Well-posedness of a class of perturbed optimization problems in Banach spaces

Let X be a Banach space and Z a nonempty subset of X. Let J : Z → R be a lower semicontinuous function bounded from below and p 1. This paper is concerned with the perturbed optimization problem of finding z0 ∈ Z such that x − z0 p + J (z0) = infz∈Z { x − z p + J (z)}, which is denoted by min J (x, Z). The notions of the J -strictly convex with respect to Z and of the Kadec with respect to Z are introduced and used in the present paper. It is proved that if X is a Kadec Banach space with respect to Z and Z is a closed relatively boundedly weakly compact subset, then the set of all x ∈ X for which every minimizing sequence of the problem min J (x, Z) has a converging subsequence is a dense Gδ-subset of X \ Z0, where Z0 is the set of all points z ∈ Z such that z is a solution of the problem min J (z, Z). If additionally p > 1 and X is J -strictly convex with respect to Z, then the set of all x ∈ X for which the problem min J (x, Z) is well-posed is a dense Gδ-subset of X \ Z0.