[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.7858/eamj.2012.28.1.037

PROXIMAL POINTS METHODS FOR GENERALIZED IMPLICIT VARIATIONAL-LIKE INCLUSIONS IN BANACH SPACES

He, Xin-Feng (College of Mathematics and Computer, Hebei University)
Lou, Jian (Computers Center, Hebei University)
He, Zhen (College of Mathematics and Computer, Hebei University)

Publication Information

East Asian mathematical journal / v.28, no.1, 2012 , pp. 37-47 More about this Journal

Abstract

In this paper, we study generalized implicit variational-like inclusions and $J^{\eta}$ -proximal operator equations in Banach spaces. It is established that generalized implicit variational-like inclusions in real Banach spaces are equivalent to fixed point problems. We also establish relationship between generalized implicit variational-like inclusions and $J^{\eta}$ -proximal operator equations. This equivalence is used to suggest a iterative algorithm for solving $J^{\eta}$ -proximal operator equations.

Keywords

generalized implicit variational-like inclusions; $J^{\eta}$ -Proximal operator; Algorithm; $J^{\eta}$ -proximal operator equations;

Citations & Related Records

Reference

1	K. R. Kazmi and F. A. Khan, Sensitivity analysis for parametric generalized implicit quasi-variational-like inclusions involving P- ${\eta}$ -accretive mappings, J. Math. Anal. Appl. 337 (2008), 1198-1210. DOI ScienceOn
2	S. B. Nadler, Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475-488. DOI
3	H. Y. Lan, ( $A,{\eta}$ )-Accretive mappings and set-valued variational inclusions with relaxed cocoercive mappings in Banach spaces, Appl. Math. Lett. 20 (2007), 571-577. DOI ScienceOn
4	R. . Agarwal, N. J. Huang and Y. J. Cho, Generalized nonlinear mixed implicit quasi-variational inclusions with setvalued mappings, J. Inequal. Appl. 7(6) (2002), 807-828.
5	R. Ahmad, A. H. Siddiqi and Z. Khan, Proximal point algorithm for generalized multivalued nonlinear quasivariational- like inclusions in Banach spaces, Appl. Math. Comput. 163 (2005), 295-308. DOI ScienceOn
6	S. S. Chang, Y. J. Cho and H. Y. Zhou, Iterative Methods for Nonlinear Operator Equations in Banach Spaces, Nova Sci. New York, 2002.
7	J. Y. Chen, N. C. Wong and J. C. Yao, Algorithm for generalized co-complementarity problems in Banach spaces, Comput. Math. Appl. 43(1) (2002), 49-54. DOI ScienceOn
8	X. P. Ding and C. L. Lou, Perturbed proximal point algorithms for general quasi-variational-like inclusions, J. Comput. Appl. Math. 210 (2000), 153-165.
9	N. J. Huang, Generlaized nonlinear variational inclusions with non-compact valued mappings, Appl. Math. Lett. 9(3) (1996), 25-29.
10	J. Lou, X. F. He and Z. He, Iterative methods for solving a system of variational inclusions involving H- ${\eta}$ -monotone operators in Banach spaces, Computers and Mathematics with Applications, Computers and Mathematics with Applications 55 (2008), 1832-1841. DOI ScienceOn
11	X. F. He, J. Lou and Z. He, Iterative methods for solving variational inclusions in Banach spaces, Journal of Computational and Applied Mathematics 203(1) (2007), 80-86. DOI ScienceOn
12	R. Ahmad and A. H. Siddiqi, Mixed variational-like inclusions and $J^{\eta}$ -proximal operator equations in Banach spaces, J. Math. Anal. Appl. 327 (2007), 515-524. DOI ScienceOn
13	Y. P. Fang and N. J.Huang, H-accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces, Appl. Math. Lett. 17 (2004), 647-653. DOI ScienceOn
14	R. P. Agarwal, Y. J. Cho and N. J. Huang, Sensitivity analysis for strongly nonlinear quasi-variational inclusions, Appl. Math. Lett. 13(6) (2000), 19-24.