit of a sequence of enriched contraction mappings.

Theorem 11. Let be a convex metric space and a family of nonempty subsets of . Suppose that is a sequence of enriched contraction mappings satisfying the property . If is a fixed point of for each and the sequence admits a subsequence converging to , then is a fixed point of .

Proof. Let be a subsequence of such that By the property , there exists a sequence in such that and For any , we have Since is an enriched contraction mapping, But . Then, we have Then, from (20), we have Letting , we deduce that which implies That is, .
But . Hence .

Proposition 12. Let be a hyperbolic metric space, a family of nonempty subsets of , and a family of mappings satisfying the property , such that, for any , is an enriched contraction mapping. Then, is also an enriched contraction mapping.

Proof. Let and be two elements in . By the property , there exist two sequences and in converging, respectively, to and such that the sequences and converge to and , respectively. For , one can set for all For any , we have Now, we prove the following claim: Given and as . Then, Further, and as . Thus, Letting in (28), (30), and (31), we have and the conclusion holds.

The following result in Proposition 4 of [10] follows from Proposition 12.

Corollary 13. Let be a metric space, a family of nonempty subsets of , and a family of mappings satisfying property such that, for any , is a -contraction. Then, is a -contraction where is a bounded (resp.convergent) sequence with .

6. Stability Results for -Convergence

Now, we present another stability result using the -convergence.

Theorem 14. Let be a hyperbolic metric space and a family of nonempty subsets of . Let be a family of mappings satisfying the property such that is an enriched contraction mapping. If for each , is a fixed point of , then the sequence converges to .

Proof. By property , there exists a sequence in such that and . is an enriched contraction which implies Given is a fixed point of for each . Then, is a fixed point of as well. That is, for each . Also, By letting , we see that .
Now, by using the triangle inequality, we get As is an enriched contraction mapping, is a contraction mapping. Hence, by letting , we obtain Thus, and hence, the conclusion follows.

Corollary 15. Let be a hyperbolic metric space and be a family of mappings satisfying the property converging to the enriched contraction mapping . If for any , is a fixed point of then the sequence converges to .

Proof. By taking for all in Theorem 14, we obtain the desired result immediately.

7. Application

We consider the following application of our results. The following theorem is motivated due to Nadler [6].

Theorem 16. Let be an open subset of , , , and . Suppose that is a sequence of real valued continuous functions defined on such that for all with a -limit a continuous function on
The set is a subset of with
Let and for all ; be a bounded sequence. For all , we have where for all

Then, the sequence converges uniformly on to where for each is the unique solution on of the initial value problem

Proof. Let be the set of all real valued continuous functions on with graph lying in and with Lipschitz constant Then, with the supremum norm is a compact Banach space. For each define For each , where and is a contraction mapping from to for all . By Proposition 12, is also an enriched contraction mapping on . For each , , Since is the -limit of , the sequence of integrands converges to zero and is uniformly bounded by The Lebesgue-bounded convergence theorem guarantees that the sequence of integrals on the R.H.S. goes to 0 as Therefore is the -limit of on Now, by Proposition 4 of [26], the -limit is equivalent to the pointwise limit. It is easy to see that is uniformly continuous on for each , and hence, the sequence is equicontinuous on the compact set Therefore, the sequence converges uniformly to on Hence, the sequence converges pointwise to on Since is a compact Banach space, the sequence has a convergent subsequence converging to . By Theorem 11, is a fixed point of . From Theorem 9, the sequence , where is the unique fixed point of for each , converges to the fixed point of . The result follows since these fixed points are the unique solutions of the initial value problem.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.