  In this paper, we establish coincidence-like results in the case when the
values of the correspondences are not convex. In order to do this, we define a
new type of correspondences, namely properly quasi-convex-like. Further, we
apply the obtained theorems to solve equilibrium problems and to establish a
minimax inequality. In the last part of the paper, we study the existence of
solutions for generalized vector variational relation problems. Our analysis is
based on the applications of the KKM principle. We establish existence theorems
involving new hypothesis and we improve the results of some recent papers.
