Strong convergence of a general algorithm for nonexpansive mappings in Banach spaces
© Wang; licensee Springer 2012
Received: 23 March 2012
Accepted: 7 November 2012
Published: 22 November 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.
They proved if and satisfy appropriate conditions, the defined by (1.1) converges strongly to a fixed point of T.
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 , .
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.
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.
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 ()
for all .
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 ()
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.
where , and satisfy the following conditions: (i) and , (ii) or , (iii) (), . Then .
Lemma 2.6 ([, Lemma 3.2])
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 .
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 ;
Proof We proceed with the following steps.
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 4. We claim that , where and is defined by .
Consequently, the weak convergence of to actually implies that . Therefore, is well defined.
From (3.4) and (3.6), we obtain .
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 , .
The strong accretivity of F implies that and the uniqueness is proved. □
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).
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