Stability results for Ekeland's ε variational principle for vector valued functions

摘要

In this paper, under the assumption that the nonconvex vector valued function f satisfies some lower semicontinuity property and bounded below, the nonconvex vector valued function sequence f_n satisfies the same lower semicontinuity property and uniformly bounded below, and f_n converges to f in the generalized sense of Mosco, we obtain the relation: ε~(1/2) - ext f = {x-bar: f(x) - f(x-bar) + ε~(1/2)||x - x-bar||e is not an element of - C, when x ≠ x-bar} is contained in lim_(n→∞) ε~(1/2) - ext f_n, where ε~(1/2)-ext f_n = {x-bar: f_n(x) - f_n(x-bar) + ε~(1/2)||x - x-bar||e is not an element of - C, where x ≠ x-bar}, C is the pointed closed convex dominating cone with nonempty interior int C, e ∈ int C. Under some conditions, we also prove the same result when f_n converges to f in the generalized sense of Painleve'-Kuratowski.