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Suzuki, we show strong convergence theorems of the CQ method for nonexpansive semigroups in Hilbert spaces by hybrid method in the mathematical programming.. In 2005, Xu [4] established

Volume 2007, Article ID 59735, 8 pages

doi:10.1155/2007/59735

Research Article

Strong Convergence Theorems of the CQ Method for

Nonexpansive Semigroups

Huimin He and Rudong Chen

Received 25 January 2007; Accepted 19 March 2007

Recommended by Jerzy Jezierski

Motivated by T Suzuki, we show strong convergence theorems of the CQ method for nonexpansive semigroups in Hilbert spaces by hybrid method in the mathematical programming The results presented extend and improve the corresponding results of Kazuhide Nakajo and Wataru Takahashi (2003)

Copyright © 2007 H He and R Chen 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

Throughout this paper, letH be a real Hilbert space with inner product ·,·and ·  We usex n  x to indicate that the sequence { x n }converges weakly tox Similarly, x n → x will

symbolize strong convergence we denote byNandR +the sets of nonnegative integers and nonnegative real numbers, respectively letC be a closed convex subset of a Hilbert

spaceH, and Let T : C → C be a nonexpansive mapping (i.e.,  Tx − T y  ≤  x − y for allx, y ∈ C) We use Fix(T) to denote the set of fixed points of T; that is, Fix(T) = { x ∈

C : x = Tx } We know that Fix(T) is nonempty if C is bounded, for more details see [1]

In [2], Shioji and Takahashi introduce in a Hilbert space the implicit iteration

x n = α n u +

1− α n1

t n

t n

0 T(s)x n ds, n ∈ N, (1.1) Where{ α n }is a sequence in (0,1),{ t n }is a sequence of positive real numbers divergent to

∞, for eacht ≥0 andu ∈ C In 2003, Suzuki [3] is the first to introduce again in a Hilbert space the following implicit iteration process:

x n = α n u +

1− α n

T

t n

x n

for the nonexpansive semigroup case In 2005, Xu [4] established a Banach space version

of the sequence (1.2) of Suzuki [3], he proved that ifE is a uniformly convex Banach space

with a weakly continuous duality map (e.g.,l pfor 1< p < ∞), ifC is a closed convex

sub-set ofE, and if { T(t) : t ∈ R+}is a nonexpansive semigroup on a closed convex subset

C such that Fix(T) , then under certain appropriate assumptions made and the se-quencesα nandt nof the parameters, he showed that the sequencex nimplicitly defined

by (1.2) for alln ≥1 converges strongly to a member ofF =t ≥0Fix(T(t)).

Recently, Chen and He [5] extend and improve the corresponding results of Suzuki [3], ifE is a reflexive Banach space which admits a weakly sequentially continuous duality

mappingJ from E to E ∗, supposeC is a nonempty closed convex subset of E Let { T(t) :

t ∈ R+}be a nonexpansive semigroup onC such that F(T) , and f : C → C is a fixed

contraction onC Let { α n }and{ t n }be sequences of real numbers satisfying 0< α n < 1,

tn > 0 and limn →∞ tn =limn →∞ αn/tn =0 Define a sequence{ xn }inC by

x n = α n f

x n

+

1− α n

T

t n

x n

Then{ x n }converges strongly toq, as n → ∞.q is the element of F, such that q is the

unique solution inF to the following variational inequality:



(f − I)q, j(x − q)

Some other results can be seen in [6–8]

Nakajo and Takahashi [9] introduced an iteration procedure for nonexpansive self-mappingsT on C as follows:

x0= x ∈ C,

yn = αnxn+

1− αn

Txn,

Cn =z ∈ C; yn − z ≤ xn − z ,

Qn =z ∈ C;

xn − z,x0− xn

≥0 ,

xn+1 = PC n ∩ Q n



x0



(1.5)

for eachn ∈ N ∪ {0}, whereαn ∈[0,a] for some a ∈[0, 1), and{ xn }converges strongly

toPFix(T) x0

Let{ T(t) : t ∈ R+}be a nonexpansive semigroup on a closed convex subsetC of a

Hilbert spaceH, that is,

(1) for eacht ∈ R+,T(t) is a nonexpansive mapping on C;

(2)T(0)x = x for all x ∈ C;

(3)T(s + t) = T(s) ◦ T(t) for all s,t ∈ R+;

(4) for each x ∈ X, the mapping T( ·)x fromR +intoC is continuous We put F =

t ≥0Fix(T(t)) We know that F is nonempty if C is bounded, see [10]

LetC be a nonempty closed convex subset of H and let { T(t) : t ∈ R+}be a nonex-pansive semigroup on a closed convex subsetC of a Hilbert space H such that F ,

Nakajo and Takahashi [9] also introduced an iteration procedure for nonexpansive semi-group{ T(t) : t ∈ R+}onC as follows:

x0= x ∈ C,

y n = α n x n+

1− α n1

t n

t n

0 T(s)x n ds,

Cn =z ∈ C; yn − z ≤ xn − z ,

Q n =z ∈ C;

x n − z,x0− x n

≥0 ,

x n+1 = P C n ∩ Q n



x0



(1.6)

for eachn ∈ N ∪ {0}, whereα n ∈[0,a] for some a ∈[0, 1) and { t n } is a positive real number divergent sequence, and the sequence{ xn }converges strongly toPFx0

In 2006, Martinez-Yanes and Xu [11] employ Nakajo-Takahashi [9] idea and prove some strong convergence theorems for nonexpansive mappings and maximal monotone operators

In this paper, we consider an iteration procedure for nonexpansive semigroups{ T(t) :

t ∈ R+}onC as follows:

x0= x ∈ C,

yn = αnxn+

1− αn

T

tn

xn,

Cn =z ∈ C; yn − z ≤ xn − z ,

Qn =z ∈ C;

xn − z,x0− xn

≥0 ,

xn+1 = PC n ∩ Q n



x0



(1.7)

for eachn ∈ N ∪ {0}, whereαn ∈[0,a] for some a ∈[0, 1) andtn ≥0 limn →∞ tn =0 then the sequence{ x n }converges strongly toP F x0

In the sequel, we will need the following definitions and results

Definition 1.1 A Banach space E is said to satisfy Opial’s condition [12] if whenever{ xn }

is a sequence inE which converges weakly to x, as n → ∞, then

lim sup

n →∞

xn − x < limsup

n →∞

xn − y , ∀ y ∈ E with x y. (1.8)

It is well known that Hilbert space andl p(1< l < ∞) space satisfy Opial’s condition [13]

Lemma 1.2 [14] Let C be a nonempty closed convex subset of a Hilbert space H Given

x ∈ H and y ∈ C, then y = P C x if and only if  x − y, y − z  ≥ 0, is satisfied for all z ∈ C.

Lemma 1.3 [14,15] Every Hilbert space H has Radon-Riesz property or Kadets-Klee prop-erty, that is, for a sequence { xn } ⊂ H with xn  x and  xn  →  x  , then there holds xn → x.

2 Main results

Lemma 2.1 Let C be a closed convex subset of a Hilbert space H Let { T(t) : t ∈ R+} be a nonexpansive semigroup on C such that F , and the sequence { xn } generated by (1.7), where α n ∈[0,a] for some a ∈ [0, 1), Then { x n } is well defined and F ⊂ C n ∩ Q n for every

n ∈ N ∪ {0}

Proof It is obvious that Cnis closed andQnis closed and convex for everyn ∈ N ∪ {0}

It follows from thatCnis convex for everyn ∈ N ∪ {0}because yn − z  ≤  xn − z is equivalent to

y

n − x n 2

+ 2

y n − x n,x n − z

So,C n ∩ Q nis closed and convex for everyn ∈ N ∪ {0} Letu ∈ F Then from

y

n − u = α n x n+

1− α n

T

t n

x n − u

≤ α n x

n − u +

1− α n T

t n

x n − u

we haveu ∈ C nfor eachn ∈ N ∪ {0} So, we haveF ⊂ C nfor alln ∈ N ∪ {0}

Next, we show by mathematical induction that{ xn }is well defined andF ⊂ Cn ∩ Qn

for everyn ∈ N ∪ {0} Forn =0, we havex0= x ∈ C and Q0= C, and hence F ⊂ C0∩ Q0 Suppose thatx k is given and F ⊂ C k ∩ Q k for some k ∈ N ∪ {0}.There exists a unique elementxk+1 ∈ Ck ∩ Qk such thatxk+1 = PC k ∩ Q k(x0) Fromxk+1 = PC k ∩ Q k(x0), it holds that



xk+1 − z,x0− xk+1

for eachz ∈ Ck ∩ Qk SinceF ⊂ Ck ∩ Qk, we getF ⊂ Qk+1, therefore we haveF ⊂ Ck+1 ∩

Q k+1

Lemma 2.2 Let C be a closed convex subset of a Hilbert space H Let { T(t) : t ∈ R+} be a nonexpansive semigroup on C such that F , and the sequence { xn } generated by (1.7), where α n ∈[0,a] for some a ∈ [0, 1), Then lim n →∞  x n+1 − x n  = 0.

Proof At first, we show that F is a closed convex subset of C Since T(t) : C → C, t > 0 is

nonexpansive, we claim thatF is closed In fact, if p n ⊂ F =t ≥0Fix(T(t)), n ≥1, such that limn →∞ pn = p, then we have

T(t)p =lim

n →∞ T(t)pn =lim

Thusp ∈ F.

Next, we show thatF is convex, we will use the following identity in Hilbert space:

tx + (1 − t)y 2

= t  x 2+ (1− t)  y 2− t(1 − t)  x − y 2, (2.5)

which holds for allx, y ∈ H and for all t ∈[0, 1] indeed,

tx + (1 − t)y 2

= t2 x 2+ (1− t)2 y 2+ 2t(1 − t)  x, y 

= t  x 2+ (1− t)  y 2+ 2t(1 − t)  x, y 

− t(1 − t)  x 2− t(1 − t)  y 2

= t  x 2+ (1− t)  y 2− t(1 − t)

 x 2+ y 2−2 x, y 

= t  x 2+ (1− t)  y 2− t(1 − t)  x − y 2.

(2.6)

Letp1,p2∈ F and for all t ∈[0, 1],p = tp1+ (1− t)p2, then

p − p1=(1− t)

p2− p1 

, p − p2=(1− t)

p1− p2 

From (2.5) and (2.7), we have

p − T(t)p 2

= t

p1− T(t)p

+ (1− t)

p2− T(t)p 2

= t p1− T(t)p 2

+ (1− t) p2− T(t)p 2

− t(1 − t) p1− p2 2

≤ t p1− p 2

+ (1− t) p2− p 2

− t(1 − t) p1− p2 2

= t(1 − t)2 p1− p2 2

+t2(1− t) p1− p2 2

− t(1 − t) p1− p2 2

= t(1 − t)(1 − t + t −1) p1− p2 2

=0.

(2.8)

Thusp = T(t)p, for all t > 0, that is, p ∈ F.

Secondly, we show that{ xn }is bounded SinceF is a nonempty closed convex subset

ofC, there exists a unique element z0∈ F such that z0= PF(x0) From xn+1 = PC n ∩ Q n(x0),

we have

x

n+1 − x0 ≤ z − x0 ∀ z ∈ C n ∩ Q n (2.9)

It follows fromLemma 2.1thatF ⊂ Cn ∩ Qnfor everyn ∈ N ∪ {0}, together withz0∈

F(T), we have

xn+1 − x0 ≤ z0− x0 ∀ n ∈ N ∪ {0} (2.10)

This implies that{ x n }is bounded, so{ T(t)x n }is also bounded, and moreover so is

{ yn }since yn  ≤ αn  xn + (1− αn) T(t)xn 

Thirdly, we show that xn+1 − xn  →0 asn → ∞ SinceQn = { z ∈ C;  xn − z,x0− xn  ≥

0},x n = P Q n(x0) Asx n+1 ∈ C n ∩ Q n ⊂ Q n, we obtain

x

n+1 − x0 ≥ x n − x0 , ∀ z ∈ C

Therefore the sequence{ xn − x0}is bounded and nondecreasing So

lim

On the other hand, fromx n+1 ∈ Q n, we get x n − x n+1,x0− x n  ≥0, and hence

xn − xn+1 2

= xn − x0

−xn+1 − x0  2

= x

n − x0 2

−2

x n − x0,x n+1 − x0



+ x

n+1 − x0 2

= x n − x0 2

+ x n+1 − x0 2

−2

xn − x0,xn+1 − xn+xn − x0 

= x

n+1 − x0 2

− x

n − x0 2

−2

x n − x n+1,x0− x n

≤ x n+1 − x0 2

− x n − x0 2

−→0 (n −→ ∞).

(2.13)

So

lim

Theorem 2.3 Let C be a closed convex subset of a Hilbert space H Let { T(t) : t ∈ R+}

be a nonexpansive semigroup on C such that F , and the sequence { xn } generated by (1.7), where α n ∈[0,a] for some a ∈ [0, 1), and t n ≥0 limn →∞ t n = 0 then the sequence

{ xn } converges strongly to PFx0.

Proof It follows from xn+1 ∈ Cnthat

T

tn

xn − xn = 1

1− αn yn − xn

≤ 1

1− α n y n − x n+1 + x

n+1 − x n 

≤ 2

1− αn xn+1 − xn

(2.15)

for everyn ∈ N ∪ {0} ByLemma 2.2, we get T(tn)xn − xn  →0

We claim that{ x n } is relatively sequentially compact Indeed, there exists a weakly convergence subsequence{ xn j } ⊆ { xn } by reflexivity ofH and boundedness of the

se-quence{ xn }, now we supposexn j  x ∈ C( j → ∞) Now we show thatx ∈ F Put xj = xn j,

β j = α n j, ands j = t n j forj ∈ N, lets j ≥0 be such that

s j −→0, T

sj

xj − x j

Fixt > 0, from

x

j − T(t)x ≤[t/sj]−1

k =0

T

(k + 1)s j

x j − T

ks j

x j

+ T t sj



sj



x j − T t

s j



sj



x + T

t/s j



sj

x − T(t)x

≤ t

s j



T

sj

x j − xj + x j − x + Tt − t

s j



s j



x − x

≤ t T

s j

x j − x j

s j + x

j − x + max T(s)x − x : 0≤ s ≤ s

j

(2.17)

For allj ∈ N ∪ {0}, as every Hilbert space satisfies Opial’s condition, then we have

lim sup

j →∞

x j − T(t)x ≤lim sup

j →∞

xj − x .

(2.18) This implies thatT(t)x = x Therefore,

Ifz0= PF(x0), it follows from (2.10), (2.19), and the lower semicontinuity of the norm that

x0− z0 ≤ x0− x ≤lim inf

j →∞ x0− xn

j ≤lim sup

j →∞

x0− xn

j ≤ x0− z0 . (2.20)

Thus, we obtain

lim

j →∞ x n

j − x0 = x0− x = x0− z0 . (2.21)

This implies that

This shows that{ xn }is relatively sequentially compact Therefore, we havexn → z0

Acknowledgment

This work is partially supported by the National Science Foundation of China, Grant 10471033

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Huimin He: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China

Email address:hehuimin20012000@yahoo.com.cn

Rudong Chen: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China

Email address:chenrd@tjpu.edu.cn

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