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 J : Z → R be an upper semicontinuous function bounded from above. This paper is concerned with the perturbed optimization problem sup z ∈ 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 G δ -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 ∈ Z such that J ( z 0 ) + ‖ x − z 0 ‖ = sup z ∈ Z { J ( z ) + ‖ x − z ‖ } is a σ-porous subset of X and the set of all points x ∈ X ∖ Z 0 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 ∖ Z 0 , where Z 0 denotes the set of all z ∈ Z such that z is in the solution set of ( z , J ) -sup.

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