Generic existence of solutions for some perturbed optimization problems (Q1974232)

This paper is concerned with the generic existence of solutions for some perturbed optimization problems containing as a particular case the problem of nearest points. A Banach space \(X\) satisfies the \textit{Kadets property} if \[ x_n@>\omega>> x\text{ (weakly) and }\|x_n\|\to\|x\|\text{ imply }x_n@>\|\;\|>> x\text{ (strongly)} \] for any sequences \((x_n)\) in \(X\) and \(x\in X\). The main result of this paper is the following: Theorem. Let \(X\) be a reflexive Banach space satisfying the Kadets property, let \(Z\) be a nonvoid closed subset of \(X\), and let \(J: Z\to\mathbb{R}\) be a lower-semicontinuous function bounded from below. Then the set of all \(x\in X\) for which the problem: \[ \text{find }z_0\in Z\text{ such that }J(z_0)+\|x- z_0\|= \inf\{J(z)+\|x-z\|;\;z\in Z\} \] has a solution, is \(G_\delta\) and dense in \(X\). This theorem extends a result of \textit{K.-S. Lau} [Indiana Univ. Math. J. 27, 791-795 (1978; Zbl 0398.41026)] on nearest points (the case \(J=0\)). Lau formulated his result for a reflexive locally uniformly convex Banach space (locally uniformly convex Banach spaces satisfy the Kadets property).