Browse > Article
http://dx.doi.org/10.4134/JKMS.2008.45.1.163

ON SOLVABILITY OF GENERALIZED NONLINEAR VARIATIONAL-LIKE INEQUALITIES  

Zhang, Lili (Department of Applied Mathematics Dalian University of Technology)
Liu, Zeqing (Department of Mathematics Liaoning Normal University)
Kang, Shin-Min (Department of Mathematics and Research Institute of Natural Science Gyeongsang National University)
Publication Information
Journal of the Korean Mathematical Society / v.45, no.1, 2008 , pp. 163-176 More about this Journal
Abstract
In this paper, we introduce and study a new class of generalized nonlinear variational-like inequalities. By employing the auxiliary principle technique we suggest an iterative algorithm to compute approximate solutions of the generalized nonlinear variational-like inequalities. We discuss the convergence of the iterative sequences generated by the algorithm in Banach spaces and prove the existence of solutions and convergence of the algorithm for the generalized nonlinear variational-like inequalities in Hilbert spaces, respectively. Our results extend, improve and unify several known results due to Ding, Liu et al, and Zeng, and others.
Keywords
generalized nonlinear variational-like inequality; auxiliary principle technique; fixed points; algorithms; reflexive Banach space; Hilbert space;
Citations & Related Records

Times Cited By Web Of Science : 4
Times Cited By SCOPUS : 1
연도 인용수 순위
1 X. P. Ding and E. Tarafdar, Existence and uniqueness of solutions for a general nonlinear variational inequality, Appl. Math. Lett. 8 (1995), no. 1, 31-36   DOI   ScienceOn
2 Q. H. Ansari and J. C. Yao, Iterative schemes for solving mixed variational-like inequalities, J. Optim. Theory Appl. 108 (2001), no. 3, 527-541   DOI   ScienceOn
3 M. S. R. Chowdhury and K. K. Tan, Generalized variational inequalities for quasimonotone operators and applications, Bull. Polish Acad. Sci. Math. 45 (1997), no. 1, 25-54
4 X. P. Ding, Existence and algorithm of solutions for nonlinear mixed variational-like inequalities in Banach spaces, J. Comput. Appl. Math. 157 (2003), no. 2, 419-434   DOI   ScienceOn
5 X. P. Ding and J. C. Yao, Existence and algorithm of solutions for mixed quasivariational-like inclusions in Banach spaces, Comput. Math. Appl. 49 (2005), no. 5-6, 857-869   DOI   ScienceOn
6 R. Glowinski, J. Lions, and R. Tremolieres, Numerical Analysis of variational inequalities, North-Holland, Amsterdam, 1987
7 Z. Liu, Z. An, S. M. Kang, and J. S. Ume, Convergence and stability of the three-step iterative schemes for a class of general quasivariational-like inequalities, Int. J. Math. Math. Sci. 70 (2004), no. 69-72, 3849-3857
8 Z. Liu, J. H. Sun, S. H. Shim, and S. M. Kang, On solvability of general nonlinear variational-like inequalities in reflexive Banach spaces, Int. J. Math. Math. Sci. (2005), no. 9, 1415-1420   DOI   ScienceOn
9 Z. Liu, J. S. Ume, and S. M. Kang, Generalized nonlinear variational-like inequalities in reflexive Banach spaces, J. Optim. Theory Appl. 126 (2005), no. 1, 157-174   DOI
10 A. H. Siddiqi and Q. H. Ansani, An algorithm for a class of quasivariational inequalities, J. Math. Anal. Appl. 145 (1990), no. 2, 413-418   DOI
11 X. P. Ding, General algorithm of solutions for nonlinear variational inequalities in Banach space, Comput. Math. Appl. 34 (1997), no. 9, 131-137   DOI   ScienceOn
12 L. C. Zeng, Iterative algorithm for finding approximate solutions of a class of mixed variational-like inequalities, Acta Math. Appl. Sin. Engl. Ser. 20 (2004), no. 3, 477-486   DOI
13 X. P. Ding, Generalized quasi-variational-like inclusions with nonconvex functionals, Appl. Math. Comput. 122 (2001), no. 3, 267-282   DOI   ScienceOn
14 Q. H. Wu, Y. Hao, Z. Liu, and S. M. Kang, Sensitivity analysis of solutions for parametric general quasivariational-like inequalities, Int. J. Pure Appl. Math. 21 (2005), no. 1, 121-130