- Research Article
- Open Access
Fixed Point Iterations of a Pair of Hemirelatively Nonexpansive Mappings
© Y. Hao and S. Y. Cho. 2010
Received: 27 September 2009
Accepted: 22 March 2010
Published: 31 March 2010
We introduce an iterative method for a pair of hemirelatively nonexpansive mappings. Strong convergence of the purposed iterative method is obtained in a Banach space.
1. Introduction and Preliminaries
where denotes the generalized duality pairing. A Banach space is said to be strictly convex if for all with and It is said to be uniformly convex if for any two sequences in such that and . Let be the unit sphere of . Then the Banach space is said to be smooth provided that
exists for each It is also said to be uniformly smooth if the limit (1.2) is attained uniformly for . It is well known that if is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of . It is also well known that is uniformly smooth if and only if is uniformly convex.
Recall that a Banach space has the Kadec-Klee property if for any sequences and with and then as ; for more details on Kadec-Klee property, the readers is referred to [1, 2] and the references therein. It is well known that if is a uniformly convex Banach space, then enjoys the Kadec-Klee property.
Let be a nonempty closed and convex subset of a Banach space and : → a mapping. The mapping is said to be closed if for any sequence such that and , then . A point is a fixed point of provided . In this paper, we use to denote the fixed point set of and use and to denote the strong convergence and weak convergence, respectively.
It is well known that if is a nonempty bounded closed and convex subset of a uniformly convex Banach space , then every nonexpansive self-mapping on has a fixed point. Further, the fixed point set of is closed and convex.
As we all know that if is a nonempty closed convex subset of a Hilbert space and : → is the metric projection of onto , then is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. In this connection, Alber  recently introduced a generalized projection operator in a Banach space which is an analogue of the metric projection in Hilbert spaces.
Observe that, in a Hilbert space , (1.4) is reduced to The generalized projection : → 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
Existence and uniqueness of the operator follow from the properties of the functional and strict monotonicity of the mapping (see, e.g., [1–4]). In Hilbert spaces, It is obvious from the definition of function that
If is a reflexive, strictly convex and smooth Banach space, then for , if and only if . It is sufficient to show that if then . From (1.6), we have . This implies that From the definition of we have . Therefore, we have see [1, 2] for more details.
Let be a nonempty closed convex subset of and a mapping from into itself. A point in is said to be an asymptotic fixed point of  if contains a sequence which converges weakly to such that . The set of asymptotic fixed points of will be denoted by . A mapping from into itself is said to be relatively nonexpansive [3, 6, 7] if and for all and . The mapping is said to be hemirelatively nonexpansive [8–12] if and for all and . The asymptotic behavior of a relatively nonexpansive mappings was studied in [3, 6, 7].
The class of hemirelatively nonexpansive mappings is more general than the class of relatively nonexpansive mappings which requires the restriction: . From Su et al. , we see that every hemirelatively nonexpansive mapping is relatively nonexpansive, but the inverse is not true. Hemirelatively nonexpansive mapping is also said to be quasi- -nonexpansive; see [13–17].
Recently, fixed point iterations of relatively nonexpansive mappings and hemirelatively nonexpansive mappings have been considered by many authors; see, for example [14–25] and the references therein. In 2005, Matsushita and Takahashi  considered fixed point problems of a single relatively nonexpansive mapping in a Banach space. To be more precise, they proved the following theorem.
Theorem 1 MT.
In 2007, Plubtieng and Ungchittrakool  further improved Theorem MT by considering a pair of relatively nonexpansive mappings. To be more precise, they proved the following theorem.
with the following restrictions:
Very recently, Su et al.  improved Theorem PU partially by considering a pair of hemirelatively nonexpansive mappings. To be more precise, they obtained the following results.
with the following restrictions:
In this paper, motivated by Theorems MT, PU, and SWX, we consider the problem of finding a common fixed point of a pair of hemirelatively nonexpansive mappings by shrinking projection methods which were introduced by Takahashi et al.  in Hilbert spaces. Strong convergence theorems of common fixed points are established in a Banach space. The results presented in this paper mainly improve the corresponding results announced in Matsushita and Takahashi , Nakajo and Takahahsi , and Su et al. .
In order to prove our main results, we need the following lemmas.
Lemma 1.3 (see ).
Lemma 1.4 (see ).
The following lemma can be deduced from Matsushita and Takahashi .
Lemma 1.6 (see ).
2. Main Results
Theorem 2.1 improves Theorem SWX in the following aspects:
from the point of view on the framework of spaces, we extend Theorem SWX from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space which enjoys the Kadec-Klee property. Note that every uniformly convex Banach space enjoys the Kadec-Klee property.
This project is supported by the National Natural Science Foundation of China (no. 10901140).
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