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: Z --> R a lower semicontinuous function bounded from below. If X is reflexive and has the Kadets property then the set of all x is an element of X for which there exists z(0) is an element of Z such that J(z(0)) + parallel to x - z(0)parallel to = inf{J(z) + parallel to x - z parallel to:z is an element of Z} is G(delta) and dense in X. This extends a result of K-S. Lau (1978, Indiana Univ. Math. J., 27, 791-795) on nearest paints (the case J = 0). (C) 2000 Academic Press.