# Some Krasnonsel'skiĭ-Mann Algorithms and the Multiple-Set Split Feasibility Problem

- Huimin He
^{1}Email author, - Sanyang Liu
^{1}and - MuhammadAslam Noor
^{2, 3}

**2010**:513956

**DOI: **10.1155/2010/513956

© Huimin He et al. 2010

**Received: **3 April 2010

**Accepted: **13 July 2010

**Published: **29 July 2010

## Abstract

Some variable Krasnonsel'skiĭ-Mann iteration algorithms generate some sequences , , and , respectively, via the formula , , , where and the mod function takes values in , , , and are sequences in and are sequences of nonexpansive mappings. We will show, in a fairly general Banach space, that the sequence , , generated by the above formulas converge weakly to the common fixed point of , respectively. These results are used to solve the multiple-set split feasibility problem recently introduced by Censor et al. (2005). The purpose of this paper is to introduce convergence theorems of some variable Krasnonsel'skiĭ-Mann iteration algorithms in Banach space and their applications which solve the multiple-set split feasibility problem.

## 1. Introduction

(See [4, 5] for details on the fixed point theory for nonexpansive mappings.)

where and are the (orthogonal) projections onto and , respectively, is any positive constant and denotes the adjoint of . Moreover, for sufficiently small , the operator which defines the fixed point equation (1.6) is nonexpansive.

where is a sequence of nonexpansive mappings in a Hilbert space , under certain conditions, they proved convergence of (1.9) essentially in a finite-dimensional Hilbert space. Furthermore, with regard to (1.9), Xu [13] extended the results of Zhao and Yang [10] in the framework of fairly general Banach space.

where and are positive integers, and are closed and convex subsets of and , respectively, and is a linear bounded operator from to .

where , , , and are sequences in , and are sequences of nonexpansive mappings. We will show, in a fairly general Banach space , that the sequences , , and generated by (1.13), (1.14), and (1.15) converge weakly to the common fixed point of , respectively. The applications of these results are used to solve the multiple-set split feasibility problem recently introduced by [15].

This paper is organized as follows. In the next section, we will prove a weak convergence theorems for the three variable K-M algorithms (1.13), (1.14), and (1.15) in a uniformly convex Banach space with a Frechet differentiable norm (the class of such Banach spaces include Hilbert space and and space for ). In the last section, we will present the applications of the weak convergence theorems for the three variable K-M algorithms (1.13), (1.14), and (1.15).

## 2. Convergence of Variable Krasnonsel'ski -Mann Iteration Algorithm

To solve the multiple-set split feasibility problem (MSSFP) in Section 3, we firstly present some theorems of the general variable Krasnonsel'ski -Mann iteration algorithms.

Theorem 2.1.

Let be a uniformly convex Banach space with a Frechet differentiable norm, let be a nonempty closed and convex subset of , and let be nonexpansive mapping, . Assume that the set of common fixed point of , , is nonempty. Let be any sequence generated by (1.13), where satisfy the conditions

;

for every and , where .

Then converges weakly to a common fixed point of .

Proof.

Since is nonexpansive mapping, for , then, the composition is nonexpansive mapping from to . Let .

Thus, is a decreasing sequence, and we have that exists. Hence, is bounded, so are , , and . Let and let .

where denotes the weak -limit set of .

To prove that is weakly convergent to a common fixed point of , it now suffices to prove that consists of exactly one point.

This is a contradiction.

The proof is completed.

Theorem 2.2.

Let be a uniformly convex Banach space with a Frechet differentiable norm, let be a nonempty closed and convex subset of , and let be nonexpansive mapping, , assume that the set of common fixed point of , , is nonempty. Let be defined by (1.14), where satisfy the following conditions

;

for every and , where .

Then converges weakly to a common fixed point of .

Proof.

Since is a nonexpansive mapping, , then, it is not hard to see that is a nonexpansive mapping from to .

The remainder of the proof is the same as Theorem 2.1.

The proof is completed.

Theorem 2.3.

Let be a uniformly convex Banach space with a Frechet differentiable norm, let be a nonempty closed convex subset of , and let be nonexpansive mapping, , assume that the set of common fixed point of , , is nonempty. Let be defined by (1.15), where satisfy the conditions

;

for every and , where .

Then converges weakly to a common fixed point of .

Proof.

Since and is a sequence of nonexpansive mappings from to , so, the proof of this theorem is similar to Theorems 2.1 and 2.2.

The proof is completed.

## 3. Applications for Solving the Multiple-Set Split Feasibility Problem (MSSFP)

A projection of onto a closed convex subset is both nonexpansive and 1-ism. It is also known that a mapping is averaged if and only if the complement is -ism for some ; see [8] for more property of averaged mappings and -ism.

To solve the MSSFP (1.10), Censor et al. [15] proposed the following projection algorithm (1.11), the algorithm (1.11) involves an additional projection . Though the MSSFP, (1.10) includes the SFP (1.5) as a special case, which does not reduced to (1.7), let alone (1.8). In this section, we will propose some new projection algorithms which solve the MSSFP (1.10) and which are the application of algorithms (1.13), (1.14), and (1.15) for solving the MSSFP. These projection algorithms can also reduce to the algorithm (1.8) when the MSSFP (1.10) is reduced to the SFP (1.5).

Theorem 3.1.

Assume that the MSSFP (1.10) is consistent. Let be the sequence generated by the algorithm (3.2), where with and satisfy the condition: . Then converges weakly to a solution of the MSSFP (1.10).

Proof.

Let ,

and is nonexpansive, it is easy to see that is -Lipschitzian, with .

Therefore, is -ism [18]. This implies that for any , is averaged. Hence, for any closed and convex subset of , the composite is averaged.

So is averaged, thus is nonexpansive.

By the position 2.2 [8], we see that the fixed point set of , , is the common fixed point set of the averaged mappings .

By Reich [3], we have converges weakly to a fixed point of which is also a common fixed point of or a solution of the MSSFP (1.10).

The proof is completed.

Theorem 3.2.

Assume that the MSSFP (1.10) is consistent. Let be any sequence generated by the algorithm (3.5), where with and satisfy the condition: . Then converges weakly to a solution of the MSSFP (1.10).

Proof.

From the proof of Theorem 3.1, it is easy to know that is averaged, so, the convex combination is also averaged.

Thus is nonexpansive.

By Reich [3], we have converges weakly to a fixed point of .

Next, we only need to prove the fixed point of is also the common fixed point of which is the solution of the MSSFP (1.10), that is, .

Indeed, it suffices to show that .

Pick an arbitrary , thus . Also pick a , thus , .

Write , with and is nonexpansive.

We claim that if is such that , then , .

This is a contradiction. Therefore, we must have , , that is, .

This proof is completed.

where .

Theorem 3.3.

Assume that the MSSFP (1.10) is consistent. Let be the sequence generated by the algorithm (3.9), where with and satisfy the following conditions:

;

and for each ,

Then converges weakly to a solution of the MSSFP (1.10).

Proof.

From the proof of application (3.2), it is easy to verify that is averaged, so, is also averaged.

Thus is nonexpansive.

Now we cam apply Theorem 2.3 to conclude that the sequence given by the projection Algorithm (3.9) converges weakly to a solution of the MSSFP (1.10).

The proof is completed.

Remark 3.4.

The algorithms (3.12), (3.13), and (3.15) of Xu [13] are some projection algorithms for solving the MSSEP (1.10), which are concrete projection algorithms. In this paper, firstly, we present some general variable K-M algorithms (1.13), (1.14), and (1.15), and prove the weak convergence for them in Section 2. Secondly, through the applications of the weak convergence for three general variable K-M algorithms (1.13), (1.14), and (1.15), we solve the MSSEP (1.10) by the algorithms (3.2), (3.5), and (3.9).

## Declarations

### Acknowledgments

The work was supported by the Fundamental Research Funds for the Central Universities, no. JY10000970006, and the National Nature Science Foundation, no. 60974082.

## Authors’ Affiliations

## References

- Krasnosel'skiĭ MA:
**Two remarks on the method of successive approximations.***Uspekhi Matematicheskikh Nauk*1955,**10:**123–127.MathSciNetGoogle 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 - 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 - Browder FE:
**Nonlinear operators and nonlinear equations of evolution in Banach spaces.**In*Nonlinear Functional Analysis*. American Mathematical Society, Providence, RI, USA; 1976:1–308.Google Scholar - Goebel K, Kirk WA:
*Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics*.*Volume 28*. Cambridge University Press, Cambridge, UK; 1990:viii+244.View ArticleMATHGoogle Scholar - Censor Y, Elfving T:
**A multiprojection algorithm using Bregman projections in a product space.***Numerical Algorithms*1994,**8**(2–4):221–239.MathSciNetView ArticleMATHGoogle Scholar - Byrne C:
**Iterative oblique projection onto convex sets and the split feasibility problem.***Inverse Problems*2002,**18**(2):441–453. 10.1088/0266-5611/18/2/310MathSciNetView 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 - Yang Q:
**The relaxed CQ algorithm solving the split feasibility problem.***Inverse Problems*2004,**20**(4):1261–1266. 10.1088/0266-5611/20/4/014MathSciNetView ArticleMATHGoogle Scholar - Zhao J, Yang Q:
**Several solution methods for the split feasibility problem.***Inverse Problems*2005,**21**(5):1791–1799. 10.1088/0266-5611/21/5/017MathSciNetView ArticleMATHGoogle Scholar - Attouchi H:
*Variational Convergence for Functions and Operatoes*. Pitman, Boston, Mass, USA; 1984.Google Scholar - Yang Q, Zhao J:
**Generalized KM theorems and their applications.***Inverse Problems*2006,**22**(3):833–844. 10.1088/0266-5611/22/3/006MathSciNetView ArticleMATHGoogle Scholar - Xu HK:
**A variable Krasnosel'skii-Mann algorithm and the multiple-set split feasibility problem.***Inverse Problems*2006,**22**(6):2021–2034. 10.1088/0266-5611/22/6/007MathSciNetView ArticleMATHGoogle Scholar - Palta RJ, Mackie TR (Eds):
**Intensity-modulated radiation therapy: the state of art**In*Medical Physics Monograph*.*Volume 29*. Medical Physics Publishing, Madison, Wis, USA; 2003. - Censor Y, Elfving T, Kopf N, Bortfeld T:
**The multiple-sets split feasibility problem and its applications for inverse problems.***Inverse Problems*2005,**21**(6):2071–2084. 10.1088/0266-5611/21/6/017MathSciNetView 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 - 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 - Baillon J-B, Haddad G:
**Quelques propriétés des opérateurs angle-bornés et**n**-cycliquement monotones.***Israel Journal of Mathematics*1977,**26**(2):137–150. 10.1007/BF03007664MathSciNetView ArticleMATHGoogle Scholar

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