Strong Convergence of Monotone Hybrid Method for Maximal Monotone Operators and Hemirelatively Nonexpansive Mappings
© C. Klin-eam and S. Suantai. 2009
Received: 20 May 2009
Accepted: 21 September 2009
Published: 11 October 2009
We prove strong convergence theorems for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a hemirelatively nonexpansive mapping in a Banach space by using monotone hybrid iteration method. By using these results, we obtain new convergence results for resolvents of maximal monotone operators and hemirelatively nonexpansive mappings in a Banach space.
Let be a real Banach space and let be the dual space of Let be a maximal monotone operator from to . It is well known that many problems in nonlinear analysis and optimization can be formulated as follows. Find a point satisfying
We denote by the set of all points such that Such a problem contains numerous problems in economics, optimization, and physics and is connected with a variational inequality problem. It is well known that the variational inequalities are equivalent to the fixed point problems. There are many authors who studied the problem of finding a common element of the fixed point of nonlinear mappings and the set of solutions of a variational inequality in the framework of Hilbert spaces see; for instance, [1–11] and the reference therein.
In 2005, Matsushita and Takahashi  proposed the following hybrid iteration method (it is also called the CQ method) with generalized projection for relatively nonexpansive mapping in a Banach space : chosen arbitrarily,
In 2008, Su et al.  modified the CQ method (1.3) for approximation a fixed point of a closed hemi-relatively nonexpansive mapping in a Banach space. Their method is known as the monotone hybrid method defined as the following. chosen arbitrarily, then
Note that the hybrid method iteration method presented by Matsushita and Takahashi  can be used for relatively nonexpansive mapping, but it cannot be used for hemi-relatively nonexpansive mapping.
Very recently, Inoue et al.  proved the following strong convergence theorem for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a relatively nonexpansive mapping by using the hybrid method.
Theorem 1.1 (Inoue et al. ).
Employing the ideas of Inoue et al.  and Su et al. , we modify iterations (1.4) and (1.5) to obtain strong convergence theorems for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a hemi-relatively nonexpansive mapping in a Banach space. Using these results, we obtain new convergence results for resolvents of maximal monotone operators and hemi-relatively nonexpansive mappings in a Banach space. The results of this paper modify and improve the results of Inoue et al. , and some others.
Throughout this paper, all linear spaces are real. Let and be the sets of all positive integers and real numbers, respectively. Let be a Banach space and let be the dual space of . For a sequence of and a point the weak convergence of to and the strong convergence of to are denoted by and , respectively.
exists for all . It is also said to be uniformly smooth if the limit exists uniformly in . A Banach space is said to be strictly convex if whenever and . It is said to be uniformly convex if for each , there exists such that whenever and . We know the following (see, ):
Following Alber , the generalized projection from onto is a map that assigns to an arbitrary point the minimum point of the functional ; that is, , where is the solution to the minimization problem
Let be a closed convex subset of a Banach space and let be a mapping from into itself. We use to denote the set of fixed points of that is, Recall that a self-mapping is hemi-relatively nonexpansive if and for all and .
A point is said to be an asymptotic fixed point of if contains a sequence which converges weakly to and . We denote the set of all asymptotic fixed points of by . A hemi-relative nonexpansive mapping is said to be relatively nonexpansive if . The asymptotic behavior of a relatively nonexpansive mapping was studied in .
We need the following lemmas for the proof of our main results.
Lemma 2.1 (Kamimura and Takahashi ).
Lemma 2.2 (Matsushita and Takahashi ).
Let be a smooth, strictly convex, and reflexive Banach space, and let be a set-valued mapping from to with graph , domain and range We denote a set-valued operator from to by is said to be monotone of A monotone operator is said to be maximal monotone if its graph is not properly contained in the graph of any other monotone operator. It is known that a monotone mapping is maximal if and only if for for every implies that . We know that if is a maximal monotone operator, then is closed and convex; see  for more details. The following result is well known.
Lemma 2.5 (Rockafellar ).
Lemma 2.6 (Kohsaka and Takahashi ).
Lemma 2.7 (Kamimura and Takahashi ).
3. Main Results
In this section, we prove a strong convergence theorem for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a hemi-relatively nonexpansive mapping in a Banach space by using the monotone hybrid iteration method.
As direct consequences of Theorem 3.1, we can obtain the following corollaries.
for each . Then, we know that is a maximal monotone operator; see  for more details.
Corollary 3.3 (Su et al. [19, Theorem 3.1]).
So, we obtain the desired result by using Theorem 3.1.
Since every relatively nonexpansive mapping is a hemi-relatively one, the following theorem is obtained directly from Theorem 3.1.
Corollary 3.5 (Su et al. [19, Theorem 3.2]).
The authors would like to thank the referee for valuable suggestions that improve this manuscript and the Thailand Research Fund (RGJ Project) and Commission on Higher Education for their financial support during the preparation of this paper. The first author was supported by the Royal Golden Jubilee Grant PHD/0018/2550 and by the Graduate School, Chiang Mai University, Thailand.
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