Technical note on the existence of solutions for generalized symmetric set-valued quasi-equilibrium problems utilizing improvement set.

In this paper, we establish some existence results for the solution of the generalized symmetric set-valued quasi-equilibrium problem (GSSQEP). The new forms of GSSQEP via improvement set and scalar generalized symmetric set-valued quasi-equilibrium problem (GSSQEP Δ for short) are also introduced. By using Kakutani–Fan–Glicksberg fixed point method, maximal element principle and nonlinear scalarization technique, we develop two classes of sufficient conditions for the existence of solutions to GSSQEP. The drawback of some existing work for this problem is overcome. Moreover, some applications to the symmetric set-valued equilibrium problem (SSEP), symmetric vector quasi-equilibrium problem (SVQEP) and the set-valued equilibrium problem (SEP) are also given in this paper. Our results improve a few existing ones in Fakhar and Zafarani [Generalized symmetric vector quasiequilibrium problems. J Optim Theory Appl. 2008;136:397–409], Farajzadeh et al. [On the existence of solutions of symmetric vector equilibrium problems via nonlinear scalarization. Bull Iran Math Soc. 2019;45:35–58], Fu [Symmetric vector quasi-equilibrium problems. J Math Anal Appl. 2003;285:708–713], Han et al. [Existence of solutions for symmetric vector set-valued quasi-equilibrium problems with applications. Pacific J Optim. 2018;14:31–49]. [ABSTRACT FROM AUTHOR]