# zbMATH — the first resource for mathematics

On a minimax problem of Ricceri. (English) Zbl 0986.49003
Summary: Let $$E$$ be a real separable and reflexive Banach space, $$X\subseteq E$$ weakly closed and unbounded, $$\Phi$$ and $$\Psi$$ two non-constant weakly sequentially lower-semicontinuous functions defined on $$X$$, such that $$\Phi+ \lambda\Psi$$ is coercive for each $$\lambda\geq 0$$. In this setting, if $\sup_{\lambda\geq 0} \inf_{x\in X} (\Phi(x)+ \lambda(\Psi(x)+ \rho))= \inf_{x\in X} \sup_{\lambda\geq 0} (\Phi(x)+ \lambda(\Psi(x)+ \rho))$ for every $$\rho\in\mathbb{R}$$ then one has $\sup_{\lambda\geq 0} \inf_{x\in X} (\Phi(x)+ \lambda\Psi(x)+ h(\lambda))= \inf_{x\in X} \sup_{\lambda\geq 0} (\Phi(x)+ \lambda\Psi(x)+ h(\lambda))$ for every concave function $$h: [0,+\infty[\to \mathbb{R}$$.

##### MSC:
 49J35 Existence of solutions for minimax problems
Full Text: