KKM-type theorems for best proximity points

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

“…We also prove fixed points of best proximity pair theorems for a family of maps with open fibers in normed linear spaces. These theorems generalize the results of [4].…”

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We generalize the R-KKM theorem in [4] to that for the intersectionally closed-valued maps and using this, we prove best proximity pair theorems for multimaps with (unionly) open fibers in normed linear spaces which generalize the results of [4].

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“…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.…”

“…The following notations are used in the sequel. Sankar Raj and Somasundaram [4] introduced R-KKM maps as follows: Let (A, B) be a proximal pair of a normed linear space M and A be convex.…”

Generalized KKM-Type Theorems for Best Proximity Points

“…Since an R-KKM map is a generalized KKM map, we obtain the following theorem which generalizes Theorem 3.2 in [14]; Theorem 2.2. Let A be a nonempty convex subset of a normed space M , and B i be a nonempty subset of M for each i ∈ I := {1, .…”