Generic Existence of Solutions for Some Perturbed Optimization Problems

Let X be a real Banach space, Z a closed nonvoid subset of X, and J: ZªR a lower semicontinuous function bounded from below. If X is reflexive and has the Kadets property then the set of all xgX for which there exists z0gZ such that J z0.q5xyz05sinf J z.q5xyz5: zgZ4 is Gd and dense in X. This extends a result of K.-S. Lau 1978, Indiana Uni¨. Math. J., 27, 791]795.on nearest points the case Js0..