On nonlinear coapproximation in Banach spaces (Q2779080)

Let \(G\) be a non-empty subset of a real Banach space \(X\) and \(f\in X\). An element \(g_f\in G\) is calledNEWLINENEWLINENEWLINE(i) a best coapproximation (best approximation) from \(G\) to \(f\) if \(\|g_f- g\|\leq \|f-g\|\) (if \(\|f- g_f\|\leq\|f-g\|\)) for all \(g\in G\). The set of best coapproximations (best approximations) to \(f\) from \(G\) is denoted by \(R_G(f)\) \((P_G(f))\).NEWLINENEWLINENEWLINE(ii) A cosun point of \(G\) if \(g_f\in R_G(f)\) implies \(g_f\in R_G(g_f+ \alpha(f- g_f))\) for every \(\alpha> 0\).NEWLINENEWLINENEWLINEIf every point in \(G\) is a cosun point of \(G\), then \(G\) is said to be a cosun. For \(f\in X\overline G\), an element \(g_f\in R_G(f)\) is called a strong coapproximation to \(f\) from \(G\) if there exists a constant \(c>0\) such that \(\|f-g\|\geq\|g_f- g\|+ c\|f- g_f\|\) for all \(g\in G\).NEWLINENEWLINENEWLINEIn the second section of the paper best coapproximations and cosun points are characterized. A counterexample is given to the following result of \textit{C. Moore} [Nonlinear Anal. Theory Methods Appl. 37 B, No. 1, 125-138 (1999; Zbl 0931.47037)].NEWLINENEWLINENEWLINEProposition 2.1. Let \(G\) be a proximinal subset (every element of the space has a best approximation in \(G\)) of a real Banach space \(X\). Then for any \(f\in X\) and \(g\in G\), \(\|g_f- g\|\leq\|f- g\|\) \(\forall g_f\in p_G(f)\).NEWLINENEWLINENEWLINEIt is shown that Proposition 2.1 is not true in general, even for a subspace in a uniformly convex and uniformly smooth space but is true if \(X\) is a Hilbert space and \(G\) is convex. It is also shown (Theorem 2.4) that if \(G\) is a nonempty cone with vertex \(g_f\) in a Hilbert space \(X\) then \(g_f\in R_G(f)\) if and only if \(g_f\in P_G(f)\).NEWLINENEWLINENEWLINESecion 3 deals with the characterization of strong coapproximations. In Section 4, the authors consider applications in two particular normed spaces \(L(T,\mu)\) and \(C(\Omega)\), give a counterexample to Theorems 3.2 and 4.3 of \textit{G. S. Rao} and \textit{R. Saravanan} [Approximation Theory Appl. 15, No. 1, 23-37 (1999; Zbl 0938.41017)], and give the correct results (Theorems 4.3 and 4.4).