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 X-0 is a convex subset in X and X-0 has approximatively Z-property (K), then the set of all points x in X-0 \ Z for which there exists z(0) is an element of Z such that J(z(0)) + parallel tox - z(0)parallel to = rho(x) and every sequence {z(n)} subset of Z satisfying lim(n-->infinity)[J(Zn) + parallel tox -Z(n)parallel to] = rho(x) forx contains a subsequence strongly convergent to an element of Z is a dense G(delta)-subset of X-0 \ Z. Moreover, under the assumption that X-0 is approximatively Z-strictly convex, we show more, namely that the set of all points x in X-0 \ Z for which there exists a unique point Z(0) is an element of Z such that J(z(0)) + parallel tox - z(0)parallel to = rho(x) and every sequence (Z(n) subset of Z satisfying lim(n-->infinity)[J(z(0)) + parallel tox - z(0)parallel to = rho(x) for x converges strongly to z(0) is a dense G(delta)-subset of X-0 \ Z. Here rho(x) = inf{J(z) + parallel tox - zparallel to; z is an element of Z}. These extend S. Cobzas's result [J. Math. Anal. Appl. 243 (2000) 344-3561. (C) 2004 Published by Elsevier Inc.