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Some Krasnonsel'skiĭ-Mann Algorithms and the Multiple-Set Split Feasibility Problem
Fixed Point Theory and Applications volume 2010, Article number: 513956 (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
The Krasnonsel'ski-Mann (K-M) iteration algorithm [1, 2] is used to solve a fixed point equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ1_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ4_HTML.gif)
(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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ6_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ7_HTML.gif)
where with
being the spectral radius of the operator
. In 2005, Zhao and Yang [10] considered the following perturbed algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ8_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ9_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ10_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ11_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ12_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ13_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ14_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ15_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ16_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ17_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ18_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ19_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ20_HTML.gif)
Since exists, by condition (ii) and (2.4), it implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ21_HTML.gif)
which further implies that by (i) , hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ22_HTML.gif)
On the other hand, it is not hard to deduce from (1.13) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ23_HTML.gif)
Since , we see that
exists. This together with (2.6) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ24_HTML.gif)
The demiclosedness principle for nonexpansive mappings (see [5, 17]) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ25_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ26_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ27_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ28_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ29_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ30_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ31_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ32_HTML.gif)
If we can show that , then we are done. So assume that
. Now since
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ33_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ34_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ35_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ36_HTML.gif)
Given , let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ37_HTML.gif)
We compute, for , such that
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ38_HTML.gif)
This shows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ39_HTML.gif)
It then follows from condition (ii) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F513956/MediaObjects/13663_2010_Article_1297_Equ40_HTML.gif)
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).
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Acknowledgments
The work was supported by the Fundamental Research Funds for the Central Universities, no. JY10000970006, and the National Nature Science Foundation, no. 60974082.
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He, H., Liu, S. & Noor, M. Some Krasnonsel'skiĭ-Mann Algorithms and the Multiple-Set Split Feasibility Problem. Fixed Point Theory Appl 2010, 513956 (2010). https://doi.org/10.1155/2010/513956
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DOI: https://doi.org/10.1155/2010/513956