Abstract

We introduce and study a new system of generalized nonlinear mixed variational inequalities in Hilbert spaces. We prove the existence and uniqueness of solutions for this system of generalized nonlinear mixed variational inequalities. We also give the convergence of the Mann iterative sequences with errors for this system of generalized nonlinear mixed variational inequalities in Hilbert spaces.

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Keywords

  • system of generalized nonlinear mixed variational inequalities
  • Mann iterative sequence with errors
  • nonlinear mapping
  • existence
  • uniqueness

Mathematics Subject Classification

49J40

References

  1. C. Baiocchi, A. Caopelo: Variational and Quasivariational Inequalities, Application to Free Boundary Problems, Wiley, New York (1984).Google Scholar
  2. A. Bensounssan, J. L. Lions: Impulse Control and Quasivariational Inequalities, Gauthiers-Villers, Bordas, Paris (1984).Google Scholar
  3. H. Brezis: Operateurs Maximaux Monotone et Semigroups de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam (1973).Google Scholar
  4. F. Giannessi, A. Maugeri: Variational Inequalities and Network Equilibrium Problems, Plenum Press, New York (1995).https://doi.org/10.1007/978-1-4899-1358-6Google Scholar
  5. R. Glowinski, P. Le Tallec: Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, SIAM Publication Co., Philadelphia (1989).https://doi.org/10.1137/1.9781611970838Google Scholar
  6. R. Glowinski, J. L. Lions, R. Trémolières: Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam (1981).Google Scholar
  7. P. T. Harker, J. S. Pang: Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Math. Programming 48 (1990) 161–220.https://doi.org/10.1007/BF01582255Google Scholar
  8. N. J. Huang: Generalized nonlinear variational inclusions with noncompact valued mapping, Appl. Math. Lett. 9(3) (1996) 25–29.https://doi.org/10.1016/0893-9659(96)00026-2Google Scholar
  9. N. J. Huang: On the generalized implicit quasi-variational inequalities, J. Math. Anal. Appl. 216 (1997) 197–210.https://doi.org/10.1006/jmaa.1997.5671Google Scholar
  10. N. J. Huang: Mann and Ishikawa type perturbed iterative algorithms for generalized nonlinear implicit quasi-variational inclusions, Comput. Math. Appl. 35(10) (1998) 1–7.https://doi.org/10.1016/S0898-1221(98)00066-2Google Scholar
  11. N. J. Huang: A new completely general class of variational inclusions with noncompact valued mappings, Comput. Math. Appl. 35(10) (1998) 9–14.https://doi.org/10.1016/S0898-1221(98)00067-4Google Scholar
  12. N. J. Huang, Y. P. Liu, Y. Y. Tang, M. R. Bai: On the generalized set-valued strongly nonlinear implicit variational inequalities, Comput. Math. Appl. 37(10) (1999) 29–36.https://doi.org/10.1016/S0898-1221(99)00123-6Google Scholar
  13. N. J. Huang: Generalized nonlinear implicit quasivariational inclusion and an application to implicit variational inequalities, Z. Angew. Math. Mech. 79 (1999) 569–575.https://doi.org/10.1002/(SICI)1521-4001(199908)79:8<569::AID-ZAMM569>3.0.CO;2-GGoogle Scholar
  14. N. J. Huang, M. R. Bai, Y. J. Cho, S. M. Kang: Generalized nonlinear mixed quasi-variational inequalities, Comput. Math. Appl. 40(2-3) (2000) 205–215.https://doi.org/10.1016/S0898-1221(00)00154-1Google Scholar
  15. M. A. Noor: Some recent advances in variational inequalities, I: Basic concepts, New Zealand J. Math. 26 (1997) 53–80.Google Scholar
  16. M. A. Noor: Some recent advances in variational inequalities, II: Other concepts, New Zealand J. Math. 26 (1997) 229–255.Google Scholar
  17. M. A. Noor: Set-valued quasi variational inclusions, Korean J. Comput. Appl. Math. 7(1) (2000) 101–113.https://doi.org/10.1007/BF03009930Google Scholar
  18. M. A. Noor, K. I. Noor, Th. M. Rassias: Set-valued resolvent equations and mixed variational inequalities, J. Math. Anal. Appl. 220 (1998) 741–759.https://doi.org/10.1006/jmaa.1997.5893Google Scholar
  19. M. A. Noor, K. I. Noor, Th. M. Rassias: Some aspects of variational inequalities, J. Comput. Appl. Math. 47 (1993) 285–312.https://doi.org/10.1016/0377-0427(93)90058-JGoogle Scholar
  20. R. U. Verma: On a new system of nonlinear variational inequalities and associated iterative algorithms, Math. Sci. Res. Hot-Line 3(8) (1999) 65–68.Google Scholar
  21. R. U. Verma: Iterative algorithms and a new system of nonlinear quasivariational inequalities, Adv. Nonlinear Var. Inequal. 4(1) (2001) 117–124.Google Scholar
  22. G. X. Z. Yuan: KKM Theory and Applications, Marcel Dekker, New York (1999).Google Scholar

Suggested citation

J. K. Kim, D. S. Kim. “A New System of Generalized Nonlinear Mixed Variational Inequalities in Hilbert Spaces.” Journal of Convex Analysis 11 (2004), No. 1, 235–243. https://doi.org/10.68381/jca11015

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Published by Heldermann Verlag, 2004. Rights now held by Banach Press LLC.