Best approximation, invariant measures, and fixed points
Let \((X,\|\cdot\|)\) be a Banach space, \(M\) be a closed convex subset of \(X\) and \(p\in X\setminus M.\) For a given semigroup \(S\) let \({\mathfrak{S}}=\{T_s: s\in S\}\) be a family of nonexpansive mappings \(T_s:M\cup\{p\}\rightarrow M\cup\{p\},\) such that \(T_s\circ T_t=T_{ts}\), \(t,s\in S.\) Under additional conditions the authors show that the set \(B_M(p)\) of best approximants of \(p\) to \(M\) is nonvoid and that \({\mathfrak {S}}\) has a common fixed point in \(B_M(p).\) The result generalizes a theorem of \textit{M. A. Al-Thagafi} [J. Approximation Theory 85, No. 3, 318-323 (1996; Zbl 0858.41022)], in terms of weakly (weak*) compactivity instead of compactity of some sets.
scientific article; zbMATH DE number 1552975