Sufficient conditions for weak continuity and weak virtual stability in banach spaces

Abstract

We give some conditions that guarantee the weak continuity and the weak virtual stability for mappings in Banach spaces by introducing the notions of functionally lipschitzian mappings and functionally uniformly lipschitzian mappings. We also show that, for finite-dimensional Banach spaces, functionally (uniformly) lipschitzian and (uniformly) lipschitzian conditions are equivalent. Some criteria for being functionally (uniformly) lipschitzian in infinite-dimensional Banach spaces are also given