Generic solutions for some perturbed optimization problem in non-reflexive Banach spaces

Let Z be a closed, boundedly relatively weakly compact, nonempty subset of a Banach space X, and J:Z→R a lower semicontinuous function bounded from below. If X0 is a convex subset in X and X0 has approximatively Z-property (K), then the set of all points x in X0⧹Z for which there exists z0∈Z such that J(z0)+‖x−z0‖=ϕ(x) and every sequence {zn}⊂Z satisfying limn→∞[J(zn)+‖x−zn‖]=ϕ(x) for x contains a subsequence strongly convergent to an element of Z is a dense Gδ-subset of X0⧹Z. Moreover, under the assumption that X0 is approximatively Z-strictly convex, we show more, namely that the set of all points x in X0⧹Z for which there exists a unique point z0∈Z such that J(z0)+‖x−z0‖=ϕ(x) and every sequence {zn}⊂Z satisfying limn→∞[J(zn)+‖x−zn‖=ϕ(x) for x converges strongly to z0 is a dense Gδ-subset of X0⧹Z. Here ϕ(x)=inf{J(z)+‖x−z‖;z∈Z}. These extend S. Cobzas's result [J. Math. Anal. Appl. 243 (2000) 344-356].