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.
The set of such solutions is denoted by .
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. ).
where is the duality mapping on . Then, converges strongly to , where is the generalized projection from onto .
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.
where denotes the generalized duality pairing. It is well known that if is uniformly convex, then is uniformly continuous on bounded subsets of . In this case, is single valued and also one to one.
If is a Hilbert space, then and is the metric projection of onto .
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, .
Let be a closed convex subset of and let be a mapping from into itself. We denote by the set of fixed points of . is called hemirelatively nonexpansive if for all and .
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 ).
Let be a smooth and uniformly convex Banach space and let and be sequences in such that either or is bounded. If . Then .
Lemma 2.4 (see Xu ).
Lemma 2.5 (see Kamimura and Takahashi ).
(A1) for all
(A2) is monotone, that is, for all
(A3) for all ;
(A4) for all is convex.
Lemma 2.6 (see Blum and Oettli ).
Lemma 2.7 (see Takahashi and Zembayashi ).
is single valued;
- (2)is a firmly nonexpansive-type mapping , that is, for all ,(211)
is closed and convex.
Lemma 2.8 (see Takahashi and Zembayashi ).
Lemma 2.9 (see Su et al. ).
Let be a strictly convex and smooth real Banach space, let be a closed convex subset of , and let be a hemirelatively nonexpansive mapping from into itself. Then, is closed and convex.
Recall that an operator in a Banach space is called closed, if , then .
3. Strong Convergence Theorem
for every , where is the duality mapping on are sequences in such that and for some . Then, converges strongly to , where is the generalized projection of onto .
First, we can easily show that and are closed and convex for each .
As , by the induction assumptions, the last inequality holds, in particular, for all . This, together with the definition of , implies that . So, is well defined.
From (3.9), we can prove that is a Cauchy sequence. Therefore, there exists a point such that converges strongly to .
Since is a closed operator and , then is a fixed point of .
From and , we have .
By the definition of , it follows that and , whence . Therefore, it follows from the uniqueness of that . This completes the proof.
for every , where is the duality mapping on and for some . Then, converges strongly to .
Putting in Theorem 3.1, we obtain Corollary 3.2.
for every , where is the duality mapping on are sequences in such that . Then, converges strongly to .
Putting for all and for all in Theorem 3.1, we obtain Corollary 3.3.
for every , where is the duality mapping on are sequences in such that and for some . Then, converges strongly to .
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 monotone operator from to . Then, is maximal if and only if for all .
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.
for every , where is the duality mapping on , is a sequences in such that and for some , Then, converges strongly to .
Since is a closed relatively nonexpansive mapping and , from Corollary 3.4, we obtain Theorem 3.6.
This work is supported by Tianjin Natural Science Foundation in China Grant no. 06YFJMJC12500.
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