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Strong convergence theorems for three-steps iterations for asymptotically nonexpansive mappings in Banach spaces
Fixed Point Theory and Applications volume 2013, Article number: 150 (2013)
Abstract
We consider the problem of the convergence of the three-steps iterative sequences for asymptotically nonexpansive mappings in a real Banach space. Under suitable conditions, it has been proved that the iterative sequence converges strongly to a fixed point, which is also a solution of certain variational inequality. The results presented in this paper extend and improve some recent results.
1 Introduction
Let X be a real Banach space with dual , denotes the normalized duality mapping from X into given by
Let C be a subset of X. A mapping is called contraction if there exists a constant such that for any . The mapping T is called nonexpansive if for any , and it is called asymptotically nonexpansive if there exists a sequence in the interval with and such that
for all and all , where N is the set of natural numbers.
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] as an important generalization of nonexpansive mappings. They proved that if C is a nonempty, bounded, closed and convex subset of a real uniformly convex Banach space and T is an asymptotically nonexpansive self-mapping of C, then T has a fixed point in C. In 2000, Noor [2] introduced a three-steps iterative scheme and studied the approximate solutions of a variational inclusion in Hilbert spaces. In 2002, Xu and Noor [3] introduced and studied a new class of three-steps iterative schemes for solving the nonlinear equation for asymptotically nonexpansive mappings T in uniformly convex Banach spaces. In 2006, Nilsrakoo and Saejung [4] defined a three-steps mean value iterative scheme and extended the results of Xu and Noor [3]. In 2007, Yao and Noor [5] made a refinement and improvement of the previously known results.
Now we define a new three-steps iteration scheme for asymptotically nonexpansive mappings as follows.
Definition 1.1 Let X be a Banach space, C be a nonempty convex subset of X, be an asymptotically nonexpansive mapping and be a contraction. For a given and , let us define the sequences , and by the iterative scheme
where , and are approximate sequences in .
If , then the iterations defined in (2) reduces to the two-steps iterations defined as follows.
Definition 1.2 For a given and , let us define the sequences and by the iterative scheme
where and are approximate sequences in .
If , then the iterations defined in (2) reduces to the one-step iterations defined as follows.
Definition 1.3 For a given and , define the sequence by the iterative scheme
where is an approximate sequence in .
The purpose of this paper is to establish a strong convergence theorem of the three-steps iterations for asymptotically nonexpansive mappings in a real Banach space equipped with a uniformly Gâteaux differentiable norm and to present some corollaries. Our results extend and improve the corresponding ones announced by Ceng et al. [6], Chang et al. [7], Lou et al. [8], Shahzad and Udomene [9] and others.
2 Preliminaries
Throughout this paper, we will use the following notions. Let X be a real Banach space with the norm and let be its dual space. When is a sequence in X, then (respectively , ) will denote the strong (respectively the weak, the weak star) convergence of the sequence to x. Of course, the weak star convergence is considered in Banach spaces X which are dual spaces. We shall denote the single-valued duality mapping and the set of fixed points for a mapping T by j and , respectively.
Definition 2.1 Let denote the unit sphere of a Banach space X. The space X is said to have a uniformly Gâteaux differentiable norm , if for each the limit
exists uniformly with respect to .
It is well known [10] that if X is equipped with a uniformly Gâteaux differentiable norm, then any duality mapping on X is single-valued and it is norm-to-weak* uniformly continuous, that is, implies that .
Lemma 2.2 [11]
Let X be a real Banach space. Then for each , the following inequality holds:
Lemma 2.3 [12]
Let , , be three nonnegative real sequences satisfying
with , , , and . Then .
Now, we start with our first result.
Lemma 2.4 Let X be a real Banach space, C be a nonempty convex subset of X and be an asymptotically nonexpansive mapping defined by (1) with and be the composite process defined by iterative scheme (2). Then the sequence is bounded.
Proof Let and . We have from (2) that
and so
Hence, it follows by induction that
Therefore, is bounded. □
In order to prove our results, we also need the following lemma; see [8].
Lemma 2.5 Let X be a real Banach space equipped with a uniformly Gâteaux differentiable norm, C a bounded, closed and convex subset of X, an asymptotically nonexpansive mapping defined by (1) with , a contraction with the contraction constant α. For any define the sequence of contractions by , where , and . Let be the unique fixed points of , that is,
Then the sequence converges strongly to the unique solution of the following variational solution p:
Lemma 2.6 Let C be a closed convex subset of a real Banach space X, be an asymptotically nonexpansive mapping and be a contraction with the contraction constant α. Let us assume that there are given three sequences , and in satisfying the following conditions:
-
(i)
, , and ;
-
(ii)
and that is the composite process defined by the iterative scheme (2). Then we have the following assertions:
-
(a)
;
-
(b)
if , then .
Proof (a) By Lemma 2.4, we know that the sequence is bounded. Hence, it follows that the sequences , , and are also bounded. Therefore, we have from (2) that
and
where it follows that
where
Obviously, by condition (i), we have and
It follows from Lemma 2.3 and condition (ii) that .
(b) By (a), , so it follows from (2) and (i) that and
Now, we will prove that . We have from (2) that
It follows from (7) that , where, by the condition , we have that
Finally, we will show that . In fact, according to (8), we have
Hence,
which implies that . □
3 Main results
Theorem 3.1 Let X be a real Banach space equipped with a uniformly Gâteaux differentiable norm, C be a bounded, closed and convex subset of X, be an asymptotically nonexpansive mapping defined by (1) with and be a contraction with the contraction constant α. Let be the sequence defined by the iterative scheme (2) with , and satisfying the following conditions:
-
(C1) , , and ;
-
(C2) ;
-
(C3) .
Then the sequence converges strongly to the unique solution p of the variational inequality:
Proof Since C is closed, by Lemma 2.4, is bounded, so , , and are also bounded. Let be the sequence defined by
It follows from Lemma 2.5 that the sequence converges strongly to a fixed point p of T and p is also the unique solution of the variational inequality (6). We will next prove that
By Lemma 2.6(b), . It is easy to show that
where if we put , then as . On the other hand, we have from (10) that
It follows by Lemma 2.2 that
Hence,
Since and the sequences and are bounded it follows that for some constant , we have
Since as and the duality mapping is norm-to-weakly* uniformly continuous, we obtain that
Finally, we will show that . We have
and so
where . By (11), we have , and . Then it follows that
If we define , and , then applying Lemma 2.3 we conclude that . Moreover, it follows from (12) that p satisfies condition (9). In order to show that p is unique, let be another solution of (9) in F. Then adding the inequalities and , we get that , which implies the equality . □
The following example gives a mapping T, which is not nonexpansive but satisfying all the assumptions of Theorem 3.1.
Example 3.2 Let B denote the unit ball in the Hilbert space and let be defined as follows:
where , . Then it is easy to verify that T is an asymptotically nonexpansive mapping with , but it is not nonexpansive. If we set , and , then the real sequences , and satisfy conditions (C1), (C2) and (C3) from Theorem 3.1, and it is easy to prove that 0 is the unique fixed point of T in B.
If in Theorem 3.1, then we have by (2) that . In fact, we have the following corollary.
Corollary 3.3 Let X be a real Banach space equipped with a uniformly Gâteaux differentiable norm, C be a bounded closed convex subset of X, be an asymptotically nonexpansive mapping defined by (1) with and be a contraction with the contraction constant α. Let be the sequence defined by the iterative scheme (3) with and satisfying the following conditions:
-
(C1) , , and ;
-
(C2) ;
-
(C3) .
Then the sequence converges strongly to the unique solution p of the variational inequality (9).
If in Theorem 3.1, then we have by (2) that . Hence, it follows that the following result is satisfied.
Corollary 3.4 Let X be a real Banach space equipped with a uniformly Gâteaux differentiable norm, C be a bounded closed convex subset of X, be an asymptotically nonexpansive mapping defined by (1) with and be a contraction with the contraction constant α. Let be the sequence defined by the iterative scheme (4) with satisfying the following conditions:
-
(C1) , and ;
-
(C2) ;
-
(C3) either or .
Then the sequence converges strongly to the unique solution of the variational inequality (9).
Remarks 3.5
-
1.
If and is a constant function in Theorem 3.1, then the iterative scheme (2) reduces to the following iterative scheme:
In consequence, Corollary 3.3 improves Theorem 1 of Chang et al. from [7].
-
2.
Let in Theorem 3.1 and the iterative scheme (3) be replaced by the following scheme:
Then by Theorem 3.1, we have the more general result than the result of Lou et al. from [8] and Corollary 3.6 of Ceng et al. in [6]. If T and are as in Corollary 3.4, assume that , and . Then the sequence defined by converges strongly to the unique solution of the variational inequality (9).
-
3.
Theorem 3.1 and Corollary 3.4 extend Theorem 3.3 of Shahzad and Udomene in [9] to a more wide class of spaces.
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Acknowledgements
The first author is supported by foundation of Heilongjiang Province education department (12521070). The second author is supported by the fund of Scientific research in Southeast University (the support project of fundamental research). The last author is supported by the State Committee for Scientific Research, Poland, Grant No. N N201 362236.
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An, C.Y., Fei, Z.Z. & Hudzik, H. Strong convergence theorems for three-steps iterations for asymptotically nonexpansive mappings in Banach spaces. Fixed Point Theory Appl 2013, 150 (2013). https://doi.org/10.1186/1687-1812-2013-150
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DOI: https://doi.org/10.1186/1687-1812-2013-150