Nonlinear Functional Analysis and Applications

Kyungnam University, Department of Mathematics Eduaction (경남대학교 수학교육과)
권호 표지
DOI QR코드

DOI QR Code

SETVALUED MIXED QUASI-EQUILIBRIUM PROBLEMS WITH OPERATOR SOLUTIONS

  • Received : 2021.04.27
  • Accepted : 2022.01.06
  • Published : 2022.03.15
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we introduce and study generalized mixed operator quasi-equilibrium problems(GMQOEP) in Hausdorff topological vector spaces and prove the existence results for the solution of (GMQOEP) in compact and noncompact settings by employing 1-person game theorems. Moreover, using coercive condition, hemicontinuity of the functions and KKM theorem, we prove new results on the existence of solution for the particular case of (GMQOEP), that is, generalized mixed operator equilibrium problem (GMOEP).

Keywords

Acknowledgement

The authors would like to thanks all the anonymous referees for their valuable comments and suggestions which have been proved helpful for the improvement of the paper.

References

  1. I. Ahmad, S.S. Irfan and R. Ahmed, Generalized composite vector equilibrium problem, Bull. Math. Anal. Appl., 9(1) (2017), 109-122.
  2. R. Ahmad, S.S. Irfan, M. Ishtyak and M. Rahman, Existence of solutions for operator mixed vector equilibrium problem, J. Ineq. Special Funct., 8(5) (2017), 66-74.
  3. E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, J. Math. Student, 63(1-4) (1994), 123-145.
  4. X.P. Ding, W.K. Kim and K.K. Tan, Equilibria of non-compact generalized game with L*-majerized preferences, J. Math. Anal. Appl., 164 (1992), 508-517. https://doi.org/10.1016/0022-247x(92)90130-6
  5. X.P. Ding, W.K. Kim and K.K. Tan, Equilibria of non-compact generalized game with L-majerized correspondences, Int. J. Math. Math. Sci., 17 (1994), 783-790. https://doi.org/10.1155/S0161171294001092
  6. A. Domokos and J. Kolumban, Variational inequalities with operator solutions, J. Global. Optim., 23 (2002), 99-110. https://doi.org/10.1023/A:1014096127736
  7. K. Fan, A minimax inequality and applications, in Inequalities III, Shisha, pp. 103-113, Academic Press 1972.
  8. S.S. Irfan, I. Ahmed, P. Shukla, Z.A. Khan and M. Aslam, Generalized implicit vector equilibrium problem, Commu. Appl. Nonlinear Anal., 28(3) (2021), 71-82.
  9. S.S. Irfan, M.F. Khan and R.U. Verma, On generalized equilibrium problem, Adv. Nonlinear Var. Ineq., 19(2) (2016), 14-26.
  10. K.R Kazmi and A. Raouf, A class of operator equilibrium problems, J. Math. Anal. Appl., 308 (2005), 554-564. https://doi.org/10.1016/j.jmaa.2004.11.062
  11. K.R. Kazmi, A variational principle for vector equilibrium problem, Proc. Indian Acad. Sci., (Math. Sci.) 111 (2001), 465-470. https://doi.org/10.1007/BF02829618
  12. K.R. Kazmi, On vector equilibrium problems, Proc. Indian Acad. Sci. (Math Sci.) 110 (2000), 213-223. https://doi.org/10.1007/BF02829492
  13. J.K. Kim and A. Raouf, A class of generalized operator equilibrium problems, Filomat 31(1) (2017), 1-8. https://doi.org/10.2298/FIL1701001K
  14. J.K. Kim and Salahuddin, Existence of solutions for multi-valued equilibrium problems, Nonlinear Funct. Anal. Appl., 23(4) (2018), 779-795. https://doi.org/10.22771/NFAA.2018.23.04.14
  15. S. Kum and W.K. Kim, Generalized vector variational and quasi-variational inequalities with operator solutions, J. Glob. Optim., 32 (2005), 581-595. https://doi.org/10.1007/s10898-004-2695-6
  16. S. Kum, A variant of generalized vector variational inequality with operator solutions, Commun. Korean Math. Soc., 21 (2006), 665-676. https://doi.org/10.4134/CKMS.2006.21.4.665
  17. S. Kum and W.K. Kim, Applications of generalized variational and quasi variational inequalities with operator solutions in a TVS, J. Optim. Theory Appl., 133 (2007), 65-75. https://doi.org/10.1007/s10957-007-9175-3
  18. G.M. Lee, D.S. Kim and B.S. Lee, On non-cooperative vector equilibrium, Indian J. Pure Appl. Math., 27 (1996), 735-739.
  19. L. Qun, Generalized vector variational-like inequalities, In Vector variational inequalities and vector-equilibria, Nonconvex Optim. Appl., 38 Kluwer Acad. Publ. Dordrecht 2000, 363-369. https://doi.org/10.1007/978-1-4613-0299-5_21
  20. O.K. Oyewole and O.T. Mewomo, Existence results for new generalized mixed equilibrium and fixed point problems in Banach spaces, Nonlinear Funct. Anal. Appl., 25(2) (2020), 273-301. https://doi.org/10.22771/NFAA.2020.25.02.06
  21. T. Ram, On existence of operator solutions of generalized vector quasi-variational inequalities, Commun. Optim. Theory, 2015 (2015), Article ID 1.
  22. T. Ram and A.K. Khanna, On generalized weak operator quasi equilibrium problems, Global J. Pure Appl. Math., 13(8) (2017), 4189-4198.
  23. T. Ram, P. Lal and J.K. Kim Operator solutions of generalized equilibrium problems in Hausdorff topological vector spaces, Nonlinear Funct. Anal. Appl., 24(1) (2019), 61-71.
  24. J. Salamon, On operator equilibrium problems, Math. Commun., 19 (2014), 581-587.