Continuity of usual operations and variational convergences
Given two variationally convergent sequences of functions \(\{f_{n} \}\) and \(\{g_{n}\}\) under what assumptions does the sequence \(\{f_{n} \intercal g_{n} \}\) also variational converge (in the same sense), where \(\intercal\) is a binary operation? This fundamental question has been investigated by many authors but mainly for the case of convex functions. The main goal of this paper is to study this question in the absence of convexity assumption, whenever possible. In doing so various asymptotic conditions are introduced within the context of general Banach spaces endowed with various additional topologies. The goal of rejecting convexity is abandoned in the final section where sums and suprema of functions are studied and convexity assumptions are required. This paper addresses a wide variety of convergence notions. These include Mosco convergence and various hybrid versions of bounded hemi--convergence (we take one half of the epi-distance convergence and marry it with the other half of epi--convergence). The operations studied are also numerous, including the general performance functions, infimal images of functions and a continuous linear operator, the infimal convolution of two functions, images of convergent sequences of sets composed with a continuous linear mapping and sums of two convergent sequences of sets. Conditions are placed on the sequential asymptotic functions for various topologies (these are the largest functions containing in their epigraph the sequential asymptotic cone of the epigraph of the original function). These conditions along with certain sequential compactness conditions are imposed in order to resolve closure issues (i.e., is the infimal convolution of the two limiting functions exact, etc).
scientific article; zbMATH DE number 1971090