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

• Journal title : East Asian mathematical journal
• Volume 28, Issue 1,  2012, pp.37-47
• Publisher : Youngnam Mathematical Society
• DOI : 10.7858/eamj.2012.28.1.037
Title & Authors
PROXIMAL POINTS METHODS FOR GENERALIZED IMPLICIT VARIATIONAL-LIKE INCLUSIONS IN BANACH SPACES
He, Xin-Feng; Lou, Jian; He, Zhen;

Abstract
In this paper, we study generalized implicit variational-like inclusions and $\small{J^{\eta}}$-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 $\small{J^{\eta}}$-proximal operator equations. This equivalence is used to suggest a iterative algorithm for solving $\small{J^{\eta}}$-proximal operator equations.
Keywords
generalized implicit variational-like inclusions;$\small{J^{\eta}}$-Proximal operator;Algorithm;$\small{J^{\eta}}$-proximal operator equations;
Language
English
Cited by
References
1.
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.

2.
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.

3.
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.

4.
S. S. Chang, Y. J. Cho and H. Y. Zhou, Iterative Methods for Nonlinear Operator Equations in Banach Spaces, Nova Sci. New York, 2002.

5.
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.

6.
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.

7.
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.

8.
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.

9.
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.

10.
N. J. Huang, Generlaized nonlinear variational inclusions with non-compact valued mappings, Appl. Math. Lett. 9(3) (1996), 25-29.

11.
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.

12.
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.

13.
S. B. Nadler, Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475-488.

14.
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.