Technical note on the existence of solutions for generalized symmetric set-valued quasi-equilibrium problems utilizing improvement set. (English) Zbl 1518.49021
Summary: 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\({}_\Delta\) 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 [M. Fakhar and J. Zafarani, J. Optim. Theory Appl. 136, No. 3, 397–409 (2008; Zbl 1148.91010); A. P. Farajzadeh et al., Bull. Iran. Math. Soc. 45, No. 1, 35–58 (2019; Zbl 1411.49006); J.-Y. Fu, J. Math. Anal. Appl. 285, No. 2, 708–713 (2003; Zbl 1031.49013); Y. Han et al., Pac. J. Optim. 14, No. 1, 31–49 (2018; Zbl 1474.49037)].
MSC:
| 49J53 | Set-valued and variational analysis |
| 49K40 | Sensitivity, stability, well-posedness |
| 90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |
| 91B50 | General equilibrium theory |
Keywords:
symmetric set-valued quasi-equilibrium problem; nonlinear scalarization; improvement set; Kakutani-Fan-Glicksberg fixed point theorem; maximal element theoryReferences:
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