- Research
- Open access
- Published:
Strong and weak convergence theorems for common zeros of finite accretive mappings
Fixed Point Theory and Applications volume 2014, Article number: 77 (2014)
Abstract
In this paper, we present a new iterative scheme to solve the problems of finding common zeros of finite m-accretive mappings in a real Hilbert space. Some strong and weak convergence theorems are established under different assumptions, which extends the corresponding works given by some authors.
MSC:47H05, 47H09.
1 Introduction and preliminaries
Let H be a real Hilbert space with the inner product and norm , respectively. Then for , and ,
We write to indicate that the sequence converges weakly to x, and implies that converges strongly to x.
Let C be a closed and convex subset of H. Then, for every point , there exists a unique nearest point in C, denoted by , such that for all . is called the metric projection of H onto C. It is well known that is characterized by the properties:
-
(i)
, for all and ;
-
(ii)
For every , ;
-
(iii)
, for every ;
-
(iv)
and imply that .
A mapping is called a contraction if there exists a constant such that , for . We use to denote the collection of mappings f verifying the above inequality. That is, .
A mapping is said to be nonexpansive if , for . We use to denote the fixed point set of T, that is, .
A mapping is called accretive if , for and it is called m-accretive if , for . An m-accretive mapping A is demi-closed, that is, if such that and , then and . Let denote the set of zeros of A, that is, . We denote by (for ) the resolvent of A, that is, . Then is nonexpansive and .
Interest in accretive mappings, which is an important class of nonlinear operators, stems mainly from their firm connection with equations of evolution. It is well known that many physically significant problems can be modeled by initial value problems of the form
where A is an accretive mapping. Typical examples where such evolution equations occur can be found in the heat, wave or Schrödinger equations. If is dependent on t, then (1.2) is reduced to
whose solutions correspond to the equilibrium of the system (1.2). Consequently, within the past 40 years or so, considerable research efforts have been devoted to methods for finding approximate solutions of (1.3). An early fundamental result in the theory of accretive operators, due to Browder [1]. One classical method for studying the problem , where A is an m-accretive mapping, is the following so-called proximal method (cf. [2]):
where . It was shown that the sequence generated by (1.4) converges weakly or strongly to a zero point of A under some conditions.
Recall that the following normal Mann iterative scheme to approximate the fixed point of a nonexpansive mapping was introduced by Mann [3]:
It was proved that under some conditions, the sequence produced by (1.5) converges weakly to a point in .
Later, many mathematicians tried to combine the ideas of proximal method and Mann iterative method to approximate the zeros of m-accretive mappings; see, e.g. [4–11] and references therein.
Especially, in 2007, Qin and Su [4] presented the following iterative scheme:
where . They showed that generated by the above scheme converges strongly to a zero of A.
Based on iterative schemes (1.4) and (1.5), Zegeye and Shahzad extended their discussion to the case of finite m-accretive mappings. They presented in [12] the following iterative scheme:
where with and . If , they proved that generated by (1.7) converges strongly to the common zeros of () under some conditions.
Later, their work was extended to the following one presented by Hu and Liu in [13]:
where with and . and . If , they proved that converges strongly to the common zeros of () under some conditions.
In this paper, based on the work of (1.6), (1.7), and (1.8), we present the following iterative scheme:
where , , and , for ; . , is a contraction, both and are finite families of m-accretive mappings. More details of iterative scheme (A) will be presented in Section 2. We shall prove a weak convergent theorem and a strong convergent theorem under different assumptions on , , , and , respectively.
In order to prove our main results, we need the following lemmas.
By using the properties of the metric projection and m-accretive mappings, we can easily prove the following two lemmas.
Lemma 1.1 For and , .
Lemma 1.2 For , and ,
Lemma 1.3 ([14])
Let and be two sequences of nonnegative real numbers satisfying
If , then exists.
Lemma 1.4 ([15])
Let H be a real Hilbert space and A be an m-accretive mapping. For and , we have
where and .
Lemma 1.5 ([16])
Let H be a real Hilbert space and C be a closed convex subset of H. Let be a nonexpansive mapping with , and . Then , defined by
converges strongly to a point in . If one defines by , , then solves the following variational inequality:
Lemma 1.6 ([17])
Let , , and be three sequences of nonnegative real numbers satisfying
where such that
-
(i)
and ,
-
(ii)
either or .
Then .
Lemma 1.7 In a Hilbert space H, we can easily get the following inequality:
2 Weak and strong convergence theorems
Lemma 2.1 Let H be a real Hilbert space, C be a nonempty closed and convex subset of H and (; ) be finitely many m-accretive mappings such that . Suppose and , where (), (), , , , and . Then and are nonexpansive.
Lemma 2.1 can easily be obtained in view of the facts that and are nonexpansive, ; .
Theorem 2.1 Let H, C, D, and (; ) be the same as those in Lemma 2.1. Suppose that . Let be generated by the iterative scheme (A). If , and are three sequences in such that , , , with and is a contraction with contractive constant . Then converges weakly to a point satisfying
Proof We split our proof into five steps.
Step 1. , and are all bounded.
We can easily know that , and . Then for , from Lemma 2.1, we have
Based on (2.2), we know that
Then (2.3) and Lemma 2.1 imply that
Using (2.4), we know that
Then Lemma 1.3 implies that exists, which ensures that is bounded. Combining with the fact that f is a contraction and noticing (2.2), (2.3), and (2.4), we can easily know that , , , , , (), and () are all bounded.
We may let = max{, , , , , , , }.
Step 2. exists.
In fact, it follows from the property of that
In view of Lemma 1.1, we know that for ,
which implies that is bounded since is bounded from step 1. Then is also bounded.
Let .
Noticing (2.5) and (2.6), we have
Therefore, in view of Lemma 1.3, exists.
Step 3. , where satisfies (2.1), as .
We first claim that there exists a unique element such that
In fact, if we let , . Then we can easily find that is proper, strictly convex and lower-semi-continuous and as . This ensures that there exists a unique element such that .
From (2.7), we know that
Therefore, , as .
Step 4. , where denotes the set consisting all of the weak limit points of .
Since is bounded, then there exists a subsequence of , for simplicity, we still denote it by , such that , as .
Since is convex, by using Lemma 1.2 and noticing (2.3), we have, for ,
Then using (2.8), we have
which implies that
Thus
Since from the proof of step 1, we know that exists, then , as .
Going back to (2.8) again, we know that
Then using (2.12), repeating the processes of (2.9)-(2.11), we know that
By using the inductive method, we have the following results:
as . Therefore, , …, , as .
Let , then , since and both and are bounded. This ensures that .
Let , then , which implies that .
By induction, let , then , which implies that . Thus .
Next, we shall show that .
From step 1, we may assume that there exists such that , and .
Now, computing the following, :
By using Lemma 1.2,
Then (2.13) and (2.14) imply that
From step 1, we know that exists, then (2.15) implies that
From the iterative scheme (A), , and the results of step 1, we know that
Then , since , as .
Thus from (2.16), we have , imitating the proof of , we can see that , and then .
Step 5. .
In fact, for ,
From step 3, we know that , as . Let be a subsequence of which is weakly convergent to . Then from step 4. Taking the limits on both sides of (2.17), we know that .
Letting , we have .
Supposing is another subsequence of such that as . Then repeating the above proof, we have . Since all of the weakly convergent subsequences of converge to the same element , then the whole sequence converges weakly to .
This completes the proof. □
Remark 2.1 To prove the strong convergence result in Theorem 2.2, we need to prove the following two lemmas first and some new proof techniques can be seen.
Lemma 2.2 Let (; ), and be the same as those in Lemma 2.1. Suppose that . Then and , for .
Proof It is easy to check and , for .
Next, we shall show that .
For , . Since , then . Thus
Then . Since , , then , . That is,
By using Lemma 1.2 and (2.18), we know that , . Thus , which implies that , . Then , for .
Finally, we shall show that .
For , then . Let , then , since . Therefore,
From (2.19), we know that
Noticing that (2.20) and (2.18) have the same form, then repeating the proof of , we know that and then .
Since , using (2.19) again, we know that
Repeating the above proof again, .
By induction, we have . Therefore, .
This completes the proof. □
Lemma 2.3 Let (; ), and be the same as those in Lemma 2.1. Suppose that . Then is nonexpansive and , for .
Proof It is easy to check that is nonexpansive. We are left to show that .
, then, from Lemma 2.2, and . Thus , which implies that .
On the other hand, let , then . Let , then , since . Then Lemma 2.1 ensures that
which implies that
Using the same method as that in Lemma 2.2, . Thus . Since , then , which implies that from Lemma 2.2. Therefore, .
This completes the proof. □
Theorem 2.2 Suppose H, D, C, , and f are the same as those in Theorem 2.1. Let be generated by the iterative scheme (A). If , and are three sequences in and satisfy the following conditions:
-
(i)
, and , as ;
-
(ii)
, , and , as ;
-
(iii)
, and , as ;
-
(iv)
, and , as .
Then converges strongly to a point , which is the unique solution of the following variational inequality:
Proof We shall split the proof into five steps:
Step 1. is bounded.
Letting . Then
By induction, , . Thus is bounded.
Step 2. and .
In fact,
Next we discuss .
If , then in view of Lemma 1.4,
For , let , then
From (2.24) and (2.25), we know that
Notice that and ; similar to (2.26), we have
Following from (2.26) and (2.27), we have
Then by induction, we can get the following result:
Putting (2.28) into (2.23), and letting ,
If , then imitating the above proof, we have
Combining (2.29) and (2.30),
Similar to the discussion of (2.24), we have
Using (2.32), then
Based on (2.31) and (2.33), and letting , we have
In view of Lemma 1.6, we know that , as . Combining with the fact that , we can easily know that , as .
Step 3. , and , as . In view of Lemma 1.4 again, we know that
and then
By induction,
as , since .
, continuing the computation of (2.15), we have
From step 2, we know that , then , , which implies that
Noticing that , and , as .
Combining with the facts of (2.34), (2.35), and step 2, we know that
Using Lemma 1.4 again, then
Since , then , as .
Step 4. , , , where satisfies (2.22).
Using Lemmas 1.5 and 2.3, we know that if we let , and , then , as . And, satisfies (2.22).
From step 3, we may choose such that , , and , as .
Using Lemma 1.7,
Then
Since , , , and are all bounded, and , and , from (2.36), .
Recalling that , then . Thus . Since , then . Then from step 2, .
Noticing that
and using (2.35), iterative scheme (A) and the result of step 2, we have .
Step 5. , which satisfies (2.22), as .
Using Lemma 1.7, we know that
We have
Letting and using (2.37) and (2.38), we have
Let , then and .
Let = + + + .
Notice that , and from the results in step 4, we have .
Using Lemma 1.6, , which satisfies (2.22), as .
This completes the proof. □
Remark 2.2 The iterative construction in this paper generalizes and extends some corresponding ones in [2, 4, 12, 13], etc., in Hilbert spaces.
References
Browder FE: Nonlinear mappings of nonexpansive and accretive type in Banach spaces. Bull. Am. Math. Soc. 1967, 73: 875–882. 10.1090/S0002-9904-1967-11823-8
Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 1976, 14: 877–898. 10.1137/0314056
Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3
Qin XL, Su YF: Approximation of a zero point of accretive operator in Banach spaces. J. Math. Anal. Appl. 2007, 329: 415–424. 10.1016/j.jmaa.2006.06.067
Mainge PE: Viscosity methods for zeroes of accretive operators. J. Approx. Theory 2006, 140: 127–140. 10.1016/j.jat.2005.11.017
Qin XL, Cho SY, Wang L: Iterative algorithms with errors for zero points of m -accretive operators. Fixed Point Theory Appl. 2013., 2013: Article ID 148
Ceng LC, Wu SY, Yao JC: New accuracy criteria for modified approximate proximal point algorithms in Hilbert spaces. Taiwan. J. Math. 2008, 12: 1691–1705.
Xu HK: Strong convergence of an iterative method for nonexpansive and accretive operators. J. Math. Anal. Appl. 2006, 314: 631–643. 10.1016/j.jmaa.2005.04.082
Cho YJ, Kang SM, Zhou HY: Approximate proximal point algorithms for finding zeroes of maximal monotone operators in Hilbert spaces. J. Inequal. Appl. 2008., 2008: Article ID 598191
Ceng LC, Khan AR, Ansari QH, Yao JC: Strong convergence of composite iterative schemes for zeros of m -accretive operators in Banach spaces. Nonlinear Anal. 2009, 70: 1830–1840. 10.1016/j.na.2008.02.083
Chen RD, Liu YJ, Shen XL: Iterative approximation of a zero of accretive operator in Banach space. Nonlinear Anal. 2009, 71: e346-e350. 10.1016/j.na.2008.11.054
Zegeye H, Shahzad N: Strong convergence theorems for a common zero of a finite family of m -accretive mappings. Nonlinear Anal. 2007, 66: 1161–1169. 10.1016/j.na.2006.01.012
Hu LG, Liu LW: A new iterative algorithm for common solutions of a finite family of accretive operators. Nonlinear Anal. 2009, 70: 2344–2351. 10.1016/j.na.2008.03.016
Tan KK, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 1993, 178: 301–308. 10.1006/jmaa.1993.1309
Barbu V: Nonlinear Semigroups and Differential Equations in Banach Space. Noordhoff, Groningen; 1976.
Xu HK: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 2004, 298: 279–291. 10.1016/j.jmaa.2004.04.059
Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 66: 240–256. 10.1112/S0024610702003332
Acknowledgements
This work was supported by the National Natural Science Foundation of China (No. 11071053), Natural Science Foundation of Hebei (No. A2014207010), Key Project of Science and Research of Hebei Education Department (ZH2012080) and Key Project of Science and Research of Hebei University of Economics and Business (2013KYZ01).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Wei, L., Tan, R. Strong and weak convergence theorems for common zeros of finite accretive mappings. Fixed Point Theory Appl 2014, 77 (2014). https://doi.org/10.1186/1687-1812-2014-77
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2014-77