Applications of Results on Abstract Convex Spaces to Topological Ordered Spaces

Abstract

Abstract. Topological semilattices with path-connected intervals are special abstract convex spaces. In this paper, we obtain generalized KKM type theorems and their analytic formulations, maximal element theorems and collectively fixed point theorems on abstract convex spaces. We also apply them to topological semilattices with path-connected intervals, and obtain generalized forms of the results of Horvath and Ciscar, Luo, and Al-Homidan et al..

Some properties of quasi-convex functions on abstract convex spaces

“…Motivated by Kassay and Kolumbán [3], generalized KKM maps on abstract convex spaces is defined in [4] as follows: Let (X, D; Γ) be an abstract convex space and I be a nonempty set. A map T : I X is called a generalized KKM map provided that for each N ∈ I , there exists a function σ :…”

“…The following equivalency of certain concavity of extended real functions and the related generalized KKM maps for abstract convex spaces is in [4]; Proposition 3.4. Let I be a nonempty set, (X, D; Γ) be an abstract convex space, f : I × X → R be a function, and γ ∈ R. Then the followings are equivalent:…”

Generalized KKM-Type Theorems for Best Proximity Points

“…The following KKM type theorem is due to the author [7]; Theorem 2.1. Let B be a nonempty set, (X, D; Γ) be an abstract convex space satisfying the partial KKM principle, and F : B ⊸ X be a multimap satisfying (1) F is a generalized KKM map.…”

“…Recently the author [7] proved the generalized KKM theorem in abstract convex spaces. The concept of abstract convex spaces is a far-reaching generalization of convex structures and it was introduced by Park [13].…”

“…In Section 2, using the generalized KKM theorem in [7] and the fact that R-KKM maps are generalized KKM maps, we shall deduce new best proximity pair theorems for a family of multimaps with unionly open fibers in normed spaces. These Theorems generalize Theorem 1.1.…”