Fixed Point Theorems for Contractive Selfmappings of a Bounded Metric Space

Abstract

The main purpose of this paper is to prove a new fixed theorem for selfmapping of a metric space . As applications, we get a new fixed point result for shrinking or contractive maps and a fixed point theorem for a new class of weakly contractive selfmappings of a bounded metric space , where the auxiliary function satisfies and .

1. Introduction

Let be a mapping of a metric space . It is well known that is called a shrinking or a contractive map if it satisfies the inequality for each with . In [1], V. V. Nemytzk was the first mathematician who studied the problem of the existence of a fixed point of these mappings. Furthermore, it is mentioned in [2] that, to obtain a fixed point of such mappings, it is necessary either to add the assumption that there exists a point for which contains a convergent subsequence, or else to assume that the space is compact.

In [3], the authors introduced the notion of weakly contractive mappings in Hilbert spaces and proved that any weakly contractive mapping defined on Hilbert spaces has a unique fixed point. Rhoades [4] proved that the corresponding result is also valid when Hilbert space is replaced by a complete metric space. Recall that a mapping of a metric space is said to be weakly contractive if , for all , where is a continuous nondecreasing function such that . Since then, weakly contractive mappings have been dealt with in a number of papers. Some of these works are noted in [1, 5–8].

On the other hand, the authors [9] have introduced the concept of -distance functions in a general topological space and mentioned that metric spaces, symmetric spaces, probabilistic metric spaces, and topological vector spaces have all such functions. Moreover, they presented an application of this new concept to the fixed point theory by giving the following result (Corollary 4.1 [9]) which generalizes the well-known Banach’s fixed point theorem as follows

Theorem 1 (Corollary 4.1 [9]). Let be a Hausdorff topological space with a -distance . Suppose that is p-bounded and S-complete. Let be a mapping satisfying: there exists such that for all , we have .
Then has a unique fixed point.

Recall that a sequence in X is p-Cauchy if it satisfies the usual metric condition with respect to p. The definition of a p-bounded S-complete space is presented in Definition 3.1 [9] as follows.

Definition 2 (Definition 3.1 [9]). Let be a topological space with a -distance p. (1)X is S-complete if for every p-Cauchy sequence , there exists x in X with .(2)X is p-Cauchy complete if for every p-Cauchy sequence , there exists x in X with with respect to .(3)X is said to be p-bounded if .

For more information, we refer the reader to [9]. Our purpose in this paper is to present a new fixed point result for shrinking maps and a fixed point theorem for a new class of weakly contractive selfmappings of a bounded metric space by using Theorem 1, where the auxiliary function satisfies and .

2. Main Results

Theorem 3. Let be a mapping of a bounded complete metric space such that . Then has a unique fixed point.

Proof. Let . It is clear that, for all , one has , and, therefore , where .
Let us consider the function defined by As mentioned in [9] (Example 2.4.), the function p is a -distance on where is the usual metric topology.
On the other hand, the mapping satisfies on the following contraction: According to Corollary 4.1. in [9], we deduce that has a unique fixed point in .

Corollary 4. Let be a shrinking mapping of a bounded complete metric space such that . Then has a unique fixed point.

Example 5. Let with the usual metric . Define by and . We haveThen satisfies all assumptions of Corollary 4 and has the unique fixed point which is equal to .

Example 6. Let , the unit closed ball of a real Banach space, with the metric .
Let us consider the mapping defined by , for all .
Then is a shrinking mapping of the complete bounded metric space andTherefore satisfies all assumptions of Corollary 4 and has the unique fixed point which is equal to .

Remarks 7. Obviously, for a shrinking mapping of a metric space , one can ask does there exist a relationship between compactness and the condition . The answer is negative. Indeed, in the first example, the space is compact and . However, for and , the mapping is shrinking and . Furthermore, in the second example, the space is not compact and .

Definition 8. Let be a mapping of a metric space . will be said a E-weakly contractive maps if , for all , where is a function satisfying and .

As a second application of Theorem 3., we get the following new fixed point for E- weakly contractive selfmappings of a bounded metric space .

Theorem 9. Let be a E-weakly contractive mapping of a bounded complete metric space . Then has a unique fixed point.

Proof. From the Definition 8., it is clear that, for all , we havewhich implies that . According to Theorem 3. the mapping has a unique fixed point in .

Example 10. Let with the usual metric . Define and by , and .
Then satisfies all assumptions of Theorem 9 and has the unique fixed point . Note that is not continuous at .

Example 11. Let with the usual metric . Define and by , , and for all .
Then is a bounded complete metric space and for all . However, is not a E-weakly contractive since and has no fixed point. Therefore the condition that is essential.

Example 12. Let with the usual metric . Define and by and for all , and w here is the derived function of .
Then is a E-weakly contractive map with no fixed point on since is a complete unbounded metric space and .

3. Application

In this section, we investigate the existence and uniqueness of a solution for the nonlinear integral equation:where and the space of all continuous functions from into , with .

is a continuous mapping and is a given function.

Letting endowed by the metric defined by obviously is a complete metric space.

Consider the mapping defined as follows:for any .

Note that (6) has a solution if and only if has a fixed point.

Under the above assumptions we have the following theorem.

Theorem 13. If there exists such that for all and such that . Then the nonlinear integral equation (6) has a unique solution.

Proof. Assuming that and , then we have hencefor all Then , which implies by Theorem 3 that there exists a unique solution of the nonlinear equation (6).

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.