- Research
- Open access
- Published:
A modified Mann iteration for zero points of accretive operators
Fixed Point Theory and Applications volume 2013, Article number: 347 (2013)
Abstract
A modified Mann iteration with computational errors is investigated. A strong convergence theorem for zero points of an m-accretive operator is established in a Banach space.
MSC:47H06, 47H09, 47J25, 65J15.
1 Introduction
In this paper, we are concerned with the problem of finding zero points of accretive operators. Interest in accretive operators stems mainly from their firm connection with equations of evolution, and this is an important class of nonlinear operators. It is known that many physically significant problems can be modelled by initial value problems of the form
where A is an accretive operator in an appropriate Banach space. Typical examples where such evolution equations occur can be found in the heat, wave or Schrödinger equations. If is dependent of t, then (1.1) is reduced to
whose solutions correspond to the equilibrium points of (1.1). An early fundamental result in the theory of accretive operators, due to Browder [1], states that the initial value problem (1.1) is solvable if A is locally Lipschitz and accretive on E. One of the most popular techniques for solving zero points of accretive operators goes back to the work of Browder [2]. One of the basic ideas in the case of a Hilbert space H is reducing the above equation (1.2) to a fixed point problem of the operator defined by , which is called the classical resolvent of A.
The paper is organized in the following way. In Section 2, we present the preliminaries that are needed in our work. In Section 3, a modified Mann iteration with computational errors is presented. A strong convergence theorem for zero points of an m-accretive operator is established in a Banach space. In Section 4, applications of the main results are discussed.
2 Preliminaries
Let E be a real Banach space E and let be the dual space of E. Let denote the pairing between E and . The normalized duality mapping is defined by
for all . Let . E is said to be smooth or is said to 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 . In the sequel, we use j to denote the single-valued normalized duality mapping. 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.
Recall that a closed convex subset C of a Banach space E is said to have the normal structure if for each bounded closed convex subset K of C which contains at least two points, there exists an element x of K which is not a diametral point of K, i.e., , where is the diameter of K. It is well known that a closed convex subset of a uniformly convex Banach space has the normal structure and a compact convex subset of a Banach space has the normal structure; see [3] for more details.
Let be a mapping. Recall that T is said to be contractive if there exits a constant such that
For such a case, we also call T an α-contraction. T is said to be nonexpansive if
Let D be a nonempty subset of C. Let . Q is said to be contraction if ; sunny if for each and , we have ; sunny nonexpansive retraction if Q is sunny, nonexpansive, and contraction. K 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 [4], describes a characterization of sunny nonexpansive retractions on a smooth Banach space.
Let E be a smooth Banach space and C be a nonempty subset of E. Let be a retraction and j be the normalized duality mapping on E. Then the following are equivalent:
-
(1)
Q is sunny and nonexpansive;
-
(2)
, ;
-
(3)
, , .
Krasnoselski-Mann iteration generates a sequence in the following manner:
It is known that the Krasnoselski-Mann iteration only has weak convergence even for nonexpansive mappings in infinite-dimensional Hilbert spaces; for more details, see [5] and the references therein. In many disciplines, including economics, image recovery, quantum physics, and control theory, problems arise in infinite dimension spaces. In such problems, strong convergence (norm convergence) is often much more desirable than weak convergence, for it translates the physically tangible property that the energy of the error between the iterate and the solution x eventually becomes arbitrarily small. To improve the weak convergence of Krasnoselski-Mann iterative process, different modified Mann iterations have been considered; see [6–30] and the references therein.
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 zeros of A.
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.
The following iteration also has been extensively investigated:
where is a real number sequence in , is a positive real number sequence, and . It is known that the sequence generated in the above iteration converges strongly to a zero point of A in a Banach space under some restrictions imposed on and .
Chen et al. [29] investigated the following iteration:
where and are real number sequences in , is a positive real number sequence, and . They proved that the sequence generated in the above iteration converges strongly to a zero point of A in a Banach space; for more details, see [29] and the references therein.
We also remark that the viscosity approximation method was first introduced by Moudafi [31] in the framework of Hilbert spaces. Moudafi proved that the desired solution is not only a fixed point of nonlinear mappings but a solution to some variational inequality; for more details, see [31] and the references therein.
Recently, Qin et al. [20] investigated the iteration (2.4) with double computational errors and established a strong convergence theorem in a real reflexive Banach space with the uniformly Gâteaux differentiable norm; for more details, see [20] and the references therein. Different regularization methods recently have been investigated for treating zero points of accretive operators. In this paper, a modified Mann iteration with computational errors is investigated. A strong convergence theorem for zero points of an m-accretive operator is established in a Banach space. The results mainly improve the corresponding results in Qin and Su [6], Hao [15], Qin et al. [20] and Chen et al. [29].
In order to state our main results, we also need the following lemmas.
Lemma 2.1 [20]
Let E be a real reflexive Banach space with the uniformly Gâteaux differentiable norm and the normal structure, and C be a nonempty closed convex subset of E. Let be a nonexpansive mapping with a fixed point, and be a fixed contraction with the coefficient . Let be a sequence generated by the following , where . Then converges strongly as to a fixed point of T, which is the unique solution in to the following variational inequality
Lemma 2.2 [32]
Let and be bounded sequences in a Banach space E, and be a sequence in with . Suppose that , and
Then .
Lemma 2.3 [33]
Let , , and be three nonnegative real sequences satisfying
where is a sequence in . Assume that the following conditions are satisfied:
-
(a)
and ;
-
(b)
.
Then .
Lemma 2.4 [34]
Let E be a Banach space and let A be an m-accretive operator. For , , and , we have , where and .
3 Main results
Theorem 3.1 Let E be a real reflexive Banach space with the uniformly Gâteaux differentiable norm. Let A be an m-accretive operator in E such that is convex and has the normal structure. Let be an α-contraction. Let be a sequence generated in the following manner: and
where and are real number sequences in , is a sequence in E, is a positive real number sequence, and . Assume that is not empty and the above control sequences satisfy the following restrictions:
-
(a)
and ;
-
(b)
;
-
(c)
;
-
(d)
for each and .
Then the sequence converges strongly to , which is the unique solution to the following variational inequality , .
Proof First, we prove that is bounded. Fixing , we see that
It follows that
This proves that the sequence is bounded. If , we see from Lemma 2.4 that
where is an appropriate constant such that . Put ; that is,
Note that
It follows that
Substituting (3.1) into (3.3), we arrive at
In view of restrictions (a), (b) and (c), we find that
By virtue of Lemma 2.2, one obtains that
It follows from (3.1) that . In view of restriction (c), we find from (3.4) that
If , we can prove that (3.5) still holds. On the other hand, we have
It follows that
In view of restriction (b), one finds that
Notice that
Since , from (3.6) we obtain that . Taking a fixed number k such that , we arrive at
Since
we therefore find that
Now, we are in a position to prove that , where , and solves the fixed point equation
Therefore, we see that
This implies that
It follows from (3.8) that
Since as and from the fact that j is strong to weak∗ uniformly continuous on bounded subsets of E, we find that
It follows that
For any , there exists such that the following inequality holds
This implies that
Note that ϵ is arbitrary. In view of (3.9), we see that . This implies that
Finally, we show that as . Notice that
It follows that
where . In view of restrictions (a) and (c), we know that . Put , , and . In view of Lemma 2.3, we find the desired conclusion. □
If , where u is a fixed element in C, for any , we find the following result.
Corollary 3.2 Let E be a real reflexive Banach space with the uniformly Gâteaux differentiable norm. Let A be an m-accretive operator in E such that is convex and has the normal structure. Let be a sequence generated in the following manner: and
where and are real number sequences in , is a positive real number sequence, and . Assume that is not empty and the above control sequences satisfy the following restrictions:
-
(a)
and ;
-
(b)
;
-
(c)
for each and .
Then the sequence converges strongly to .
4 Applications
In this section, we consider solutions of variational inequalities. Let C be a nonempty, closed, and convex subset of a Banach space E. Let be a single-valued monotone operator which is hemicontinuous; that is, continuous along each line segment in C with respect to the weak∗ topology of . Consider the following variational inequality problem of finding a point such that
In this section, we use to denote the solution set of the variational inequality involving A. The symbol stands for the normal cone for C at a point ; that is,
Theorem 4.1 Let E be a real reflexive Banach space with the uniformly Gâteaux differentiable norm. Let C be a nonempty, closed, and convex subset of E. Let be a single-valued, monotone, and hemicontinuous operator. Assume that is not empty and C has the normal structure. Let be an α-contraction. Let be a sequence generated in the following manner: and
where and are real number sequences in , is a sequence in E, is a positive real number sequence, and . Assume that is not empty and the above control sequences satisfy the following restrictions:
-
(a)
and ;
-
(b)
;
-
(c)
for each and .
Then the sequence converges strongly to , which is the unique solution to the following variational inequality , .
Proof Define a mapping by
By Rockafellar [35], we know that T is maximal monotone, and . For each and , we see that there exists a unique such that , where . Notice that
which is equivalent to
that is, . This implies that . Following the proof of Theorem 3.1, we can immediately conclude the desired conclusion. □
If , where u is a fixed element in C, for any , we find the following result.
Corollary 4.2 Let E be a real reflexive Banach space with the uniformly Gâteaux differentiable norm. Let C be a nonempty, closed, and convex subset of E. Let be a single-valued, monotone, and hemicontinuous operator. Assume that is not empty and C has the normal structure. Let be a sequence generated in the following manner: and
where and are real number sequences in , is a sequence in E, is a positive real number sequence, and . Assume that is not empty and the above control sequences satisfy the following restrictions:
-
(a)
and ;
-
(b)
;
-
(c)
for each and .
Then the sequence converges strongly to .
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
Browder FE: Existence and approximation of solutions of nonlinear variational inequalities. Proc. Natl. Acad. Sci. USA 1966, 56: 1080–1086. 10.1073/pnas.56.4.1080
Takahashi W: Nonlinear Functional Analysis: Fixed Point Theory and Its Application. Yokohama Publishers, Yokohama; 2000.
Bruck RE: Nonexpansive projections on subsets of Banach spaces. Pac. J. Math. 1973, 47: 341–355. 10.2140/pjm.1973.47.341
Genel A, Lindenstrass J: An example concerning fixed points. Isr. J. Math. 1975, 22: 81–86. 10.1007/BF02757276
Qin X, Su Y: 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
Yang S: Zero theorems of accretive operators in reflexive Banach spaces. J. Nonlinear Funct. Anal. 2013., 2013: Article ID 2
Cho SY, Kang SM: Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl. Math. Lett. 2011, 24: 224–228. 10.1016/j.aml.2010.09.008
Cho SY, Qin X, Kang SM: Iterative processes for common fixed points of two different families of mappings with applications. J. Glob. Optim. 2013, 57: 1429–1446. 10.1007/s10898-012-0017-y
Yang S: A proximal point algorithm for zeros of monotone operators. Math. Finance Lett. 2013., 2013: Article ID 7
Zhang H: Implicit iterative methods for zeros of accretive operators. J. Fixed Point Theory 2013., 2013: Article ID 2
Song J, Chen M: On generalized asymptotically quasi- Ï• -nonexpansive mappings and a Ky Fan inequality. Fixed Point Theory Appl. 2013., 2013: Article ID 237
Cho SY, Qin X, Kang SM: Hybrid projection algorithms for treating common fixed points of a family of demicontinuous pseudocontractions. Appl. Math. Lett. 2012, 25: 854–857. 10.1016/j.aml.2011.10.031
Wu C, Lv S: Bregman projection methods for zeros of monotone operators. J. Fixed Point Theory 2013., 2013: Article ID 7
Hao Y: Zero theorems of accretive operators. Bull. Malays. Math. Sci. Soc. 2011, 34: 103–112.
Zegeye H, Shahzad N: Strong convergence theorem for a common point of solution of variational inequality and fixed point problem. Adv. Fixed Point Theory 2012, 2: 374–397.
Cho SY, Li W, Kang SM: Convergence analysis of an iterative algorithm for monotone operators. J. Inequal. Appl. 2013., 2013: Article ID 199
Wang ZM, Lou W: A new iterative algorithm of common solutions to quasi-variational inclusion and fixed point problems. J. Math. Comput. Sci. 2013, 3: 57–72.
Wang G, Sun S: Hybrid projection algorithms for fixed point and equilibrium problems in a Banach space. Adv. Fixed Point Theory 2013, 3: 578–594.
Qin X, Cho SY, Wang L: Iterative algorithms with errors for zero points of m -accretive operators. Fixed Point Theory Appl. 2013., 2013: Article ID 148
Wu C: Viscosity iterative algorithms for variational inequality. Adv. Inequal. Appl. 2014., 2014: Article ID 8
Wu C, Wang G: Hybrid projection algorithms for asymptotically quasi- Ï• -nonexpansive mappings. Commun. Optim. Theory 2013., 2013: Article ID 2
Kim JK: Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi- Ï• -nonexpansive mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 10
Qin X, Cho SY, Kim JK: Convergence theorems on asymptotically pseudocontractive mappings in the intermediate sense. Fixed Point Theory Appl. 2010., 2010: Article ID 186874
Lv S, Wu C: Convergence of iterative algorithms for a generalized variational inequality and a nonexpansive mapping. Eng. Math. Lett. 2012, 1: 44–57.
Cho SY, Kang SM: Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math. Sci. 2012, 32: 1607–1618. 10.1016/S0252-9602(12)60127-1
Qin X, Su Y: Strong convergence theorems for relatively nonexpansive mappings in a Banach space. Nonlinear Anal. 2007, 67: 1958–1965. 10.1016/j.na.2006.08.021
Zhu ZJ: Strong convergence theorems for fixed points of nonlinear mappings. Eng. Math. Lett. 2014., 2014: Article ID 3
Chen R, Liu Y, Shen X: Iterative approximation of a zero of accretive operator in Banach space. Nonlinear Anal. 2009, 71: e346-e350. 10.1016/j.na.2008.11.054
Qin X, Cho YJ, Kang SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 2009, 225: 20–30. 10.1016/j.cam.2008.06.011
Moudafi A: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 2000, 241: 46–55. 10.1006/jmaa.1999.6615
Suzuki T: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochne integrals. J. Math. Anal. Appl. 2005, 305: 227–239. 10.1016/j.jmaa.2004.11.017
Liu L: Ishikawa-type and Mann-type iterative processes with errors for constructing solutions of nonlinear equations involving m -accretive operators in Banach spaces. Nonlinear Anal. 1998, 34: 307–317. 10.1016/S0362-546X(97)00579-8
Barbu V: Nonlinear Semigroups and Differential Equations in Banach Space. Noordhoff, Groningen; 1976.
Rockafellar TT: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 1970, 149: 75–88. 10.1090/S0002-9947-1970-0282272-5
Acknowledgements
This work is supported by the Department of Hebei Education (Z2013110). The authors are grateful to Professor Jin Wang and the anonymous reviewers for useful suggestions which improved the contents of the article.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to this manuscript. All 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
Song, J., Chen, M. A modified Mann iteration for zero points of accretive operators. Fixed Point Theory Appl 2013, 347 (2013). https://doi.org/10.1186/1687-1812-2013-347
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2013-347