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Best proximity pair theorems for multifunctions with open fibres. (English) Zbl 0965.41020
Let $A$ and $B$ be non-empty subsets of a normed linear space $E$, and let $T:A\to 2^B$ be a convex multi-valued function with open fibres $T^{-1}(y)$ (i.e.) $\{x\in X:y\in Tx\}$. For an element $x_0\in A$ sufficient conditions are found so that $\text{dist}(x_0, Tx_0)= \text{dist}(A,B)$. This is the case if, say, $A$ is a non-empty approximately compact, and convex proximinal subset of $E$, and $B$ is a non-empty, closed and convex subset of $E$, and $A_0$ is compact, while $T(A_0)\subset B_0$. Here $A_0=\{a\in A: \text{dist}(a,b)= \text{dist}(A,B)$ for some $b\in B\}$. Consequences include special cases of the Brouwer’s fixed point theorem.

MSC:
41A65Abstract approximation theory
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References:
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