An equilibrium version of set-valued Ekeland variational principle and its applications to set-valued vector equilibrium problems
Let \(X\) be a topological space, \(Y\) a Hausdorff topological vector space ordered by a closed cone \(K\) and \(f:X\to 2^Y\setminus\{\emptyset\}.\) The author considers various semicontinuity and boundedness properties of \(f\), expressed in terms of the topologies of the spaces \(X,Y\) and of the order induced by the cone \(K\), and proves several Ekeland-type variational principles for these kinds of maps. He also considers set-valued bimaps \(f:X\times X\to 2^Y\setminus\{\emptyset\}\) satisfying a so-called \(K\)-sequential lower monotony (\(K\)-slm) condition: if \(x_n\to \bar x\) in \(X\) is such that \(f(x_n,x_{n+1})\subset -K\) for all \(n\in\mathbb{N}\), then \(f(x_n,\bar x)\subset -K\) for all \(n\in\mathbb{N}\). An equilibrium version of the Ekeland variational principle, extending one proved by \textit{X.-H. Gong} [Acta Math. Sci., Ser. B, Engl. Ed. 34, No. 4, 1179--1192 (2014; Zbl 1324.49014)], is proved and applied to equilibrium problems, i.e., the existence of \(\bar x\in X\) s.t. \(f(\bar x,y)\nsubseteq -\mathrm{int}(K)\) for all \(y\in X\). Stronger results are obtained in the case when \(X\) is a Fréchet space (i.e., a complete metrizable locally convex space). The obtained results extend those obtained for vector mappings \(f:X\to Y\) and, at the same time, improve some set-valued variational and equilibrium results.
scientific article; zbMATH DE number 6707181
scientific article