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.
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