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Volume 2008, Article ID 484050, 13 pagesdoi:10.1155/2008/484050 Research Article Composite Implicit General Iterative Process for a Nonexpansive Semigroup in Hilbert Space Lihua Li, 1 Su

Volume 2008, Article ID 484050, 13 pages

doi:10.1155/2008/484050

Research Article

Composite Implicit General Iterative Process for

a Nonexpansive Semigroup in Hilbert Space

Lihua Li, 1 Suhong Li, 1 and Yongfu Su 2

1 Department of Mathematic and Physics, Hebei Normal University of Science and Technology

Qinhuangdao, Hebei 066004, China

2 Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China

Correspondence should be addressed to Lihua Li,lilihua103@eyou.com

Received 19 March 2008; Accepted 14 August 2008

Recommended by H´el`ene Frankowska

Let C be nonempty closed convex subset of real Hilbert space H Consider C a nonexpansive

semigroupI  {Ts : s ≥ 0} with a common fixed point, a contraction f with coefficient 0 < α < 1, and a strongly positive linear bounded operator A with coefficient γ > 0 Let 0 < γ < γ/α.

It is proved that the sequence{x n } generated iteratively by x n  I − α n A1/t nt n

0Tsy n ds 

α n γfx n , y n  I − β n Ax n  β n γfx n  converges strongly to a common fixed point x∗ ∈ FI

which solves the variational inequalityγf − Ax∗, z − x∗ ≤ 0 for all z ∈ FI.

Copyrightq 2008 Lihua Li 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 C be a closed convex subset of a Hilbert space H, recall that T : C → C is nonexpansive

if Tx − Ty ≤ x − y for all x, y ∈ C Denote by FT the set of fixed points of T, that is,

FT : {x ∈ C : Tx  x}.

Recall that a familyI  {Ts | 0 ≤ s < ∞} of mappings from C into itself is called a nonexpansive semigroup on C if it satisfies the following conditions:

i T0x  x for all x ∈ C;

ii Ts  t  TsTt for all s, t ≥ 0;

iii Tsx − Tsy ≤ x − y for all x, y ∈ C and s ≥ 0;

iv for all x ∈ C, s | → Tsx is continuous.

We denote by FI the set of all common fixed points of I, that is, FI 

∩0≤s<∞FTs It is known that FI is closed and convex.

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problemssee, e.g., 1 5 and the references therein A typical problem

is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping

on a real Hilbert space H:

min

x∈C

1

2Ax, x − x, b, 1.1

where C is the fixed point set of a nonexpansive mapping T on H, and b is a given point in

H Assume that A is strongly positive, that is, there is a constant γ > 0 with the property

Ax, x ≥ γ x 2 ∀x ∈ H. 1.2

It is well known that FT is closed convex cf 6 In3 see also 4, it is proved that the sequence{x n } defined by the iterative method below, with the initial guess x0 ∈ H chosen

arbitrarily,

x n1I − α n A

converges strongly to the unique solution of the minimization problem1.1 provided that the sequence{α n} satisfies certain conditions

On the other hand, Moudafi 7 introduced the viscosity approximation method for nonexpansive mappingssee 8 for further developments in both Hilbert and Banach spaces Let f be a contraction on H Starting with an arbitrary initial x0 ∈ H, define a

sequence{x n} recursively by

x n11− σ n



Tx n  σ n f

x n



where{σ n } is a sequence in 0, 1 It is proved 7,8 that under certain appropriate conditions imposed on {σ n }, the sequence {x n} generated by 1.4 strongly converges to the unique

solution x∗in C of the variational inequality



I − fx∗, x − x∗

Recently, Marino and Xu9 combined the iterative method 1.3 with the viscosity approximation method1.4 considering the following general iteration process:

x n1I − α n A

Tx n  α n γf

x n



and proved that if the sequence{α n } satisfies appropriate conditions, then the sequence {x n}

generated by1.6 converges strongly to the unique solution of the variational inequality



A − γfx∗, x − x∗

which is the optimality condition for the minimization problem

min

x∈C

1

2Ax, x − hx, 1.8

where h is a potential function for γf i.e., h x  γfx, for x ∈ H.

In this paper, motivated and inspired by the idea of Marino and Xu9, we introduce the composite implicit general iteration process1.9 as follows:

x nI − α n A 1

t n

t n

0

Tsy n ds  α n γf

x n



,

y nI − β n A

x n  β n γf

x n



,

1.9

where{α n }, {β n } ⊂ 0, 1, and investigate the problem of approximating common fixed point

of nonexpansive semigroup {Ts : s ≥ 0} which solves some variational inequality The

results presented in this paper extend and improve the main results in Marino and Xu9, and the methods of proof given in this paper are also quite different

In what follows, we will make use of the following lemmas Some of them are known; others are not hard to derive

Lemma 1.1 Marino and Xu 9 Assume that A is a strongly positive linear bounded operator on

a Hilbert space H with coefficient γ > 0 and 0 < ρ ≤ A −1 Then I − ρA ≤ 1 − γ.

Lemma 1.2 Shimizu and Takashi 10 Let C be a nonempty bounded closed convex subset of H

and let I  {Ts : 0 ≤ s < ∞} be a nonexpansive semigroup on C, then for any h ≥ 0,

lim

t → ∞sup

x∈C



1tt

0

Tsxds − Th

1

t

t

0

Lemma 1.3 Let C be a nonempty bounded closed convex subset of a Hilbert space H and let I 

{Tt : 0 ≤ t < ∞} be a nonexpansive semigroup on C If {x n } is a sequence in C satisfying the

following properties:

i x n  z;

ii lim supt → ∞lim supn → ∞ Ttx n − x n  0,

where x n  z denote that {x n } converges weakly to z, then z ∈ FI.

Proof This lemma is the continuous version of Lemma 2.3 of Tan and Xu 11 This proof given in11 is easily extended to the continuous case

2 Main results

Lemma 2.1 Let H be a Hilbert space, C a closed convex subset of H, let I  {Ts : s ≥ 0} be a

nonexpansive semigroup on C, {t n } ⊂ 0, ∞ is a sequence, then I − 1/t nt n

0Tsds is monotone Proof In fact, for all x, y ∈ H,

x − y,

I − 1

t n

t n

0

Tsds x −

I − 1

t n

t n

0

 x − y 2−

x − y, 1

t n

t n

0

Tsxds − 1

t n

t n

0

Tsyds

≥ x − y 2− x − y 1

t n

t n

0



Tsx − Tsyds

≥ x − y 2− x − y 2  0.

2.1

Theorem 2.2 Let C be nonempty closed convex subset of real Hilbert space H, suppose that f :

C → C is a fixed contractive mapping with coefficient 0 < α < 1, and I  {Ts : s ≥ 0} is a nonexpansive semigroup on C such that FI is nonempty, and A is a strongly positive linear bounded operator with coefficient γ > 0, {α n }, {β n } ⊂ 0, 1, {t n } ⊂ 0, ∞ are real sequences such that

lim

n → ∞ α n  0, β n ◦α n



then for any 0 < γ < γ/α, there is a unique {x n } ∈ C such that

x nI − α n A 1

t n

t n

0

Tsy n ds  α n γf

x n



,

y nI − β n A

x n  β n γf

x n



,

2.3

and the iteration process {x n } converges strongly to the unique solution x∗∈ FI of the variational

inequality γf − Ax∗, z − x∗ ≤ 0 for all z ∈ FI.

Proof Our proof is divided into five steps.

Since α n → 0, β n → 0 as n → ∞, we may assume, with no loss of generality, that α n <

A −1, β n < A −1for all n ≥ 1.

i {x n} is bounded

Firstly, we will show that the mapping T n f : C → C defined by

T n f I − α n A 1

t n

t n

0

Ts I − β n A

 β n γf

is a contraction Indeed, fromLemma 1.1, we have for any x, y ∈ C that

T f

n x − T n f y  ≤ I − α n A 1

t n

t n

0

Ts I − β n A

x  β n γfx

− Ts I − β n A

y  β n γfyds  α n γfx − fy

≤1− α n γ I − β n A

x  β n γfx

− I − β n A

y  β n γfy   α n γα x − y

≤1− α n γ I − β n A  x − y  β n γα x − y 

 α n γαx − y

≤1− α n γ 1− β n



γ − γα

x − y  α n γα x − y

1− α n γ 1− β n



γ − γα

 α n γα

x − y

 1− α n



γ − γα

−1− α n γ

β n



γ − γα

x − y

< 1− α n



γ − γα

x − y < x − y

2.5

Let x n ∈ C be the unique fixed point of T f

n Thus,

x nI − α n A 1

t n

t n

0

Tsy n ds  α n γf

x n



,

y nI − β n A

x n  β n γf

x n

is well defined Next, we will show that{x n} is bounded

Pick any z ∈ FI to obtain

x n − z 

I − α n A 1

t n

t n

0

Tsy n ds − z  α n



γf

x n



− Az

≤I − α n A 1

t n

t n

o

Tsy n − zds  α n γf

x n

− fz  γfz − Az

≤1− α n γy n − z  α n γf

x n

− fz  γfz − Az,

2.7

x n − z ≤ 1 − α n γy n − z  α n γαx n − z  α nγfz − Az. 2.8

Also

y n − z ≤ I − β n Ax n − z  β nγf

x n



− Az

≤1− β n γx n − z  β n γαx n − z  β nγfz − Az

 1− β n



γ − γαx n − z  β nγfz − Az.

2.9

Substituting2.9 into 2.8, we obtain that

x n − z ≤ 1 − α n γ 1− β n



γ − γαx n − z  1 − α n γ

β nγfz − Az

 α n γαx n − z  α nγfz − Az

1− α n γ 1− β n



γ − γα

 α n γαx n − z

 1− α n γ

β n  α nγfz − Az

1−γ − γα α n1− α n γ

β nx n − z

 α n1− α n γ

β nγfz − Az,



γ − γα α n1− α n γ

β nx n − z n

1− α n γ

β nγfz − Az,

x n − z ≤ 1

γ − γαγfz − Az.

2.10

Thus{x n} is bounded

ii limn → ∞ x n − Tsx n  0

Denote that z n : 1/tnt n

0Tsy n ds, since {x n } is bounded, z n − z ≤ y n − z and {Az n }, {fx n} are also bounded, From 2.6 and limn → ∞ α n 0, we have

x n − z n   α nγf

x n



− Az n  −→ 0 n −→ ∞. 2.11

Let K  {w ∈ C : w − z ≤ 1/γ − γα γfz − Az }, then K is a nonempty bounded closed convex subset of C and Ts-invariant Since {x n } ⊂ K and K is bounded, there exists r > 0 such that K ⊂ B r, it follows fromLemma 1.2that

lim

From2.11 and 2.12, we have

lim

iii There exists a subsequence {x n k } of {x n } such that x n k  x∗ ∈ FI and x∗ is the unique solution of the following variational inequality:



A − γfx∗, x∗− z≤ 0 ∀z ∈ FI. 2.14

Firstly since

y n − x n   β nγf

x n



− Ax n. 2.15

From condition β n → 0 and the boundedness of {x n }, we obtain that y n − x n → 0 Again by boundedness of{x n }, we know that there exists a subsequence {n k } of {n} such that x n k  x∗

Then y n k  x∗ FromLemma 1.3and stepii, we have that x∗∈ FI.

Next we will prove that x∗solves the variational inequality2.14 Since

x nI − α n A 1

t n

t n

0

Tsy n ds  α n γf

x n



we derive that

A − γfx n − 1

α n



I − α n A

I − 1

t n

t n

0

Tsds y n 1

α n I − α n A

y n−I − α n A

x n



.

2.17

It follows that, for all z ∈ FI,



A − γf

x n , y n − z − 1

α n



I − α n A

I − 1

t n

t n

0

Tsds y n , y n − z

 1

α n



I − α n A

y n−I − α n A

x n , y n − z

 − 1

α n

I − 1

t n

t n

0

Tsds y n−

I − 1

t n

t n

0

Tsds z, y n − z



A

I − 1

t n

t n

0

Tsds y n , y n − z

 1

α n



I − α n A

y n−I − α n A

x n , y n − z.

2.18

UsingLemma 2.1, we have from2.18 that



A − γfx n , y n − z

≤

A

I − 1

t n

t n

0

Tsds y n , y n − z

 1

α n



I − α n A

y n − x n



, y n − z

≤

A

I − 1

t n

t n

0

Tsds y n , y n − z

 1

α n β nγf

x n



− Ax ny n − z.

2.19

Now replacing n in 2.19 with n k and letting k → ∞, we notice that

I − 1

t n

t

nk

0

and from condition β n  ◦α n  and boundedness of {x n}, we have

1

α n k

β n kγf

x n k



− Ax n ky n

k − z −→ 0. 2.21

For x∗∈ FI, we obtain



A − γfx∗, x∗− z≤ 0. 2.22

From 9, Theorem 3.2, we know that the solution of the variational inequality 2.14 is

unique That is, x∗∈ FI is a unique solution of 2.14

iv

lim sup

n → ∞

1

t n

t n

0

Tsy n ds − x∗, γf

x∗

− Ax∗

≤ 0, 2.23

where x∗is obtained in stepiii

To see this, there exists a subsequence{n i } of {n} such that

lim sup

n → ∞

1

t n

t n

0

Tsy n ds − x∗, γf

x∗

− Ax∗

 lim

i → ∞

1

t n i

t ni

0

Tsy n i ds − x∗, γf

x∗

− Ax∗

,

2.24

we may also assume that x n i  z, then 1/t n it ni

0 Tsy n i ds  z, note from step ii that

z ∈ FI in virtue ofLemma 1.2 It follows from the variational inequality2.14 that

lim sup

n → ∞

1

t n

t n

0

Tsx n ds − x∗, γf

x∗

− Ax∗

z − x∗, γf

x∗

− Ax∗

≤ 0. 2.25

So2.23 holds thank to 2.14

v x n → x∗ n → ∞.

Finally, we will prove x n → x∗ Since

y n − x∗2 I − β n A

x n − x∗

 β n



γf

x n



− Ax∗2

≤I − β n Ax n − x∗2 β nγf

x n



− Ax∗2

≤1− β n γx n − x∗2 β nγf

x n



− Ax∗2

.

2.26

Next, we calculate

x n − x∗2 

I − α n A 1

t n

t n

0

Tsy n ds − x∗  α n



γf

x n



− Ax∗2



I − α n A 1

t n

t n

0

Tsy n ds − x∗ 2 α2

nγf

x n



− Ax∗2

 2α nI − α n A 1

t n

t n

0

Tsy n ds − x∗ , γf

x n



− Ax∗

≤1− α n γ2y n − x∗2 α2

nγf

x n



− Ax∗2

 2α nI − α n A 1

t n

t n

0

Tsy n ds − x∗ , γf

x n



− Ax∗.

2.27

Thus it follows from2.26 that

x n − x∗2≤1− α n γ2

1− β n γx n − x∗21− α n γ2

β nγf

x n



− Ax∗2

 α2

nγf

x n



− Ax∗2 2α n

1

t n

t n

0

Tsy n ds − x∗, γf

x n



− Ax∗

− 2α2

n

A

1

t n

t n

0

Tsy n ds − x∗ , γf

x n



− Ax∗

≤1− α n γ2

1− β n γx n − x∗21− α n γ2

β nγf

x n



− Ax∗2

 α2

nγf

x n



− Ax∗2 2α n γ

1

t n

t n

0

Tsy n ds − x∗, f

x n



− fx∗

 2α n

1

t n

t n

0

Tsy n ds − x∗, γf

x∗

− Ax∗

− 2α2

n

A

1

t n

t n

0

Tsy n ds − x∗ , γf

x n



− Ax∗

≤1− α n γ2

1− β n γx n − x∗2 2α n γαy n − x∗x n − x∗

1− α n γ2

β nγf

x n



− Ax∗2

 α n

 2

1

t n

t n

0

Tsy n ds − x∗, γf

x∗

− Ax∗

 α n



γfx n



− Ax∗2 2

A t1nt n

0

Tsy n ds − x∗ ·γf

x n



− Ax∗

≤1− α n γ2

1− β n γx n − x∗2 2α n γα 1− β n



γ − γαx n − x∗2

 2α n γαβ nγf

x∗

− Ax∗x n − x∗  1 − α n γ2

β nγf

x n



− Ax∗2

 α n

 2

1

t n

t n

0

Tsy n ds − x∗, γf

x∗

− Ax∗

 α nγf

x n



− Ax∗2 2

A t1nt n

0

Tsy n ds − x∗ ·γf

x n



− Ax∗

1− α n γ2

1− β n γx n − x∗2 2α n γα

1− β n γx n − x∗

 2α n β n α2γ2x n − x∗2 2α n γαβ nγf

x∗

− Ax∗x n − x∗

1− α n γ2

β nγf

x n



− Ax∗2

 α n

 2

1

t n

t n

0

Tsy n ds − x∗, γf

x∗

− Ax∗

 α n

γfx

n



− Ax∗2 2

A t1nt n

0

Tsy n ds − x∗ ·γf

x n



− Ax∗

 1− α n γ2 2α n γα

1− β n γx n − x∗2 2α n β n α2γ2x n − x∗2

 2α n γαβ nγf

x∗

− Ax∗x n − x∗  1 − α n γ2

β nγf

x n



− Ax∗2

 α n

 2

1

t n

t n

0

Tsy n ds − x∗, γf

x∗

− Ax∗

 α n

γfx

n



− Ax∗2 2

A t1nt n

0

Tsy n ds − x∗ ·γf

x n



− Ax∗

< 1− α n γ2

 2α n γαx n − x∗2 2α n β n α2γ2x n − x∗2

 2α n γαβ nγf

x∗

− Ax∗x n − x∗  1 − α n γ2

β nγf

x n



− Ax∗2

 α n

 2

1

t n

t n

0

Tsy n ds − x∗, γf

x∗

− Ax∗

 α n

γfx

n



− Ax∗2 2α n



A t1nt n

0

Tsy n ds − x∗ 

·γf

x n



− Ax∗

 1− 2γ − γα

α nx n − x∗2 2α n β n α2γ2x n − x∗2

 2α n γαβ nγf

x∗

− Ax∗x n − x∗  1 − α n γ2

β nγf

x n



− Ax∗2

 α n

 2

1

t n

t n

0

Tsy n ds − x∗, γf

x∗

− Ax∗

 α n

γfx

n



− Ax∗2 2

A t1nt n

0

Tsy n ds − x∗ 

·γf

x n



− Ax∗  γ2x n − x∗2

.

2.28

 β n γf

is a contraction Indeed, fromLemma 1.1,...

nk

0