Bounded convergence for perturbed minimization problems

Abstract

Bounded convergence results are established for sequences of convex functions of the form F(· ,0). As particular cases, one obtains convergence results for functions of the following types: F(· ,T(·)), f+g○ T, max {f,g○ T}, g○ T, f + g, max {f,g}. In the case of the sum we obtain the well-known theorem of Beer and Lucchetti as well as the recent result obtained by Eberhard and Wenczel (and reduce to that last result when the space is complete).