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

Abstract

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${}_{\mathrm{\Delta }}$ 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].

Keywords:

• Symmetric set-valued quasi-equilibrium problem
• nonlinear scalarization
• improvement set
• Kakutani–Fan–Glicksberg fixed point theorem
• maximal element theory

2000 Mathematics Subject Classifications:

• 49K40
• 90C33
• 91B50

Acknowledgments

The authors would like to thank the editor and anonymous referees for valuable comments which helped improve the paper. The work of the first author was completed during his visit to the Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong, to which he is grateful to the hospitality received.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author was partially supported by the Chongqing Natural Science Foundation [grant number cstc2021jcyj-msxmX0080], the Group Building Scientific Innovation Project for Universities in Chongqing [grant number CXQT21021] and the Education Committee Project Foundation of Bayu Scholar; The third author was supported by PolyU grant G-UAHF and the Guangdong Basic and Applied Basic Research Foundation [grant number 2020A1515010463]; The fourth author was supported by the National Natural Science Foundation of China [grant numbers 12071307].