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].  2004 Published by Elsevier Inc