Open Access

Approximate Fixed Point Theorems for the Class of Almost - Mappings in Abstract Convex Uniform Spaces

Fixed Point Theory and Applications20092009:791514

https://doi.org/10.1155/2009/791514

Received: 25 February 2009

Accepted: 11 June 2009

Published: 16 June 2009

Abstract

We use a concept of abstract convexity to define the almost - property, al- - family, and almost -spaces. We get some new approximate fixed point theorems and fixed point theorems in almost -spaces. Our results extend some results of other authors.

1. Introduction and Preliminaries

In 1929, Knaster et al. [1] proved the well-known theorem for an -simplex. Ky Fan's generalization of the theorem to infinite dimensional topological vector spaces in 1961 [2] proved to be a very versatile tool in modern nonlinear analysis with many far-reaching applications.

Chang and Yen [3] undertook a systematic study of the property, and Chang et al. [4] generalized this property as well as the notion of a family of [4] to the wider concepts of the - property and its related - family.

Among the many contributions in the study of the property and related topics, we mention the work by Amini et al. [5] where the classes of and - mappings have been introduced in the framework of abstract convex spaces. The authors of [5] also define a concept of convexity that contains a number of other concepts of abstract convexities and obtain fixed point theorems for multifunctions verifying the - property on -spaces that extend results of Ben-El-Mechaiekh et al. [6] and Horvath [7], motivated by the works of Ky Fan [2] and Browder [8]. We refer for the study of these notions to Ben-El-Mechaiekh et al. [9], and more recently, to Park [10], and Kim and Park [11].

In this paper, we use a concept of abstract convexity to define the almost - property, the corresponding notion of almost - family as well as the concept of almost -spaces.

Let and be two sets, and let be a set-valued mapping. We will use the following notations in the sequel;

(i)

(ii)

(iii)

(iv) and

(v)if is a nonempty subset of , then denotes the class of all nonempty finite subsets of .

For the case where and are two topological spaces, a set-valued map 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 .

Definition 1.1.

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. This kind of abstract convexity was widely studied; see [5, 9, 12, 13].

Suppose that is a nonempty subset of an abstract convex space . Then

(i)a natural definition of -convex hull of is
(1.1)

(ii)we say that is -convex if for each , .

Remark 1.2.

It is clear that if , then is -convex. That is, each member of is -convex.

Definition 1.3.

We list some properties of a uniform space. A uniformity [14] for a set is a nonempty family of subsets of such that

(i)each member of contains the diagonal where the diagonal denotes the set of all pairs for in ;

(ii)if , then ;

(iii)if , then for some ;

(iv)if , then ;

(v)if and , then .

The pair is called a uniform space. Every member in is called an entourage. An entourage is said to be symmetric if whenever .

Definition 1.4.

If is an abstract convex space with a uniformity , then we say that is an abstract convex uniform space.

Definition 1.5.

Let be a nonempty subset of an abstract convex uniform space which has a uniformity , and has a symmetric basis . Then is called almost -convex if, for any and for any , there exists a mapping such that for all and . Moreover, we call the mapping a -convex-inducing mapping.

Remark 1.6.

It is clear that every -convex set must be almost -convex, but the converse is not true. And in general, the -convex-inducing mapping is not unique. If and , then can be regarded as . If , then can be regarded as .

Recently, Amini et al. [5] introduced the class of multifunctions with the property in abstract convex spaces.

Definition 1.7 (see [5]).

Let be a nonempty set, an abstract convex space, and a topological space. If , and are three multifunctions satisfying
(1.2)
then is called a - mapping with respect to . If the multifunction satisfies the requirement that for any - mapping with respect to , the family has the finite intersection property where denotes the closure of , then is said to have the - property with respect to . We define
(1.3)

We extended the property to the almost property, as follows.

Definition 1.8.

Let be a nonempty set, let be an almost -convex subset of an abstract convex uniform space which has a uniformity and has a symmetric basis , and let be a topological space. If , and are three multifunctions satisfying for each , each , and each , there exists a -convex-inducing mapping such that
(1.4)
then is called an almost - mapping with respect to . If the multifunction satisfies the requirement that for any almost - mapping with respect to , the family has the finite intersection property, then is said to have the almost - property with respect to . We define
(1.5)

From the above definitions, we have the following proposition of the family.

Proposition 1.9.

Let be a nonempty set, let be an almost -convex subset of an abstract convex uniform space , let and be two topological spaces, and let be a multifunction. If and if is continuous, then

The -mappings and the -spaces, in an abstract convex space setting, were also introduced by Amini et al. [5].

Definition 1.10 (see [5]).

Let be an abstract convex space, and a topological space. map is called a -mapping if there exists a multifunction such that

(i)for each , implies , and

(ii) .

The mapping is called a companion mapping of .

Furthermore, if the abstract convex space which has a uniformity and has a symmetric basis , then is called a -space if for each entourage , there exists a -mapping such that .

Remark 1.11.

(i)If is a -mapping, then for each nonempty subset of , is also a -mapping.

(ii)It is easy to see that if and , then is also a -space.

In order to establish the main result of this paper for the multifunctions with the almost property, we need the following definitions concerning the almost -mappings and the almost -spaces.

Definition 1.12.

Let be an almost -convex subset of an abstract convex uniform space which has a uniformity and has a symmetric base family , and a topological space. A map is called an almost -mapping if there exists a multifunction such that

(i)for each , and , there exists a -convex-inducing such that , and

(ii)

The mapping is called an almost companion mapping of .

Furthermore, is called an almost -space, if, for each entourage , there exists an almost -mapping such that .

Definition 1.13.

Let be an almost -space, and let . We say that has the approximate fixed point property if, for each , there exists such that .

2. Main Results

Using the above introduced concepts and definitions, we now state our main theorem.

Theorem 2.1.

Let be an almost -space, and let be a surjective single-valued function. If is compact, then has the approximate fixed point property.

Proof.

Let be a symmetric basis of the uniform structure, and let . Take such that . Then, by the definition of the almost -space, there exists an almsot -mapping such that . Since is an almsot -mapping, there exists an almost companion mapping such that .

Let . Then is compact, since is compact. Hence there exists such that . Since is surjective, there exists a finite subset of such that .

Now, we define by

(2.1)
By the definition of , we obtain that is not an almost mapping with respect to . Hence, there exist and such that for any -convex-inducing , we have
(2.2)

So, for any -convex-inducing , there exist and such that . Consequently, , and so for all . Since is an almost -mapping, there exists a -convex-inducing such that . So and . Thus, there exists such that . Since is an almost -space, we have , and so , that is, .

Therefore, . The proof is finished.

Remark 2.2.

In the case, if is a -space and , then the above theorem reduces to Amini et al. [5, Theorem 2.5]

From Theorem 2.1 above, we obtain immediately the following fixed point theorem.

Theorem 2.3.

Suppose that all of the assumptions of Theorem 2.1 hold. If is closed, then has a fixed point in .

Proof.

By Theorem 2.1, for each , there exist such that . Since is compact, without loss of generality, we may assume that converges to some in ; then also converges to since is a Hausdorff uniform space and for each . By the closedness of , we have that .

Corollary 2.4.

Let be an almost -space, and let be a surjective single-valued function. Suppose such that is totally bounded. Then has the approximate fixed point property.

Corollary 2.5.

Suppose that all of the assumptions of the above Corollary 2.5 hold. If is closed, then has a fixed point in .

In case is an almost convex subset of Hausdorff topological vector spaces and for each , we have

(i) , and

(ii) .

This allows us to state the following results.

Theorem 2.6.

Let be a Hausdorff locally convex space, let be an almost convex subset of , and let be a surjective function. Assume that is compact and closed, then has a fixed point in .

Proof.

Let be the family of all convex subsets of , and let be a local basis of such that each is symmetric and convex for each . For each , we set . Noting that . Set
(2.3)
Then is a basis of a uniformity of . For each , , we define the two set-valued mappings by for each . Then we have
  1. (i)

    for each , , and

     

(ii) .

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

All conditions of Theorems 2.1 and 2.3 are therefore fulfilled; the result follows from an argument similar to those in the proofs of Theorems 2.1 and 2.3.

Theorem 2.7.

Let be a topological vector space, let be an almost convex subset of , and let be a surjective function. Suppose that is compact, then for any symmetric convex neighborhood of in , there is such that .

Proof.

Let be the family of all convex subsets of , and let be a new local basis of . We will use to construct a weaker topology on such that becomes a new topological vector space. For each , we set . Noting that . Set
(2.4)

Then is a basis of a uniformity of . In vein of the reasonings similar to those of Theorems 2.1 and 2.6, we complete the proof.

Authors’ Affiliations

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

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Copyright

© Tong-Huei Chang et al. 2009

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