Well Posedness of Generalized Mutually Minimization 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. Let B(X) be the family of nonempty bounded closed subset of X endowed with the Hausdorff distance. A generalized mutually minimization problem min_C(F,G) is said to be well posed if it has a unique solution (x. z) and every minimizing sequence converges strongly to (x. z). Under the assumption that C is both strictly convex and Kadec, G is a nonempty closed, relatively boundedly weakly compact subset of X, using the concept of the admissible family M of B(X) , we prove the generic result that the set E of all subsets F such that the generalized mutually minimization problem minC(F,G) is well posed is a dense subset of M. These extend and sharpen some recent results due to De Blasi, Myjak and Papini, Li, Li and Xu, and Ni, etc.