On a minimax problem of Ricceri. (English) Zbl 0986.49003

Cordaro, Giuseppe

J. Inequal. Appl. 6, No. 3, 261-285 (2001).

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

Keywords:

minimax problem; concave function; weak coerciveness; weakly sequentially lower-semicontinuity