Summary: 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 B. Alleche and V. D. Rădulescu in their final remark of [ibid. 12, No. 8, 1789–1810 (2018; Zbl 1403.26004)]. 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.