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

摘要

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 z(0) is an element of Z such that parallel to x - z(0)parallel to(p) + J(z(0)) = inf(z is an element of Z){parallel to X - Z parallel to(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 is an element of X for which every minimizing sequence of the problem min(J)(x, Z) has a converging subsequence is a dense G(delta)-subset of X Z(0), where Z(0) is the set of all points z is an element of Z such that z is a solution of the problem min(J)(z, Z). If additionally p > I and X is J-strictly convex with respect to Z, then the set of all x is an element of X for which the problem min(J)(x, Z) is well-posed is a dense G(delta)-subset of X Z(0). (C) 2008 Elsevier Inc. All rights reserved.