Weak and strong convergence theorems for common zeros of accretive operators
© Wang and Li; licensee Springer 2014
Received: 23 March 2014
Accepted: 16 July 2014
Published: 15 August 2014
In this paper, we propose two proximal point algorithms for investigating common zeros of a family of accretive operators. Weak and strong convergence of the two algorithms are obtained in a Banach space.
MSC:47H06, 47H09, 47J25.
1 Introduction and preliminaries
where denotes the generalized duality pairing. In the case that , we write J for and call J the normalized duality mapping.
Following Browder , 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 .
Let . E is said to be smooth or is said to be have a Gâteaux differentiable norm if the limit exists for each . E is said to have a uniformly Gâteaux differentiable norm if for each , the limit is attained uniformly for all . E is said to be uniformly smooth or is said to have a uniformly Fréchet differentiable norm if the limit is attained uniformly for .
It is well known that Fréchet differentiability of the norm of E implies Gâteaux differentiability of the norm of E. It is known that if the norm of E is uniformly Gâteaux differentiable, then the duality mapping J is single-valued and uniformly norm-to-weak∗ continuous on each bounded subset of E.
a contraction if ;
sunny if for each and , we have ;
a sunny nonexpansive retraction if is sunny, nonexpansive and a contraction.
D is said to be a nonexpansive retract of C if there exists a nonexpansive retraction from C onto D. The following result, which was established in [2–4], describes a characterization of sunny nonexpansive retractions on a smooth Banach space.
is sunny and nonexpansive;
, , .
It is well known that if E is a Hilbert space, then a sunny nonexpansive retraction is coincident with the metric projection from E onto C. Let C be a nonempty closed convex subset of a smooth Banach space E, let and let . Then we have from the above that if and only if for all , where is a sunny nonexpansive retraction from E onto C.
for and implies that .
It is known that a uniformly convex Banach space is reflexive and strictly convex.
Let C be a nonempty closed convex subset of E. Let be a mapping. In this paper, we use to denote the set of fixed points of S. Recall that S is said to be nonexpansive iff , . For the existence of fixed points of a nonexpansive mapping, we refer the readers to . Let x be a fixed element in C, and let S be a nonexpansive mapping with a nonempty fixed point set. For each , let be the unique solution of the equation . In the framework of uniformly smooth Banach spaces, Reich  proved that converges strongly to a fixed point , where is the unique sunny nonexpansive retraction from C onto , of S as . Xu  further extended the results to the framework of reflexive Banach spaces; for more details, see  and  and the references therein.
where ∂ denotes the sub-differential in the sense of convex analysis.
Zero problems of accretive operators recently have been extensively studied (see [6–21] and the references therein) because of their important applications in real world. Proximal point algorithm, which was proposed by Martinet [22, 23] and generalized by Rockafellar [24, 25], is a classical method for investigating zeros of monotone operators. In this paper, we propose two proximal point algorithms for investigating common zeros of a family of m-accretive operators. Weak and strong convergence theorems are established in Banach spaces.
The first part of the next lemma is an immediate consequence of the subdifferential inequality and the proof of the second part can be found in .
- (i)For all , the following inequality holds:
Assume that a sequence in E converges weakly to a point .
Lemma 1.2 
Let E be a reflexive Banach space and have a weakly continuous duality map with gauge φ. Let C be a closed convex subset of E, and let be a nonexpansive mapping. Fix and . Let be the unique fixed point of the mapping . Then S has a fixed point if and only if remains bounded as , and in this case, converges as strongly to a fixed point of S. Define a mapping by . Then is the sunny nonexpansive retraction from C onto .
Lemma 1.3 
Let C be a closed convex subset of a strictly convex Banach space E. Let be some positive integer, and let be a nonexpansive mapping. Suppose that is nonempty. Then the mapping , where is a real number sequence in such that , is nonexpansive with .
Lemma 1.4 
Let , and be three nonnegative real sequences satisfying , , where is some positive integer, is a number sequence in such that , is a number sequence such that . Then .
Lemma 1.5 
for all and such that .
Lemma 1.6 
Let E be a uniformly convex Banach space. Let C be a nonempty closed convex subset of E, and let be a nonexpansive mapping. Then is demiclosed at zero.
2 Main results
where a, b and c are real numbers. Then the sequence converges weakly to .
In view of restriction (a), we find that . It follows that . Using restriction (b), we arrive at for each . Since is bounded, we see that there exists a subsequence of converging weakly to . Using Lemma 1.6, we obtain that . This proves that .
This is a contradiction. Hence . This completes the proof. □
, and ;
, and .
Then the sequence converges strongly to , where is the unique sunny nonexpansive retract from onto .
This completes the proof. □
In this section, we give an application of Theorem 2.1 in the framework of Hilbert spaces.
To study the equilibrium problem (3.1), we may assume that F satisfies the following restrictions:
(A1) for all ;
(A2) F is monotone, i.e., for all ;
(A3) for each , ;
(A4) for each , is convex and lower semi-continuous.
Lemma 3.1 
where a, b and c are real numbers. Then the sequence converges weakly to .
, and ;
, and .
Then the sequence converges strongly to , where is the metric projection from C onto .
This first author thanks the Fundamental Research Funds for the Central Universities (2014ZD44). The authors are grateful to the reviewers’ suggestions which improved the contents of the article.
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