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Strong convergence theorems for common solutions of a family of nonexpansive mappings and an accretive operator
Fixed Point Theory and Applications volume 2013, Article number: 172 (2013)
Abstract
In this paper, common solutions of a family of nonexpansive mappings and an accretive operator are investigated based on a viscosity iterative method. Strong convergence theorems for common solutions are established in a Banach space.
MSC:47H09, 47J05.
1 Introduction
Fixed point theory has emerged as an effective and powerful tool for studying a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity and optimization; see [1–12] and the references therein. The computation of solutions is important in the study of many real world problems. The well-known convex feasibility problem which captures applications in various disciplines such as image restoration and radiation therapy treatment planning is to find a point in the intersection of common fixed point sets of a family of nonlinear mappings; see, for example, [13–24] and the references therein.
The aim of this paper is to investigate a common solution problem of a family of nonexpansive mappings and an accretive operator based on a viscosity iterative method. The organization of this article is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, a viscosity iterative method is discussed. Strong convergence theorems of common solutions are established in a reflexive and strictly convex Banach space E which enjoys weakly continuous duality mappings.
2 Preliminaries
Throughout this paper, we assume that E is a real Banach space. Let C be a nonempty, closed and convex subset of E, and let be a mapping. A point is a fixed point of T provided . Denote by the set of fixed points of T; that is, .
Recall that is nonexpansive iff
is a contraction iff there exists a constant such that
We use to denote the collection of all contractions on C. That is, .
The Picard iterative algorithm is an efficient algorithm to study contractions. However, the Picard iterative algorithm fails to converge to fixed points of nonexpansive mappings even that their fixed point sets are not empty. One classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping. More precisely, take and define a mapping by
where is a fixed element and f is a contraction on C with the constant α. It is easy to see that is a contraction with the constant α. Indeed, we have the following:
Banach’s contraction mapping principle guarantees that has a unique fixed point. We denote the unique fixed point by . Reich [25] proved that if E is a uniformly smooth Banach space, then strongly converges to a fixed point of T, and the limit defines the (unique) sunny nonexpansive retraction from onto . Recently, Xu [26] further proved that the above results still hold in reflexive Banach spaces which have weakly continuous duality mappings.
Recall that the normal Mann iterative algorithm was introduced by Mann in 1953. Since then the construction of fixed points for nonexpansive mappings via the normal Mann iterative algorithm has been extensively investigated by many authors.
The normal Mann iterative algorithm generates a sequence in the following manner:
where the sequence is in the interval . If T is a nonexpansive mapping with a fixed point and the control sequence is chosen so that , then the sequence generated by the normal Mann iterative algorithm converges weakly to a fixed point of T (this is also valid in a uniformly convex Banach space with the Fréchet differentiable norm). Since the Mann iterative algorithm only has weak convergence in infinite dimension spaces, many authors tried to modify the normal Mann iteration algorithm to have strong convergence for nonexpansive mappings.
Kim and Xu [27] considered the following iterative algorithm.
where T is a nonexpansive mapping of C into itself, is a given point, and are two real number sequences in . They proved that the sequence generated by the above iterative algorithm strongly converges to a fixed point of the mapping T provided that the control sequences and satisfy appropriate conditions.
Recently, many authors have studied the following convex feasibility problem (CFP): , where is an integer, and each is assumed to be the fixed point set of a nonexpansive mapping , . There is a considerable investigation on CFP in the setting of Hilbert spaces which captures applications in various disciplines such as image restoration [28], computer tomography [29] and radiation therapy treatment planning [30].
In this paper, we consider the mapping defined by
where are real numbers such that and are nonexpansive mappings of C into itself. Nonexpansivity of each ensures the nonexpansivity of .
We have the following lemmas which are important to prove our main results.
Lemma 2.1 [31]
Let C be a nonempty closed convex subset of a strictly convex Banach space E. Let be nonexpansive mappings of C into itself such that and let be real numbers such that , where b is some real number, for any . Then, for every and , the limit exists.
Using Lemma 2.1, one can define the mapping W of C into itself as follows.
Such a mapping W is called the W-mapping generated by and . Throughout this paper, we will assume that for all .
Lemma 2.2 [31]
Let C be a nonempty closed convex subset of a strictly convex Banach space E. Let be nonexpansive mappings of C into itself such that and let be real numbers such that for any . Then .
Let I denote the identity operator on E. An operator with domain and range is said to be accretive if for each and , , there exists such that
An accretive operator A is said to be m-accretive if for all . In a real Hilbert space, an operator A is m-accretive if and only if A is maximal monotone. In this paper, we use to denote the set of zero points of A. Interest in accretive operators, which is an important class of nonlinear operators, stems mainly from their firm connection with equations of evolution.
For an accretive operator A, we can define a nonexpansive single-valued mapping by for each , which is called the resolvent of A. One of classical methods of studying the problem , where is an accretive operator, is the following:
where and is a sequence of positive real numbers. Recently, different regularization iterative methods have been employed to treat zero points of accretive operators in the framework of Banach spaces; see [32–36] and the references therein.
In this paper, we investigate common fixed point problems of a family of nonexpansive mappings generated in (2.1) and a zero point problem of an accretive operator based on a viscosity approximation method. Strong convergence theorems of common fixed points are established in a Banach space. In order to prove our main results, we need the following definitions and lemmas.
Recall that if C and D are nonempty subsets of a Banach space E such that C is nonempty closed convex and , then a map is sunny provided that for all and whenever . A sunny nonexpansive retraction is a sunny retraction, which is also nonexpansive. Sunny nonexpansive retractions play an important role in our argument. They are characterized as follows: If E is a smooth Banach space, then is a sunny nonexpansive retraction if and only if the following inequality holds:
Chen and Zhu [34] showed that if E is a reflexive Banach space and has a weakly continuous duality, then there is a sunny nonexpansive retraction from onto and it can be constructed as follows.
Lemma 2.3 [32]
Let E be a reflexive Banach space which has a weakly continuous duality mapping . Let C be closed convex subset of E and let be a nonexpansive mapping. Let be a contractive mapping with . For any , let be defined by , where T is a nonexpansive mapping. Then T has a fixed point if and only if remains bounded as and, in this case, converges, as , strongly to a fixed point of T.
Lemma 2.4 Under the condition of Lemma 2.3, we define the mapping by
Then the mapping Q is a sunny nonexpansive retraction from onto .
Proof From Theorem 3.1 of [34], for all and , we have
Letting , we have
Since for any , we have
This completes the proof. □
Recall that a gauge is a continuous strictly increasing function such that and as . The duality mapping associated to a gauge φ is defined by
Following Browder [37], we say that a Banach space E has a weakly continuous duality mapping if there exists a gauge φ for which the duality mapping is single-valued and weak-to-weak∗ sequentially continuous (i.e., if is a sequence in E weakly convergent to a point x, then the sequence converges weakly∗ to ). It is known that has a weakly continuous duality mapping with a gauge function for all . Set
Then
where ∂ denotes the sub-differential in the sense of convex analysis.
The first part of the next lemma is an immediate consequence of the sub-differential inequality and the proof of the second part can be found in [38].
Lemma 2.5 Assume that a Banach space E has a weakly continuous duality mapping with a gauge φ.
-
(i)
For all , the following inequality holds:
In particular, for all
-
(ii)
Assume that a sequence in E converges weakly to a point .
Then the following identity holds:
Lemma 2.6 [39]
Let and be bounded sequences in a Banach space X and let be a sequence in with . Suppose that for all integers and
Then .
Lemma 2.7 [40]
Assume that is a sequence of nonnegative real numbers such that
where is a sequence in and is a sequence such that
-
(i)
;
-
(ii)
or .
Then .
The following lemma can be obtained from Chang et al. [23]. For the sake of completeness, we still give the proof.
Lemma 2.8 Let C be a nonempty closed convex subset of a strictly convex Banach space E, let be a family of infinitely nonexpansive mappings with , let be a real sequence such that for each . If K is any bounded subset of C, then .
Proof Let . Since K is a bounded subset of C, there exists an such that . It follows that
Since , for any given , there exists a positive integer such that
For any positive integers , we find that
Letting , we find that
This implies that . □
3 Main results
Theorem 3.1 Let E be a reflexive and strictly convex Banach space E which enjoys a weakly continuous duality map with gauge φ and let A be an m-accretive operator in E with the domain . Assume that is convex. Let be a nonexpansive mapping from into itself for . Let with the coefficient () and for some . Assume that , where W is a mapping defined by (2.2). Let be a sequence generated in the following iterative algorithm:
where is generated in (2.1), and are real number sequences in satisfying the following restrictions:
-
(a)
, ;
-
(b)
.
Then strongly converges to , where is defined by (2.3).
Proof First we prove that sequences and are bounded. Fixing , we see that
It follows that
This in turn implies that
which gives that the sequence is bounded, so is .
Next, we prove that as . Putting , we have
In the light of
we obtain that
Since and are nonexpansive, we have
where is an appropriate constant such that
for all . Substituting (3.4) into (3.3), we have
In view of conditions (a) and (b), we get that
We can obtain from Lemma 2.6 that easily. On the other hand, we see from (3.1) that
This implies that
Next, we prove that . In view of
we obtain that
On the other hand, we have
In view of (3.5) and (3.6), we have
Notice that
This implies that
From condition (b) and (3.7), we obtain that
On the other hand, we have
In view of Lemma 2.8, we find that
This in turn implies that
Next, we show as . To show it, we first prove that
In view of Lemma 2.4, we have the sunny nonexpansive retraction . Take a subsequence of such that
Since E is reflexive, we may further assume that for some . Since is weakly continuous, we obtain from Lemma 2.5 that
Put
It follows that
With the aid of (3.9), we arrive at
Notice that
From (3.12) and (3.13), we find that
This implies that . And hence . That is, . Since Q is the sunny nonexpansive retraction from onto F, we have from (3.11)
This shows that (3.10) holds. It follows from Lemma 2.5 that
We find that as from Lemma 2.7. That is, . This completes the proof. □
Remark 3.2 Taking , the identity mapping, , we see that . Then the strict convexity of E in Theorem 3.1 may not be needed.
Corollary 3.3 Let E be a reflexive Banach space E which enjoys a weakly continuous duality map with gauge φ and A be an m-accretive operator in E with the domain . Assume that is convex. Let with the coefficient () and for some . Assume that . Let be a sequence generated in the following iterative algorithm:
where and are real number sequences in satisfying the following restrictions:
-
(a)
, ;
-
(b)
.
Then strongly converges to , where is defined by (2.3).
If , where u is a fixed element in , then Theorem 3.1 is reduced to the following.
Corollary 3.4 Let E be a reflexive and strictly convex Banach space E which enjoys a weakly continuous duality map with gauge φ and A be an m-accretive operator in E with the domain . Assume that is convex. Let be a nonexpansive mapping from into itself for . Let for some . Assume that , where W is a mapping defined by (2.2). Let be a sequence generated in the following iterative algorithm:
where is generated in (2.1), and are real number sequences in satisfying the following restrictions:
-
(a)
, ;
-
(b)
.
Then strongly converges to , where is defined by (2.3).
If , then Theorem 3.1 is reduced to the following.
Corollary 3.5 Let E be a reflexive and strictly convex Banach space E which enjoys a weakly continuous duality map with gauge φ and let C be a closed and convex subset of E. Let be a nonexpansive mapping from C into itself for . Let with the coefficient (). Assume that . Let be a sequence generated in the following iterative algorithm:
where is generated in (2.1), and are real number sequences in satisfying the following restrictions:
-
(a)
, ;
-
(b)
.
Then strongly converges to , where is defined by (2.3).
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Cheng, P., Wu, H. Strong convergence theorems for common solutions of a family of nonexpansive mappings and an accretive operator. Fixed Point Theory Appl 2013, 172 (2013). https://doi.org/10.1186/1687-1812-2013-172
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DOI: https://doi.org/10.1186/1687-1812-2013-172