Best proximity pair theorems for multifunctions with open fibres (Q1971649)

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.