On controlled extensions of functions

For any numerical function E :R2→R we give sufficient conditions for resolving the controlled extension problem for a closed subset A of a normal space X. Namely, if the functions f :A→R, g :A→R and h:X →R satisfy the equality E(f (a), g(a)) = h(a), for every a ∈ A, then we are interested to find the extensions fˆ and gˆ of f and g, respectively, such that E(fˆ(x), gˆ(x)) = h(x), for every x ∈ X. We generalize earlier results concerning E(u, v) = u · v by using the techniques of selections of paraconvex-valued LSC mappings and soft single-valued mappings.  2003 Elsevier Inc. All rights reserved.