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Webb We modified the classic Mann iterative process to have strong convergence theorem for a finite family of nonexpansive mappings in the framework of Hilbert spaces.. 1.1 Iterative met

Volume 2007, Article ID 76971, 9 pages

doi:10.1155/2007/76971

Research Article

Strong Convergence Theorems for a Finite Family of

Nonexpansive Mappings

Meijuan Shang, Yongfu Su, and Xiaolong Qin

Received 23 May 2007; Accepted 2 August 2007

Recommended by J R L Webb

We modified the classic Mann iterative process to have strong convergence theorem for a finite family of nonexpansive mappings in the framework of Hilbert spaces Our results improve and extend the results announced by many others

Copyright © 2007 Meijuan Shang 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

1 Introduction and preliminaries

Let H be a real Hilbert space, C a nonempty closed convex subset of H, and T : C →

C a mapping Recall that T is nonexpansive if  Tx − T y  ≤  x − y for allx, y ∈ C A

pointx ∈ C is called a fixed point of T provided Tx = x Denote by F(T) the set of fixed

points ofT, that is, F(T) = { x ∈ C : Tx = x } Recall that a self-mapping f : C → C is a

contraction onC, if there exists a constant α ∈(0, 1) such that f (x) − f (y)  ≤ α  x − y 

for allx, y ∈ C We use Π Cto denote the collection of all contractions onC, that is, Π C = { f | f : C → C a contraction } An operatorA is strongly positive if there exists a constant

γ > 0 with the property

 Ax,x  ≥ γ  x 2 ∀ x ∈ H. (1.1) Iterative methods for nonexpansive mappings have recently been applied to solve con-vex minimization problems (see, e.g., [1,2] and the references therein) A typical prob-lem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert spaceH:

min

x ∈ C

1

whereC is the fixed point set of a nonexpansive mapping S, and b is a given point in H.

In [2], it is proved that the sequence{ x n }defined by the iterative method below, with the initial guessx0∈ H chosen arbitrarily,

x n+1 =I − α n ASx n+α n b, n ≥0, (1.3)

converges strongly to the unique solution of the minimization problem (1.2) provided the sequence{ α n }satisfies certain conditions Recently, Marino and Xu [1] introduced a new iterative scheme by the viscosity approximation method

x n+1 =I − α n ASx n+α n γ fx n

They proved that the sequence{ x n }generated by the above iterative scheme converges strongly to the unique solution of the variational inequality(A − γ f )x ∗,x − x ∗  ≥0,

x ∈ C, which is the optimality condition for the minimization problem

min

x ∈ C

1

whereC is the fixed point set of a nonexpansive mapping S, and h is a potential function

forγ f (i.e., h (x) = γ f (x) for x ∈ H.)

Mann’s iteration process [3] is often used to approximate a fixed point of a nonexpan-sive mapping, which is defined as

x n+1 = α n x n+

1− α n

where the initial guess x0 is taken in C arbitrarily and the sequence { α n } ∞

n =0 is in the interval [0, 1] But Mann’s iteration process has only weak convergence, in general For example, Reich [4] shows that ifE is a uniformly convex Banach space and has a Frehet

differentiable norm and if the sequence { α n }is such that 

α n(1− α n)= ∞, then the sequence{ x n }generated by process (1.6) converges weakly to a point inF(T) Therefore,

many authors try to modify Mann’s iteration process to have strong convergence Kim and Xu [5] introduced the following iteration process:

x0= x ∈ C arbitrarily chosen,

y n = β n x n+

1− β n

Tx n,

x n+1 = α n u +1− α n

y n

(1.7)

They proved that the sequence{ x n }defined by (1.7) converges strongly to a fixed point

ofT provided the control sequences { α n }and{ β n }satisfy appropriate conditions Recently, Yao et al [6] also modified Mann’s iterative scheme (1.7) and got a strong convergence theorem They improved the results of Kim and Xu [5] to some extent

In this paper, we study the mappingW ndefined by

U n0 = I,

U n1 = γ n1 T1U n0+

1− γ n1

I,

U n2 = γ n2 T2U n1+

1− γ n2

I,

U n,N −1= γ n,N −1T N −1U n,N −2+

1− γ n,N −1



I,

W n = U nN = γ nN T N U n,N −1+

1− γ nN

I,

(1.8)

where{ γ n1 },{ γ n2 }, , { γ nN } ∈(0, 1] Such a mappingW nis called theW n-mapping gen-erated byT1,T2, ,T Nand{ γ n1 },{ γ n2 }, , { γ nN } Nonexpansivity of eachT iensures the nonexpansivity ofW n It follows from [7, Lemma 3.1] thatF(W n)= ∩ N i =1F(T i)

Very recently, Xu [2] studied the following iterative scheme:

x n+1 = α n u +I − α n AT n+1 x n, n ≥0, (1.9) whereA is a linear bounded operator, T n = T n mod N and the mod function takes values in

{1, 2, ,N } He proved that the sequence{ x n }generated by the above iterative scheme converges strongly to the unique solution of the minimization problem (1.2) providedT n

satisfy

FT N ··· T2T1



= FT1T N ··· T3T2



= ··· = FT N −1··· T1T n

, (1.10) and{ α n } ∈(0, 1) satisfying the following control conditions:

(C1) limn →∞ α n =0;

(C2)∞

n =1α n = ∞;

(C3)∞

n =1| α n − α n+N | < ∞or limn →∞ α n /α n+N =0

Remark 1.1 There are many nonexpansive mappings, which do not satisfy (1.10) For example, if X =[0, 1] and T1,T2 are defined byT1x = x/2 + 1/2 and T2x = x/4, then F(T1T2)= {4/7 }, whereasF(T2T1)= {1/7 }

In this paper, motivated by Kim and Xu [5], Marino and Xu [1], Xu [2], and Yao et

al [6], we introduce a composite iteration scheme as follows:

x0= x ∈ C arbitrarily chosen,

y n = β n x n+

1− β n

W n x n,

x n+1 = α n γ fx n

+

I − α n Ay n,

(1.11)

where f ∈ΠCis a contraction, andA is a linear bounded operator We prove, under

cer-tain appropriate assumptions on the sequences{ α n }and{ β n }, that{ x n }defined by (1.11) converges to a common fixed point of the finite family of nonexpansive mappings, which solves some variation inequality and is also the optimality condition for the minimization problem (1.5)

Now, we consider some special cases of iterative scheme (1.11) WhenA = I and γ =1

in (1.11), we have that (1.11) collapses to

x0= x ∈ C arbitrarily chosen,

y n = β n x n+

1− β n

W n x n,

x n+1 = α n fx n

+

1− α n

y n

(1.12)

WhenA = I and γ =1 in (1.11),N =1 and{ γ n1 } =1 in (1.8), we have that (1.11) col-lapses to the iterative scheme which was considered by Yao et al [6] WhenA = I and

γ =1 in (1.11),N =1 and{ γ n1 } =1 in (1.8), andf (y) = u ∈ C for all y ∈ C in (1.11), we have that (1.11) reduces to (1.7), which was considered by Kim and Xu [5]

In order to prove our main results, we need the following lemmas

Lemma 1.2 In a Hilbert space H, there holds the inequality

 x + y 2≤  x 2+ 2

Lemma 1.3 (Suzuki [8]) Let{ x n } and { y n } be bounded sequences in a Banach space X and let β n be a sequence in [0, 1] with 0 < liminf n →∞ β n ≤lim supn →∞ β n < 1 Suppose x n+1 =

(1− β n)y n+β n x n for all integers n ≥ 0 and

lim sup

n →∞

y n+1 − y n− x n+1 − x n ≤0. (1.14)

Then lim n →∞  y n − x n  = 0.

Lemma 1.4 (Xu [2]) Assume that{ α n } is a sequence of nonnegative real numbers such that

α n+1 ≤1− γ n

where γ n is a sequence in (0, 1) and { δ n } is a sequence such that

(i)∞

n =1γ n = ∞;

(ii) lim supn →∞ δ n /γ n ≤ 0 or∞

n =1| δ n | < ∞ Then lim n →∞ α n = 0.

Lemma 1.5 (Marino and Xu [1]) Assume thatA is a strongly positive linear bounded oper-ator on a Hilbert space H with coefficient γ > 0 and 0 < ρ ≤  A  −1, then  I − ρA  ≤1− ργ.

Lemma 1.6 (Marino and Xu [1]) LetH be a Hilbert space Let A be a strongly positive linear bounded selfadjoint operator with coe fficient γ > 0 Assume that 0 < γ < γ/α Let T : C → C

be a nonexpansive mapping with a fixed point x t ∈ C of the contraction C  x → tγ f (x) +

(1− tA)Tx Then { x t } converges strongly as t → 0 to a fixed point x of T, which solves the variational inequality



2 Main results

Theorem 2.1 Let C be a closed convex subset of a Hilbert space H Let A be a strongly pos-itive linear bounded operator with coefficient γ > 0 and W n is defined by ( 1.8 ) Assume that

0< γ < γ/α and F = ∩ N i =1F(T i)= ∅ Given a map f ∈ΠC , the initial guess x0∈ C is chosen arbitrarily and given sequences { α n } ∞

n =0and { β n } ∞

n =0in (0, 1), the following conditions are satisfied:

(C1)∞

n =0α n = ∞ ;

(C2) limn →∞ α n = 0;

(C3) 0< liminf n →∞ β n ≤lim supn →∞ β n < 1;

(C4) limn →∞ | γ n,i − γ n −1,i | = 0, for all i =1, 2, ,N.

Let { x n } ∞

n =1be the composite process defined by ( 1.11 ) Then { x n } ∞

n =1converges strongly to

q ∈ F, which also solves the following variational inequality:



Proof First, we observe that { x n } ∞

n =0is bounded Indeed, take a pointp ∈ F and notice

that

y n − p  ≤ β nx n − p+

1− β nW n x n − p  ≤  x n − p. (2.2)

It follows that

x n+1 − p  =  α n

γ fx n

− Ap+

I − α n Ay n − p

≤1− α n(γ − γα) x n − p+α nγ f (p) − Ap. (2.3)

By simple inductions, we have x n − p  ≤max{ x0− p , Ap − γ f (p)  /(γ − γα) }, which gives that the sequence{ x n }is bounded, so are{ y n }and{ z n }

Next, we claim that

x n+1 − x n −→0. (2.4) Putl n =(x n+1 − β n x n)/(1 − β n) Now, we computel n+1 − l n, that is,

x n+1 =1− β n

l n+β n x n, n ≥0. (2.5) Observing that

l n+1 − l n = α n+1 γ fx n+1

+

I − α n+1 Ay n+1 − β n+1 x n+1

1− β n+1

− α n γ fx n

+

I − α n Ay n − β n x n

1− β n

= α n+1

γ fx n+1

− Ay n+1

1− β n+1 − α n

γ fx n

− Ay n

1− β n

+W n+1 x n+1 − W n x n,

(2.6)

we have

l n+1 − l n ≤ α n+1

1− β n+1

γ f

x n+1

− Ay n+1+ α n

1− β n

Ay n − γ f

x n

+x n+1 − x n+W n+1 x n − W n x n. (2.7)

Next, we will useM to denote the possible different constants appearing in the following

reasoning It follows from the definition ofW nthat

W n+1 x n − W n x n

=γ n+1,N T N U n+1,N −1x n+

1− γ n+1,N

x n − γ n,N T N U n,N −1x n −1− γ n,N

x n

≤ γ n+1,N − γ n,N x n+γ n+1,N T N U n+1,N −1x n − γ n,N T N U n,N −1x n

≤ γ n+1,N − γ n,N x n+γ n+1,N

T N U n+1,N −1x n − T N U n,N −1x n

+ γ n+1,N − γ n,N T N U n,N −1x n

≤2M γ n+1,N − γ n,N +γ n+1,NU n+1,N −1x n − U n,N −1x n.

(2.8) Next, we consider

U n+1,N −1x n − U n,N −1x n

=γ n+1,N −1T N −1U n+1,N −2x n+

1− γ n+1,N −1 

x n

− γ n,N −1T N −1U n,N −2x n −1− γ n,N −1



x n

≤ γ n+1,N −1− γ n,N −1 x n+γ n+1,N −1T N −1U n+1,N −2y n − T N −1U n,N −2x n

+ γ n+1,N −1− γ n,N −1 T N −1U n,N −2x n

≤2M γ n+1,N −1− γ n,N −1 +U n+1,N −2x n − U n,N −2x n.

(2.9)

It follows that

U n+1,N −1x n − U n,N −1x n

≤2M γ n+1,N −1− γ n,N −1 + 2M γ n+1,N −2− γ n,N −2 +U n+1,N −3x n − U n,N −3x n

≤2M N −

1

i =2

γ n+1,i − γ n,i +U n+1,1 x n − U n,1 x n

≤2M N−1

i =1

γ n+1,i − γ n,i

(2.10)

Substituting (2.10) into (2.8) yields that

W n+1 x n − W n x n ≤2M γ n+1,N − γ n,N + 2γ n+1,N M N −

1

i =1

γ n+1,i − γ n,i

≤2MN

i =1

γ n+1,i − γ n,i

(2.11)

It follows that

l n+1 − l n− x n − x n −1 

≤ α n+1

1− β n+1

γ f

x n+1

− Ay n+1+ α n

1− β n

Ay n − γ f

x n+ 2MN

i =1

γ n+1,i − γ n,i

(2.12) Observing conditions (C1), (C4) and takeing the limits asn → ∞, we get

lim sup

n →∞

l n+1 − l n− x n+1 − x n ≤0. (2.13)

We can obtain limn →∞  l n − x n  =0 easily byLemma 1.3 Sincex n+1 − x n =(1− β n)(l n −

x n), we have that (2.4) holds Observing thatx n+1 − y n = α n(γ f (x n)− Ay n), we can easily get limn →∞  y n − x n+1  =0, which implies that

y n − x n ≤  x n − x n+1+x n+1 − y n, (2.14)

that is,

lim

n →∞y n − x n =0. (2.15)

On the other hand, we have

W n x n − x n ≤  x n − y n+y n − W n x n ≤  x n − y n+β nx n − W n x n, (2.16)

which implies (1− β n) W n x n − x n  ≤  x n − y n  From condition (C3) and (2.15), we obtain

W n x n − x n −→0. (2.17) Next, we claim that

lim sup

n →∞



whereq =limt →0x t withx t being the fixed point of the contractionx → tγ f (x) + (I − tA)W n x Then, x tsolves the fixed point equationx t = tγ f (x t) + (I − tA)W n x t Thus, we

have x t − x n  = (I − tA)(W n x t − x n) +t(γ f (x t)− Ax n) It follows fromLemma 1.2 that

x t − x n 2

=(I − tA)

W n x t − x n

+tγ fx t

− Ax n 2

≤(1− γt)2 W n x t − x n 2

+ 2tγ fx t

− Ax n,x t − x n

≤1−2γt + (γt)2 x t − x n 2

+f n(t)

+ 2tγ fx t

− Ax t,x t − x n

+ 2tAx t − Ax n,x t − x n

,

(2.19)

where

f n(t) =2x t − x n+x n − W n x nx n − W n x n −→0, asn −→0. (2.20)

It follows that



Ax t − γ fx t

,x t − x n

≤ γt

2



Ax t − Ax n,x t − x n

+ 1

2t f n(t). (2.21)

Letn → ∞in (2.21) and note that (2.20) yields

lim sup

n →∞



Ax t − γ fx t

,x t − x n

≤ t

whereM > 0 is a constant such that M ≥ γ  Ax t − Ax n,x t − x n for allt ∈(0, 1) andn ≥1 Takingt →0 from (2.22), we have lim supt →0lim supn →∞  Ax t − γ f (x t),x t − x n  ≤0 Since

H is a Hilbert space, the order of limsup t →0and lim supn →∞is exchangeable, and hence (2.18) holds Now fromLemma 1.2, we have

x n+1 − q 2

=I − α n A

y n − q+α n

γ fx n

− Aq 2

≤I − α n A

y n − q 2

+ 2α n

γ fx n

− Aq,x n+1 − q

≤1− α n γ2 x n − q 2

+α n γα

x n − q 2

+x n+1 − q 2 + 2α n

γ f (q) − Aq,x n+1 − q,

(2.23)

which implies that

x n+1 − q 2

≤



1− α n γ2

+α n γα

1− α n γα x n − q 2

+ 2α n

1− α n γα



γ f (q) − Aq,x n+1 − q

≤1−2α n(γ − αγ)

1− α n γα



x n − q 2

+2α n(γ − αγ)

1− α n γα

 1

γ − αγ



γ f (q) − Aq,x n+1 − q+ α n γ2

2(γ − αγ)M



.

(2.24)

Putl n =2α n(γ − α n γ)/(1 − α n αγ) and t n =1/(γ − αγ)  γ f (q) − Aq,x n+1 − q +α n γ2/(2(γ − αγ))M, that is,

x n+1 − q 2

≤1− l nx n − q+l n t n . (2.25)

It follows from conditions (C1), (C2), and (2.22) that limn →∞ l n =0, ∞

n =1l n = ∞, and lim supn →∞ t n ≤0 Apply Lemma 1.4to (2.25) to conclude thatx n → q This completes

Remark 2.2 Our results relax the condition of Kim and Xu [1] imposed on control se-quences and also improve the results of Yao et al [6] from one single mapping to a finite family of nonexpansive mappings, respectively

References

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[5] T.-H Kim and H.-K Xu, “Strong convergence of modified Mann iterations,” Nonlinear Analysis:

Theory, Methods & Applications, vol 61, no 1-2, pp 51–60, 2005.

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Mann iteration,” to appear in Nonlinear Analysis: Theory, Methods & Applications

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feasibil-ity problems,” Mathematical and Computer Modelling, vol 32, no 11–13, pp 1463–1471, 2000.

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Ap-plications, vol 305, no 1, pp 227–239, 2005.

Meijuan Shang: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China; Department of Mathematics, Shijiazhuang University, Shijiazhuang 050035, China

Email address:meijuanshang@yahoo.com.cn

Yongfu Su: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China

Email address:suyongfu@tjpu.edu.cn

Xiaolong Qin: Department of Mathematics, Gyeongsang National University, Chinju 660-701, Korea

Email address:qxlxajh@163.com

Mathematical Analysis and Applications, vol 67, no 2, pp 274–276, 1979.... & Applications

[7] W Takahashi and K Shimoji, ? ?Convergence theorems for nonexpansive mappings and

feasibil-ity problems,” Mathematical... − αγ)M



.

(2.24)

Putl n