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

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

The Krasnonsel'ski-Mann (K-M) iteration algorithm [1, 2] is used to solve a fixed point equation

(1.1)

where is a self-mapping of closed convex subset of a Banach space . The K-M algorithm generates a sequence according to the recursive formula

(1.2)

where is a sequence in the interval and the initial guess is chosen arbitrarily. It is known [3] that if is a uniformly convex Banach space with a Frechet differentiable norm (in particular, a Hilbert space), if is nonexpansive, that is, satisfies the property

(1.3)

and if has a fixed point, then the sequence generated by the K-M algorithm (1.2) converges weakly to a fixed point of provided that fulfils the condition

(1.4)

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

Many problems can be formulated as a fixed point equation (1.1) with a nonexpansive and thus K-M algorithm (1.2) applies. For instance, the split feasibility problem (SFP) introduced in [6–8], which is to find a point

(1.5)

where and are closed convex subsets of Hilbert spaces and , respectively, and is a linear bounded operator from to . This problem plays an important role in the study of signal processing and image reconstruction. Assuming that the SFP (1.5) is consistent (i.e., (1.5) has a solution), it is not hard to see that solves (1.5) if and only if it solves the fixed point equation

(1.6)

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.

To solve the SFP (1.5), Byrne [7, 8] proposed his CQ algorithm (see also [9]) which generates a sequence by

(1.7)

where with being the spectral radius of the operator . In 2005, Zhao and Yang [10] considered the following perturbed algorithm:

(1.8)

where and are sequences of closed and convex subsets of and , respectively, which are convergent to and , respectively, in the sense of Mosco (c.f. [11]). Motivated by (1.8), Zhao and Yang [10, 12] also studied the following more general algorithm which generates a sequence according to the recursive formula

(1.9)

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.

The multiple-set split feasibility problem (MSSFP) which finds application in intensity-modulated radiation therapy [14] has recently been proposed in [15] and is formulated as finding a point

(1.10)

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

Assuming consistency of the MSSFP (1.10), Censor et al. [15] introduced the following projection algorithm:

(1.11)

where is another closed and convex subset of , with and being the spectral radius of , and for all and for all . They studied convergence of the algorithm (1.11) in the case where both and are finite dimensional. In 2006, Xu [13] demonstrated some projection algorithms for solving the MSSFP (1.10) in Hilbert space as follows:

(1.12)

where , , and and the mod function takes values in . This is a motivation for us to study the following more general algorithm which generate the sequences , , and , respectively, via the formulas

(1.13)

(1.14)

(1.15)

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].

Note that, letting be a nonempty subset of Banach space and , are self-mappings of , we use to denote , that is,

(1.16)

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 .

Take () to deduce that

(2.1)

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

Now since is uniformly convex, by [16, Theorem ], there exists a continuous strictly convex function , with , so that

(2.2)

for all , such that and and for all . Let , , be replaced by (note that ), and taking a constant so that , by the above (2.2), we obtain that

(2.3)

It follows that

(2.4)

Since exists, by condition (ii) and (2.4), it implies that

(2.5)

which further implies that by (i) , hence,

(2.6)

On the other hand, it is not hard to deduce from (1.13) that

(2.7)

Since , we see that exists. This together with (2.6) implies that

(2.8)

The demiclosedness principle for nonexpansive mappings (see [5, 17]) implies that

(2.9)

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.

Indeed, if there are , , since and exist, if , then

(2.10)

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)

Recall that a mapping in a Hilbert space is said to be averaged if can be written as , where and is nonexpansive. Recall also that an operator in is said to be inverse strongly monotone (-ism) for a given constant if

(3.1)

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).

The first one is a K-M type successive iteration method which produces a sequence by

(3.2)

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 ,  

Hence,

(3.3)

Since

(3.4)

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.

The second algorithm is also a K-M type method which generates a sequence by

(3.5)

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 , .

Indeed, we have

(3.6)

If we can show that , then we are done. So assume that . Now since , we have

(3.7)

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

This proof is completed.

We now apply Theorem 2.3 to solve the MSSFP (1.10). Recall that the -distance between two closed and convex subsets and of a Hilbert space is defined by

(3.8)

The third method is a K-M type cyclic algorithm which produces a sequence in the following manner: apply to the initial guess to get , next apply to to get , and continue this way to get ; then repeat this process to get , and so on. Thus, the sequence is defined and we write it in the form

(3.9)

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.

The projection iteration algorithm (3.9) can also be written as

(3.10)

Given , let

(3.11)

We compute, for , such that ,

(3.12)

This shows that

(3.13)

It then follows from condition (ii) that

(3.14)

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).