Volume 2010, Article ID 513956, 12 pagesdoi:10.1155/2010/513956 Research Article Some Krasnonsel’ski˘ı-Mann Algorithms and the Multiple-Set Split Feasibility Problem 1 Department of Math
Trang 1Volume 2010, Article ID 513956, 12 pages
doi:10.1155/2010/513956
Research Article
Some Krasnonsel’ski˘ı-Mann Algorithms and
the Multiple-Set Split Feasibility Problem
1 Department of Mathematics, Xidian University, Xi’an 710071, China
2 Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan
3 Mathematics Department, College of Science, King Saud University, Riyadh 11451, Saudi Arabia
Correspondence should be addressed to Huimin He,huiminhe@126.com
Received 3 April 2010; Revised 7 July 2010; Accepted 13 July 2010
Academic Editor: S Reich
Copyrightq 2010 Huimin He et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
Some variable Krasnonsel’ski˘ı-Mann iteration algorithms generate some sequences {x n }, {y n}, and{z n }, respectively, via the formula x n1 1 − α n x n α n T N · · · T2T1x n , y n1 1 − β n y n
β nN
i1 λ i T i y n , z n1 1 − γ n1 z n γ n1 T n1 z n , where T n T n mod N and the mod function takes values in{1, 2, , N}, {α n }, {β n }, and {γ n } are sequences in 0, 1, and {T1, T2, , T N} are sequences of nonexpansive mappings We will show, in a fairly general Banach space, that the sequence{x n }, {y n }, {z n} generated by the above formulas converge weakly to the common fixed point of{T1, T2, , T N}, 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’ski˘ı-Mann iteration algorithms in Banach space and their applications which solve the multiple-set split feasibility problem
1 Introduction
The Krasnonsel’ski˘ı-Mann K-M iteration algorithm 1, 2 is used to solve a fixed point equation
where T is a self-mapping of closed convex subset C of a Banach space X The K-M algorithm
generates a sequence{x n} according to the recursive formula
Trang 2where{α n } is a sequence in the interval 0, 1 and the initial guess x0 ∈ C is chosen arbitrarily.
It is known3 that if X is a uniformly convex Banach space with a Frechet differentiable norm
in particular, a Hilbert space, if T : C → C is nonexpansive, that is, T satisfies the property
and if T has a fixed point, then the sequence {x n} generated by the K-M algorithm 1.2
converges weakly to a fixed point of T provided that {α n} fulfils the condition
∞
n0
See 4,5 for details on the fixed point theory for nonexpansive mappings.
Many problems can be formulated as a fixed point equation1.1 with a nonexpansive
introduced in6 8, which is to find a point
where C and Q are closed convex subsets of Hilbert spaces H1and H2, respectively, and A is
a linear bounded operator from H1to H2 This problem plays an important role in the study
of signal processing and image reconstruction Assuming that the SFP1.5 is consistent i.e.,
1.5 has a solution, it is not hard to see that x ∈ C solves 1.5 if and only if it solves the fixed point equation
where P C and P Q are theorthogonal projections onto C and Q, respectively, γ > 0 is any positive constant and A∗denotes the adjoint of A Moreover, for sufficiently small γ > 0, the operator P C I − γA∗I − P Q A which defines the fixed point equation 1.6 is nonexpansive
To solve the SFP1.5, Byrne 7,8 proposed his CQ algorithm see also 9 which generates a sequence{x n} by
where γ ∈ 0, 2/λ with λ being the spectral radius of the operator A∗A In 2005, Zhao and
Yang10 considered the following perturbed algorithm:
x n1 1 − α n x n α n P C n
where C n and Q n are sequences of closed and convex subsets of H1 and H2, respectively,
which are convergent to C and Q, respectively, in the sense of Mosco c.f 11 Motivated
Trang 3by1.8, Zhao and Yang 10,12 also studied the following more general algorithm which generates a sequence{x n} according to the recursive formula
where {T n } is a sequence of nonexpansive mappings in a Hilbert space H, under certain
conditions, they proved convergence of1.9 essentially in a finite-dimensional Hilbert space Furthermore, with regard to1.9, 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
x ∈ C
N
i1
C i such that Ax ∈ Q
M
j1
where N and M are positive integers, {C1, C2, , C N } and {Q1, Q2, , Q M} are closed and
convex subsets of H1 and H2, respectively, and A is a linear bounded operator from H1 to
Assuming consistency of the MSSFP1.10, Censor et al 15 introduced the following projection algorithm:
⎛
⎝x n − γ
⎛
⎝N
i1
α i x n − P C i x n M
j1
⎞
⎠
⎞
where Ω is another closed and convex subset of H1, 0 < γ < 2/L with L N
i1 α i
j1 β j and ρA∗A being the spectral radius of A∗A, and α i > 0 for all i and β j > 0 for all j They studied convergence of the algorithm 1.11 in the case where both H1and H2
are finite dimensional In 2006, Xu13 demonstrated some projection algorithms for solving the MSSFP1.10 in Hilbert space as follows:
· · ·P C1
i1
λ i P C i
⎛
⎝y n − γM
j1
⎞
⎠, n ≥ 0,
z n1 P C n1
⎛
⎝z n − γM
j1
⎞
⎠, n ≥ 0,
1.12
where qx 1/2M
j1 β j Q j
2,∇qx M
j1 β j A∗I − P Q j Ax, x ∈ C, and C n
C n mod N and the mod function takes values in {1, 2, , N} This is a motivation for us to
Trang 4study the following more general algorithm which generate the sequences{x n }, {y n}, and
{z n}, respectively, via the formulas
x n1 1 − α n x n α n T N · · · T2T1x n , 1.13
N
i1
z n γ n1 T n1 z n , 1.15
where T n T n mod N,{α n }, {β n }, and {γ n } are sequences in 0, 1, and {T1, T2, , T N} are
sequences of nonexpansive mappings We will show, in a fairly general Banach space X, that
the sequences{x n }, {y n }, and {z n} generated by 1.13, 1.14, and 1.15 converge weakly to the common fixed point of{T1, T2, , T N}, respectively The applications of these results are used to solve the multiple-set split feasibility problem recently introduced by15
Note that, letting C be a nonempty subset of Banach space X and A, B are self-mappings of C, we use D ρ
This paper is organized as follows In the next section, we will prove a weak convergence theorems for the three variable K-M algorithms1.13, 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 L p and l p space for 1 < p < ∞ In the last section, we
will present the applications of the weak convergence theorems for the three variable K-M algorithms1.13, 1.14, and 1.15
2 Convergence of Variable Krasnonsel’ski˘ı-Mann Iteration Algorithm
To solve the multiple-set split feasibility problemMSSFP in Section 3, we firstly present some theorems of the general variable Krasnonsel’ski˘ı-Mann iteration algorithms
Theorem 2.1 Let X be a uniformly convex Banach space with a Frechet differentiable norm, let C
1, 2, , N Assume that the set of common fixed point of {T1, T2, , T N },N
i1FixTi , is nonempty Let {x n } be any sequence generated by 1.13, where 0 < α n < 1 satisfy the conditions
i∞n0 α n 1 − α n ∞;
ii∞n0 α n D ρ T N · · · T1, T i < ∞ for every ρ > 0 and i 1, 2, , N, where D ρ T N · · · T1,
T i N · · · T1x − T i
Then {x n } converges weakly to a common fixed point p of {T1, T2, , T N }.
Trang 5Proof Since T i : C → C is nonexpansive mapping, for i 1, 2, , N, then, the composition
T N · · · T2T1is nonexpansive mapping from C to C Let U : T N · · · T2T1
Take x ∈ N
j1FixTj x ∈ FixU to deduce that
n
2.1
{x n } is bounded, so are {T i x n }, i 1, 2, , N, and {Ux n n n − T i x n
Now since X is uniformly convex, by 16, Theorem 2, there exists a continuous strictly
convex function ϕ, with ϕ0 0, so that
λx 1 − λy2 2
1 − λy2
− λ1 − λϕx − y, 2.2
n − T i x n , i
1, 2, , N, be replaced by e n,i n,i ρ U, T i , and taking a constant M so that
n1 2 n x n − x α n e n,i α n T i x n − x α n e n,i 2
≤ 1 − α n n − x α n e n,i 2 α n i x n − x α n e n,i 2
− α n 1 − α n n − T i x n
α n i x n 2 2α n n,i i x n 2n n,i 2
− α n 1 − α n n − T i x n
n 2 Mα n D ρ U, T i − α n 1 − α n n − T i x n
2.3
It follows that
∞
n1
α n 1 − α n ϕx n − T i y n< ∞ 2.5
Trang 6which further implies that byi lim infn → ∞ n − T i x n
lim inf
On the other hand, it is not hard to deduce from1.13 that
n1 − T i x n1 n x n α n Ux n − T i x n1
n x n α n Ux n − T i x n T i x n − T i x n1
≤ 1 − α n n − T i x n n n − T i x n n1 − x n
1 − α n n − T i x n n n − T i x n n n − Ux n
≤ 1 − α n n − T i x n n n − T i x n
α n n − T i x n n i x n − Ux n
n − T i x n n n − T i x n
n − T i x n n D ρ T i , U .
2.7
Since∞
n1 α n D ρ U, T i < ∞, we see that lim n → ∞ n − T i x n 2.6 implies that
lim
The demiclosedness principle for nonexpansive mappingssee 5,17 implies that
i1
where ω w x n {x : ∃x n j x} denotes the weak ω-limit set of {x n}
To prove that{x n } is weakly convergent to a common fixed point p of {T1, T2, , T N},
it now suffices to prove that ωw x n consists of exactly one point
Trang 7Indeed, if there are x, x ∈ ω w x n x n i x, x m j x, since lim n → ∞ n
limn → ∞ n
lim
n → ∞ n
2 lim
j → ∞
x m j − x x − x2
lim
j → ∞
x m j − x2 2
2 lim
j → ∞
x m j − x, x − x
lim
j → ∞
x m j − x2 2
> lim
i → ∞ m i
2 lim
i → ∞ n i
2
lim
i → ∞ n i
2
lim
i → ∞ n i
j → ∞ n i − x, x − x
lim
i → ∞ n i
> lim
i → ∞ n i
2 lim
n n
2
.
2.10
This is a contradiction
The proof is completed
Theorem 2.2 Let X be a uniformly convex Banach space with a Frechet differentiable norm, let C
1, 2, , N, assume that the set of common fixed point of {T1, T2, , T N },N
i1FixTi , is nonempty Let {y n } be defined by 1.14, where 0 < β n < 1 satisfy the following conditions
i∞
n0 β n 1 − β n ∞;
ii∞
n0 β n D ρN
i1 λ i T i , T i < ∞ for every ρ > 0 and i 1, 2, , N, where
i1 λ i T i , T i
N
i1 λ i T i x − T i
Then {y n } converges weakly to a common fixed point q of {T1, T2, , T N }.
thatN
i1 λ i T i is a nonexpansive mapping from C to C.
The remainder of the proof is the same asTheorem 2.1
The proof is completed
Theorem 2.3 Let X be a uniformly convex Banach space with a Frechet differentiable norm, let C be
i1FixTi , is nonempty Let {z n } be
i∞
n0 γ n 1 − γ n ∞;
ii∞
n0 γ n D ρ T n1 , T i < ∞ for every ρ > 0 and i 1, 2, , N, where D ρ T n1 , T i sup n1 x − T i
Then {z n } converges weakly to a common fixed point w of {T1, T2, , T N }.
Trang 8Proof Since T n T n mod Nand{T1, T2, , T N} is a sequence of nonexpansive mappings from
The proof is completed
3 Applications for Solving the Multiple-Set Split
Feasibility Problem (MSSFP)
Recall that a mapping T in a Hilbert space H is said to be averaged if T can be written as
1 − λI λS, where λ ∈ 0, 1 and S is nonexpansive Recall also that an operator A in H is said to be γ-inverse strongly monotone γ-ism for a given constant γ > 0 if
A projection P K of H onto a closed convex subset K is both nonexpansive and 1-ism It is also known that a mapping T is averaged if and only if the complement I − T is γ-ism for some
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 PΩ 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 MSSFP1.10 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 algorithm1.8 when the MSSFP1.10 is reduced to the SFP 1.5
The first one is a K-M type successive iteration method which produces a sequence
{x n} by
x n1 1 − α n x n α n
P C N
· · ·P C1
Theorem 3.1 Assume that the MSSFP 1.10 is consistent Let {x n } be the sequence generated by
∞
n0 α n 1 − α n ∞ Then {x n } converges weakly to a solution of the MSSFP 1.10.
Hence,
· · ·P C1
Since
∇qx M
j1
j1 β j Therefore, ∇q is 1/L-ism 18 This implies that for any 0 < γ < 2/L, I − γ∇q is averaged Hence, for any closed and convex subset K of H1, the composite P K I − γ∇q is
averaged
So U T N · · · T1 P C I−γ∇q · · · P C I−γ∇q is averaged, thus U is nonexpansive.
Trang 9By the position 2.28, we see that the fixed point set of U, FixU, is the common fixed
point set of the averaged mappings{T N · · · T1}
By Reich3, we have {x n } converges weakly to a fixed point of U which is also a
common fixed point of{T N · · · T1} 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{y n} by
N
i1
λ i P C i
⎛
⎝y n − γM
j1
⎞
Theorem 3.2 Assume that the MSSFP 1.10 is consistent Let {x n } be any sequence generated by
∞
n0 β n 1 − β n ∞ Then {y n } converges weakly to a solution of the MSSFP 1.10.
so, the convex combination S :N
i1 λ i T iis also averaged
Thus S is nonexpansive.
By Reich3, we have {y n } converges weakly to a fixed point of S.
Next, we only need to prove the fixed point of S is also the common fixed point of {T N · · · T1} which is the solution of the MSSFP 1.10, that is, FixS N
i1FixTi
Indeed, it suffices to show thatN
n1FixTi ⊃ FixN
i1 λ i T i
Pick an arbitrary x ∈ FixN
i1 λ i T i, thusN
n1 T i,
thus T i y y, i 1, 2, , N.
Write T i 1 − β i I β i T i , i 1, 2, , N with β i ∈ 0, 1 and T iis nonexpansive
We claim that if z is such that T i z / i
Indeed, we have
T i z − y2
1− β i
β i T i z − y 2
1− β iz − y2 β i T
i z − y2
− β i
1− β iz − T
i z2
i z 2
<z − y2
3.6
If we can show that T i x x, then we are done So assume that Tx / x Now sinceN
i1 λ i T i x
x / Tx, we have
x − y N
i1
λ i T i x − y
≤N
i1
λ iT i x − y
<x − y.
3.7
Trang 10This is a contradiction Therefore, we must have T i x x, i 1, 2, , N, that is,N
n1FixTi x x.
This proof is completed
We now apply Theorem 2.3 to solve the MSSFP 1.10 Recall that the ρ-distance between two closed and convex subsets E1and E2of a Hilbert space H is defined by
The third method is a K-M type cyclic algorithm which produces a sequence{z n} in
the following manner: apply T1 to the initial guess z0 to get z1 1 − γ1z0 γ1P C1z0 −
j1 β j A∗I − P Q j Az0, next apply T2to z1to get z2 1 − γ2z1 γ2P C2z1− γM
j1 β j A∗I −
P Q j Az1, and continue this way to get z N 1 − γ N z0 γ N P C N z N−1 − γM
j1 β j A∗I −
P Q j Az N−1 ; then repeat this process to get z N1 1 − γ N1 z0 γ N P C1z N − γM
j1 β j A∗I −
P Q j Az N , and so on Thus, the sequence {z n} is defined and we write it in the form
z0 γ n1 P C n1
⎛
⎝z n − γM
j1
⎞
where C n C n mod N
Theorem 3.3 Assume that the MSSFP 1.10 is consistent Let {x n } be the sequence generated by
conditions:
i∞n0 γ n 1 − γ n ∞;
ii∞n0 γ n d ρ C n1 , C i < ∞ and ∞n0 γ n d ρ Q n1 , Q i < ∞ for each ρ > 0, i
1, 2, , N.
Then {z n } converges weakly to a solution of the MSSFP 1.10.
so, T n1: Tn1 mod Nis also averaged
Thus T n1is nonexpansive
The projection iteration algorithm3.9 can also be written as
z n γ n1 T n1 z n 3.10
Given ρ > 0, let
ρ supmax x − γA∗