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‖=?(x) and every sequence {z_n}⊂Z satisfying lim_n→∞J(z_n)+‖x−z_n‖]=?(x) for x contains a subsequence strongly convergent to an element of Z is a dense G_δ-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‖=?(x) and every sequence {z_n}⊂Z satisfying lim_n→∞J(z_n)+‖x−z_n‖=?(x) for x converges strongly to z₀ is a dense G_δ-subset of X₀?Z. Here . These extend S. Cobzas's result J. Math. Anal. Appl. 243 (2000) 344-356].