Existence of solutions of set-valued equilibrium problems in topological vector spaces with applications

Abstract

In this paper, we first introduce the new notions of $\overline{{\mathbb {R}}}_+$-local-intersection property, $\overline{{\mathbb {R}}}_+$-local-inclusion property, $\overline{{\mathbb {R}}}_+$-strongly transfer lower semicontinuity, ${\mathbb {R}}_-$-weakly transfer lower semicontinuity and ${\mathbb {R}}_-$-strongly transfer lower semicontinuity for a set-valued mapping $\Phi$ in topological vector spaces. Then, using these notions, we characterize the existence of set-valued equilibrium without assuming any form of convexity of function and/or convexity and compactness of sets. Furthermore, we apply the basic results obtained in the paper to Browder variational inclusion, with weekend conditions on the involved set-valued operators and provide an affirmative answer to some open question posed by Alleche and Rǎdulescu in their final remark of (Optim Lett, 2018. https://doi.org/10.1007/s11590-1233-2). Some application of our results to characterize the existence of set-valued pure strategy Nash equilibrium in games with discontinuous and non-convex payoff functions and nonconvex and/or noncompact strategy spaces is presented. Finally, we characterize the existence of single-valued equilibrium problem without assuming any form of convexity of function and/or convexity and compactness of sets in the setting of topological vector spaces. Our results improve and generalize many known results in the current literature.