Preserving concavity in initial-boundary value problems of parabolic type and in its numerical solution

Abstract

The notion of a weakly absolute extensor for the class of bicompacts is intro- duced in [5]. In this paper, the notion of a weakly absolute retract for bicompacts is introduced. It is shown that the class of weakly absolute retracts for bicompacts coincides with the class of weakly absolute extensors for bicompacts. 1. I n t r o d u c t i o n Here "compact Hausdorff space" means a bicompact space [5], "mapping" means a continuous function [3]. Let us recall that a subset A of a topological space X is functionally closed (resp. functionally open) [2] if there exists a mapping f : X --* I such that A = f l ( 0 ) (resp. A = f l ( 0 , 1 ] ) . Let us recall that a bicompact X is a weakly absolute extensor for bicompacts (abbreviated WAE for bicompacts) [5], if for every bicompact B and every closed subspace A of it, every mapping f : A ---, X can be extended over some functionally closed subspace W of B which contains A. A bicompact X is an absolute neighbourhood extensor for bicompacts (abbreviated ANE for bicompacts) [3], if for every bicompact B and every closed subspace A of it, every mapping f : A --* X can be extended over some open subspace U of B which contains A. A bicompact X is an absolute neighbourhood retract for bicompacts (abbreviated ANR for bicompacts) [3], if given a bicompact Y having a closed subspace Y0 homeomorphic to X, then Y0 is a neighbourhood retract of Y. A subspace A of a space X is said to be an L-retract of X [6], if A is a retract of a functionally closed subspace W of X which contains A. Mathematics subject classification numbers, 1991. Primary 54C55, 54C99; Sec- ondary 54B17, 54C15, 54C50.