On a generalized sup-inf problem

In this paper, necessary and sufficient conditions for solvability of nonlinear inequality systems are given using certain generalized convexity concepts. Our results imply some theorems of Kirszbraun, Fan, Minty, Simons, Sebestyén, and Gwinner-Oettli.The authors are grateful to the referees for their helpful comments. They thank one of the referees, who emphasized the connection between the Wald minimax theorem and Theorem 2.1, and suggested an alternative proof of Theorem 2.1 in Section 4.