Minimax theorems for limits of parametrized functions having at most one local minimum lying in a certain set. (English) Zbl 1101.49008
Summary: We establish some minimax theorems, of purely topological nature, that, through the variational methods, can be usefully applied to nonlinear differential equations. Here is a (simplified) sample: Let \(X\) be a Hausdorff topological space, \(I\subseteq\mathbb R\) an interval and \(\Psi:X\times I\to]-\infty,+\infty]\). Assume that the function \(\Psi(x,\cdot)\) is lower semicontinuous and quasi-concave in \(I\) for all \(x\in X\), while the function \(\Psi(\cdot,q)\) has compact sublevel sets and one local minimum at most for each \(q\) in a dense subset of \(I\). Then, one has
\[
\sup_{q\in I} \inf_{x\in X} \Psi(x,q)= \inf_{x\in X} \sum_{q\in I} \Psi(x,q).
\]
MSC:
49J35 | Existence of solutions for minimax problems |
90C47 | Minimax problems in mathematical programming |
54C30 | Real-valued functions in general topology |
54D05 | Connected and locally connected spaces (general aspects) |
References:
[1] | Ricceri, B., Some topological mini-max theorems via an alternative principle for multifunctions, Arch. Math. (Basel), 60, 367-377 (1993) · Zbl 0778.49008 |
[2] | Ricceri, B., On a topological minimax theorem and its applications, (Ricceri, B.; Simons, S., Minimax Theory and Applications (1998), Kluwer Academic: Kluwer Academic Dordrecht), 191-216 · Zbl 0953.49008 |
[3] | Ricceri, B., A further improvement of a minimax theorem of Borenshtein and Shul’man, J. Nonlinear Convex Anal., 2, 279-283 (2001) · Zbl 1045.49002 |
[4] | Ricceri, B., Sublevel sets and global minima of coercive functionals and local minima of their perturbations, J. Nonlinear Convex Anal., 5, 157-168 (2004) · Zbl 1083.49004 |
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