ON A CLASS OF GENERALIZED NONLINEAR IMPLICIT QUASIVARIATIONAL INCLUSIONS

摘要/Abstract

摘要： 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.

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.

中图分类号:

丁协平. ON A CLASS OF GENERALIZED NONLINEAR IMPLICIT QUASIVARIATIONAL INCLUSIONS[J]. Applied Mathematics and Mechanics (English Edition), 1999, 20(10): 1087-1098.

Ding Xieping . ON A CLASS OF GENERALIZED NONLINEAR IMPLICIT QUASIVARIATIONAL INCLUSIONS[J]. Applied Mathematics and Mechanics (English Edition), 1999, 20(10): 1087-1098.

参考文献

[1] Bensoussan A, Gourst M, Lions J I. Control impulsinel et inequations quasivariationrlles stationaries[J]. C R Acad Sci Paris,1973,276:1279~1284.
[2] Bensoussan A, Lions J I. Impulse control and quasivariational inequalities, Bordas, Paris: Gauthiers-Villes, 1984.
[3] Baiocchi C, Capelo A. Variational and Quasivariational Inequalities, Applications to Free Boundary Problems[M]. New York: Wiley, 1984.
[4] Mosco U. Implicit variational problems and quasivariational inequalities[J]. Lecture Notes in Math,Berlin: Springer-Verlag,1976,543:83~156.
[5] Isac G. Fixed point theory, coincidence equations on convex cones and complementarity problems[J]. Contemp Math,1988,72:139~155.
[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.
[7] Noor M A. An iterative scheme for a class of quasivariational inequalities[J]. J Math Anal Appl,1985,110:462~468.
[8] Noor M A. Quasi variational inequalities[J]. Appl Math Lett,1988,1(4):367~370.
[9] Noor M A. An iterative algorithm for variational inequalities[J]. J Math Anal Appl,1991,158(2):448~455.
[10] Noor M A. General algorithm and sensitivity analysis for variational inequalities[J]. J Appl Math Stoch Anal,1992,5(1):29~42.
[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.
[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.
[13] Siddiqi A H, Ansari Q H. Strongly nonlinear quasivariational inequalities[J]. J Math Anal Appl,1990,149(2):444~450.
[14] Siddiqi A H, Ansari Q H. General strongly nonlinear variational inequalities[J].J Math Anal Appl,1992,166(2):386~392.
[15] Ding X P, Tarafdar E. Generalized nonlinear variational inequalities with nonmonotone set-valued mappings[J]. Appl Math Lett,1994,7(4):5~11.
[16] Ding X P, Tarafdar E. On existence and uniqueness of solutions for a general nonlinear variational inequality[J]. Appl Math Lett,1995,8(1):31~36.
[17] Gao J S, Yao J C. Extension of strongly nonlinear quasivariational inequalities[J]. Appl Math Lett,1992,5(3):35~38.
[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.
[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.
[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.
[21] Saigal R. Extension of the generalized complementarity problem[J]. Math Oper Res,1976,1(3):260~266.
[22] Fang S C. An iterative method for generalized complementarity problems[J]. IEEE Trans Automat Control,1980,25:1225~1227.
[23] Fang S C, Peterson E L. Generalized variational inequalities[J]. J Optim Theory Appl,1982,38(2):363~383.
[24] Chan D, Pang J S. The generalized quasivariational inequality[J]. Math Oper Res,1982,7(1):211~222.
[25] Siddiqi A H, Ansari Q H. An iterative method for generalized variational inequalities[J]. Math Japon,1990,34(3):475~481.
[26] Ding Xieping. Generalized strongly nonlinear quasivariational inequalities[J]. J Math Anal Appl,1993,173(2):577~587.
[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.
[28] Ding Xieping. Set-valued implicit Wiener-Hopf equations and generalized strongly nonlinear quasivariational inequalities[J]. J Appl Anal,1998,4(1):59~71.
[29] Ding X P, Tarafdar E. Monotone generalized variational inequalities and generalized complementarity problems[J]. J Optim Theory Appl,1996,88(1):107~122.
[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.
[31] Hassouni A, Moudafi A. A perturbed algorithm for variational inclusions[J]. J Math Anal Appl,1994,185(3):706~712.
[32] Adly S. Perturbed algorithms and sensitivity analysis for a general class of variational inclusions[J]. J Math Anal Appl,1996,201(2):609~630.
[33] Huang N J. Generalized nonlinear variational inclusions with noncompact valued mapping[J]. Appl Math Lett,1996,9(3):25~29.
[34] Kazmi K R. Mann and Ishikawa perturbed iterative algorithms for generalized quasivariational inclusions[J]. J Math Anal Appl,1997,209(2):572~583.
[35] Ding Xieping. Perturbed proximal point algorithm for generalized quasivariational inclusions[J]. J Math Anal Appl,1997,210(1):88~101.
[36] Ding Xieping. Proximal point algorithm w ith errors for generalized strongly nonlinear quasivariational inclusions [ J]. Applied Mathematics and Mechanics ( English Edition ), 1998, 19( 7): 637~ 643.
[37] Pascali D, Shurlan S. Nonlinear Mappings of Monotone Type[M]. Romania: Sijthoff & Noordhoff Inter Pub,1978.
[38] Huang N J. On the generalized implicit quasivaritational inequalities[J].J Math Anal Appl,1997,216(1):197~210.
[39] Noor M A. Generalized multivalued quasivariational inequalities[J]. Comput Math Appl,1996,31(12):1~13.
[40] Nadler S B, Jr. Multi-valued contraction mappings[J]. Pacific J Math,1969,30(2):475~488.