© Tong-Huei Chang et al. 2009
Received: 25 February 2009
Accepted: 11 June 2009
Published: 16 June 2009
1. Introduction and Preliminaries
In 1929, Knaster et al.  proved the well-known theorem for an -simplex. Ky Fan's generalization of the theorem to infinite dimensional topological vector spaces in 1961  proved to be a very versatile tool in modern nonlinear analysis with many far-reaching applications.
Chang and Yen  undertook a systematic study of the property, and Chang et al.  generalized this property as well as the notion of a family of  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.  where the classes of and - mappings have been introduced in the framework of abstract convex spaces. The authors of  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.  and Horvath , motivated by the works of Ky Fan  and Browder . We refer for the study of these notions to Ben-El-Mechaiekh et al. , and more recently, to Park , and Kim and Park .
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].
We list some properties of a uniform space. A uniformity  for a set is a nonempty family of subsets of such that
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.
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.  introduced the class of multifunctions with the property in abstract convex spaces.
Definition 1.7 (see ).
The -mappings and the -spaces, in an abstract convex space setting, were also introduced by Amini et al. .
Definition 1.10 (see ).
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
2. Main Results
Using the above introduced concepts and definitions, we now state our main theorem.
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 .
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, .
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.
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 .
This allows us to state the following results.
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.
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 .
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