Fixed Point Theorems for Multivalued Nonself -Almost Contractions in Banach Spaces Endowed with Graphs

Abstract

In this paper, we prove some fixed point theorems for multivalued nonself -almost contractions in Banach spaces with a directed graph and give some examples to illustrate our main results. The main results in this paper extend and generalize many known results in the literature therein.

1. Introduction

Fixed point theory plays a very important role in nonlinear analysis, economics, and applications. The important tool for solving existence problems in many branches of mathematics and applied sciences is Banach contraction principle [1]. It has been generalized in many directions by many authors ([2–15]).

On the other hand, the fixed point theory for nonself mappings has been studied by many authors ([3–5]). In 2013, Berinde and Păcurar [5] introduced the new type of contraction for nonself single valued mappings.

Definition 1. Let be a nonempty subset of a normed space . A mapping is said to be almost contractive if there exist and such that for all

By the above definition, they proved some fixed point theorem for almost contractive mappings in Banach spaces.

Later, Alghamdi et al. [4] extended the almost contractive mappings to a multivalued nonself almost contractions.

Definition 2. Let be a complete convex metric space and a nonempty subset of . A mapping is said to be a multivalued almost contraction if there exist and such that for all , where is the Pompeiu-Hausdorff metric induced by .

They proved a fixed point theorem for multivalued nonself almost contraction on convex metric spaces.

Let be a directed graph, where is a set of vertices of graph and is a set of its edges. Assume that has no parallel edges. We denote by the directed graph obtained from by reversing the direction of edges: that is,

Let and be two vertices in ; a path in from to of length is a sequence of vertices such that , , for each . A graph is said to be connected if there exists a (directed) path between any two vertices of . We denote

On the other hand, in 2008, Jachymski [6] extended Banach contraction principle by introducing the notion of -contraction and proved a fixed point theorem for -contractions in a metric space endowed with a directed graph. These results have been generalized by some authors in several ways (see [7–14]).

Motivated by the previous results, we introduced the concept of multivalued nonself -almost contractions in normed spaces and establish some fixed point theorems for this this type of mappings in normed spaces endowed with a directed graph. We give some examples to illustrate our main results.

2. Preliminaries

In this section, we give some basic and useful definitions and known results that are useful for the main results in this paper.

Let be a normed space. We denote by the class of all nonempty closed and bounded subsets of . For any , define the function by where

Note that is called the Pompeiu-Hausdorff metric induced by metric .

The following lemma is very useful for our main results.

Lemma 3 (see [2]). Let be a metric space. If and , then, for each , there exists such that .

Let be a normed space, a nonempty closed subset of , and a nonself mapping. If with , then we can always choose an such that , where and , which expresses the fact that We denote

The following property was introduced by Jachymski [6]. It is very useful for our main results.

Property A (see [7]). With any sequence in , if and for all , then for any .

In order to obtain our main results, we need the following definition of domination in graphs [16, 17].

Let be a directed graph. A set is called a dominating set if, for all , there exists such that and we say that dominates or is dominated by . For any , a set is dominated by if for any and we say that dominates if for all .

3. Main Results

In this section, we prove some fixed point theorems for multivalued nonself -almost contractions in a Banach space endowed with a directed graph. First, we recall some definitions which are useful for our main results.

Definition 4. Let be a nonempty subset of a normed space and be a directed graph such that . A mapping is said to be -almost contraction(1)if there exist and with such that, for any , whenever ,(2)if and are such that ; then .

First, we prove a fixed point theorem for a -almost contraction in a Banach space endowed with a directed graph.

Theorem 5. Let be a nonempty closed subset of a Banach space and be a directed graph such that . Let be a -almost contraction. Suppose that(1) satisfies Rothe’s boundary condition: that is, ,(2) has Property A,(3)there is such that for some (4) is dominated by for all with .Then has a fixed point in .

Proof. Let be such that , where . If , we denote : that is, . If , there exists such that Since is dominated by , we obtain . If then . Suppose ; we can choose such that By Lemma 3, we can choose such that Hence, . If , we denote . If , then there exists such that If then . Suppose ; we can choose with such that By Lemma 3, we can choose such that Hence, . By induction, we obtain two sequences and such that (i);(ii), where(a) if ,(b) whenever , and then is such that (c) is a sequence of positive integers with .Suppose that Note that and for all . Moreover, if , then both and belong to the set . Since and are not in the set for all , we cannot have two consecutive terms of in the set .
Now, we claim that is a Cauchy sequence. We conclude that there are three possibilities.
Case  1. If , then and . Since is a -almost contraction and , we have Case  2. If and , then and : that is, Since is a -almost contraction and , we have Case  3. If and , then and such that Moreover, and . Since is a -almost contraction, we have We denote , so Therefore, combining Cases 1, 2, and 3, we obtain for all . By inductivity, it follows that, for any , Now, for any , we obtain This implies that is a Cauchy sequence. Since is closed and , there exists such that . Let be an infinite subsequence of . Since has Property A, for all . Then we have Letting , we obtain and hence : that is, is a fixed point of . This completes the proof.

As a consequence of Theorem 5, by putting , we obtain the following corollary.

Corollary 6. Let be a nonempty closed subset of a Banach space and be an almost contractive mapping with Rothe’s boundary condition. Then has a fixed point.

Remark 7. (i) In Theorem 5, if we take , we obtain Theorem  9 of Alghmdi et al. [4] in Banach spaces.
(ii) In Theorem 5, if we take and , we obtain the existence result which is similar to Berinde and Păcurar’s result [5].
(iii) In Theorem 5, if we take and , we obtain the existence result which is similar to Jachymski’s result [5] in Banach spaces.
(iv) In Theorem 5, if we take and , we obtain the existence result which is similar to Nadler’s theorem [5] in Banach spaces.

Next, we give an example to illustrate Theorem 5.

Example 8. Let and with the usual norm and be such that , Notice that has Property A. Define a mapping by We see that the only with is , and we see that and ; hence, is dominated by . By direct computation, we can show that is -almost contraction with and . Moreover, we see that which implies satisfies Rothe’s boundary condition. Choose , we see that . Therefore, all conditions of Theorem 5 are satisfied and we see that .

Competing Interests

The authors declare no competing interests.

Acknowledgments

The authors would like to thank the Thailand Research Fund under Project RTA5780007 and Chiang Mai University, Chiang Mai, Thailand, for the financial support. The first author was supported by Chiang Mai Rajabhat University.