East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
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SYSTEM OF GENERALIZED NONLINEAR MIXED VARIATIONAL INCLUSIONS INVOLVING RELAXED COCOERCIVE MAPPINGS IN HILBERT SPACES

  • Received : 2014.09.29
  • Accepted : 2015.03.17
  • Published : 2015.05.31
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Abstract

We considered a new system of generalized nonlinear mixed variational inclusions in Hilbert spaces and define an iterative method for finding the approximate solutions of this class of system of generalized nonlinear mixed variational inclusions. We also established that the approximate solutions obtained by our algorithm converges to the exact solutions of a new system of generalized nonlinear mixed variational inclusions.

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References

  1. M. K. Ahmad and Salahuddin, Perturbed three step approximation processes with errors for a general implicit nonlinear variational inequalities, Int. J. Math. Math. Sci. Article ID 43818, (2006), 1-14.
  2. M. K. Ahmad and Salahuddin, A stable perturbed algorithms for a new class of generalized nonlinear implicit quasi variational inclusions in Banach spaces, Advances in Pure Math. 2 (2) (2012), 139-148. https://doi.org/10.4236/apm.2012.23021
  3. D. P. Bertsekas and J. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods, Prentice Hall Englewood Cliffs, NJ USA, 1989.
  4. H. Brezis, Operateurs Maximaux Monotone er Semi Groupes de Contractions dans les Espaces de Hilbert, North-Holland Mathematices Studies 5 Notes de Matematica (50), North-Holland Amsterdam, 1973.
  5. L. C. Ceng, C. Y. Wang and J. C. Yao, Strong convergence theorem by a relaxed extragradient method for a general system of variational inequalities, Math. Methods Oper. Research 67 (2008), 375-390, doi:10.1007/500186-007-0207-4.
  6. S. S. Chang, Fixed Point Theory with Applications, Chongqing Publishing House, Chongqing China, 1984.
  7. S. S. Chang, H. W. J. Lee and C. K. Chan, Generalized system for relaxed cocoercive variational inequalities in Hilbert space, Appl. Math. Lett. 20 (3) (2007), 329-334. https://doi.org/10.1016/j.aml.2006.04.017
  8. Y. J. Cho, Y. P. Fang, N. J. Huang and H. J. Hwang, Algorithms for systems of nonlinear variational inequalities, J. Kor. Math. Soc. 41 (3) (2004), 489-499. https://doi.org/10.4134/JKMS.2004.41.3.489
  9. Y. J. Cho and X. Qin, Generalized system for relaxed cocoercive variational inequalities and projection methods in Hilbert spaces, Math. Inequal. Appl. 12 (2) (2009), 365-375.
  10. M. C. Ferris and J. S. Pang, Engineering and economics applications of complementarity problems, SIAM Review 39 (4) (1997), 669-713. https://doi.org/10.1137/S0036144595285963
  11. F. Giannessi and A. Maugeri, Variational Inequalities and Network Equilibrium Problems, Plenum Press, NY USA, 1995.
  12. A. Hassouni and A. Moudafi, A perturbed algorithms for variational inequalities, J. Math. Anal. Appl. 185 (2001), 706-712.
  13. I. Inchan and N. Petrot, System of general variational inequalities involving different nonlinear operators related to fixed point problems and its applications, Fixed point Theory Vol. 2011, Article ID 689478, pages 17, doi 10.1155/2011/689478.
  14. J. U. Jeong, A system of parametric generalized nonlinear mixed quasi variational inclusions in Lp-spaces, J. Appl. Math. Comput. 19 (1-2) (2005), 493-506.
  15. G. Kassay and J. Kolumban, System of multivalued variational inequalities, Pub. Math. Deb. 56 (1-2) (2000), 185-195.
  16. J. K. Kim and D. S. Kim, A new system of generalized nonlinear mixed variational inequalities in Hilbert spaces, J. Convex Analysis 11 (1) (2004), 235-243.
  17. B. S. Lee, M. F. Khan and Salahuddin, Generalized nonlinear quasi variational inclusions in Banach spaces, Comput. Math. Appl. 56 (5) (2008), 1414-1422. https://doi.org/10.1016/j.camwa.2007.11.053
  18. Jr. S. B. Nadler, Multivalued contraction mappings, Pacific J. Math. 30 (1969), 475-487. https://doi.org/10.2140/pjm.1969.30.475
  19. H. Nie, Z. Liu, K. H. Kim and S. M. Kang, A system of nonlinear variational inequalities strongly monotone and pseudo contractive mappings, Adv. Nonlinear Var. Inequal. 6 (2) (2003), 91-99.
  20. J. S. Pang, Asymmetric variational inequality problem over product sets, Application and iterative methods, Math. Prog. 31 (1985), 206-219 https://doi.org/10.1007/BF02591749
  21. S. Suantai and N. Petrot, Existence and stability of iterative algorithms for the system of nonlinear quasi mixed equilibrium problems, Appl. Math. Lett. 24 (2011), 308-313. https://doi.org/10.1016/j.aml.2010.10.011
  22. R. U. Verma, On a new system of nonlinear variational inequalities and associated iterative algorithms, Math-Sci Res Hotline 3 (8) (1999), 65-68.
  23. R. U. Verma, Iterative algorithms and a new system of nonlinear quasivariational inequalities, Adv. Nonlinear Var. Inequal. 4 (1) (2001), 117-124.
  24. R. U. Verma, Projection method, Algorithm and a new system of nonlinear variational inequalities, Comput. Math. Appl. 41 (7-8) (2001), 1025-1031. https://doi.org/10.1016/S0898-1221(00)00336-9
  25. R. U. Verma, Generalized system for relaxed cocoercive variational inequalities and projection methods, J. Optim. Th. Appl. 121 (1) (2004), 203-210. https://doi.org/10.1023/B:JOTA.0000026271.19947.05