Characterization of nonlinear coapproximation in normed linear spaces and its application (Q2772890)

Let \(X\) be a normed linear space, \(G\) a nonvoid subset of \(X\) and \(f\in X \backslash \overline{G}\). An element \(g_{f}\in G\) is called an element of best coapproximation of \(f\) by elements in \(G\) if \(\left\|g-g_{f}\right\|\leq\left\|f-g\right\|\) for every \(g\in G\). The set of all best coapproximation elements to \(f\) in \(G\) is denoted by \(R_{G}\left( f\right) \). The element \(g_{f}\in G\) is called a strongly unique best coapproximation to \(f\) from \(G\) if there exists \(K_{f}>0\) such that \(\left\|f-g\right\|\geq\left\|g_{f}-g\right\|+K_{f}\left\|f-g_{f}\right\|\) for every \(g\in G\). Finally, it is called a cosun point of \(G\) if for any \(f\in X \backslash \overline{G}\) such that \(g_{f}\in R_{G}\left( f\right) \) we have \(g_{f}\in R_{G}\left( g_{f}+\alpha\left( f-g_{f}\right) \right) \) for all \(\alpha\geq 0\). The author proves theorems of characterizations for the above defined elements of \(G\) and applies the general results to the spaces \(C_{R}\left( \Omega\right) \), \(L_{1}\left( T,\nu\right) \) and to 2 - normed spaces.