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Volume 2007, Article ID 37513, 10 pagesdoi:10.1155/2007/37513 Research Article Strong Convergence to Common Fixed Points of a Finite Family of Nonexpansive Mappings Yeong-Cheng Liou, Yon

Volume 2007, Article ID 37513, 10 pages

doi:10.1155/2007/37513

Research Article

Strong Convergence to Common Fixed Points of a Finite Family

of Nonexpansive Mappings

Yeong-Cheng Liou, Yonghong Yao, and Kenji Kimura

Received 24 December 2006; Accepted 2 May 2007

Recommended by Yeol Je Cho

We suggest and analyze an iterative algorithm for a finite family of nonexpansive map-pings T1,T2, ,T r Further, we prove that the proposed iterative algorithm converges strongly to a common fixed point ofT1,T2, ,T r

Copyright © 2007 Yeong-Cheng Liou et al This is an open access article distributed un-der the Creative Commons Attribution License, which permits unrestricted use, distri-bution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

LetC be a closed convex subset of a Banach space E A mapping T of C into itself is called

nonexpansive if Tx − T y  ≤  x − y for allx, y ∈ C We denote by F(T) the set of fixed

points ofT Let T1,T2, ,T rbe a finite family of nonexpansive mappings satisfying that the setF =r i =1F(T i) of common fixed points ofT1,T2, ,T ris nonempty The problem

of finding a common fixed point has been investigated by many researchers; see, for ex-ample, Atsushiba and Takahashi [1], Bauschke [2], Lions [3], Shimizu and Takahashi [4], Takahashi et al [5], Zeng et al [6] To solve this problem, the iterative schemex1∈ C and

x n+1 = α n x1+

1− α n

whereT n+r = T nand 0< α n, is used Wittmann [7] dealt with the iterative scheme for the caser =1; see originally Halpern [8] Bauschke [2] dealt with the iterative scheme for a finite family of nonexpansive mappings under the restriction that

F = FT r T r −1··· T1



= FT1T r ··· T2



= ··· = FT r −1··· T1T r

Recently, Kimura et al [9] dealt with an iteration scheme which is more general than that of Wittmann’s result They proved the following theorems

Theorem 1.1 (see [9, Theorem 4]) Let E be a uniformly convex Banach space whose norm

is uniformly Gˆateaux di fferentiable and let C be a closed convex subset of E Let T1,T2, ,T r

be nonexpansive mappings of C into itself such that the set F =r i =1F(T i ) of common fixed points of T1,T2, ,T r is nonempty Let { α n } and { β n } be two sequences in [0, 1] which satisfy the following control conditions:

(C1) limn →∞ α n = 0;

(C2)∞

n =1α n = ∞ ;

(C3)∞

n =1| α n+1 − α n | < ∞ ;

(C4) limn →∞ β i

n = β i andr

i =1β i

n = 1, n ∈ N for some β i ∈ (0, 1);

(C5)∞

n =1

r

i =1| β i

n+1 − β i

n | < ∞ Let x ∈ C and define a sequence { x n } by x1∈ C and

x n+1 = α n x +1− α n r

i =1

β i

Then { x n } converges strongly to the point Px, where P is a sunny nonexpansive retraction of

C onto F.

Theorem 1.2 (see [9, Theorem 5]) Let E be a uniformly convex Banach space whose norm

is uniformly Gˆateaux differentiable and let C be a closed convex subset of E Let S, T be nonexpansive mappings of C into itself such that the set F(S) ∩ F(T) of common fixed points

of S and T is nonempty Let x ∈ C and let { x n } be a sequence generated by

x n+1 = α n x +1− α n

β n Sx n+

1− β n

Tx n

Assume (C1) and (C2) hold and the following conditions are satisfied:

(C3) limn →∞(α n /α n+1)= 1;

(C4) limn →∞ β n = β ∈ (0, 1);

(C5)∞

n =1| β n+1 − β n | < ∞

Then { x n } converges strongly to the point Px, where P is a sunny nonexpansive retraction

of C onto F(S) ∩ F(T).

We remark that the control conditions (C3) and (C3) were introduced initially by Wittmann [7] and Xu [10], respectively On the other hand, we have to remark that con-ditions (C1) and (C2) are necessary for the strong convergence of algorithms (1.3) and (1.4) for nonexpansive mappings It is unclear if they are sufficient

The objective of this paper is to show another generalization of Mann and Halpern iterative algorithm to a setting of a finite family of nonexpansive mappings We deal with the iterative schemex0∈ C and

x n+1 = α n fx n

+β n x n+γ n

r



i =1

τ i

Using this iterative scheme, we can find a common fixed point of a finite family of non-expansive mappings under some type of control conditions

2 Preliminaries

LetE be a Banach space with norm  · and letE ∗be the dual ofE Denote by ,·the duality product The normalized duality mappingJ from E to E ∗is defined by

J(x) =x ∗ ∈ E ∗:

x,x ∗

=  x 2= x ∗ 2

(2.1) forx ∈ E.

A Banach spaceE is said to be strictly convex if (x + y)/2  < 1 for all x, y ∈ E with

 x  =  y  =1 andx = y It is also said to be uniformly convex if lim n →∞  x n − y n  =0 for any two sequences{ x n },{ y n }inE such that  x n  =  y n  =1 and limn →∞ (x n+y n)/2  =

1 LetU = { x ∈ E :  x  =1}be the unit sphere ofE Then the Banach space E is said to

be smooth provided that

lim

t →0

 x + ty  −  x 

exists for eachx, y ∈ U In this case, the norm of E is said to be Gˆateaux differentiable It

is said to be uniformly smooth if the limit is attained uniformly forx, y ∈ U The norm of

E is said to be uniformly Gˆateaux differentiable if for any y ∈ U the limit exists uniformly

for allx ∈ U It is known that if the norm of E is uniformly Gˆateaux differentiable, then

the normalized duality mapping J is norm to weak star uniformly continuous on any

bounded subsets ofE.

LetC be a closed convex subset of a Banach space E and let D be a subset of C Recall

that a self-mapping f : C → C is a contraction on C if there exists a constant α ∈(0, 1) such that f (x) − f (y)  ≤ α  x − y ,x, y ∈ C A mapping P : C → D is said to be sunny

ifP(Px + t(x − Px)) = Px whenever Px + t(x − Px) ∈ C for x ∈ C and t ≥0 IfP2= P,

thenP is called a retraction We know that a retraction P of C onto D is sunny and

nonexpansive if and only if x − Px,J(y − Px) ≤0 for all y ∈ D From this inequality,

it is easy to show that there exists at most one sunny nonexpansive retraction ofC onto

D If there is a sunny nonexpansive retraction of C onto D, then D is said to be a sunny

nonexpansive retraction ofC.

Now, we introduce several lemmas for our main results in this paper

Lemma 2.1 (see [11]) Let C be a nonempty closed convex subset of a strictly convex Banach space For each r ∈ N , let T r be a nonexpansive mapping of C into E Let { τ r } be a sequence

of positive real numbers such that∞

r =1τ r = 1 If∞

r =1F(T r ) is nonempty, then the mapping

T =∞ r =1τ r T r is well-defined and F(T) =∞ r =1F(T r ).

Lemma 2.2 (see [12]) 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 Then, lim n →∞  y n − x n  = 0.

Lemma 2.3 (see [10]) Assume { a n } is a sequence of nonnegative real numbers such that

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

(1)∞

n =1γ n = ∞ ;

(2) lim supn →∞ δ n /γ n ≤ 0 or∞

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

3 Main results

First, we consider the following iterative scheme:

x n+1 = α n fx n

+β n x n+γ n

τ n Sx n+

1− τ n

Tx n

where{ α n },{ β n },{ γ n }, and{ τ n }are sequences in [0, 1]

Theorem 3.1 Let E be a strictly convex Banach space whose norm is uniformly Gˆateaux differentiable and let C be a closed convex subset of E Let S and T be nonexpansive mappings

of C into itself such that F(S) ∩ F(T) = ∅ Let f : C → C be a fixed contractive mapping Assume that { z t } converges strongly to a fixed point z of U as t → 0, where z t is the unique element of C which satisfies z t = t f (z t) + (1− t)Uz t , U = τS + (1 − τ)T, 0 < τ < 1 Let { α n } ,

{ β n } , { γ n } , and { τ n } be four real sequences in [0, 1] such that α n+β n+γ n = 1 Assume { α n }

satisfies conditions (C1) and (C2) and assume the following control conditions hold:

(D3) 0< liminf n →∞ β n ≤lim supn →∞ β n < 1;

(D4) limn →∞ τ n = τ.

For arbitrary x0∈ C, then the sequence { x n } defined by ( 3.1 ) converges strongly to a common fixed point of S and T.

Proof We show first that { x n }is bounded To end this, by taking a fixed element p ∈

F(S) ∩ F(T) and using (3.1), we have

x n+1 − p α n fx n

− p +β n x n − p +γ n

τ n Sx n − p +

1− τ n Tx n − p

≤ α n α x n − p +α n f (p) − p +

β n+γ n x n − p

=1− α n+αα n x n − p +α n f (p) − p

≤max x n − p , 1

1− α f (p) − p .

(3.2)

By induction, we get

x n − p max x0− p , 1

1− α f (p) − p (3.3)

for alln ≥0 This shows that{ x n }is bounded, so are{ Tx n },{ Sx n }, and{ f (x n)}

We show then that x n+1 − x n  →0(n → ∞)

Define a sequence{ y n }which satisfies

x n+1 =1− β n

Observe that

y n+1 − y n = α n+1

1− β n+1



fx n+1

− fx n

+

 α n+1

1− β n+1 − α n

1− β n



fx n

+γ n+1 τ n+1

1− β n+1



Sx n+1 − Sx n

+

γ n+1 τ n+1

1− β n+1 − γ n τ n

1− β n



Sx n

+γ n+1

1− τ n+1)

1− β n+1



Tx n+1 − Tx n

+

γ n+1

1− τ n+1

1− β n+1 − γ n

1− τ n

1− β n



Tx n

= α n+1

1− β n+1



fx n+1

− fx n

+

 α n+1

1− β n+1 − α n

1− β n



fx n

+γ n+1 τ n+1

1− β n+1



Sx n+1 − Sx n

+γ n+1

1− τ n+1

1− β n+1



Tx n+1 − Tx n

+ γ n+1

1− β n+1



τ n+1 − τ n

Sx n+

 γ n+1

1− β n+1 − γ n

1− β n



τ n Sx n

+ γ n+1

1− β n+1



τ n − τ n+1

Tx n+

 γ n+1

1− β n+1 − γ n

1− β n



1− τ n

Tx n

(3.5)

It follows that

y n+1 − y n x n+1 − x n

≤ αα n+1

1− β n+1 x n+1 − x n +

 α n+1

1− β n+1 − α n

1− β n



 fx n

+

 γ n+1

1− β n+1 −1

 x n+1 − x n +τ n

 γ n+1

1− β n+1 − γ n

1− β n



 Sx n

+

1− τ n γ n+1

1− β n+1 − γ n

1− β n



 Tx n

+ γ n+1

1− β n+1

τ n − τ n+1 Sx n + Tx n

≤(1 +α)α n+1

1− β n+1 x n+1 − x n + γ n+1

1− β n+1

τ n − τ n+1 Sx n + Tx n

+

 α n+1

1− β n+1 − α n

1− β n



 fx n +τ n Sx n +

1− τ n Tx n

(3.6)

Since{ x n },{ Tx n },{ Sx n }, and{ f (x n)}are bounded, we obtain

lim sup

Hence, byLemma 2.2we know that y n − x n  →0 asn → ∞ Consequently, limn →∞  x n+1

− x n  =limn →∞(1− β n) y n − x n  =0

DefineU = τS + (1 − τ)T Then, byLemma 2.1,F(U) = F(S) ∩ F(T).

Observing that

x n − Ux n x n+1 − x n + x n+1 − Ux n

≤ x n+1 − x n +α n fx n

− Ux n +β n x n − Ux n

and using control conditions (C1), (D3), and (D4) on{ α n },{ β n }, and{ τ n }, we conclude that limn →∞  Ux n − x n  =0

We next show that

lim sup

n →∞



z − f (z), jz − x n

Letx tbe the unique fixed point of the contraction mappingU tgiven by

U t x = t f (x) + (1 − t)Ux. (3.10) Then

x t − x n = tfx t

− x n

+ (1− t)Ux t − x n

We compute as follows:

x t − x n 2≤(1− t)2 Ux t − x n 2+ 2tfx t

− x n,jx t − x n

≤(1− t)2 Ux t − Ux n + Ux n − x n 2

+ 2tfx t

− x t,jx t − x n

+ 2t x t − x n 2

≤(1− t)2 x t − x n 2+a n(t) + 2t x t − x n 2

+ 2tfx t

− x t,jx t − x n

,

(3.12)

wherea n(t) =  Ux n − x n (2 x t − x n + Ux n − x n )→0 asn → ∞

The last inequality implies



x t − fx t

,jx t − x n

≤ t

2 x t − x n 2+ 1

It follows that

lim sup

n →∞



x t − fx t

,jx t − x n

≤ t

Lettingt →0, we obtain

lim sup

t →0

lim sup

n →∞



x t − fx t

,jx t − x n

Moreover, we have



z − f (z), jz − x n

=z − f (z), jz − x n

−z − f (z), jx t − x n

+

z − f (z), jx t − x n

−x t − f (z), jx t − x n

+

x t − f (z), jx t − x n

x t − fx t

,jx t − x n

+

x t − fx t

,jx t − x n

=z − f (z), jz − x n

− jx t − x n

+

z − x t,jx t − x n

+

fx t

− f (z), jx t − x n

+

x t − fx t

,jx t − x n

.

(3.16)

Then, we obtain

lim sup

n →∞



z − f (z), jz − x n

≤sup

n ∈N



z − f (z), jz − x n

− jx t − x n

+ z − x t lim sup

n →∞ x t − x n

+ fx t

− f (z) lim sup

n →∞ x t − x n + lim sup

n →∞



x t − fx t

,jx t − x n

≤sup

n ∈N



z − f (z), jz − x n

− jx t − x n

+ (1 +α) z − x t lim sup

n →∞ x t − x n

+ lim sup

n →∞



x t − fx t

,jx t − x n

.

(3.17)

By hypothesisx t → z ∈ F(S) ∩ F(T) as t →0 andj is norm-to-weak ∗uniformly continuous

on bounded subset ofE, we obtain

lim

t →0sup

n ∈N



z − f (z), jz − x n

− jx t − x n

Therefore, we have

lim sup

n →∞



z − f (z), jz − x n

=lim sup

t →0

lim sup

n →∞



z − f (z), jz − x n

≤lim sup

t →0

lim sup

n →∞



x t − fx t

,jx t − x n

≤0. (3.19)

Finally, we have

x n+1 − z 2

≤ β n

x n − z+γ n

τ n Sx n+

1− τ n

Tx n − z 2 + 2α n

fx n

− z, jx n+1 − z

≤ β2

n x n − z 2

+γ2

n τ n

Sx n − z+

1− τ n

Tx n − z 2 + 2β n γ n x n − z τ n

Sx n − z+

1− τ n

Tx n − z

+ 2α n

fx n

− f (z), jx n+1 − z + 2α n

f (z) − z, jx n+1 − z

≤ β2

n x n − z 2

+γ2

n x n − z 2

+ 2β n γ n x n − z x n − z

+ 2α n

fx n

− f (z), jx n+1 − z + 2α n

f (z) − z, jx n+1 − z

≤1− α n 2

x n − z 2

+αα n x n − z 2

+ x n+1 − z 2 

+ 2α n

f (z) − z, jx n+1 − z .

(3.20)

It follows that

x n+1 − z 2

≤1−(2− α)α n

1− αα n x n − z 2

+ 2α n

1− αα n



f (z) − z, jx n+1 − z

n

1− αα n x n − z 2

≤1−2(1− α)α n x n − z 2 + 2(1− α)α n



1 (1− α)1− αα n 

f (z) − z, jx n+1 − z +α n

2 x n − z 2

.

(3.21) Noting that∞

n =0[2(1− α)α n]= ∞and

lim sup

n →∞

(1− α)1− αα n 

f (z) − z, jx n+1 − z +α n

2 x n − z 2

≤0. (3.22)

ApplyLemma 2.3to (3.21) to conclude thatx n → z as n → ∞ This completes the proof



Remark 3.2 We note that every uniformly smooth Banach space has a uniformly Gˆateaux

differentiable norm By Xu [13, Theorem 4.1], we know that{ z t }converges strongly to a fixed point ofU as t →0, wherez tis the unique element ofC which satisfies z t = t f (z t) + (1− t)Uz t

Corollary 3.3 Let E be a strictly convex and uniformly smooth Banach space whose norm

is uniformly Gˆateaux di fferentiable and let C be a closed convex subset of E Let S and T be nonexpansive mappings of C into itself such that F(S) ∩ F(T) = ∅ Let f : C → C be a fixed contractive mapping Let { α n } , { β n } , { γ n } , and { τ n } be four real sequences in [0, 1] such that

α n+β n+γ n = 1 Assume the control conditions (C1), (C2), (D3), and (D4) are satisfied For arbitrary x0∈ C, then the sequence { x n } defined by ( 3.1 ) converges strongly to a common fixed point of S and T.

We can obtain the following results from Takahashi and Ueda [14] which is related to the existence of sunny nonexpansive retractions

Corollary 3.4 Let E be a uniformly convex Banach space whose norm is uniformly Gˆateaux differentiable and let C be a closed convex subset of E Let S and T be nonexpansive mappings of C into itself such that F(S) ∩ F(T) = ∅ Let u ∈ C be a given point Let { α n } ,

{ β n } , { γ n } , and { τ n } be four real sequences in [0, 1] such that α n+β n+γ n = 1 Assume the control conditions (C1), (C2), (D3), and (D4) are satisfied For arbitrary x0∈ C, let the sequence { x n } be defined by

x n+1 = α n u + β n x n+γ n

τ n Sx n+

1− τ n

Tx n

Then { x n } converges strongly to the point Pu, where P is a sunny nonexpansive retraction of

C onto F(S) ∩ F(T).

We can also obtain the following theorems for a finite family of nonexpansive map-pings The proof is similar to that ofTheorem 3.1, the details of the proof, therefore, are omitted

Theorem 3.5 Let E be a strictly convex Banach space whose norm is uniformly Gˆateaux

di fferentiable and let C be a closed convex subset of E Let T1,T2, ,T r be a finite family

of nonexpansive mappings of C into itself such that the set F =r i =1F(T i ) of common fixed points of T1,T2, ,T r is nonempty Let f : C → C be a fixed contractive mapping Assume that { z t } converges strongly to a fixed point z of U as t → 0, where z t is the unique element

of C which satisfies z t = t f (z t) + (1− t)Uz t , U =r i =1τ i T i , 0 < τ i < 1, andr i =1τ i

n = 1 Let

{ α n } , { β n } , { γ n } , and { τ i

n } be real sequences in [0, 1] such that α n+β n+γ n = 1 Assume the control conditions (C1), (C2), and (D3) hold Assume { τ i

n } satisfies the condition (D4  ):

lim

n →∞ τ i

n = τ i, i =1, 2, ,r, r

i =1

τ i

For arbitrary x0∈ C, let the sequence { x n } be defined by

x n+1 = α n fx n

+β n x n+γ n

r



i =1

τ i

Then { x n } converges strongly to a common fixed point of T1,T2, ,T r

Theorem 3.6 Let E be a strictly convex and uniformly smooth Banach space and let C be

a closed convex subset of E Let T1,T2, ,T r be a finite family of nonexpansive mappings

of C into itself such that the set F =r i =1F(T i ) of common fixed points of T1,T2, ,T r is nonempty Let f : C → C be a fixed contractive mapping Let { α n } , { β n } , { γ n } , and { τ i

n } be real sequences in [0, 1] such that α n+β n+γ n = 1 Assume the control conditions (C1), (C2), (D3), and (D4  ) are satisfied For arbitrary x0∈ C, then the sequence { x n } defined by ( 3.25 ) converges strongly to a common fixed point of T1,T2, ,T r

Acknowledgment

The research was partially supposed by Grant NSC 95-2622-E-230-005CC3

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Yeong-Cheng Liou: Department of Information Management, Cheng Shiu University,

Kaohsiung 833, Taiwan

Email addresses:simplex liou@hotmail.com ; ycliou@csu.edu.tw

Yonghong Yao: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China

Email address:yuyanrong@tjpu.edu.cn

Kenji Kimura: Department of Applied Mathematics, National Sun Yat-Sen University,

Kaohsiung 804, Taiwan

Email address:kimura@math.nsysu.edu.tw

...

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