Browse > Article
http://dx.doi.org/10.4134/JKMS.2008.45.5.1323

ON GENERALIZED NONLINEAR QUASI-VARIATIONAL-LIKE INCLUSIONS DEALING WITH (h,η)-PROXIMAL MAPPING  

Liu, Zeqing (Department of Mathematics Liaoning Normal University)
Chen, Zhengsheng (School of Management Dalian University of Technology)
Shim, Soo-Hak (The Research Institute of Natural Science Gyeongsang National University)
Kang, Shin-Min (Department of Mathematics and The Research Institute of Natural Science Gyeongsang National University)
Publication Information
Journal of the Korean Mathematical Society / v.45, no.5, 2008 , pp. 1323-1339 More about this Journal
Abstract
In this paper, a new class of $(h,{\eta})$-proximal for proper functionals in Hilbert spaces is introduced. The existence and Lip-schitz continuity of the $(h,{\eta})$-proximal mappings for proper functionals are proved. A class of generalized nonlinear quasi-variational-like inclusions in Hilbert spaces is introduced. A perturbed three-step iterative algorithm with errors for the generalized nonlinear quasi-variational-like inclusion is suggested. The existence and uniqueness theorems of solution for the generalized nonlinear quasi-variational-like inclusion are established. The convergence and stability results of iterative sequence generated by the perturbed three-step iterative algorithm with errors are discussed.
Keywords
generalized nonlinear quasi-variational-like inclusion$(h,{\eta})$-proximalmapping; perturbed three-step iterative algorithm with errors; strongly monotone mapping; generalized pseudocontractive mapping; mixed Lipschitz mapping; relaxed coercive mapping;
Citations & Related Records
Times Cited By KSCI : 2  (Citation Analysis)
Times Cited By Web Of Science : 1  (Related Records In Web of Science)
Times Cited By SCOPUS : 1
연도 인용수 순위
1 C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities. Applications to free boundary problems, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984
2 P. T. Harker and J. S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Math. Programming 48 (1990), no. 2, (Ser. B), 161-220   DOI
3 Y. J. Cho, J. H. Kim, N. J. Huang, and S. M. Kang, Ishikawa and Mann iterative processes with errors for generalized strongly nonlinear implicit quasi-variational inequalities, Publ. Math. Debrecen 58 (2001), no. 4, 635-649
4 X. P. Ding, Perturbed proximal point algorithms for generalized quasivariational inclusions, J. Math. Anal. Appl. 210 (1997), no. 1, 88-101   DOI   ScienceOn
5 X. P. Ding, Proximal point algorithm with errors for generalized strongly nonlinear quasivariational inclusions, Appl. Math. Mech. (English Ed.) 19 (1998), no. 7, 637-643; translated from Appl. Math. Mech. 19 (1998), no. 7, 597-602
6 J. S. Guo and J. C. Yao, Extension of strongly nonlinear quasivariational inequalities, Appl. Math. Lett. 5 (1992), no. 3, 35-38
7 A. M. Harder and T. L. Hicks, Fixed point theorems and stability results for fixed point iteration procedures, Indian J. Pure Appl. Math. 21 (1990), no. 1, 1-9
8 A. Hassouni and A. Moudafi, A perturbed algorithm for variational inclusions, J. Math. Anal. Appl. 185 (1994), no. 3, 706-712   DOI   ScienceOn
9 L. S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995), no. 1, 114-125   DOI   ScienceOn
10 K. R. Kazmi, Mann and Ishikawa type perturbed iterative algorithms for generalized quasivariational inclusions, J. Math. Anal. Appl. 209 (1997), no. 2, 572-584   DOI   ScienceOn
11 Z. Liu and S. M. Kang, Convergence and stability of perturbed three-step iterative algorithm for completely generalized nonlinear quasivariational inequalities, Appl. Math. Comput. 149 (2004), no. 1, 245-258   DOI   ScienceOn
12 A. H. Siddiqi and Q. H. Ansari, Strongly nonlinear quasivariational inequalities, J. Math. Anal. Appl. 149 (1990), no. 2, 444-450   DOI
13 A. H. Siddiqi and Q. H. Ansari, General strongly nonlinear variational inequalities, J. Math. Anal. Appl. 166 (1992), no. 2, 386-392   DOI
14 L. Zhang, Z. Liu and S. M. Kang, On solvability of generalized nonlinear variational-like inequalities, J. Korean Math. Soc. 45 (2008), no. 1, 163-176   과학기술학회마을   DOI   ScienceOn
15 R. U. Verma, Generalized pseudo-contractions and nonlinear variational inequalities, Publ. Math. Debrecen 53 (1998), no. 1-2, 23-28
16 J. C. Yao, Existence of generalized variational inequalities, Oper. Res. Lett. 15 (1994), no. 1, 35-40   DOI   ScienceOn
17 J. C. Yao, The generalized quasi-variational inequality problem with applications, J. Math. Anal. Appl. 158 (1991), no. 1, 139-160   DOI
18 X. P. Ding and C. L. Luo, Perturbed proximal point algorithms for general quasivariational-like inclusions, J. Comput. Appl. Math. 113 (2000), no. 1-2, 153-165   DOI   ScienceOn
19 X. P. Ding and K. K. Tan, A minimax inequality with applications to existence of equilibrium point and fixed point theorems, Colloq. Math. 63 (1992), no. 2, 233-247   DOI
20 F. Giannessi and A. Mauger, Variational Inequalities and Network Equilibrium Problems, Proceedings of the conference held in Erice, June 19-25, 1994. Edited by F. Giannessi and A. Maugeri. Plenum Press, New York, 1995
21 R. Glowinski, J. Lions and R. Tremolieres, Numerical Analysis of Variational Inequalities, North-Holland Publishing Co., Amsterdam-New York, 1981
22 Z. Liu, J. S. Ume, and S. M. Kang, General strongly nonlinear quasivariational inequalities with relaxed Lipschitz and relaxed monotone mappings, J. Optim. Theory Appl. 114 (2002), no. 3, 639-656   DOI   ScienceOn
23 Z. Liu, L. Debnath, S. M. Kang, and J. S. Ume, Generalized mixed quasivariational inclusions and generalized mixed resolvent equations for fuzzy mappings, Appl. Math. Comput. 149 (2004), no. 3, 879-891   DOI   ScienceOn
24 Z. Liu, L. Debnath, S. M. Kang, and J. S. Ume, Sensitivity analysis for parametric completely generalized nonlinear implicit quasivariational inclusions, J. Math. Anal. Appl. 277 (2003), no. 1, 142-154   DOI   ScienceOn
25 Z. Liu, J. S. Ume, and S. M. Kang, General variational inclusions and general resolvent equations, Bull. Korean Math. Soc. 41 (2004), no. 2, 241-256   DOI   ScienceOn
26 J. X. Zhou and G. Chen, Diagonal convexity conditions for problems in convex analysis and quasi-variational inequalities, J. Math. Anal. Appl. 132 (1988), no. 1, 213-225   DOI
27 S. Adly, Perturbed algorithms and sensitivity analysis for a general class of variational inclusions, J. Math. Anal. Appl. 201 (1996), no. 2, 609-630   DOI   ScienceOn