Equivalents of an approximate variational principle for vector-valued functions and applications

Abstract.

In [1], we gave a unified variational principle for vector valued functions. In this paper, we give four equivalents of a corollary of this principle, generalizing the equivalence results of [3] to the vector case and improving the equivalence results of [5]. We also applied one of the equivalents to derive the vector form of the “Drop Theorem” ([3]). Applying the unified variational principle, we obtained a fixed point theorem for directional contractions as an application of our vector variational principle, generalizing a fixed point theorem for directional contractions in [9].