Generalized KKM-Type Theorems for Best Proximity Points

Abstract

Abstract. This paper is concerned with best proximity points for multimaps in normed spaces and in hyperconvex metric spaces. Using the generalized KKM theorem, we deduce new best proximity pair theorems for a family of multimaps with unionly open fibers in normed spaces. And we prove a new best proximity point theorem for quasi-lower semicontinuous multimaps in hyperconvex metric spaces.

KKM-type theorems for best proximal points in normed linear space

“…The author [1] generalized R-KKM maps in [4] as follows: For I = {1, · · · , n}, let (A, B i ) be a proximal pair of a normed linear space M , A be convex, and B := I B i . The map F : B…”

“…Recently the author [1] showed that R-KKM maps are generalized KKM maps, applied R-KKM property to the map defined on product spaces, and proved new best proximal point theorems in normed linear spaces using the generalized KKM theorem.…”

“…In this paper, we just generalize the R-KKM theorem of [4] to that for the intersectionally closed-valued maps and prove fixed points of best proximity pair theorem for maps with unionly open fibers in normed linear spaces in [1] with different method. We also prove fixed points of best proximity pair theorems for a family of maps with open fibers in normed linear spaces.…”