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 supz∈Z{J(z) + x − z }, 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 ∈ X for which the problem (x, J )-sup has a solution is a dense Gδ-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 z0 ∈ Z such that J(z0)+ x −z0 =supz∈Z{J(z)+ x −z } is a σ-porous subset of X and the set of all points x ∈ X \ Z0 such that there exists a maximizing sequence of the problem (x, J )-sup which has no convergent subsequence is a σ-porous subset of X \ Z0, where Z0 denotes the set of all z ∈ Z such that z is in the solution set of (z, J )-sup. © 2006 Elsevier Inc. All rights reserved.