STABILITY RESULTS FOR EKELAND EPSILON-VARIATIONAL PRINCIPLE AND CONE EXTREMAL SOLUTIONS

Given X a Banach space and f: X --> R or {+infinity} a proper lower semicontinuous function which is bounded from below, the Ekeland's epsilon-variational principle asserts the existence of a point xBAR in X, which we call epsilon-extremal with respect to f, which satisfies f(u) > f (xBAR)- epsilon\\u - xBAR\\ for all u is-an-element-of X, u not-equal xBAR. By using set convergence notions (Kuratowski-Painleve, Mosco, bounded Hausdorff) and their epigraphical versions we study the (semi) continuity properties of the mapping which to f associates epsilon-ext f the set of such epsilon-extremal points. The key for the geometrical understanding of such properties is to consider the equivalent Phelps extremization principle which, given a closed set D in X and a partial ordering with respect to a pointed cone, associates the set of elements of D maximal with respect to this order. Direct or potential applications are given in various fields (multicriteria optimization, numerical algorithmic, calculus of variations).