Semicontinuous Mappings into T.V.S. with Applications to Mixed Vector Variational-Like Inequalities

Abstract

Let $$\cal L$$ (X, Z) be the space of continuous linear mappings between topological vector spaces, where Z is Hausdorff and preordered by a closed convex cone C. In this paper, we introduce a notion of semicontinuity to any function from a topological space into X. A notion of semicontinuity is also introduced to any function from a topological space into $$\cal L$$ (X, Z). These two notions of semicontinuity are related by the embedding of X into $$\cal L$$ (X, Z). Their basic properties are given. As an application, we derive some existence results for the mixed vector variational-like inequality.