Existence and porosity for a class of perturbed optimization problems in Banach spaces

Let X be a Banach space and Z a nonempty closed subset of X. Let J : Z -> R be an upper semicontinuous function bounded from above. This paper is concerned with the perturbed optimization problem sup(z is an element of Z){J(z) + parallel to x - z parallel to}, which is denoted by (x, J)-sup. We shall prove in the present paper that if Z is a closed boundedly relatively weakly compact nonempty subset, then the set of all x E X for which the problem (x, J)-sup has a solution is a dense G(delta)-subset of X. In the case when X is uniformly convex and J is bounded, we will show that the set of all points x in X for which there does not exist z(0) is an element of Z such that J(z(0)) + parallel to x - z(0)parallel to = sup(z is an element of Z){J(z) + parallel to x - z parallel to} is a sigma-porous subset of X and the set of all points x is an element of X \ Z(0) such that there exists a maximizing sequence of the problem (x, J)-sup which has no convergent subsequence is a sigma-porous subset of X \ Z(0), where Z(0) denotes the set of all z is an element of Z such that z is in the solution set of (z, J)-sup. (c) 2006 Elsevier Inc. All rights reserved.