East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
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ON NONLINEAR VARIATIONAL INCLUSIONS WITH ($A,{\eta}$)-MONOTONE MAPPINGS

  • Hao, Yan (School of Mathematics, Physics and Information Science Zhejiang Ocean University)
  • Received : 2008.10.14
  • Accepted : 2008.10.22
  • Published : 2009.06.30
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Abstract

In this paper, we introduce a generalized system of nonlinear relaxed co-coercive variational inclusions involving (A, ${\eta}$)-monotone map-pings in the framework of Hilbert spaces. Based on the generalized resol-vent operator technique associated with (A, ${\eta}$)-monotonicity, we consider the approximation solvability of solutions to the generalized system. Since (A, ${\eta}$)-monotonicity generalizes A-monotonicity and H-monotonicity, The results presented this paper improve and extend the corresponding results announced by many others.

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References

  1. R. P. Agarwal, Y. J. Cho and N. J. Huang, Sensitivity analysis for strongly nonlinear quasi-variational inclusions, Appl. Math. Lett., 13 (2000), 19-24.
  2. Y. J. Cho and H. Y. Lan, Generalized nonlinear random (A, $\eta$)-accretive equations with random relaxed cocoercive mappings in Banach spaces, Comput. Math. Appl., 55 (2008), 2173-2182. https://doi.org/10.1016/j.camwa.2007.09.002
  3. Y. J. Cho, J. Li and N. J. Huang, Solvability of implicit complementarity problems, Math. Comput. Model., 45 (2007), 1001-1009. https://doi.org/10.1016/j.mcm.2006.09.007
  4. Y. J. Cho and H. Y. Lan, Generalized nonlinear random (A, $\eta$)-accretive equations with random relaxed cocoercive mappings in Banach spaces, Comput. Math. Appl., 55 (2008), 2173-2182. https://doi.org/10.1016/j.camwa.2007.09.002
  5. 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
  6. Y. P. Fang and N. J. Huang, H-monotone operator and resolvent operator technique for variational incluisons, Appl. Maht. Comput., 145 (2003), 795-803. https://doi.org/10.1016/S0096-3003(03)00275-3
  7. Y. P. Fang and N. J. Huang, H-monotone operators and system of variational inclusions, Commun. Appl. Nonlinear Anal., 11 (2004), 93-101.
  8. Y. P. Fang, N. J. Huang and H. B. Thompson, A new system of variational inclusions with (H, $\eta$)-monotone operators in Hilbert spaces, Comput. Math. Appl., 49 (2005), 365-374. https://doi.org/10.1016/j.camwa.2004.04.037
  9. Y. P. Fang and N. J. Huang, H-accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces, Appl. Math. Lett., 17 (2004), 647-653. https://doi.org/10.1016/S0893-9659(04)90099-7
  10. N. J. Huang and Y. P. Fang, A new clas of general variational inclusions involving maximal $\eta$-monotone mappings, Publ. Math. Debrecen, 62 (2003), 83-98.
  11. N. J. Huang, A new class of generalized set-valued implicit variational inclusions in banach spaces with an application, Comput. Math. Appl., 41 (2001), 937-943. https://doi.org/10.1016/S0898-1221(00)00331-X
  12. H. Y. Lan, J. H. Ki and Y. J. Cho, On a new system of nonlinear A-monotone multi-valued variational inclusions, J. Math. Anal. Appl., 327 (2007), 481-493. https://doi.org/10.1016/j.jmaa.2005.11.067
  13. M. A. Noor and Z. Y. Huang, Some resolvent iterative methods for variational inclusions and nonexpansive mappings, Appl. Math. Appl., 194 (2007), 267-275.
  14. M. A. Noor, An iterative scheme for a class of quasi variational inequalities, J. Math. Anal. Appl., 110 (1985) 463-468. https://doi.org/10.1016/0022-247X(85)90308-7
  15. X. Qin and M. A. Noor, General Wiener-Hopf equation technique for nonexpansive mappings and general variational inequalities in Hilbert spaces, Appl. Math. Comput., 201 (2008), 716-722. https://doi.org/10.1016/j.amc.2008.01.007
  16. X. Qin and M. Shang, Generalized variational inequalities involving relaxed monotone mappings and nonexpansive mappings, J. Inequal. Appl., 2007 (2007) 20457. https://doi.org/10.1155/2007/20457
  17. X. Qin, M. Shang and Y. Su, Generalized variational inequalities involving relaxed monotone mappings in Hilbert spaces, PamAmer. Math. J., 17 (2007), 81-88.
  18. X. Qin, S. M. Kang and M. Shang, Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces, Appl. Anal., 87 (2008), 421-430. https://doi.org/10.1080/00036810801952953
  19. X. Qin, M. Shang and H. Zhou, Strong convergence of a general iterative method for variational inequality problems and fixed point problems in Hilbert spaces, Appl. Math. Comput., 200 (2008), 242-253. https://doi.org/10.1016/j.amc.2007.11.004
  20. R. U. Verma, Sensitivity analysis for generalized strongly monotone variatonal inclusions based on the (A, $\eta$)-resolvent operator technique, Appl. Math. Lett., 19 (2006), 1409-1413. https://doi.org/10.1016/j.aml.2006.02.014
  21. R. U. Verma, Sensitivity analysis for relaxed cocoercive nonlinear quasivariational inclusions, J. Appl. Math. Stoch. Anal., 2006 (2006) 52041.
  22. R. U. Verma, A-monotonicity and applications to nonlinear variational inclusion problems, J. Appl. Math. Stoch. Anal., 17 (2004), 193-195.
  23. R. U. Verma, Approximation solvability of a class of A-monotone variational inclusion problems, J. KSIAM., 8 (2004), 55-66.
  24. R. U. Verma, A-monotonicity and its role in nonlinear variational inclusions, J. Optim. Theory Appl., 129 (2006), 457-467. https://doi.org/10.1007/s10957-006-9079-7
  25. R. U. Verma, A-monotone nonlinear relaxed cocoercive variational inclusions, Central European J. Math., 5 (2007), 386-396. https://doi.org/10.2478/s11533-007-0005-5
  26. R. U. Verma, Approximation solvability of a class of nonlinear set-valued variational inclusions involving (A, $\eta$)-monotone mappings, J. Math. Anal. Appl., 337 (2007), 969-975. https://doi.org/10.1016/j.jmaa.2007.01.114