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On a class of generalized nonlinear implicit quasivariational inclusions

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Abstract

In this paper, a new class of generalized nonlinear implicit quasivariational inclusions involving a set-valued maximal monotone mapping are studied. A existence theorem of solutions for this class of generalized nonlinear implicit quasivariational inclusions is proved without compactness assumptions. A new iterative algorithm for finding approximate solutions of the generalized nonlinear implicit quasivariational inclusions is suggested and analysed and the convergence of, iterative sequence generated by the new algorithm is also given. As special cases, some known results in this field are also discussed.

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References

  1. Bensoussan A, Gourst M, Lions J I. Control impulsinel et inequations quasivariationrlles stationaries[J].C R Acad Sci. Paris, 1973,276, 1279–1284.

    MATH  Google Scholar 

  2. Bensoussan A, Lions J I. Impulse control and quasivariational inequalities Bordas, Paris: Gauthiers-Villes, 1984.

    Google Scholar 

  3. Baiocchi C, Capelo A.Variational and Quasivariational Inequalities, Applications to Free Boundary Problems[M]. New York: Wiley, 1984.

    Google Scholar 

  4. Mosco U. Implicit variational problems and quasivariational inequalities[J].Lecture Notes in Math, Berlin: Springer-Verlag, 1976,543: 83–156.

    Google Scholar 

  5. Isac G. Fixed point theory, coincidence equations on convex cones and complementarity problems[J].Contemp Math, 1988,72: 139–155.

    MATH  MathSciNet  Google Scholar 

  6. Isac G. A special variational inequality and implicit complementarity problem[J].J Fac Sci Univ Tokyo Sect IA Math, 1990,37(1): 109–127.

    MATH  MathSciNet  Google Scholar 

  7. Noor M A. An iterative scheme for a class of quasivariational inequalities[J].J Math Anal Appl, 1985,110: 462–468.

    Article  MathSciNet  Google Scholar 

  8. Noor M A. Quasi variational inequalities[J].Appl Math Lett, 1988,1(4): 367–370.

    Article  MATH  MathSciNet  Google Scholar 

  9. Noor M A. An iterative algorithm for variational inequalities[J].J Math Anal Appl, 1991,158(2): 448–455.

    Article  MATH  MathSciNet  Google Scholar 

  10. Noor M A. General algorithm and sensitivity analysis for variational inequalities[J].J Appl Math Stoch Anal, 1992,5(1): 29–42.

    MATH  MathSciNet  Google Scholar 

  11. Pang J S. The implicit complementarity problem[A]. In: Mangasarian, Mayer, Robinson Eds.Nonlinear Programming 4[C]. New York: Acad Press, 1981, 487–518.

    Google Scholar 

  12. Pang J S. On the convergence of a basic iterative method for the implicit complementarity problems[J].J Optim Theory Appl, 1982,37(1): 149–162.

    Article  MATH  MathSciNet  Google Scholar 

  13. Siddiqi A H, Ansari Q H. Strongly nonlinear quasivariational inequalities[J].J Math Anal Appl, 1990,149(2): 444–450.

    Article  MATH  MathSciNet  Google Scholar 

  14. Siddiqi A H, Ansari Q H. General strongly nonlinear variational inequalities[J].J Math Anal Appl, 1992,166(2): 386–392.

    Article  MATH  MathSciNet  Google Scholar 

  15. Ding X P, Tarafdar E. Generalized nonlinear variational inequalities with nonmonotone set-valued mappings[J].Appl Math Lett, 1994,7(4): 5–11.

    Article  MATH  MathSciNet  Google Scholar 

  16. Ding X P, Tarafdar E. On, existence and uniqueness of solutions, for a genral nonlinear variational inequality[J].Appl Math Lett, 1995,8(1): 31–36.

    Article  MATH  MathSciNet  Google Scholar 

  17. Gao J S, Yao J C. Extension of strongly nonlinear quasivariational inequalities[J].Appl Math Lett, 1992,5(3): 35–38.

    Article  MathSciNet  Google Scholar 

  18. Zeng L. Iterative algorithm for finding approximate solutions for general strongly nonlinear variational inequalities[J].J Math Anal Appl, 1994,187(2): 352–360.

    Article  MATH  MathSciNet  Google Scholar 

  19. Harker P T, Pang J S. Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications[J].Math Programming, 1990,48(1): 161–220.

    Article  MATH  MathSciNet  Google Scholar 

  20. Noor M A, Noor K I, Rassias T M. Some aspects of variational inequalities[J].J Comput Appl Math, 1993,47(1): 285–312.

    Article  MATH  MathSciNet  Google Scholar 

  21. Saigal R. Extension of the generalized complementarity problem[J].Math Oper Res, 1976,1(3): 260–266.

    MATH  MathSciNet  Google Scholar 

  22. Fang S C. An iterative method for generalized complementarity problems[J].IEEE Trans Automat Control, 1980,25: 1225–1227.

    Article  MATH  MathSciNet  Google Scholar 

  23. Fang S C, Peterson E L. Generalized variational inequalities[J].J Optim Theory Appl, 1982,38(2): 363–383.

    Article  MATH  MathSciNet  Google Scholar 

  24. Chan D, Pang J S. The generalized quasivariational inequality[J].Math Oper Res, 1982,7(1): 211–222.

    Article  MATH  MathSciNet  Google Scholar 

  25. Siddiqi A H, Ansari Q H. An iterative method for generalized variational inequalities[J].Math Japan, 1990,34(3): 475–481.

    MathSciNet  Google Scholar 

  26. Ding Xieping. Generalized strongly nonlinear quasivariational inequalities[J].J Math Anal Appl, 1993,173(2): 577–587.

    Article  MathSciNet  MATH  Google Scholar 

  27. Ding Xieping. A new class of generalized strongly nonlinear quasivariational inequalities and quasi-complementarity problems[J].Indian J Pure Appl Math, 1994,25(11) 1115–1128.

    MathSciNet  Google Scholar 

  28. Ding Xieping. Set-valued implicit Wiener-Hopf equations and generalized strongly nonlinear quasivariational inequalities[J].J Appl Anal, 1998,4(1): 59–71.

    Google Scholar 

  29. Ding X P, Tarafdar E. Monotone generalized variational inequalities and generalized complementarity problems[J]J Optim Theory Appl, 1996,88(1): 107–122.

    Article  MATH  MathSciNet  Google Scholar 

  30. Zeng L. Iterative algorithm for finding approximate solutions to completely generalized strongly nonlinear quasivariational inequalities[J].J Math Anal Appl, 1996,201(1): 180–194.

    Article  MATH  MathSciNet  Google Scholar 

  31. Hassouni A, Moudafi A. A perturbed algorithm for variational inclusions[J].J Math Anal Appl, 1994,185(3): 706–712.

    Article  MATH  MathSciNet  Google Scholar 

  32. Adly S. Pertubed algorithms and sensitivity analysis for a general class of variational inclusions[J].J Math Anal Appl, 1996,201(2): 609–630.

    Article  MATH  MathSciNet  Google Scholar 

  33. Huang N J. Generalized nonlinear variational inclusions with noncompact valued mapping [J].Appl Math Lett, 1996,9(3): 25–29.

    Article  MATH  MathSciNet  Google Scholar 

  34. Kazmi K R. Mann and Ishikawa perturbed iterative algorithms for generalized quasivariational inclusions[J].J Math Anal Appl, 1997,209(2): 572–583.

    Article  MATH  MathSciNet  Google Scholar 

  35. Ding Xieping. Perturbed proximal point algorithm for generalized quasivariational inclusions[J].J Math Anal Appl, 1997,210(1): 88–101.

    Article  MathSciNet  Google Scholar 

  36. Ding Xieping. Proximal point algorithm with errors for generalized, strongly nonlinear quasivariational inclusions [J].Applied mathematics and Mechanics (English Edition), 1998,19(7): 647–643.

    Article  Google Scholar 

  37. Pascali D, Shurlan S.Nonlinear Mappings of Monotone Type[M]. Romaina: Sijthoff & Noordhoff Inter Pub, 1978.

    Google Scholar 

  38. Huang N J. On the generalized implicit quasivaritational inequalities[J].J Math Anal Appl, 1997,216(1): 197–210.

    Article  MATH  MathSciNet  Google Scholar 

  39. Noor M A. Generalized multivalued quasivarariational inequalities[J].Comput Math Appl, 1996,31(12) 1–13.

    Article  MATH  MathSciNet  Google Scholar 

  40. Nadler S B, Jr. Multi-valued contraction mappings[J].Pacific J Math,1969,30(2): 475–488.

    Google Scholar 

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Authors and Affiliations

  1. Department of Matheamtics, Sichuan Normal University, 610066, Chengdu, P R China

    Ding Xieping

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  1. Ding Xieping
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Foundation item: the National Natural Science Foundation of China(19871059)

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Xieping, D. On a class of generalized nonlinear implicit quasivariational inclusions. Appl Math Mech 20, 1087–1098 (1999). https://doi.org/10.1007/BF02460325

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  • DOI: https://doi.org/10.1007/BF02460325

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