Open Access

Fixed Point Theorems for a Weaker Meir-Keeler Type -Set Contraction in Metric Spaces

Fixed Point Theory and Applications20092009:129124

https://doi.org/10.1155/2009/129124

Received: 25 March 2009

Accepted: 19 June 2009

Published: 14 July 2009

Abstract

We define a weaker Meir-Keeler type function and establish the fixed point theorems for a weaker Meir-Keeler type -set contraction in metric spaces.

1. Introduction and Preliminarie

In 1929, Knaster et al. [1] had proved the well-known theorem on -simplex. Besides, in 1961, Fan [2] had generalized the theorem to an infinite dimensional topological vector space. Later, Amini et al. [3] had introduced the class of -type mappings on metric spaces and established some fixed point theorems for this class. In this paper, we define a weaker Meir-Keeler type function and establish the fixed point theorems for a weaker Meir-Keeler type -set contraction in metric spaces.

Throughout this paper, by we denote the set of all real nonnegative numbers, while is the set of all natural numbers. We digress briefly to list some notations and review some definitions. Let and be two Hausdorff topological spaces, and let be a set-valued mapping. Then is said to be closed if its graph is closed. is said to be compact if the image of under is contained in a compact subset of . If is a nonempty subset of , then denotes the class of all nonempty finite subsets of . And, the following notations are used:

(i) ,

(ii) ,

(iii) and

(iv) .

Let be a metric space, and . Let , and let .

Suppose that is a bounded subset of a metric space . Then we define the following

(i) , and

(ii) is said to be subadmissible [3], if for each , .

In 1996, Chang and Yen [4] introduced the family on the topological vector spaces and got results about fixed point theorems, coincidence theorems, and its applications on this family. Later, Amini et al. [3] introduced the following concept of the property on a subadmissible subset of a metric space .

Let be an nonempty subadmissible subset of a metric space , and let a topological space. If are two set-valued mappings such that for any , , then is called a generalized mapping with respect to . If the set-valued mapping satisfies the requirement that for any generalized mapping with respest to , the family has finite intersection property, then is said to have the property. The class is denoted to be the set has the property .

Recall the notion of the Meir-Keeler type function. A function is said to be a Meir-Keeler type function (see [5]), if for each , there exists such that for with , we have .

We now define a new weaker Meir-Keeler type function as follows.

Definition 1.1.

We call a weaker Meir-Keeler type function, if for each , there exists such that for with , and there exists such that .

A function is said to be upper semicontinuous, if for each , . Recall also that is said to be a comparison function (see [6]) if it is increasing and . As a consequence, we also have that for each , , and , is continuous at . We generalize the comparison function to be the other form, as follows.

Definition 1.2.

We call a generalized comparison function, if is upper semicontinuous with and for all .

Proposition 1.3.

If is a generalized comparison function, then there exists a strictly increasing, continuous function such that , for all .

Proof.

Let . Since is an upper semicontinuous function, hence it attains its minimum in any closed bounded interval of .

For each , we first define four sequences , and as follows:

(i) ,

(ii) ,

(iii) ,

(iv) for , and

(v) for .

And, we next let a function satisfy the following:

(1)

(2)if   , then
(1.1)
(3)if   , then
(1.2)

Then by the definition of the function , we are easy to conclude that is strictly increasing, continuous. We complete the proof by showing that for all .

If , then
(1.3)
If , then
(1.4)

So for all .

Since and for all , so for all

Proposition 1.4.

If is a generalized comparison function, then there exists a strictly increasing, continuous function such that
(1.5)

Proof.

By Proposition 1.3, there exists a strictly increasing, continuous function such that , for all . So, we may assume that , by letting for all .

Remark 1.5.

In the above case, the function is invertible. If for each , we let and for all , then we have that ; that is, .

Proof.

We claim that , for . Suppose that for some positive real number . Then
(1.6)

which is a contradiction. So .

We now are going to give the axiomatic definition for the measure of noncompactness in a complete metric space.

Definition 1.6.

Let be a metric space, and let the family of bounded subsets of . A map
(1.7)

is called a measure of noncompactness defined on if it satisfies the following properties:

(i) if and only if is precompact, for each ,

(ii) , for each ,

(iii) , for each ,

(iv) , for each .

The above notion is a generalization of the set measure of noncompactness in metric spaces. The following -measure is a well-known measure of noncompactness.

Definition 1.7.

Let be a complete metric space, and let the family of bounded subsets of . For each , we define the set measure of noncompactness by:
(1.8)

Definition 1.8.

Let be a nonempty subset of a metric space . If a mapping with for each , and are bounded, then is called

(i)a -set contraction, if for each , , where ,

(ii)a weaker Meir-Keeler type -set contraction, if for each , , where is a weaker Meir-Keeler type function,

(iii)a generalized comparison (comparison) type -set contraction, if for each , , where is a generalized comparison (comparison) function.

Remark 1.9.

It is clear that if is a -set contraction, then is a weaker Meir-Keeler type -set contraction, but the converse does not hold.

2. Main Results

Using the conception of the weaker Meir-Keeler type function, we establish the following important theorem.

Theorem 2.1.

Let be a nonempty bounded subadmissible subset of a metric space . If is a weaker Meir-Keeler type -set contraction with for each , is nonicreasing, then contains a precompact subadmissible subset with .

Proof.

Take , and let
(2.1)

Then

(1) is a subadmissible subset of , for each ;

(2) , for each .

Since is a weaker Meir-Keeler type -set contraction, then and . Hence, we conclude that .

Since is nonincreasing, it must converge to some with ; that is, . We now claim that . On the contrary, assume that .Then by the definition of the weaker Meir-Keeler type function, there exists such that for each with , there exists such that .Since , there exists such that , for all . Thus, we conclude that . So we get a contradiction. So , and so .

Let . Then is a nonempty precompact subadmissible subset of , and by (2), we have .

Remark 2.2.

In the process of the proof of Theorem 2.1, we call the set a Meir-Keeler type precompact-inducing subadmissible subset of .

Applying Proposition 1.3, 1.4, and Remark 1.5, we are easy to conclude the following corollary.

Corollary 2.3.

Let be a nonempty bounded subadmissible subset of a metric space . If is a generalized comparison (comparison) type -set contraction, then contains a precompact subadmissible subset with .

Proof.

The proof is similar to the proof of Theorem 2.1; we omit it.

Remark 2.4.

In the process of the proof of Corollary 2.3, we also call the set a generalized comparison type precompact-inducing subadmissible subset of .

Corollary 2.5.

Let be a nonempty bounded subadmissible subset of a metric space . If is a -set contraction, then contains a precompact subadmissible subset with .

Following the concepts of the family (see [3]), we immediately have the following Lemma 2.6.

Lemma 2.6.

Let be a nonempty subadmissible subset of a metric space , and let a topological spaces. Then whenever , and is a nonempty subadmissible subset of .

We now concern a fixed point theorem for a weaker Meir-Keeler type -set contraction in a complete metric space, which needs not to be a compact map.

Theorem 2.7.

Let be a nonempty bounded subadmissible subset of a metric space . If is a weaker Meir-Keeler type -set contraction with for each , is nonicreasing, and closed with , then has a fixed point in .

Proof.

By the same process of Theorem 2.1, we get a weaker Meir-Keeler type precompact-inducing subadmissible subset of . Since and for each , we have for each . Since as , by the above Lemma 2.6, we have that is a nonempty compact subset of .

Since and is a nonempty subadmissible subset of , by Lemma 2.6, .

We now claim that for each , there exists an such that . If the above statement is not true, then there exists such that , for all . Let . Then we now define by
(2.2)

Then

(1) is compact, for each , and

(2) is a generalized mapping with respect to .

We prove (2) by contradiction. Suppose is not a generalized mapping with respect to . Then there exists such that
(2.3)

Choose and such that . From the definition of , it follows that , for each . Since , , we have , which implies that . Therefore, . This contradicts to . Hence, is a generalized mapping with respect to .

Since , the family has the finite intersection property, and so we conclude that . Choose , then for all . But, since and , so there exists an such that . So, we have reached a contradiction.

Therefore, we have proved that for each , there exists an such that . Let . Since and is compact, we may assume that converges to some , then also converges to . Since is closed, we have . This completes the proof.

Corollary 2.8.

Let be a nonempty bounded subadmissible subset of a metric space . If is a generalized composion type -set contraction and closed with , then has a fixed point in .

Corollary 2.9.

Let be a nonempty bounded subadmissible subset of a metric space . If is a -set contraction and closed with , then has a fixed point in .

The -spaces, in an abstract convex space setting, were introduced by Amini et al.[7]. An abstract convex space consists of a nonempty topological space and a family of subsets of such that and belong to and is closed under arbitrary intersection. Let be an abstract convex space, and let a topological space. A map is called a -mapping if there exists a multifunction such that

(i)for each , implies ;

(ii) .

The mapping is called a companion mapping of . Furthermore, if the abstract convex space which has a uniformity and has an open symmetric base family , then is called a -space if for each entourage , there exists a -mapping such that . Following the conceptions of the abstract convex space and the -space, we are easy to know that a bounded metric space is an important example of the abstract convex space, and if and , then is also a -space.

Applying Theorem of Amini et al. [7], we can deduce the following theorem in metric spaces.

Theorem 2.10.

Let be a nonempty subadmissible subset of a metric space . If is compact, then for each , there exists such that .

Proof.

Consider the family of all subadmissible subsets of and for each , , we set . Let
(2.4)

Then is a basis of a uniformity of . For each , we define two set-valued mappings by for each . Then we have

(i)for each , ;

(ii) .

So, is a companion mapping of . This implies that is a -mapping such that . Therefore, is a -space.

Now we let be an identity mapping, all of the the conditions of Theorem of Amini et al. [7] are fulfilled, and we can obtain the results.

Applying Theorems 2.1 and 2.10, we can conclude the following fixed point theorems.

Theorem 2.11.

Let be a nonempty bounded subadmissible subset of a metric space . If is a weaker Meir-Keeler type -set contraction with for each is noincreasing, and closed with , then has a fixed point in .

Theorem 2.12.

Let be a nonempty bounded subadmissible subset of a metric space . If is a generalized comparison (comparison) type -set contraction and closed with , then has a fixed point in .

Authors’ Affiliations

(1)
Department of Applied Mathematics, National Hsinchu University of Education

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Copyright

© C.-M. Chen and T.-H. Chang. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.