Strong Convergence Theorems of Common Fixed Points for a Family of Quasi- -Nonexpansive Mappings
© Xiaolong Qin et al. 2010
Received: 31 August 2009
Accepted: 19 November 2009
Published: 24 November 2009
We consider a modified Halpern type iterative algorithm for a family of quasi- -nonexpansive mappings in the framework of Banach spaces. Strong convergence theorems of the purposed iterative algorithms are established.
Let be a Banach space, a nonempty closed and convex subset of , and a nonlinear mapping. Recall that is nonexpansive if
A point is a fixed point of provided . Denote by the set of fixed points of , that is, .
where is a fixed element. Banach Contraction Mapping Principle guarantees that has a unique fixed point in . It is unclear, in general, what the behavior of is as even if has a fixed point. However, in the case of having a fixed point, Browder  proved the following well-known strong convergence theorem.
Theorem 1 B.
Let be a bounded closed convex subset of a Hilbert space and a nonexpansive mapping on . Fix and define as for any . Then converges strongly to an element of nearest to .
Motivated by Theorem B, Halpern  considered the following explicit iteration:
and obtained the following theorem.
Theorem 1 H.
Let be a bounded closed convex subset of a Hilbert space and a nonexpansive mapping on . Define a real sequence in by , . Then the sequence defined by (1.3) converges strongly to the element of nearest to .
In , Lions improved the result of Halpern , still in Hilbert spaces, by proving the strong convergence of to a fixed point of provided that the control sequence satisfies the following conditions:
It was observed that both the Halpern's and Lion's conditions on the real sequence excluded the canonical choice . This was overcome by Wittmann , who proved, still in Hilbert spaces, the strong convergence of to a fixed point of if satisfies the following conditions:
In , Shioji and Takahashi extended Wittmann's results to the setting of Banach spaces under the assumptions (C1), (C2), and (C4) imposed on the control sequences . In , Xu remarked that the conditions (C1) and (C2) are necessary for the strong convergence of the iterative sequence defined in (1.3) for all nonexpansive self-mappings. It is well known that the iterative algorithm (1.3) is widely believed to have slow convergence because the restriction of condition (C2). Thus, to improve the rate of convergence of the iterative process (1.3), one cannot rely only on the process itself.
Recently, hybrid projection algorithms have been studied for the fixed point problems of nonlinear mappings by many authors; see, for example, [8–24]. In 2006, Martinez-Yanes and Xu  proposed the following modification of the Halpern iteration for a single nonexpansive mapping in a Hilbert space. To be more precise, they proved the following theorem.
Theorem 1 MYX.
converges strongly to
Theorem 1 QS.
where is the single-valued duality mapping on . If is nonempty, then converges to
In this paper, motivated by Kimura and Takahashi , Martinez-Yanes and Xu , Qin and Su , and Qin et al. , we consider a hybrid projection algorithm to modify the iterative process (1.3) to have strong convergence under condition (C1) only for a family of closed quasi- -nonexpansive mappings.
Let be a Banach space with the dual space . We denote by the normalized duality mapping from to defined by
where denotes the generalized duality pairing. It is well known that, if is strictly convex, then is single-valued and, if is uniformly convex, then is uniformly continuous on bounded subsets of .
We 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.
A Banach space is said to be strictly convex if for all with and . The space is said to be uniformly convex if for any two sequences and in such that and . Let be the unit sphere of . Then the space is said to be smooth if
exists for each It is also said to be uniformly smooth if the limit 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 .
In a smooth Banach space , we consider the functional defined by
Observe that, in a Hilbert space , (2.3) reduces to for all The generalized projection is a mapping that assigns to an arbitrary point the minimum point of the functional that is, where is the solution to the minimization problem:
The existence and uniqueness of the operator follows from some properties of the functional and the strict monotonicity of the mapping (see, e.g., [25–28]). In Hilbert spaces, It is obvious from the definition of the function that
If is a reflexive, strictly convex, and smooth Banach space, then, for any , if and only if . In fact, it is sufficient to show that, if , then . From (2.5), we have . This implies From the definition of one has . Therefore, we have (see [27, 29] for more details).
Let be a nonempty closed and convex subset of and a mapping from into itself. A point 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 ([27, 31, 32]) if and for all and . The asymptotic behavior of a relatively nonexpansive mapping was studied by some authors ([27, 31, 32]).
The class of quasi- -nonexpansive mappings is more general than the class of relatively nonexpansive mappings, which requires the strong restriction: .
In order to prove our main results, we need the following lemmas.
Lemma 2.3 (see ).
Let be a uniformly convex and smooth Banach space and , two sequences of . If and either or is bounded, then
Let be a uniformly convex and smooth Banach space, a nonempty, closed, and convex subset of and a closed quasi- -nonexpansive mapping from into itself. Then is a closed and convex subset of .
3. Main Results
From now on, we use to denote an index set. Now, we are in a position to prove our main results.
then the sequence defined by (3.1) converges strongly to .
This shows that is closed and convex for each and Therefore, we obtain that is convex for each .
Next, we show that for all . For each and , we have
which yields that for all and It follows that . This proves that for all .
Next, we prove that for all We prove this by induction. For we have Assume that for some . Next, we show that for the same . Since is the projection of onto we obtain that
Taking the limit as in (3.8), we get that From Lemma 2.3, one has as It follows that is a Cauchy sequence in . Since is a Banach space and is closed and convex, we can assume that as .
Finally, we show that To end this, we first show . By taking in (3.8), we have
On the other hand, we have By the assumption on , we see that for each Since is also uniformly norm-to-norm continuous on bounded sets, we obtain that
On the other hand, we have
From (3.10)–(3.13), we obtain From the closedness of , we get
Finally, we show that From , we see that
and hence by Lemma 2.4. This completes the proof.
Comparing the hybrid projection algorithm (3.1) in Theorem 3.1 with algorithm (1.5) in Theorem QS, we remark that the set is constructed based on the set instead of for each We obtain that the sequence generated by the algorithm (3.1) is a Cauchy sequence. The proof is, therefore, different from the one presented in Qin and Su .
As a corollary of Theorem 3.1, for a single quasi- -nonexpansive mapping, we have the following result immediately.
then the sequence converges strongly to .
Corollary 3.3 mainly improves Theorem of Qin and Su  from the class of relatively nonexpansive mappings to the class of quasi- -nonexpansive mappings, which relaxes the strong restriction:
In the framework of Hilbert spaces, Theorem 3.1 is reduced to the following result.
then the sequence converges strongly to .
Corollary 3.5 includes the corresponding result of Martinez-Yanes and Xu  as a special case. To be more precise, Corollary 3.5 improves Theorem 3.1 of Martinez-Yanes and Xu  from a single mapping to a family of mappings and from nonexpansive mappings to quasi-nonexpansive mappings, respectively.
This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).
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