A strict minimax inequality criterion and some of its consequences

Abstract

In this paper, we point out a very flexible scheme within which a strict minimax inequality occurs. We then show the fruitfulness of this approach presenting a series of various consequences. Here is one of them: Let Y be a finite-dimensional real Hilbert space, J : Y → R a C1 function with locally Lipschitzian derivative, and \({\varphi : Y \to [0, + \infty[}\) a C1 convex function with locally Lipschitzian derivative at 0 and \({\varphi^{-1}(0) = \{0\}}\) . Then, for each \({x_0 \in Y}\) for which J′(x0) ≠ 0, there exists δ > 0 such that, for each \({r \in ]0, \delta[}\) , the restriction of J to B(x0, r) has a unique global minimum ur which satisfies $$J(u_r)\leq J(x)-\varphi(x-u_r)$$ for all \({x \in B(x_0, r)}\) , where \({B(x_0, r) = \{x \in Y : \|x-x_0\|\leq{r}\}.}\)