New existence theorems for vector-valued functions with essential vectorial semicontinuity and adjustability convexity

In this paper, the new concepts of essential vectorial lower semicontinuity, essential vectorial upper semicontinuity and essential vectorial continuity are introduced and their properties and auxiliary results are established.We present new existence theorems for C-adjustability convex vector-valued functions in the weakly compact setting and apply them to obtain new fixed point theorem and existence theorems for root-finding problem, eigenvector problem and minimization problem.