Well Posedness of Generalized Mutually Maximization Problem

Let C be a closed bounded convex subset of a Banach space X with 0 being an interior point of C and p_C (.) be the Minkowski functional with respect to C. A generalized mutually maximization problem max_C (F, G) is said to be well posed if it has a unique solution (x, z) and every maximizing sequence converges strongly to (x, z). Under the assumption that C is both strictly convex and Kadec, G is a nonempty closed, bounded relatively weakly compact subset of X, using the concept of the admissible family D of B (X), we prove the generic result that the set E of all subsets F (in D) such that the generalized mutually maximization problem max_C (F, G) is well posed is a residual subset of D. These extend and sharpen some recent results due to De Blasi, Myjak and Papini, Li, Li and Ni, Li and Xu, and Ni, etc.