THE SHRINKING PROJECTION METHODS FOR HEMI-RELATIVELY NONEXPANSIVE MAPPINGS, VARIATIONAL INEQUALITIES AND EQUILIBRIUM PROBLEMS

Title & Authors
THE SHRINKING PROJECTION METHODS FOR HEMI-RELATIVELY NONEXPANSIVE MAPPINGS, VARIATIONAL INEQUALITIES AND EQUILIBRIUM PROBLEMS
Wang, Zi-Ming; Kang, Mi Kwang; Cho, Yeol Je;

Abstract
In this paper, we introduce the shrinking projection method for hemi-relatively nonexpansive mappings to find a common solution of variational inequality problems and equilibrium problems in uniformly convex and uniformly smooth Banach spaces and prove some strong convergence theorems to the common solution by using the proposed method.
Keywords
variational inequality;equilibrium problem;hemi-relatively non-expansive mapping;shrinking projection method;
Language
English
Cited by
References
1.
Ya. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, Theory and applications of nonlinear operators of accretive and monotone type, 1550, Lecture Notes in Pure and Appl. Math., 178, Dekker, New York, 1996.

2.
Ya. Alber and S. Reich, An iterative method for solving a class of nonlinear operator equations in Banach spaces, Panamer. Math. J. 4 (1994), no. 2, 39-54.

3.
E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student 63 (1994), no. 1-4, 123-145.

4.
D. Butnariu, S. Reich, and A. J. Zaslavski, Weak convergence of orbits of nonlinear operators in reflexive Banach spaces, Numer. Funct. Anal. Optim. 24 (2003), no. 5-6, 489-508.

5.
S. S. Chang, On Chidume's open questions and approximate solutions of multivalued strongly accretive mapping equations in Banach spaces, J.Math. Anal. Appl. 216 (1997), no. 1, 94-111.

6.
P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005), no. 1, 117-136.

7.
J. Fan, A Mann type iterative scheme for variational inequalities in noncompact subsets of Banach spaces, J. Math. Anal. Appl. 337 (2008), no. 2, 1041-1047.

8.
S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147-150.

9.
S. Kamimura and W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim. 13 (2002), no. 3, 938-945.

10.
J. Li, On the existence of solutions of variational inequalities in Banach spaces, J. Math. Anal. Appl. 295 (2004), no. 1, 115-126.

11.
Y. Liu, Strong convergence theorems for variational inequalities and relatively weak nonexpansive mappings, J. Global Optim. 46 (2010), no. 3, 319-329.

12.
S. Matsushita and W. Takahashi, Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. 2004 (2004), no. 1, 37-47.

13.
A. Moudafi, Second-order differential proximal methods for equilibrium problems, J. Inequal. Pure Appl. Math. 4 (2003), no. 1, 1-7.

14.
X. Qin, S. S. Chang, and Y. J. Cho, Iterative methods for generalized equilibrium problems and fixed point problems with applications, Nonlinear Anal. Real World Appl. 11 (2010), no. 4, 2963-2972.

15.
X. Qin, Y. J. Cho, and S. M. Kang, Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces, J. Comput. Appl. Math. 225 (2009), no. 1, 20-30.

16.
X. Qin, S. Y. Cho, and S. M. Kang, Strong convergence of shrinking projection methods for quasi-${\phi}$-nonexpansive mappings and equilibrium problems, J. Comput. Appl. Math. 234 (2010), no. 3, 750-760.

17.
S. Reich, Book Review: Geometry of Banach spaces, duality mappings and nonlinear problems, Bull. Amer. Math. Soc. (N.S.) 26 (1992), no. 2, 367-370.

18.
Y. Su, Z. Wang, and H. Xu, Strong convergence theorems for a common fixed point of two hemi-relatively nonexpansive mappings, Nonlinear Anal. 71 (2009), no. 11, 5616-5628.

19.
W. Takahashi, Nonlinear Functional Analysis, Yokohama-Publishers, 2000.

20.
W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118 (2003), no. 2, 417-428.

21.
W. Takahashi and K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal. 70 (2009), no. 1, 45-57.