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

Title & Authors
ON GENERALIZED NONLINEAR QUASI-VARIATIONAL-LIKE INCLUSIONS DEALING WITH (h,η)-PROXIMAL MAPPING
Liu, Zeqing; Chen, Zhengsheng; Shim, Soo-Hak; Kang, Shin-Min;

Abstract
In this paper, a new class of $\small{(h,{\eta})}$-proximal for proper functionals in Hilbert spaces is introduced. The existence and Lip-schitz continuity of the $\small{(h,{\eta})}$-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$\small{(h,{\eta})}$-proximalmapping;perturbed three-step iterative algorithm with errors;strongly monotone mapping;generalized pseudocontractive mapping;mixed Lipschitz mapping;relaxed coercive mapping;
Language
English
Cited by
1.
Perturbed Mann iterative method with errors for a new system of generalized nonlinear variational-like inclusions, Mathematical and Computer Modelling, 2010, 51, 1-2, 63
References
1.
S. Adly, Perturbed algorithms and sensitivity analysis for a general class of variational inclusions, J. Math. Anal. Appl. 201 (1996), no. 2, 609-630

2.
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

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

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.
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

7.
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

8.
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

9.
R. Glowinski, J. Lions and R. Tremolieres, Numerical Analysis of Variational Inequalities, North-Holland Publishing Co., Amsterdam-New York, 1981

10.
J. S. Guo and J. C. Yao, Extension of strongly nonlinear quasivariational inequalities, Appl. Math. Lett. 5 (1992), no. 3, 35-38

11.
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

12.
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

13.
A. Hassouni and A. Moudafi, A perturbed algorithm for variational inclusions, J. Math. Anal. Appl. 185 (1994), no. 3, 706-712

14.
K. R. Kazmi, Mann and Ishikawa type perturbed iterative algorithms for generalized quasivariational inclusions, J. Math. Anal. Appl. 209 (1997), no. 2, 572-584

15.
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

16.
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

17.
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

18.
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

19.
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

20.
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

21.
A. H. Siddiqi and Q. H. Ansari, Strongly nonlinear quasivariational inequalities, J. Math. Anal. Appl. 149 (1990), no. 2, 444-450

22.
A. H. Siddiqi and Q. H. Ansari, General strongly nonlinear variational inequalities, J. Math. Anal. Appl. 166 (1992), no. 2, 386-392

23.
R. U. Verma, Generalized pseudo-contractions and nonlinear variational inequalities, Publ. Math. Debrecen 53 (1998), no. 1-2, 23-28

24.
J. C. Yao, Existence of generalized variational inequalities, Oper. Res. Lett. 15 (1994), no. 1, 35-40

25.
J. C. Yao, The generalized quasi-variational inequality problem with applications, J. Math. Anal. Appl. 158 (1991), no. 1, 139-160

26.
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

27.
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