On controlled extensions of functions. (Q1408774)

The aim of the present paper is to offer conditions for resolving the controlled extension problem for a closed subset \(A\) of a normal space \(X\) given any numerical functions \(E: \mathbb{R}^2\to \mathbb{R}\). More precisely, if the functions \(f: A\to \mathbb{R}\), \(g: A\to \mathbb{R}\) and \(h: X\to\mathbb{R}\) satisfy the equality \(E(f(a),g(a))= h(a)\), for every \(a\in A\), then the authors are interested in finding the extensions \(\widehat f\) and \(\widehat g\) of \(f\) and \(g\), respectively, such that \(E(\widehat f(x),\widehat g(x))= h(x)\), for every \(x\in X\). They also give interesting generalizations of earlier results concerning \(E(u,v)= u\cdot v\) by using the techniques of selections of paraconvex-valued LSC mappings and soft single-valued mappings.