A strict minimax inequality criterion and some of its consequences

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, $J:Y\to \mathbf{R}$ a $C^1$ function with locally Lipschitzian derivative, and $\phi :Y\to [0,+\mathrm{\infty }[$ a $C^1$ convex function with locally Lipschitzian derivative at 0 and ${\phi }^{-1}(0)=\{0\}$. Then, for each $x_0\in Y$ for wich $J^{\mathrm{\prime }}(x_0)\mathrm{\ne }0$, there exists $\delta >0$ such that, for each $r\in ]0,\delta [$, the restriction of $J$ to $B(x_0,r)$ has a unique global minimum $u_r$ which satisfies $$J(u_r)\le J(x)-\phi (x-u_r)$$ for all $x\in B(x_0,r)$, where $B(x_0,r)={x\in Y: |x-x_0|\leq r}\ .$