2012/01/07 by Biagio Ricceri, Ricceri, Biagio
Computer Science · Mathematics · #Optimization and Variational Analysis #Mathematical Inequalities and Applications #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1201.1574
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→ \bf R a C1 function with locally Lipschitzian derivative, and φ:Y→ [0,+∞[ a C1 convex function with locally Lipschitzian derivative at 0 and φ-1(0)=\0\. Then, for each x0∈ Y for wich J'(x0)≠ 0, there exists δ>0 such that, for each r∈ ]0,δ[, the restriction of J to B(x0,r) has a unique global minimum ur which satisfies J(ur)≤ J(x)-φ(x-ur) for all x∈ B(x0,r), where B(x0,r)=\x∈ Y: ‖x-x0‖≤ r\ .