- Research Article
- Open Access
Strong Convergence Theorem by Monotone Hybrid Algorithm for Equilibrium Problems, Hemirelatively Nonexpansive Mappings, and Maximal Monotone Operators
© Y. Cheng and M. Tian. 2008
Received: 17 June 2008
Accepted: 11 November 2008
Published: 18 November 2008
We introduce a new hybrid iterative algorithm for finding a common element of the set of fixed points of hemirelatively nonexpansive mappings and the set of solutions of an equilibrium problem and for finding a common element of the set of zero points of maximal monotone operators and the set of solutions of an equilibrium problem in a Banach space. Using this theorem, we obtain three new results for finding a solution of an equilibrium problem, a fixed point of a hemirelatively nonexpnasive mapping, and a zero point of maximal monotone operators in a Banach space.
In 2006, Martinez-Yanes and Xu  obtained strong convergence theorems for finding a fixed point of a nonexpansive mapping by a new hybrid method in a Hilbert space. In particular, Takahashi and Zembayashi  established a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a uniformly convex and uniformly smooth Banach space. Very recently, Su et al.  proved the following theorem by a monotone hybrid method.
Theorem 1.1 (see Su et al. ).
In this paper, motivated by Su et al. , we prove a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a hemirelatively nonexpansive mapping and for finding a common element of the set of zero points of maximal monotone operators and the set of solutions of an equilibrium problem in a Banach space by using the monotone hybrid method. Using this theorem, we obtain three new strong convergence results for finding a solution of an equilibrium problem, a fixed point of a hemirelatively nonexpnasive mapping, and a zero point of maximal monotone operators in a Banach space.
If is a reflexive strict convex and smooth Banach space, then for if and only if . It is sufficient to show that if , then . From (2.4), we have . This implies . From the definition of , we have , that is, .
A point in is said to be an asymptotic fixed point of  if contains a sequence which converges weakly to such that the strong . The set of asymptotic fixed points of will be denoted by . A hemirelatively nonexpansive mapping from into itself is called relatively nonexpansive [1, 5, 6] if .
We need the following lemmas for the proof of our main results.
Lemma 2.1 (see Alber ).
Lemma 2.2 (see Alber ).
Lemma 2.3 (see Kamimura and Takahashi ).
Lemma 2.4 (see Xu ).
Lemma 2.5 (see Kamimura and Takahashi ).
Lemma 2.6 (see Blum and Oettli ).
Lemma 2.7 (see Takahashi and Zembayashi ).
- (2)is a firmly nonexpansive-type mapping , that is, for all ,
Lemma 2.8 (see Takahashi and Zembayashi ).
Lemma 2.9 (see Su et al. ).
3. Strong Convergence Theorem
Since every relatively nonexpansive mapping is a hemirelatively one, Corollary 3.4 is implied by Theorem 3.1.
Remark 3.5 (see Rockafellar ).
Let be a reflexive, strictly convex, and smooth Banach space and let be a maximal monotone operator from to . Using Remark 3.5 and strict convexity of , we obtain that for every and , there exists a unique such that If , then we can define a single-valued mapping by , and such a is called the resolvent of . We know that for all and is relatively nonexpansive mapping (see  for more details). Using Theorem 3.1, we can consider the problem of strong convergence concerning maximal monotone operators in a Banach space.
This work is supported by Tianjin Natural Science Foundation in China Grant no. 06YFJMJC12500.
- Martinez-Yanes C, Xu H-K: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Analysis: Theory, Methods & Applications 2006, 64(11):2400-2411. 10.1016/j.na.2005.08.018MATHMathSciNetView ArticleGoogle Scholar
- Takahashi W, Zembayashi K: Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings. Fixed Point Theory and Applications 2008, 2008:-11.Google Scholar
- Su Y, Wang D, Shang M: Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings. Fixed Point Theory and Applications 2008, 2008:-8.Google Scholar
- Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathematics. Volume 178. Edited by: Kartsatos AG. Marcel Dekker, New York, NY, USA; 1996:15-50.Google Scholar
- Butnariu D, Reich S, Zaslavski AJ: Asymptotic behavior of relatively nonexpansive operators in Banach spaces. Journal of Applied Analysis 2001, 7(2):151-174. 10.1515/JAA.2001.151MATHMathSciNetView ArticleGoogle Scholar
- Matsushita S-Y, Takahashi W: A strong convergence theorem for relatively nonexpansive mappings in a Banach space. Journal of Approximation Theory 2005, 134(2):257-266. 10.1016/j.jat.2005.02.007MATHMathSciNetView ArticleGoogle Scholar
- Kamimura S, Takahashi W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM Journal on Optimization 2002, 13(3):938-945. 10.1137/S105262340139611XMathSciNetView ArticleGoogle Scholar
- Xu HK: Inequalities in Banach spaces with applications. Nonlinear Analysis: Theory, Methods & Applications 1991, 16(12):1127-1138. 10.1016/0362-546X(91)90200-KMATHMathSciNetView ArticleGoogle Scholar
- Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994, 63(1–4):123-145.MATHMathSciNetGoogle Scholar
- Takahashi W, Zembayashi K: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2009, 70(1):45-57. 10.1016/j.na.2007.11.031MATHMathSciNetView ArticleGoogle Scholar
- Kohsaka F, Takahashi W: Existence and approximation of fixed points of firmly nonexpansive type mappings in Banach spaces. to appear in SIAM Journal on OptimizationGoogle Scholar
- Rockafellar RT: On the maximality of sums of nonlinear monotone operators. Transactions of the American Mathematical Society 1970, 149: 75-88. 10.1090/S0002-9947-1970-0282272-5MATHMathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.