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

Abstract: Let X be a Banach space and Z a nonempty closed subset of X. Let be an upper semicontinuous function bounded from above. This paper is concerned with the perturbed optimization problem , which is denoted by -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 for which the problem -sup has a solution is a dense -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 such that is a σ-porous subset of X and the set of all points such that there exists a maximizing sequence of the problem -sup which has no convergent subsequence is a σ-porous subset of , where denotes the set of all such that z is in the solution set of -sup. [Copyright &y& Elsevier]