Nonlinear Functional Analysis and Applications

The Basic Sciences Research Institute, Kyungnam University (경남대학교 기초과학연구소)
권호 표지
DOI QR코드

DOI QR Code

PARAMETRIC GENERALIZED MULTI-VALUED NONLINEAR QUASI-VARIATIONAL INCLUSION PROBLEM

  • Khan, F.A. (Department of Mathematics, University of Tabuk) ;
  • Alanazi, A.M. (Department of Mathematics, University of Tabuk) ;
  • Ali, Javid (Department of Mathematics, Aligarh Muslim University) ;
  • Alanazi, Dalal J. (Department of Mathematics, University of Tabuk)
  • Received : 2020.08.14
  • Accepted : 2021.04.10
  • Published : 2021.12.15
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we investigate the behavior and sensitivity analysis of a solution set for a parametric generalized multi-valued nonlinear quasi-variational inclusion problem in a real Hilbert space. For this study, we utilize the technique of resolvent operator and the property of a fixed-point set of a multi-valued contractive mapping. We also examine Lipschitz continuity of the solution set with respect to the parameter under some appropriate conditions.

Keywords

Acknowledgement

The authors would like to thank the anonymous referee for his/her comments that helped us to improve this paper.

References

  1. S. Adly, Perturbed algorithms and sensitivity analysis for a general class of variational inclusions, J. Math. Anal. Appl., 201(3) (1996), 609-630. https://doi.org/10.1006/jmaa.1996.0277
  2. R.P. Agarwal, Y.J. Cho and N.J. Huang, Sensitivity analysis for strongly nonlinear quasi-variational inclusions, Appl. Math. Lett., 13 (2002), 19-24.
  3. R.P. Agarwal, N.J. Huang and Y.J. Cho, Generalized nonlinear mixed implicit quasi-variational inclusions with set-valued mappings, J. Inequal. Appl., 7(6) (2002), 807-828.
  4. L. Ciric, Some recent results in metrical fixed point theory, University of Belgrade, Beograd, Serbia, 2003.
  5. S. Dafermos, Sensitivity analysis in variational inequalities, Math. Oper. Res., 13 (1998), 421-434. https://doi.org/10.1287/moor.13.3.421
  6. X.P. Ding, Sensitivity analysis for generalized nonlinear implicit quasi-variational inclusions, Appl. Math. Lett., 17 (2004), 225-235. https://doi.org/10.1016/S0893-9659(04)90036-5
  7. X.P. Ding, Parametric completely generalized mixed implicit quasi-variational inclusions involving h-maximal monotone mappings, J. Comput. Appl. Math., 182 (2005), 252-269. https://doi.org/10.1016/j.cam.2004.11.048
  8. X.P. Ding and C.L. Luo, On parametric generalized quasi-variational inequalities, J. Optim. Theory Appl., 100 (1999), 195-205. https://doi.org/10.1023/A:1021777217261
  9. Y.P. Fang and N.J. Huang, H-monotone operator and resolvent operator technique for variational inclusions, Appl. Math. Comput., 145 (2003), 795-803. https://doi.org/10.1016/S0096-3003(03)00275-3
  10. A. Hassouni and A. Moudafi, A perturbed algorithm for variational inclusion, J. Math. Anal. Appl., 185 (1994), 706-721. https://doi.org/10.1006/jmaa.1994.1277
  11. N.J. Huang, A new completely general class of variational inclusions with noncompact set-valued mappings, Comput. Math. Appl., 35(10) (1998), 9-14. https://doi.org/10.1016/S0898-1221(98)00067-4
  12. K.R. Kazmi and F.A. Khan, Iterative approximation of a solution of multi-valued variational-like inclusion in Banach spaces: A P-η-proximal-point mapping approach, J. Math. Anal. Appl., 325 (2007), 665-674. https://doi.org/10.1016/j.jmaa.2006.02.013
  13. K.R. Kazmi and F.A. Khan, Sensitivity analysis for parametric general set-valued mixed variational-like inequality in uniformly smooth Banach space, Math. Inequal. Appl., 10(2) (2007), 403-415.
  14. K.R. Kazmi and F.A. Khan, Sensitivity analysis for parametric generalized implicit quasi-variational-like inclusions involving P-η-accretive mappings, J. Math. Anal. Appl., 337 (2008), 1198-1210. https://doi.org/10.1016/j.jmaa.2007.01.115
  15. J.K. Kim, Sensitivity analysis for general nonlinear nonconvex set-valued variational inequalities in Banach spaces, J. Comput. Anal. Appl., 22(2) (2017), 327-335.
  16. J.K. Kim, H.Y. Lan and Y.J. Cho, Solution sensitivity of generalized nonlinear parametric (A, η, m)- proximal operator system of equations in Hilbert spaces, J. Inequ. Appl., 2014:362 (2014), doi: 10,1186/1029-242X-2014-362. https://doi.org/10.1186/1029-242X-2014-362
  17. J.K. Kim and Salahuddin, Solution sensitivity for a system of generalized nonlinear equations in Banach spaces, Nonlinear Funct. Anal. Appl., 22(1) (2017), 1-22. https://doi.org/10.22771/NFAA.2017.22.01.01
  18. T.C. Lim, On fixed point stability for set-valued contractive mappings with applications to generalized differential equation, J. Math. Anal. Appl., 110 (1985), 436-441. https://doi.org/10.1016/0022-247x(85)90306-3
  19. Z. Liu, L. Debnath, S.M. Kang and J.S. Ume, Sensitivity analysis for parametric completely generalized nonlinear implicit quasi-variational inclusions, J. Math. Anal. Appl., 277(1) (2003), 142-154. https://doi.org/10.1016/S0022-247X(02)00518-8
  20. V. Manojlovic and V. Vuorinen, On quasiconformal maps with identity boundary values, Trans. Amer. Math. Soc., 363(5) (2011), 2467-2479. https://doi.org/10.1090/s0002-9947-2010-04994-3
  21. A. Mukheimer, J. Vujakovic, A. Hussain, H. Aydi, S. Radenovic and S. Yaqoob, A new approach to multivalued nonlinear weakly Picard operators, J. Inequal. Appl., 2019:288 (2019), doi:org/10.1186/s13660-019-2244-y.
  22. R.N. Mukherjee and H.L. Verma, Sensitivity analysis of generalized variational inequalities, J. Math. Anal. Appl., 167 (1992), 299-304. https://doi.org/10.1016/0022-247x(92)90207-t
  23. S.B. Nadler(Jr.), Multi-valued contractive mappings, Pacific J. Math., 30 (1969), 475-488. https://doi.org/10.2140/pjm.1969.30.475
  24. M.A. Noor, Sensitivity analysis for quasi-variational inclusions, J. Math. Anal. Appl., 236 (1999), 290-299. https://doi.org/10.1006/jmaa.1999.6424
  25. M.A. Noor, Sensitivity analysis framework for general quasi-variational inclusions, Comput. Math. Appl., 44 (2002), 1175-1181. https://doi.org/10.1016/S0898-1221(02)00224-9
  26. J.Y. Park and J.U. Jeong, Parametric generalized mixed variational inequalities, Appl. Math. Lett., 17 (2004), 43-48. https://doi.org/10.1016/S0893-9659(04)90009-2
  27. J.W. Peng and X.J. Long, Sensitivity analysis for parametric completely generalized strongly nonlinear implicit quasi-variational inclusions, Comput. Math. Appl., 50 (2005), 869-880. https://doi.org/10.1016/j.camwa.2005.02.017
  28. T. Ram, Parametric generalized nonlinear quasi-variational inclusion problems, Int. J. Math. Arch., 3(3) (2012), 1273-1282.
  29. V. Todorcevic, Harmonic quasiconformal mappings and hyperbolic type metrics, Springer Nature Switzerland AG, 2019.
  30. N.D. Yen, Holder continuity of solutions to a parametric variational inequality, Appl. Math. Optim., 31 (1995), 245-255. https://doi.org/10.1007/BF01215992