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THE SHRINKING PROJECTION METHODS FOR HEMI-RELATIVELY NONEXPANSIVE MAPPINGS, VARIATIONAL INEQUALITIES AND EQUILIBRIUM PROBLEMS
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 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;
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 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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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

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. crossref(new window)

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

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

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)