Composite Implicit General Iterative Process for a Nonexpansive Semigroup in Hilbert Space

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Volume 2008, Article ID 484050,13pages 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,[email protected]

Received 19 March 2008; Accepted 14 August 2008

Recommended by H´el`ene Frankowska

LetC be nonempty closed convex subset of real Hilbert spaceH. ConsiderC a nonexpansive semigroupI{Ts:s≥0}with a common fixed point, a contractionfwith coeﬃcient 0< α <1, and a strongly positive linear bounded operatorAwith coeﬃcientγ > 0. Let 0 < γ < γ/α. It is proved that the sequence{xn}generated iteratively byxn I−αnA1/tn

tn

0Tsynds

αnγfxn, yn I−βnAxnβnγfxnconverges strongly to a common fixed pointx∗ ∈ FI

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

Copyrightq2008 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

LetCbe a closed convex subset of a Hilbert spaceH, recall thatT :C→Cis nonexpansive if Tx−Ty ≤ x−y for allx, y ∈ C. Denote byFTthe set of fixed points ofT, that is,

FT:{x∈C:Txx}.

Recall that a familyI {Ts |0 ≤s <∞}of mappings fromCinto itself is called a nonexpansive semigroup onCif it satisfies the following conditions:

iT0xxfor allx∈C;

iiTst TsTtfor alls, t≥0;

iii Tsx−Tsy ≤ x−y for allx, y∈Cands≥0; ivfor allx∈C, s| →Tsxis continuous.

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

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Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problemssee, e.g.,1–5and 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 spaceH:

min

x∈C

1

2Ax, x − x, b, 1.1

whereCis the fixed point set of a nonexpansive mappingT onH, andbis a given point in

H. Assume thatAis strongly positive, that is, there is a constantγ >0 with the property

Ax, x ≥γ x 2 ∀x∈H. 1.2

It is well known thatFTis closed convexcf.6. In3 see also4, it is proved that the sequence{xn}defined by the iterative method below, with the initial guessx0 ∈ Hchosen arbitrarily,

xn1

I−αnA

Txnαnb, n≥0 1.3

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

On the other hand, Moudafi 7 introduced the viscosity approximation method for nonexpansive mappingssee8 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{xn}recursively by

xn1

1−σn

Txnσnf

xn

, n≥0, 1.4

where{σn}is a sequence in0,1. It is proved7,8that under certain appropriate conditions

imposed on {σn}, the sequence{xn} generated by 1.4 strongly converges to the unique

solutionx∗inCof the variational inequality

I−fx∗, x−x∗≥0, x∈C. 1.5

Recently, Marino and Xu9combined the iterative method 1.3 with the viscosity approximation method1.4considering the following general iteration process:

xn1

I−αnA

Txnαnγf

xn

, n≥0, 1.6

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

A−γfx∗, x−x∗≥0, x∈C, 1.7

which is the optimality condition for the minimization problem

min

x∈C

1

2Ax, x −hx, 1.8

wherehis a potential function forγfi.e.,hx γfx, forx∈H.

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

xn

I−αnA

1

tn

0

Tsyndsαnγf

xn

,

yn

I−βnA

xnβnγf

xn

,

1.9

where{αn},{βn} ⊂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 1.1Marino and Xu9. Assume thatAis a strongly positive linear bounded operator on a Hilbert spaceHwith coeﬃcientγ >0and0< ρ≤ A −1_{. Then} _I₋_ρA_≤₁₋_γ_.

Lemma 1.2Shimizu and Takashi10. LetCbe a nonempty bounded closed convex subset ofH and letI{Ts: 0≤s <∞}be a nonexpansive semigroup onC, then for anyh≥0,

lim

t→ ∞sup_x_∈_C

1_tt

0

Tsxds−Th 1 t

_t

0

Tsxds0. 1.10

Lemma 1.3. LetCbe a nonempty bounded closed convex subset of a Hilbert spaceHand let I

{Tt : 0 ≤ t < ∞} be a nonexpansive semigroup onC. If{xn}is a sequence inCsatisfying the following properties:

ixn z;

iilim sup_t_{→ ∞}lim sup_n_{→ ∞} Ttxn−xn 0,

wherexn zdenote that{xn}converges weakly toz, thenz∈FI.

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2. Main results

Lemma 2.1. LetHbe a Hilbert space,Ca closed convex subset ofH, letI {Ts : s ≥0}be a nonexpansive semigroup onC,{tn} ⊂0,∞is a sequence, thenI−1/tn

tn

0Tsdsis monotone.

Proof. In fact, for allx, y∈H,

x−y, I− 1 tn

_t_n

0

Tsds

x− I− 1

tn

_t_n

0

Tsds

y

x−y 2−

x−y, 1 tn

tn

0

Tsxds− 1 tn

tn

0

Tsyds

≥ x−y 2₋_x₋_y 1

tn

_t_n

0

Tsx−Tsyds

≥ x−y 2− x−y 2 0.

2.1

Theorem 2.2. Let Cbe nonempty closed convex subset of real Hilbert spaceH, suppose thatf :

C→Cis a fixed contractive mapping with coeﬃcient 0 < α < 1, andI {Ts : s ≥ 0}is a nonexpansive semigroup onCsuch thatFIis nonempty, andAis a strongly positive linear bounded operator with coeﬃcientγ >0,{αn},{βn} ⊂0,1,{tn} ⊂0,∞are real sequences such that

lim

n→ ∞αn0, βn◦

αn

, lim

n→ ∞tn∞, 2.2

then for any0< γ < γ/α, there is a unique{xn} ∈Csuch that

xn

I−αnA

1

tn

_t_n

0

Tsyndsαnγf

xn

,

yn

I−βnA

xnβnγf

xn

,

2.3

and the iteration process{xn}converges strongly to the unique solutionx∗∈FIof the variational inequalityγf−Ax∗, z−x∗ ≤0for allz∈FI.

Proof. Our proof is divided into five steps.

Sinceαn→0,βn→0 as n→ ∞, we may assume, with no loss of generality, thatαn < A −1_,_β

n< A −1for alln≥1.

i{xn}is bounded.

Firstly, we will show that the mappingTnf :C→Cdefined by

Tnf

I−αnA

1

tn

_t_n

0

TsI−βnA

βnγf

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is a contraction. Indeed, fromLemma 1.1, we have for anyx, y∈Cthat

_Tf

nx−Tnfy≤I−αnA1 tn

tn

0

_T_s

I−βnA

xβnγfx

−TsI−βnA

yβnγfydsαnγfx−fy

≤1−αnγI−βnA

xβnγfx

−I−βnA

yβnγfyαnγα x−y

≤1−αnγI−βnA x−y βnγα x−y

αnγαx−y

≤1−αnγ

1−βn

γ−γα x−y αnγα x−y

1−αnγ

1−βn

γ−γααnγα

x−y

1−αn

γ−γα−1−αnγ

βn

γ−γα x−y

<1−αn

γ−γα x−y < x−y .

2.5

Letxn∈Cbe the unique fixed point ofTnf. Thus,

xn

I−αnA

1

tn

0

Tsyndsαnγf

xn

,

yn

I−βnA

xnβnγf

xn

2.6

is well defined. Next, we will show that{xn}is bounded.

Pick anyz∈FIto obtain

_x_n₋_z

I−αnA

1

tn

_t_n

0

Tsynds−z

αn

γfxn

−Az

≤I−αnA

1

tn

_t_n

o

_T_s_y_n₋_z_ds_α_n

γfxn

−fzγfz−Az

≤1−αnγyn−zαn

γfxn

−fzγfz−Az,

2.7

_x_n₋_z_≤

1−αnγyn−zαnγαxn−zαnγfz−Az. 2.8

Also

_y_n₋_z_≤_I₋_β_n_A_x_n₋_z_β_n_γf

xn

−Az

≤1−βnγxn−zβnγαxn−zβnγfz−Az

1−βn

γ−γαxn−zβnγfz−Az.

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Substituting2.9into2.8, we obtain that

_x_n₋_z_≤ 1−αnγ

1−βn

γ−γαxn−z

1−αnγ

βnγfz−Az

αnγαxn−zαnγfz−Az

1−αnγ

1−βn

γ−γααnγαxn−z

1−αnγ

βnαnγfz−Az

1−γ−γααn

1−αnγ

βnxn−z

αn

1−αnγ

βnγfz−Az,

γ−γααn

1−αnγ

βnxn−z≤

αn

1−αnγ

βnγfz−Az,

_x_n₋_z_≤ 1

γ−γαγfz−Az.

2.10

Thus{xn}is bounded.

iilimn→ ∞ xn−Tsxn 0.

Denote thatzn : 1/tn

tn

0Tsynds, since{xn}is bounded, zn−z ≤ yn−z and {Azn},{fxn}are also bounded, From2.6and limn→ ∞αn0, we have

_x_n₋_z_n_α_n_γf

xn

−Azn−→0 n−→ ∞. 2.11

LetK{w∈C: w−z ≤1/γ−γα γfz−Az }, thenKis a nonempty bounded closed convex subset ofCandTs-invariant. Since{xn} ⊂KandKis bounded, there existsr >0

such thatK⊂Br, it follows fromLemma 1.2that

lim

n→ ∞zn−Tszn0 ∀s≥0. 2.12

From2.11and2.12, we have

lim

n→ ∞xn−Tsxn0. 2.13

iiiThere exists a subsequence{xnk}of{xn}such thatxnk x∗ ∈FIandx∗ 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_n_γf

xn

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From conditionβn→0 and the boundedness of{xn}, we obtain that yn−xn →0. Again by

boundedness of{xn}, we know that there exists a subsequence{nk}of{n}such thatxnk x∗.

Thenynk x∗. FromLemma 1.3and stepii, we have thatx∗∈FI.

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

xn

I−αnA

1

tn

_t_n

0

Tsyndsαnγf

xn

, 2.16

we derive that

A−γfxn−

1

αn

I−αnA I−

1

tn

_t_n

0

Tsds

yn

1

αn

I−αnA

yn−

I−αnA

xn

.

2.17

It follows that, for allz∈FI,

A−γfxn, yn−z

− 1

αn

I−αnA I− 1 tn

tn

0

Tsds

yn, yn−z

1

αn

I−αnA

yn−

I−αnA

xn, yn−z

− 1

αn I−

1

tn

_t_n

0

Tsds

yn− I−

1

tn

_t_n

0

Tsds

z, yn−z

A I− 1

tn

_t_n

0

Tsds

yn, yn−z

1

αn

I−αnA

yn−

I−αnA

xn, yn−z

.

2.18

UsingLemma 2.1, we have from2.18that

A−γfxn, yn−z

≤

A I− 1

tn

_t_n

0

Tsds

yn, yn−z

1

αn

I−αnA

yn−xn

, yn−z

≤

A I− 1

tn

0

Tsds

yn, yn−z

1

αnβn

_γf

xn

−Axnyn−z.

2.19

Now replacingnin2.19withnkand lettingk→ ∞, we notice that

I− 1 tnk

_t

nk

0

Tsds

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and from conditionβn ◦αnand boundedness of{xn}, we have

1

αnk

βnkγf

xnk

−Axnkynk−z−→0. 2.21

Forx∗∈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∗∈FIis a unique solution of2.14.

iv

lim sup

n→ ∞

1

tn

0

Tsynds−x∗, γf

x∗−Ax∗

≤0, 2.23

wherex∗is obtained in stepiii.

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

lim sup

n→ ∞

1

tn

0

x∗−Ax∗

lim

i→ ∞

1

tni

t_ni

0

Tsynids−x∗, γf

x∗−Ax∗

,

2.24

we may also assume thatxni z, then1/tni

t_ni

0 Tsynids z, note from step ii that

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

lim sup

n→ ∞

1

tn

_t_n

0

Tsxnds−x∗, γf

x∗−Ax∗

z−x∗, γfx∗−Ax∗≤0. 2.25

So2.23holds thank to2.14.

vxn→x∗ n→ ∞.

Finally, we will provexn→x∗. Since

_y_n₋_x∗2

I−βnA

xn−x∗

βn

γfxn

−Ax∗2

≤I−βnAxn−x∗2βnγf

xn

−Ax∗2

≤1−βnγxn−x∗2βnγf

xn

−Ax∗2.

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Next, we calculate

_x_n₋_x∗2

I−αnA

1

tn

0

Tsynds−x∗

αn

γfxn

−Ax∗

2

I−αnA

1

tn

_t_n

0

Tsynds−x∗

2α2nγf

xn

−Ax∗2

2αn

I−αnA 1 tn

tn

0

Tsynds−x∗

, γfxn

−Ax∗

≤1−αnγ

2

yn−x∗2α2nγf

xn

−Ax∗2

2αn

I−αnA

1

tn

_t_n

0

Tsynds−x∗

, γfxn

−Ax∗.

2.27

Thus it follows from2.26that

_x_n₋_x∗2_≤ 1−αnγ

2

1−βnγxn−x∗2

1−αnγ

2

βnγf

xn

−Ax∗2

α2nγf

xn

−Ax∗22αn

1

tn

0

xn

−Ax∗

−2α2n

A 1

tn

0

Tsynds−x∗

, γfxn

−Ax∗

≤1−αnγ

2

1−βnγxn−x∗2

1−αnγ

2

βnγf

xn

−Ax∗2

α2nγf

xn

−Ax∗22αnγ

1

tn

_t_n

0

Tsynds−x∗, f

xn

−fx∗

2αn

1

tn

_t_n

0

x∗−Ax∗

−2α2

n A 1 tn _t_n 0

Tsynds−x∗

, γfxn

−Ax∗

≤1−αnγ

2

1−βnγxn−x∗22αnγαyn−x∗xn−x∗

1−αnγ

2

βnγf

xn

−Ax∗2

αn 2 1 tn tn 0

x∗−Ax∗

αn γf

xn

−Ax∗22A 1 tn

_t_n

0

Tsynds−x∗

·γfxn

−Ax∗

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≤1−αnγ

2

1−βnγxn−x∗22αnγα

1−βn

γ−γαxn−x∗2

2αnγαβnγf

x∗−Ax∗xn−x∗

1−αnγ

2

βnγf

xn

−Ax∗2

αn 2 1 tn _t_n 0

x∗−Ax∗

αnγf

xn

−Ax∗22A 1 tn

tn

0

Tsynds−x∗

·γfxn

−Ax∗

1−αnγ

2

1−βnγxn−x∗22αnγα

1−βnγxn−x∗

2αnβnα2γ2xn−x∗22αnγαβnγf

1−αnγ

2

βnγf

xn

−Ax∗2

αn 2 1 tn _t_n 0

x∗−Ax∗

αn γf

xn

−Ax∗22A 1 tn

tn

0

Tsynds−x∗

·γfxn

−Ax∗

1−αnγ

2

2αnγα

1−βnγxn−x∗22αnβnα2γ2xn−x∗2

2αnγαβnγf

1−αnγ

2

βnγf

xn

−Ax∗2

αn 2 1 tn _t_n 0

x∗−Ax∗

αn γf

xn

−Ax∗22A 1 tn

_t_n

0

Tsynds−x∗

·γfxn

−Ax∗

<1−αnγ

2₂

αnγαxn−x∗22αnβnα2γ2xn−x∗2

2αnγαβnγf

1−αnγ

2

βnγf

xn

−Ax∗2

αn 2 1 tn tn 0

x∗−Ax∗

αn γf

xn

−Ax∗22αn

A 1

tn

_t_n

0

Tsynds−x∗

·γfxn

−Ax∗

1−2γ−γααnxn−x∗22αnβnα2γ2xn−x∗2

2αnγαβnγf

1−αnγ

2

βnγf

xn

−Ax∗2

αn 2 1 tn tn 0

x∗−Ax∗

αn γf

xn

−Ax∗22A 1 tn

tn

0

Tsynds−x∗

·γfxn

−Ax∗γ2xn−x∗2

.

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Thus

2γ−γαxn−x∗2≤2βnα2γ2xn−x∗22γαβnγf

x∗−Ax∗·xn−x∗

1−αnγ

2βn αn

_γf

xn

−Ax∗2

2

1

tn

_t_n

0

x∗−Ax∗

αn γf

xn

−Ax∗22A 1 tn

_t_n

0

Tsynds−x∗

·γfxn

−Ax∗γ2_x

n−x∗2

≤βn

2α2γ2xn−x∗22γαγf

x∗−Ax∗·xn−x∗

βn αn

_γf

xn

−Ax∗2

2

1

tn

_t_n

0

x∗−Ax∗

αn γf

xn

−Ax∗22A 1 tn

_t_n

0

Tsynds−x∗

·γfxn

.

2.29

Since{xn}is bounded, we can take a constantL1, L2, L3>0 such that

L1 ≥2α2γ2xn−x∗22γαγf

x∗−Ax∗·xn−x∗,

L2 ≥γf

xn

−Ax∗2,

L3 ≥γf

xn

−Ax∗22A 1 tn

tn

0

Tsynds−x∗

·γfxn

2.30

for alln≥0. It then follows from2.29that

2γ−γαxn−x∗2 ≤βnL1

βn αnL2

2

1

tn

_t_n

0

x∗−Ax∗

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Then

2γ−γαlim sup

n→ ∞

_x_n₋_x∗2

≤lim sup

n→ ∞

βnL1

βn

αnL2αnL3

2 lim sup

n→ ∞

1

tn

_t_n

0

x∗−Ax∗

.

2.32

From conditionαn→0,βn◦αnand2.23, we conclude that

lim sup

n→ ∞

_x_n₋_x∗2_≤

0. 2.33

Soxn→x∗. This completes the proof of theTheorem 2.2.

It follows from the above proof thatTheorem 2.2is valid for nonexpansive mappings. Thus, we have that Corollaries2.3and2.4are two special cases ofTheorem 2.2.

Corollary 2.3. LetTbe a nonexpansive mapping from nonempty closed convex subsetCof a Hilbert spaceHtoC,{xn}is generated by the following algorithm:

xn

I−αnA

Tynαnγf

xn

,

yn

I−βnA

xnβnγf

xn

, 2.34

where{αn},{βn} ⊂0,1are real sequences such that

lim

n→ ∞αn0, βn◦

αn

, 2.35

then for any0 < γ < γ/α, the sequence{xn}above converges strongly to the unique solutionx∗ ∈ FTof the variational inequalityγf−Ax∗, z−x∗ ≤0for allz∈FT.

Corollary 2.4. LetTbe a nonexpansive mapping from nonempty closed convex subsetCof a Hilbert spaceHtoC,{xn}is generated by the following algorithm:

xn

I−αnA

Tynαnγf

xn

,

yn

I−βnA

xnβnγf

xn

,

2.36

where{αn}is a sequence in (0,1) satisfying the following condition:limn→ ∞αn0, then the sequence

{xn}converges strongly to the unique solutionx∗∈FTof the variational inequalityI−fx∗, x∗− z ≤0for allz∈FT.

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