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Strong convergence of a general algorithm for nonexpansive mappings in Banach spaces
Fixed Point Theory and Applications volume 2012, Article number: 207 (2012)
In this work, we consider a general algorithm for a countable family of nonexpansive mappings in Banach spaces. We proved that the proposed algorithm converges strongly to a common fixed point of a countable family of nonexpansive mappings which solves uniquely the corresponding variational inequality. It is worth pointing out that our proofs contain some new techniques. Our results improve and extend the corresponding ones announced by many others.
MSC:47H05, 47H09, 47H10.
Let X be a real Banach space and let C be a nonempty closed convex subset of X. Recall that a mapping is said to be nonexpansive if , . We denote by the set of fixed points of T.
In 2009, Yao et al.  considered the following algorithm in a Hilbert space. For an arbitrary point ,
They proved if and satisfy appropriate conditions, the defined by (1.1) converges strongly to a fixed point of T.
Recently, motivated and inspired by the above results, Wang and Hu  introduced the following algorithm in a Hilbert space. For an arbitrary point ,
where is a metric projection, is a β-Lipschitzian and η-strongly monotone operators. They proved that the proposed algorithm converges strongly to , which solves the variational inequality , .
On the other hand, Aoyama et al.  considered the following algorithm in a uniformly convex and 2-uniformly smooth Banach space. For ,
where is a sunny nonexpansive retraction, and A is a β-Lipschitzian and η-inverse strongly accretive operator. They proved that generated by (1.3) converges weakly to a unique element z of , where .
The results of Yao et al.  and Wang and Hu  both are obtained when the underlying space is a Hilbert space. Meanwhile, Aoyama et al.  just obtained a weak convergence theorem for strongly accretive and Lipschitzian operators. So, the above results bring us the following natural question.
In this work, motivated and inspired by the above results, we introduce a general algorithm (3.1) (defined below) for a countable family of nonexpansive mappings in a uniformly convex and 2-uniformly smooth Banach space. We prove that the sequence defined by (3.1) converges strongly to , which solves uniquely the variational inequality , . Furthermore, we provide an affirmative answer to Question 1.1. It is worth pointing out that our proofs contain some new techniques.
Let X be a real Banach space with the norm and let be its dual space. The value of and will be denoted by . For the sequence in X, we write to indicate that the sequence converges weakly to x. means that converges strongly to x.
Let . A mapping F from C into X is said to be η-strongly accretive if there exists such that
for all . A mapping F from C into X is said to be β-Lipschitzian if, for ,
for all . It is well known that the η-strongly accretive operators are the extension of the η-strongly monotone operators from Hilbert spaces to Banach spaces.
Let . A Banach space X is said to be uniformly convex if, for each , there exists such that, for any ,
It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space X is said to be smooth if the limit
exists for all . It is said to be uniformly smooth if the limit (2.1) is attained uniformly for . Also, we define a function called the modulus of smoothness of X as follows:
It is known that X is uniformly smooth if and only if . Let q be a fixed real number with . Then a Banach space X is said to be q-uniformly smooth if there exists a constant such that for all . One should note that no Banach space is q-uniformly smooth for ; see  for more details. So, in this paper, we focus on a 2-uniformly smooth Banach space. It is well known that Hilbert spaces and Lebesgue () spaces are uniformly convex and 2-uniformly smooth.
In order to prove our main results, we need the following lemmas.
Lemma 2.1 ()
Let q be a given real number with and let X be a q-uniformly smooth Banach space. Then
for all , where K is the q-uniformly smooth constant of X and is the generalized duality mapping from X into defined by
for all .
Let D be a subset of C and let Q be a mapping of C into D. Then Q is said to be sunny if
whenever for and . A mapping Q of C into itself is called a retraction if . If a mapping Q of C into itself is a retraction, then for every , where is the range of Q. A subset D of C is called a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction from C onto D. The following lemma concerns the sunny nonexpansive retraction.
Lemma 2.2 ()
Let C be a closed convex subset of a smooth Banach space X, let D be a nonempty subset of C and Q be a retraction from C onto D. Then Q is sunny and nonexpansive if and only if
for all and .
Remark 2.3 It is well known that if X is a Hilbert space, then a sunny nonexpansive retraction is coincident with the metric projection from X onto C.
Lemma 2.4 ()
Let C be a nonempty bounded closed convex subset of a uniformly convex Banach space X and let T be a nonexpansive mapping of C into itself. If is a sequence of C such that and , then x is a fixed point of T.
Let be a sequence of non-negative real numbers satisfying
where , and satisfy the following conditions: (i) and , (ii) or , (iii) (), . Then .
Lemma 2.6 ([, Lemma 3.2])
Let C be a nonempty closed convex subset of a Banach space E. Suppose that
Then, for each , converges strongly to some point of C. Moreover, let T be a mapping of C into itself defined by for all . Then .
Furthermore, we need the following extension of Lemma 2.5 in Wang and Hu  in a 2-uniformly smooth Banach space.
Lemma 2.7 Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space X. Let be a β-Lipschitzian and η-strongly accretive operator with and . Then is a contraction with a contraction coefficient .
Proof Using Lemma 2.1, we have
for all . From and , we have
where . Hence, S is a contraction with a contraction coefficient . □
3 Main results
We now state and prove the main results of this paper.
Theorem 3.1 Let C be a nonempty closed convex subset and sunny nonexpansive retract of a uniformly convex and 2-uniformly smooth Banach space X. Let be a β-Lipschitzian and η-strongly accretive operator with . Let be a sequence of nonexpansive mappings from C into itself such that for each bounded subset B of C. Suppose, in addition, that , where is the nonexpansive mapping defined by . Let be a sunny nonexpansive retraction from X onto C. Let and be two real sequences in and satisfy the following conditions:
(A1) and ;
For given arbitrarily, let the sequence be generated by
Then the sequence strongly converges to a point which solves uniquely the variational inequality
Proof We proceed with the following steps.
Step 1. We claim that is bounded. From , we may assume, without loss of generality, that for all n, where ϵ is an arbitrarily small positive number. In fact, let , from (3.1) and using Lemma 2.7, we have
where . Then from (3.1) and (3.2), we obtain
Thus, we have is continuous, . Therefore, we obtain . By induction, we have
Therefore, is bounded. We also obtain that , and are bounded. Without loss of generality, we may assume that , , and , where B is a bounded set of C.
Step 2. We claim that . Using the same method as in Step 2 of the proof in [, Theorem 3.5], we have . Observe that
Step 3. We claim that . Observe that
Hence, from Step 2 and using Lemma 2.6, we have
Step 4. We claim that , where and is defined by .
From , we may assume, without loss of generality, that . Using Lemma 2.7, it is easy to see that is a contraction. Thus, is well defined. Next, we show that is well defined. Let , using Lemma 2.7, we have
Thus, we have is bounded and so is . On the other hand, we have
Assume that such that as . Put . It follows from (3.3) that (). Since is bounded, without loss of generality, we may assume that . We can use Lemma 2.4 to get . Therefore, using Lemma 2.2 and Lemma 2.7, we have
Consequently, the weak convergence of to actually implies that . Therefore, is well defined.
Since is bounded, there exists a subsequence of which converges weakly to ω. From Step 3 and using Lemma 2.4, we have . Observe that
On the other hand, we have
Therefore, for , we can use Lemma 2.2 to get
where . Now replacing t in (3.5) with and letting , we have
From (3.4) and (3.6), we obtain .
Step 5. We claim that converges strongly to . From (3.1) and using Lemma 2.2, we have
By (3.1) and (3.7), we have
where , . It is easy to see that and . Hence, by Lemma 2.5, the sequence converges strongly to .
Step 6. We claim that is a unique solution of the variational inequality , .
From Step 4, we have shown that is a solution of the variational inequality , . Without loss of generality, we assume that is also a solution of the variational inequality. Therefore, we have
Adding up (3.8) and (3.9), we get
The strong accretivity of F implies that and the uniqueness is proved. □
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The author thanks the editor and the referees for their useful comments and suggestions. This study was supported by the Natural Science Foundation of Yancheng Teachers University under Grant (12YCKL001) and UNSF of Jiangsu province, China (09KJD110005).
The author declares that she has no competing interests.
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Cite this article
Wang, S. Strong convergence of a general algorithm for nonexpansive mappings in Banach spaces. Fixed Point Theory Appl 2012, 207 (2012). https://doi.org/10.1186/1687-1812-2012-207
- strong convergence
- variational inequality
- fixed points
- η-strongly accretive
- Banach space