# Robustness of Mann Type Algorithm with Perturbed Mapping for Nonexpansive Mappings in Banach Spaces

- LC Ceng
^{1, 2}, - YC Liou
^{3}and - JC Yao
^{4}Email author

**2010**:734181

https://doi.org/10.1155/2010/734181

© L. C. Ceng et al. 2010

**Received: **30 October 2009

**Accepted: **10 January 2010

**Published: **1 February 2010

## Abstract

The purpose of this paper is to study the robustness of Mann type algorithm in the sense that approximately perturbed mapping does not alter the convergence of Mann type algorithm. It is proven that Mann type algorithm with perturbed mapping remains convergent in a Banach space setting where , a nonexpansive mapping, , , errors and a strongly accretive and strictly pseudocontractive mapping.

## Keywords

## 1. Introduction

Let be a nonempty closed convex subset of a real Banach space , and a nonexpansive mapping (i.e., for all ). We use to denote the set of fixed points of ; that is, . Throughout this paper it is assumed that . Construction of fixed points of nonlinear mappings is an important and active research area. In particular, iterative methods for finding fixed points of nonexpansive mappings have received vast investigation since these methods find applications in a variety of applied areas of variational inequality problems, equilibrium problems, inverse problems, partial differential equations, image recovery, and signal processing (see, e.g., [1–17]).

In 1953, Mann [18] introduced an iterative algorithm which is now referred to as Mann's algorithm. Most of the literature deals with the special case of the general Mann's algorithm; that is, for an arbitrary initial guess , the sequence is generated by the recursive manner

where is a convex subset of a Banach space is a mapping and is a sequence in the interval . It is well known that Mann's algorithm can be employed to approximate fixed points of nonexpansive mappings and zeros of (strongly) accretive mappings in Hilbert spaces and Banach spaces. Many convergence theorems have been announced and published by a large number of authors. A typical convergence result in connection with the Mann's algorithm is the following theorem proved by Ishikawa [19].

Theorem IS (see [19])

Let be a nonempty subset of a Banach space and let be a nonexpansive mapping. Let be a real sequence satisfying the following control conditions:

Let be defined by (1.1) such that for all . If is bounded then as .

The interest and importance of Theorem IS lie in the fact that strong or weak convergence of the sequence can be achieved under certain appropriate assumptions imposed on the mapping , the domain or the space . In an infinite-dimensional space , Mann's algorithm has only weak convergence, in general. In fact, it is known that if the sequence is such that , then Mann's algorithm converges weakly to a fixed point of provided that the underlying space is a Hilbert space or more general, a uniformly convex Banach space which has a Fréchet differentiable norm or satisfies Opial's property. See, for example, [20, 21].

The study of the robustness of Mann's algorithm is initiated by Combettes [22] where he considered a parallel projection method algorithm in signal synthesis (design and recovery) problems in a real Hilbert space as follows:

where for each , is the (nearest point) projection of a signal onto a closed convex subset of [23] ( is interpreted as the th constraint set of the signals), is a sequence of relaxation parameters in are strictly positive weights such that , and stands for the error made in computing the projection onto at iteration . Then he proved the following robustness result of algorithm (1.2).

Theorem 1.1 (see [22]).

Then the sequence generated by (1.2) converges weakly to a point in .

where is given by (1.3). Note that can be written as and thus is nonexpansive. Note also that . Furthermore, conditions (i) and (ii) in Theorem 1.1 can be stated as

Very early, some authors had considered Mann iterations in the setting of uniformly convex Banach spaces and established strong and weak convergence results for Mann iterations; see, e.g., [24, 25]. Recently, Kim and Xu [26] studied the robustness of Mann's algorithm for nonexpansive mappings in Banach spaces and extended Combettes' robustness result (Theorem 1.1 above) for projections from Hilbert spaces to the setting of uniformly convex Banach spaces.

Theorem 1.2 (see [26, Theorem 3.3]).

where and satisfy the following properties:

Then the sequence converges weakly to a fixed point of .

Further, Kim and Xu [26] also extended the robustness to nonexpansive mappings which are defined on subsets of a Hilbert space and to accretive operators.

Theorem 1.3 (see [26, Theorem 4.1]).

where the sequences and are such that

Then converges weakly to a fixed point of .

Theorem 1.4 (see [26, Theorem 5.1]).

Let be a uniformly convex Banach space. Assume in addition that either has the KK- property or satisfies Opial's property. Let be an -accretive operator in such that . Moreover, assume that and satisfy the following properties:

(iii) , where and are two constants;

converges weakly to a point of .

Let be a real reflexive Banach space. Let be a nonexpansive mapping with . Assume that is -strongly accretive and -strictly pseudocontractive with where . In this paper, inspired by Combettes' robustness result (Theorem 1.1 above) and Kim and Xu's robustness result (Theorem 1.2 above) we will consider the robustness of Mann type algorithm with perturbed mapping, which generates, from an arbitrary initial guess , a sequence by the recursive manner

where and are sequences in and in , respectively, such that

More precisely, we will prove under conditions (i)–(iii) the weak convergence of the algorithm (1.9) in a uniformly convex Banach space
which either has the *KK-* property or satisfies Opial's property. This theorem extends Kim and Xu's robustness result (Theorem 1.2 above) from Mann's algorithm (1.6) with errors to Mann type algorithm (1.9) with perturbed mapping
. On the other hand, we also extend Kim and Xu's robustness results (Theorems 1.3 and 1.4 above) for nonexpansive mappings which are defined on subsets of a Hilbert space and accretive operators in a uniformly convex Banach space from Mann's algorithm with errors to Mann type algorithm with perturbed mapping.

Throughout this paper, we use the following notations:

## 2. Preliminaries

Let be a real Banach space. Recall that the norm of is said to be Fréchet differentiable if, for each , the unit sphere of , the limit

exists and is attained uniformly in . In this case, we have

for all , where is the normalized duality map from to defined by

is the duality pairing between and , and is a function defined on such that . Examples of Banach spaces which have a Fréchet differentiable norm include and for (these spaces are actually uniformly smooth).

It is known that a Banach space is Fréchet differentiable if and only if the duality map is single-valued and norm-to-norm continuous.

We need the concept of the *KK*-property. A Banach space
is said to have the *KK*-property (the Kadec-Klee property) if, for any sequence
in
, the conditions
and
imply that
. It is known [27, Remark 3.2] that the dual space of a reflexive Banach space with a Fréchet differentiable norm has the *KK*-property.

Recall now that satisfies Opial's property [28] provided that, for each sequence in , the condition implies

It is known [28] that each enjoys this property, while does not unless . It is known [29] that any separable Banach space can be equivalently renormed so that it satisfies Opial's property.

Recall that a Banach space is said to be uniformly convex if, for each , the modulus of convexity of defined by

is positive.

We need an inequality characterization of uniform convexity.

Lemma 2.1 (see [30]).

A mapping with domain and range in is called -strongly accretive if for each ,

for some . is called -strictly pseudocontractive if for each ,

for some . It is easy to see that (2.8) can be rewritten as

The following proposition will be used frequently throughout this paper. For the sake of completeness, we include its proof.

Proposition 2.2.

Let be a real Banach space and a mapping.

(i)If is a -strictly pseudocontractive, then is Lipschitz continuous with constant

(ii)If is -strongly accretive and -strictly pseudocontractive with , then for each fixed , the mapping has the following property:

This shows that inequality (2.10) holds.

Proposition 2.3.

Let be a uniformly convex Banach space and a nonempty closed convex subset of .

(i)Reference [31] (demiclosedness principle). If is a nonexpansive mapping and if is a sequence in such that and , then .

(ii)Reference [32]. If is also bounded, then there exists a continuous, strictly increasing, and convex function (depending only on the diameter of ) with and such that

for all , and nonexpansive mappings .

We also use the following elementary lemma.

Lemma 2.4 (see [33]).

Let and be sequences of nonnegative real numbers such that and for all . Then exists.

## 3. Robustness of Mann Type Algorithm with Perturbed Mapping

Let be a real reflexive Banach space. Let be a nonexpansive mapping with . Assume that is -strongly accretive and -strictly pseudocontractive with . We now discuss the robustness of Mann type algorithm with perturbed mapping, which generates, from an initial guess , a sequence as follows:

where and are sequences in and in , respectively, such that

We remark that Mann type algorithm with perturbed mapping is based on Mann iteration method and steepest-descent method. Indeed, in algorithm (3.1), one iteration step " " is taken from Mann iteration method, and another iteration step " " is taken from steepest-descent method.

We first discuss some properties of algorithm (3.1).

Lemma 3.1.

Let be generated by algorithm (3.1) and let Then exists.

Proof.

The conclusion of the lemma is a consequence of Lemma 2.4.

Proposition 3.2.

Let be a uniformly convex Banach space.

(ii)If, in addition, the dual space of has the -property, then the weak -limit set of , , is a singleton.

Thus is nonexpansive (due to ) and so is .

Second, let us show that for each ,

This shows that inequality (3.5) holds for the case of . Thus, by induction, we know that inequality (3.5) holds for all .

Now set

- (ii)
This is Lemma 3.2 of [27].

Now we can state and prove the main result of this section.

Theorem 3.3.

where and satisfy the following properties:

Then the sequence converges weakly to a fixed point of .

Proof.

Hence to prove that
converges weakly to a fixed point of
, it suffices to show that
is a singleton. We distinguish two cases. First assume that
has the *KK*-property. Then that
is a singleton is guaranteed by Proposition 3.2 no. (ii).

Next assume that satisfies Opial's property. Take and let and be subsequences of such that and , respectively. If , we reach the following contradiction:

This shows that is a singleton. The proof is therefore complete.

## 4. The Case Where Mappings Are Defined on Subsets

We observe that if the domain is a proper closed convex subset of , then the vectors and may not belong to . In this case the next iterate may not be well defined by (3.13). In order to consider this situation, we will use the nearest projection and for the projection to be nonexpansive, we have to restrict our spaces to be Hilbert spaces.

Let be a real Hilbert space with inner product and norm . Given a closed convex subset of . Recall that the (nearest point) projection from onto assigns each point with its (unique) nearest point in which is denoted by . Namely, is the unique point in with the property

Let be a nonexpansive mapping with and -strongly monotone and -strictly pseudocontractive with . Starting with and after in is defined, we have two ways to define the next iterate : either applying the projection to the vectors and and defining as the convex combination of and , or projecting a convex combination of and onto to define . More precisely, we define as follows:

or

Theorem 4.1.

Let be a nonempty closed convex subset of a Hilbert space . Let be a nonexpansive mapping with and -strongly monotone and -strictly pseudocontractive with . Let be generated by either (4.2) or (4.3) where the sequences and are such that

Then converges weakly to a fixed point of .

Proof.

Hence exists; in particular, is bounded. Let be a constant such that for all .

We compute

Similarly, if is generated by algorithm (4.3), then relations (4.4)–(4.11) still hold.

It is now readily seen that (4.11) together with Lemma 2.4 implies that exists, which together with (4.8) further implies that

Equation (4.12) implies that , due to the demiclosedness principle. Finally, repeating the last part of the proof of Theorem 3.3 in the case of Opial's property, we see that converges weakly to a fixed point of . The proof is therefore complete.

Finally in this section, we consider the case of accretive operators. Recall that a multivalued operator with domain and range in a Banach space is said to be accretive if, for each and , there is such that

where is the duality map from to the dual space . An accretive operator is -accretive if for all .

Denote by the zero set of ; that is,

Throughout the rest of this paper it is always assumed that is -accretive and is nonempty.

Denote by the resolvent of for :

It is known that is a nonexpansive mapping from to which will be assumed convex (this is so if is uniformly convex). It is also known that for .

Now consider the problem of finding a zero of an -accretive operator in a Banach space ,

We will study the convergence of the following algorithm:

where is a perturbed mapping, the initial guess is arbitrary, and are two sequences in is a sequence of positive numbers, and is an error sequence in .

Theorem 4.2.

Let be a uniformly convex Banach space. Assume in addition that either has the -property or satisfies Opial's property. Let be an -accretive operator in such that and let be -strongly accretive and -strictly pseudocontractive with . Moreover, assume that and satisfy the following properties:

(iv) , where and are two constants;

Then the sequence generated by algorithm (4.17) converges weakly to a point of .

Proof.

The proof is a refinement of that of Theorem 3.3 given in Section 3 and [34, Theorem 3.3] together with Proposition 3.2. So we only sketch it.

By Lemma 2.4, we see that exists.

With slight modifications of the proof of Theorem 3.3 (replacing by ), we can obtain that

By the demiclosedness principle, (4.25) ensures that . Repeating the last part of the proof of Theorem 3.3, we conclude that converges weakly to a point of .

## Declarations

### Acknowledgments

This research was partially supported by Grant no. NSC 98-2923-E-110-003-MY3 and was also partially supported by the Leading Academic Discipline Project of Shanghai Normal University (DZL707), Innovation Program of Shanghai Municipal Education Commission Grant (09ZZ133), National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), Science and Technology Commission of Shanghai Municipality Grant (075105118), and Shanghai Leading Academic Discipline Project (S30405).

## Authors’ Affiliations

## References

- Browder FE, Petryshyn WV:
**Construction of fixed points of nonlinear mappings in Hilbert space.***Journal of Mathematical Analysis and Applications*1967,**20:**197–228. 10.1016/0022-247X(67)90085-6MathSciNetView ArticleMATHGoogle Scholar - Byrne C:
**A unified treatment of some iterative algorithms in signal processing and image reconstruction.***Inverse Problems*2004,**20**(1):103–120. 10.1088/0266-5611/20/1/006MathSciNetView ArticleMATHGoogle Scholar - Engl HW, Leitão A:
**A Mann iterative regularization method for elliptic Cauchy problems.***Numerical Functional Analysis and Optimization*2001,**22**(7–8):861–884. 10.1081/NFA-100108313MathSciNetView ArticleMATHGoogle Scholar - Engl HW, Scherzer O:
**Convergence rates results for iterative methods for solving nonlinear ill-posed problems.**In*Surveys on Solution Methods for Inverse Problems*. Springer, Vienna, Austria; 2000:7–34.View ArticleGoogle Scholar - Magnanti TL, Perakis G:
**Solving variational inequality and fixed point problems by line searches and potential optimization.***Mathematical Programming, Series A*2004,**101**(3):435–461. 10.1007/s10107-003-0476-5MathSciNetView ArticleMATHGoogle Scholar - Podilchuk CI, Mammone RJ:
**Image recovery by convex projections using a least-squares constraint.***Journal of the Optical Society of America A*1990,**7:**517–521.View ArticleGoogle Scholar - Sezan MI, Stark H:
**Applications of convex projection theory to image recovery in tomography and related areas.**In*Image Recovery: Theory and Application*. Edited by: Stark H. Academic Press, Orlando, Fla, USA; 1987:415–462.Google Scholar - Tan K-K, Xu HK:
**Fixed point iteration processes for asymptotically nonexpansive mappings.***Proceedings of the American Mathematical Society*1994,**122**(3):733–739. 10.1090/S0002-9939-1994-1203993-5MathSciNetView ArticleMATHGoogle Scholar - Yamada I, Ogura N:
**Adaptive projected subgradient method for asymptotic minimization of sequence of nonnegative convex functions.***Numerical Functional Analysis and Optimization*2004,**25**(7–8):593–617.MathSciNetView ArticleMATHGoogle Scholar - Yamada I, Ogura N:
**Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings.***Numerical Functional Analysis and Optimization*2004,**25**(7–8):619–655.MathSciNetView ArticleMATHGoogle Scholar - Youla D:
**Mathematical theory of image restoration by the method of convex projections.**In*Image Recovery: Theory and Application*. Edited by: Stark H. Academic Press, Orlando, Fla, USA; 1987:29–77.Google Scholar - Youla D:
**On deterministic convergence of iterations of related projection operators.***Journal of Visual Communication and Image Representation*1990,**1:**12–20. 10.1016/1047-3203(90)90013-LView ArticleGoogle Scholar - Ceng L-C, Ansari QH, Yao J-C:
**Mann-type steepest-descent and modified hybrid steepest-descent methods for variational inequalities in Banach spaces.***Numerical Functional Analysis and Optimization*2008,**29**(9–10):987–1033. 10.1080/01630560802418391MathSciNetView ArticleMATHGoogle Scholar - Zeng L-C, Yao J-C:
**Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems.***Taiwanese Journal of Mathematics*2006,**10**(5):1293–1303.MathSciNetMATHGoogle Scholar - Ceng L-C, Yao J-C:
**Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings.***Applied Mathematics and Computation*2008,**198**(2):729–741. 10.1016/j.amc.2007.09.011MathSciNetView ArticleMATHGoogle Scholar - Ceng L-C, Yao J-C:
**A hybrid iterative scheme for mixed equilibrium problems and fixed point problems.***Journal of Computational and Applied Mathematics*2008,**214**(1):186–201. 10.1016/j.cam.2007.02.022MathSciNetView ArticleMATHGoogle Scholar - Ceng L-C, Schaible S, Yao J-C:
**Implicit iteration scheme with perturbed mapping for equilibrium problems and fixed point problems of finitely many nonexpansive mappings.***Journal of Optimization Theory and Applications*2008,**139**(2):403–418. 10.1007/s10957-008-9361-yMathSciNetView ArticleMATHGoogle Scholar - Mann WR:
**Mean value methods in iteration.***Proceedings of the American Mathematical Society*1953,**4:**506–510. 10.1090/S0002-9939-1953-0054846-3MathSciNetView ArticleMATHGoogle Scholar - Ishikawa S:
**Fixed points and iteration of a nonexpansive mapping in a Banach space.***Proceedings of the American Mathematical Society*1976,**59**(1):65–71. 10.1090/S0002-9939-1976-0412909-XMathSciNetView ArticleMATHGoogle Scholar - Reich S:
**Weak convergence theorems for nonexpansive mappings in Banach spaces.***Journal of Mathematical Analysis and Applications*1979,**67**(2):274–276. 10.1016/0022-247X(79)90024-6MathSciNetView ArticleMATHGoogle Scholar - Nevanlinna O, Reich S:
**Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces.***Israel Journal of Mathematics*1979,**32**(1):44–58. 10.1007/BF02761184MathSciNetView ArticleMATHGoogle Scholar - Combettes PL:
**On the numerical robustness of the parallel projection method in signal synthesis.***IEEE Signal Processing Letters*2001,**8**(2):45–47. 10.1109/97.895371View ArticleGoogle Scholar - Combettes PL:
**The convex feasibility problem in image recovery.**In*Advances in Imaging and Electron Physics*.*Volume 95*. Edited by: Hawkes P. New York Academic, New York, NY, USA; 1996:155–270.Google Scholar - Reich S:
**Weak convergence theorems for nonexpansive mappings in Banach spaces.***Journal of Mathematical Analysis and Applications*1979,**67:**274–276. 10.1016/0022-247X(79)90024-6MathSciNetView ArticleMATHGoogle Scholar - Nevanlinna O, Reich S:
**Strong convergence of contraction semi-groups and of iterative methods for accretive operators in Banach spaces.***Israel Journal of Mathematics*1979,**32:**44–58. 10.1007/BF02761184MathSciNetView ArticleMATHGoogle Scholar - Kim T-H, Xu H-K:
**Robustness of Mann's algorithm for nonexpansive mappings.***Journal of Mathematical Analysis and Applications*2007,**327**(2):1105–1115. 10.1016/j.jmaa.2006.05.009MathSciNetView ArticleMATHGoogle Scholar - García Falset J, Kaczor W, Kuczumow T, Reich S:
**Weak convergence theorems for asymptotically nonexpansive mappings and semigroups.***Nonlinear Analysis: Theory, Methods & Applications*2001,**43**(3):377–401. 10.1016/S0362-546X(99)00200-XMathSciNetView ArticleMATHGoogle Scholar - Opial Z:
**Weak convergence of the sequence of successive approximations for nonexpansive mappings.***Bulletin of the American Mathematical Society*1967,**73:**591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleMATHGoogle Scholar - van Dulst D:
**Equivalent norms and the fixed point property for nonexpansive mappings.***The Journal of the London Mathematical Society. Second Series*1982,**25**(1):139–144. 10.1112/jlms/s2-25.1.139MathSciNetView ArticleMATHGoogle Scholar - Xu HK:
**Inequalities in Banach spaces with applications.***Nonlinear Analysis: Theory, Methods & Applications*1991,**16**(12):1127–1138. 10.1016/0362-546X(91)90200-KMathSciNetView ArticleMATHGoogle Scholar - Browder FE:
**Convergence theorems for sequences of nonlinear operators in Banach spaces.***Mathematische Zeitschrift*1967,**100:**201–225. 10.1007/BF01109805MathSciNetView ArticleMATHGoogle Scholar - Bruck RE:
**A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces.***Israel Journal of Mathematics*1979,**32**(2–3):107–116. 10.1007/BF02764907MathSciNetView ArticleMATHGoogle Scholar - Tan K-K, Xu HK:
**Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process.***Journal of Mathematical Analysis and Applications*1993,**178**(2):301–308. 10.1006/jmaa.1993.1309MathSciNetView ArticleMATHGoogle Scholar - Marino G, Xu H-K:
**Convergence of generalized proximal point algorithms.***Communications on Pure and Applied Analysis*2004,**3**(4):791–808.MathSciNetView ArticleMATHGoogle Scholar

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