Abstract
In the framework of a real Banach space with uniformly Gateaux differentiable norm, some new viscosity iterative sequences are introduced for an infinite family of asymptotically nonexpansive mappings in this paper. Under some appropriate conditions, we prove that the iterative sequences converge strongly to a common fixed point of the mappings , which is also a solution of a variational inequality. Our results extend and improve some recent results of other authors.
1. Introduction
Let be a real Banach space, a nonempty subset of , and a nonlinear mapping. Denote by the set of fixed points of . Recall that is said to be nonexpansive, if is said to be contraction, if there exists a constant , such that We use to denote the collection of all contractions on .
is said to be asymptotically nonexpansive, if there exists a sequence with as such that is said to be uniform Lipschitzian with the coefficient , if for any , there holds
It is clear that the class of contraction mappings must be included in the class of nonexpansive mappings and the class of nonexpansive mappings in that of asymptotically nonexpansive mappings.
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] in 1972. They proved that if is a nonempty bounded closed convex subset of a uniformly convex Banach space , then every asymptotically nonexpansive self-mapping has a fixed point in . Further, the set of fixed points of is closed and convex.
In 2001, Khan and Takahasi [2] used the modified Ishikawa process to approximate common fixed points of two asymptotically nonexpansive mappings. In 2003, Sun [3] studied an implicit iterative scheme initiated by Xu and Ori [4] for a finite family of asymptotically quasi-nonexpansive mappings. Shahzad and Udomene [5], in 2006, proved some convergence theorems for the modified Ishikawa iterative process of two asymptotically quasi-nonexpansive mappings to a common fixed point. Shahzad and Zegeye [6] introduced a new concept of generalized asymptotically nonexpansive mappings and proved some strong convergence theorems for fixed points of finite family of this class. In 2008, Khan et al. [7] introduced an iterative sequence for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces. Meanwhile, Zhao [8] proved the following conclusion.
Let be a uniformly smooth Banach space, , and let be a finite family of nonexpansive mappings from into itself, such that the set is nonempty. Under some sufficient conditions, the iterative sequence defined by (5) converges strongly to a common fixed point of . Consider
Common fixed points of nonlinear mappings play an important role in solving systems of equations and inequalities. Many researchers [9–19] are interested in studying approximation method for finding common fixed points of nonlinear mapping in recent years.
Motivated and inspired by the above results, in this paper, for an infinite family of asymptotically nonexpansive mappings from into itself, we introduce a new viscosity iterative process defined by Under appropriate conditions on , , , and in , we prove that the converges strongly to , which is also a solution of the following variational inequality: Our results extend and improve some results of other authors (e.g., see [8, 17, 19]) from nonexpansive mappings to the more general class of asymptotically nonexpansive mappings and from finite family mappings to infinite family mappings.
2. Preliminaries
In order to prove our results, we need the following definitions and lemmas.
Assume that is a real Banach space and is the dual space of .
is the normalized duality mapping defined by The space is said to be with Gateaux differentiable norm, if the limit exists for each and any in its unit sphere .
In a Banach space whose norm is uniformly Gateaux differentiable, the duality mapping is single-valued and uniformly continuous on any bounded sets of .
Lemma 1 (see [18]). In a Banach space , there holds the inequality
Lemma 2 (see [20]). Let be a sequence of nonnegative real numbers satisfying the property
where and are two sequences such that
Then .
In order to prove the main results of this paper, the following lemmas should be used.
Lemma 3 (see [21]). Let , be bounded sequences in a Banach space X, , satisfying , and suppose Then .
3. Main Results
Now, we are ready to give the main results.
Theorem 4. Let K be a nonempty closed convex subset of a real Banach space E with uniformly Gateaux differentiable norm. Let be an infinite family of asymptotically nonexpansive mappings with the coefficients and uniform Lipschitzian with the coefficient from K into itself, and with the coefficient . Assume that the sequences satisfy the following conditions:(C1);
(C2);(C3).
Then the sequence defined by (6) converges strongly to if and only if, for any i, holds. And is a solution of the following variational inequality:
Proof.
Sufficiency. The sufficient proof is divided into five steps.
Step 1. We observe that is bounded. Indeed, let , . Taking a fixed point of , we have
Using an introduction, we have
Hence, is bounded so are the sets and .
Step 2. We claim that . Setting , it follows from (C1) and (C2) that
Define , and observe that
We have
Using the conclusion of step 1, by (C1), and , and we obtain that .
Hence, by Lemma 3, we have . Consequently,
Step 3. We prove . From (6), we arrive at
Since and are bounded, by (C1) and , we get . As , then .
Step 4. We show that . Same as [15], let with being the fixed point of the contraction , where . That is, . Thanks to Lemma 1, we have
where . It follows from step 2 that . Then from
we see that
where , such that
Then
So for any , there exists . When , we get
On the other hand, because and is norm-to-norm uniformly continuous on bounded subsets of , there exists , such that when , we have
Choosing , , we have
Since is chosen arbitrarily, we get
Hence, we have
Step 5. We prove that . Setting , we have
So,
which implies that
where
Since is bounded, by (C1), (C3), and step 3, we have
According to Lemma 2, we deduce that .
Necessity. Since , , is uniform Lipschitzian,
Hence, the proof of Theorem 4 is completed.
Theorem 5. Under the same conditions as those in Theorem 4, then the sequence defined by converges strongly to , if and only if holds. And is a solution of the following variational inequality:
Proof. The proof is the same as that of Theorem 4. We can just use to take the place of .
Theorem 6. Let be a finite family of asymptotically nonexpansive mappings with the coefficients and . The rest of the conditions are the same as those in Theorem 4. Then the sequence defined by (38) converges strongly to , if and only if holds. And is a solution of the following variational inequality:
Proof. This is a special case of Theorem 5.
Remark 7. If is a finite family of nonexpansive mappings in Theorem 6, then it is the main result in reference [8]. So our results in this paper extend and improve the recent results of many other authors (e.g., see [1, 8, 17, 19]) in different ways, such as from nonexpansive mappings to the more general class of asymptotically nonexpansive mappings and from finite family mappings to infinite family mappings. And the methods here are different from the previous ones.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author would like to thank the editors and referees for many useful comments and suggestions for the improvement of the paper. This work was partially supported by the Natural Science Foundation of Zhejiang Province (LY14A010011) and the National Natural Science Foundation of China (11271330).