Some results on Rockafellar-type iterative algorithms for zeros of accretive operators
© Jung; licensee Springer. 2013
Received: 15 November 2012
Accepted: 3 May 2013
Published: 20 May 2013
We provide a new proof technique to obtain strong convergence of the sequences generated by viscosity iterative methods for a Rockafellar-type iterative algorithm and a Halpern-type iterative algorithm to a zero of an accretive operator in Banach spaces. By using a new method different from previous ones, the main results improve and develop the recent well-known results in this area.
MSC:47H06, 47H09, 47H10, 47J25, 49M05, 65J15.
If , then the inclusion is solvable.
where , , and is an error sequence, and obtained strong and weak convergence of sequences generated by these algorithms.
where , and .
where the sequences and satisfy the conditions: (i) ; (ii) ; (iii) ; and (iv) . In particular, in order to obtain strong convergence of the sequence generated by (1.2) to a zero of an accretive operator A, they utilized the well-known inequality in uniformly convex Banach spaces (see Xu ). The results of Zhang and Song  in a Banach space with a uniformly Gâteaux differentiable norm and the corresponding results of Song  are mutually complementary since Zhang and Song  assumed uniform convexity on the space instead of reflexivity on the space in Song  and relaxed the conditions and , on sequences and in Song . Yu  filled the gaps in the result of Zhang and Song  for the Halpern-type iterative algorithm (1.2) by utilizing the result on the sequence of real numbers in , which is of fundamental importance for the techniques of analysis. Also, Zhang and Song  studied the Rockafellar-type iterative algorithm (1.1) in a uniformly convex Banach space with a weakly continuous normalized duality mapping or with a uniformly Gâteaux differentiable norm.
In this paper, motivated by the above mentioned results, we consider viscosity iterative methods for the Rockafellar-type iterative algorithm (1.1) and Halpern-type iterative algorithm (1.2). By using a new method different from ones in [18, 20] which recover the gaps in  as in , we establish results on strong convergence of the sequences generated by the proposed iterative methods to a zero of an accretive operator A, which solves a certain variational inequality in a uniformly convex Banach space having a weakly continuous duality mapping with gauge function φ or having a uniformly Gâteaux differentiable norm. Our results improve, develop and complement the corresponding results of Song , Zhang and Song , Yu  and Song et al.  as well as many existing ones.
2 Preliminaries and lemmas
exists for each x, y in its unit sphere . Such an E is called a smooth Banach space. The norm is said to be uniformly Gâteaux differentiable if for , the limit is attained uniformly for .
for all , where . For more detail, see Xu .
where ℝ is the set of all real numbers; in particular, , . It is well known that E is smooth if and only if the normalized duality mapping is single-valued, and that in a Hilbert space H, the normalized duality mapping is the identity.
We say that a Banach space E has a weakly continuous duality mapping if there exists a gauge function φ such that the duality mapping is single-valued and continuous from the weak topology to the weak∗ topology, that is, for any with , . For example, every space () has a weakly continuous duality mapping with gauge function .
for all , implies ,
for any fixed positive integer N,
for all .
We need the following lemmas for the proofs of our main results.
Lemma 2.2 
Let be a real number and let a sequence satisfy the condition for all Banach limit LIM. If , then .
Lemma 2.3 
Also, we will use the next lemma which is of fundamental importance for our proof.
Lemma 2.4 
3 Main results
Throughout this section, it is assumed that is an accretive operator satisfying the range condition with ; C is a nonempty closed convex subset of E such that ; is a contractive mapping with a constant ; and and are sequences satisfying the conditions:
(C3) for some a;
We need the following result for the existence of solutions of a certain variational inequality.
Using Theorem J, we have the following result.
where with being defined by for each .
and hence is bounded. Also, by Theorem J, converges as strongly to a point in , which is denoted by .
This completes the proof. □
By using Theorem 3.1, we establish the strong convergence of the Rockafellar-type iterative algorithm (3.1).
where with being defined by for each and . From now, we put for all .
We know that for all and all and , , and are bounded by the proof of Theorem 3.1.
In order to prove that , we consider two possible cases as in the proof of Yu .
Now, we proceed with the following steps.
Step 1. We know from (3.5) that .
from Lemma 2.3, we conclude that and .
Moreover, by using the same argument as in Step 1-Step 4 of Case 1 with , and , we obtain the following:
Step 1′ ;
Step 2′ ;
Step 3′ ;
Therefore, . This completes the proof. □
By taking in Theorem 3.2, we obtain the following result, which is an extension of Corollary 3.4 of Zhang and Song  to the viscosity iteration method.
Then converges strongly to , where q is the unique solution of the variational inequality (3.10).
where with being defined by for each and . From now, we put for .
We also know that , , and are bounded by the proof of Theorem 3.1.
As in the proof of Theorem 3.2, we divide the proof into several steps. We only include the differences.
Step 1. By considering two cases as in the proof of Theorem 3.2, we have that and , where is as in Case 2 in the proof of Theorem 3.2.
In the case when , by using the same argument with and , we also have .
- (2)In the case when , by using the same argument with , and and (2) of Step 2, we can obtain
Therefore, . This completes the proof. □
By taking , we also have the following.
Then converges strongly to , where q is the unique solution of the variational inequality (3.15).
By taking in Corollary 3.3, we also have the following.
Then converges strongly to , where q is the unique solution of the variational inequality (3.20).
Proof Since A is maximal monotone, A is monotone and satisfies the range condition for all . Putting in Corollary 3.3, we can obtain the desired result. □
By using arguments similar to those in the proofs of Theorems 3.1, 3.2 and 3.3 and , we can obtain the following theorems for the Halpern-type iterative algorithm (3.2).
where with being defined by for each .
Theorem 3.5 Let E be a uniformly convex Banach space having a weakly continuous duality mapping with gauge function φ. Then the sequence generated by (3.2) converges strongly to , where q is the unique solution of the variational inequality (3.10).
Theorem 3.6 Let E be a uniformly convex Banach space having a uniformly Gâteaux differentiable norm. Then the sequence generated by (3.2) converges strongly to , where q is the unique solution of the variational inequality (3.15).
Corollary 3.5 Let H be a Hilbert space. Assume that is a maximal monotone operator with . Let be a sequence generated by (3.2). Then converges strongly to , where q is the unique solution of the variational inequality (3.20).
- (a)The following gaps, which authors in  overlooked, are corrected: there exist two subsequences and of satisfying
The case of an iterative scheme in [, Theorem 3.7] is extended to the case of a viscosity iterative scheme , where is a contractive mapping with a constant .
We utilize the weakly continuous duality mapping with gauge function φ instead of the weakly continuous normalized duality mapping in [, Theorem 3.7].
Theorem 3.2 and Theorem 3.3 improve Theorem 3.3 and Theorem 3.4 of Yu , which were given without proofs, to the case of the viscosity iterative method together with our proofs. Theorem 3.3 also develops and complements Theorem 4.2 of Song . In particular, the limit point of the sequence in Theorem 3.3 is the unique solution of the variational inequality (3.15) in comparison with [, Theorem 4.2].
As in [22, 31, 32], we can replace the contractive mapping f in our algorithms by the weakly contractive mapping g (recall that a mapping is said to be weakly contractive  if , , where is a continuous and strictly increasing function such that ψ is positive on and ).
The author would like to thank the anonymous referees for their valuable comments and suggestions, which improved the presentation of this manuscript. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012000895).
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