General Viscosity Approximation Methods for Common Fixed Points of Nonexpansive Semigroups in Hilbert Spaces

Volume 2011, Article ID 783502,12pages doi:10.1155/2011/783502

Research Article

General Viscosity Approximation Methods for

Common Fixed Points of Nonexpansive Semigroups

in Hilbert Spaces

Xue-song Li,

_{Nan-jing Huang,}

_{and Jong Kyu Kim}

1_{Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China}

2_{Department of Mathematics, Kyungnam University, Masan, Kyungnam 631-701, Republic of Korea}

Correspondence should be addressed to Jong Kyu Kim,[email protected]

Received 12 November 2010; Accepted 17 December 2010

Academic Editor: Jen Chih Yao

Copyrightq2011 Xue-song 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.

This paper is devoted to the strong convergence of two kinds of general viscosity iteration processes for approximating common fixed points of a nonexpansive semigroup in Hilbert spaces. The results presented in this paper improve and generalize some corresponding results inX. Li et al., 2009, S. Li et al., 2009, and Marino and Xu, 2006.

1. Introduction

LetH be a real Hilbert space andA be a linear bounded operator onH. Throughout this paper, we always assume thatAis strongly positive; that is, there exists a constantγ >0 such that

Ax, x ≥γx2, ∀x∈H. 1.1

We recall that a mapping T : H → H is said to be contractive if there exists a constant

α∈0,1such thatTx−Ty ≤αx−yfor allx, y∈H.T:H → His said to be

inonexpansive if

iiL-Lipschitzian if there exists a constantL >0 such that

_{Tx−Ty≤Lx−y, ∀x, y∈H; 1.3}

iiipseudocontractive if

Tx−Ty, x−y≤x−y2, ∀x, y∈H; 1.4

ivφ-strongly pseudocontractive if there exists a strictly increasing function φ :

0,∞ → 0,∞withφ0 0 such that

Tx−Ty, x−y≤x−y2−φx−yx−y, ∀x, y∈H. 1.5

It is obvious that pseudocontractive mapping is more general thanφ-strongly pseudocon-tractive mapping. Ifφr αrwith 0 < α ≤1, thenφ-strongly pseudocontractive mapping reduces to β-strongly pseudocontractive mapping with 1−α β ∈ 0,1, which is more general than contractive mapping.

A nonexpansive semigroup is a family

Γ:{Ts:s≥0} 1.6

of self-mappings onHsuch that

1T0 I, whereIis the identity mapping onH;

2TstxTsTtxfor allx∈Hands, t≥0;

3Tsis nonexpansive for eachs≥0;

4for eachx∈H, the mappingT·xfromRintoHis continuous.

We denote byFΓthe common fixed points set of nonexpansive semigroupΓ, that is,

FΓ s≥0

FTs {x∈H:Tsxx for eachs≥0}. _1.7

In the sequel, we always assume thatFΓ_/∅.

In recent decades, many authors studied the convergence of iterative algorithms for nonexpansive mappings, nonexpansive semigroup, and pseudocontractive semigroup in Banach spacessee, e.g.,1–15. Letf :H → Hbe a contractive mapping with coefficient

α∈0,1,T :H → Hbe a nonexpansive mapping, andAbe a strongly positive and linear bounded operator with coefficientγ > 0. LetF denote the fixed points set of T. Recently, Marino and Xu6considered the general viscosity approximation process as follows:

wheret ∈ 0,1such thatt < A−1and 0 < γ < γ/α. Marino and Xu6proved that the sequence{xt}generated by1.8converges strongly ast → 0 to the unique solution of the variational inequality

A−γfx∗, x−x∗≥0, ∀x∈F, 1.9

which is the optimality condition for the minimization problem

min

x∈F

1

2Ax, xhx, 1.10

wherehis a potential function forγf, that is,hx γfxfor allx∈H.

LetΓ : {Ts : s ≥ 0}be a nonexpansive semigroup on Hand f : H → H be a contractive mapping with coefficientα ∈0,1. Very recently, S. Li et al.5considered the following general viscosity iteration process:

xn I−αnA1 tn

tn

0

Tsxndsαnγfxn, ∀n≥1, 1.11

where{αn} ⊂ 0,1and{tn}are two sequences satisfying certain conditions. S. Li et al.5

claimed that the sequence{xn}generated by1.11converges strongly astn → ∞tox∗ ∈ FΓwhich solves the following variational inequality:

A−γfx∗, x−x∗≥0, ∀x∈FΓ. 1.12

More research work related to general viscosity iteration processes for nonexpansive mapping and nonexpansive semigroup can be foundsee, e.g.,5,6,12.

An interesting work is to extend some results involving general viscosity approx-imation processes for nonexpansive mapping, nonexpansive semigroup, and contractive mapping to nonexpansive semigroup and φ-strongly pseudocontractive mapping pseu-docontractive mapping, resp.. Motivated by the works mentioned above, in this paper, on one hand we study the convergence of general implicit viscosity iteration process

1.11 constructed from the nonexpansive semigroup Γ : {Tt : t ≥ 0} and φ-strongly pseudocontractive mappingpseudocontractive mapping, resp.in Hilbert spaces. On the other hand, we consider the convergence of the following general viscosity iteration process:

xn I−αnATtnxnαnγfxn, ∀n≥1, 1.13

where αn ∈ 0,1, γ > 0, Ttn ∈ Γ and f is a φ-strongly pseudocontractive mapping

2. Preliminaries

A mappingT with domainDTand rangeRTin H is said to be demiclosed at a point

p∈Hif{xn}is a sequence inDTwhich converges weakly tox∈DTand{Txn}converges strongly top, thenTxp.

For the sake of convenience, we restate the following lemmas that will be used.

Lemma 2.1see6. LetAbe a strongly positive and linear bounded operator on a real Hilbert space

Hwith coefficientγ >0and0< ρ≤ A−1. ThenI−ρA ≤1−ργ.

Lemma 2.2see16. LetEbe a Banach space andT :E → Ebe aφ-strongly pseudocontractive and continuous mapping. ThenThas a unique fixed point inE.

Lemma 2.3see9. LetEbe a uniformly convex Banach space,Ka nonempty closed convex subset ofEandT :K → Ea nonexpansive mapping. ThenI−T is demiclosed at zero.

Lemma 2.4see10. LetCbe a nonempty bounded closed convex subset of a real Hilbert spaceH

andΓ {Ts:s≥0}be a nonexpansive semigroup onH. Then for anyh≥0,

lim

t→ ∞sup_x∈C

1

t

_t

0

Tsxds−Th 1 t

_t

0

Tsxds0. 2.1

3. Main Results

We first discuss the convergence of general implicit viscosity iteration process 1.11

constructed from a nonexpansive semigroupΓ:{Ts:s≥0}.

Theorem 3.1. LetΓ : {Ts:s≥0}be a nonexpansive semigroup onHandf :H → Hbe an

Lf-Lipschitzianφ-strongly pseudocontractive mapping withlimt→∞φt ∞. LetAbe a strongly

positive and linear bounded operator onHwith coefficientγ. Then for any0 < γ ≤ γ, the sequence {xn}generated by1.11is well defined. Suppose that

lim

t→ ∞αn0, nlim→ ∞tn∞. 3.1

Then the sequence{xn}converges strongly asn → ∞to a common fixed pointx∗∈FΓthat is the unique solution inFΓto variational inequality (VI):

γfx∗−Ax∗, x∗−p≥0, ∀p∈FΓ. 3.2

Proof. Since limn→ ∞αn 0, we may assume without loss of generality thatαn < A−1, for

anyn≥1. Let us define a mappingTn:H → Hprovided by

Tnx:αnγfx I−αnA1

tn

_tn

0

An application ofLemma 2.1yields that

Tnx−Tny, x−y

I−αnA1 tn

_tn

0

Tsx−Tsyds, x−y

αnγfx−fy, x−y

≤ I−αnAx−y2αnγx−y2−φx−yx−y

≤1−αnγ−γx−y2−αnγφx−yx−y

≤x−y2−αnγφx−yx−y,

3.4

and thus Tn is φ-strongly pseudocontractive and strongly continuous. It follows from

Lemma 2.2thatTnhas a unique fixed pointsayxn ∈H, that is,{xn}generated by1.11is well defined.

Takingp∈FΓ, we have

_xn−p2

αnγfxn−Ap, xn−p

I−αnA1 tn

_tn

0

Tsxn−pds, xn−p

≤αnγfxn−γfp, xn−pαnγfp−Ap, xn−pI−αnAxn−p2

≤1−αnγ−γxn−p2−αnγφxn−pxn−pαnγfp−Apxn−p 3.5

and so

γ−γxn−pγφxn−p≤γfp−Ap. 3.6

This implies thatxn−p ≤φ−1γfp−Ap/γand{xn}is bounded.

We denotezn 1/tn₀tnTsxndsand havezn−p ≤ xn−p, for anyp ∈ FΓ. Since{xn}and{zn}are bounded, it follows from the Lipschitzian conditions ofΓandfthat

{Azn}and{fxn}are two bounded sequences. Therefore,

xn−znαnγfxn−Azn−→0. 3.7

Let

C

x∈H:x−p≤φ−1 γf

p−Ap γ

. 3.8

Sincetn → ∞,Cis a nonempty bounded closed convex subset andTs-invarianti.e.,TsC

is a subset ofC, it follows fromLemma 2.4that

lim

For eachs≥0, we know that

xn−Tsxn ≤ xn−znzn−TsznTszn−Tsxn

≤2xn−znzn−Tszn.

3.10

Consequently, we have from formulas3.7and3.9that

lim

n→ ∞xn−Tsxn0, ∀s≥0. 3.11

Because{xn}is bounded, there exists a subsequence{xnk} ⊂ {xn}which converges weakly to somex∗. It is known fromLemma 2.3thatI−Tsis demiclosed at zero for eachs≥0, where

Iis the identity mapping onH. Thus,x∗∈FΓfollows readily. In addition, by1.11andLemma 2.1, we observe

xn−x∗2αnγfxn−Ax∗, xn−x∗

I−αnA1 tn

_tn

0

Tsxn−x∗ds, xn−x∗

≤αnγfxn−γfx∗, xn−x∗αnγfx∗−Ax∗, xn−x∗I−αnAxn−x∗2

≤1−αnγ−γxn−x∗2−αnγφxn−x∗xn−x∗αnγfx∗−Ax∗, xn−x∗, 3.12

which implies that

γφxn−x∗xn−x∗ ≤γfx∗−Ax∗, xn−x∗. 3.13

This means that{xnk}converges strongly tox∗. If there exists another subsequence{xnj} ⊂

{xn}which converges weakly toy∗, then from3.11and3.13we know that{xnj}converges strongly toy∗∈FΓ. For anyp∈FΓ, it follows from1.11that

Azn−γfxn, xn−p 1 αn

zn−xn, xn−p

1 αn

1

tn

tn

0

Tsxn−pds, xn−p

−xn−p2

≤0.

3.14

The convergence of sequences{xnk}and{xnj}yields that

Ax∗−γfx∗, x∗−y∗≤0,

Thus,

γx∗−y∗2≤Ax∗−y∗, x∗−y∗

≤γfx∗−fy∗, x∗−y∗

≤γx∗−y∗2−γφx∗−y∗x∗−y∗.

3.16

This implies thatx∗ y∗. Therefore,{xn}converges strongly tox∗ ∈FΓ. From3.14and the deduction above, we know thatx∗is also the unique solution to VI3.2. This completes the proof.

Theorem 3.2. LetΓ : {Ts:s≥0}be a nonexpansive semigroup onHandf :H → Hbe an

Lf-Lipschitzian pseudocontractive mapping. LetAbe a strongly positive and linear bounded operator onHwith coefficientγ. Then for any0< γ < γ, the sequence{xn}generated by1.11is well defined. Suppose that

lim

t→ ∞αn0, nlim→ ∞tn∞. 3.17

Then the sequence{xn}converges strongly asn → ∞to a common fixed pointx∗∈FΓthat is the unique solution inFΓto VI3.2.

Proof. Similar to the proof ofTheorem 3.1, we can verify that the sequence{xn}generated by

1.11is well defined,

_xn−p≤ 1

γ−γγf

p−Ap for a fixedp∈FΓ,

lim

n→ ∞xn−Tsxn0, ∀s≥0.

3.18

Thus, {xn} is bounded and so there exists a subsequence {xnk} ⊂ {xn} which converges weakly to somex∗. It is obvious thatx∗∈FΓ.

In addition, by1.11andLemma 2.1, we can show that

xn−x∗2≤ 1 γ−γ

γfx∗_−Ax∗_{, xn−x}∗_{. 3.19}

This means that{xnk}converges strongly tox∗. The rest of the proof is almost the same as

Theorem 3.1. This completes the proof.

Remark 3.3. 1 Theorems 3.1 and 3.2 improve and generalize Theorem 3.1 of 5 from contractive mapping to φ-strongly pseudocontractive mapping and pseudocontractive mapping, respectively.2Theorems3.1and3.2also improve and generalize Theorem 3.2 of

6from nonexpansive mapping to nonexpansive semigroup, and from contractive mapping toφ-strongly pseudocontractive mapping and pseudocontractive mapping, respectively.

that Theorems3.1and3.2are valid for nonexpansive mappings. Thus, we have the following mean ergodic assertions of general viscosity iteration process for nonexpansive mappings in Hilbert spaces.

Corollary 3.4. LetH, f, Abe as inTheorem 3.1,T :H → Hbe a nonexpansive mapping such that the fixed points setFofTis nonempty. Let{αn} ⊂0,1be a real sequence such thatlimn→ ∞αn0.

Then for any0< γ≤γ, there exists a unique{xn}such that

xn I−αnA 1 n1

n

j0

Tjxnαnγfxn, ∀n≥0. 3.20

Moreover, the sequence{xn}generated by3.20converges strongly asn → ∞to a common fixed pointx∗∈Fthat is the unique solution inFto variational inequality (VI):

γfx∗_−Ax∗_{, x}∗_{−p≥0, ∀p∈F. 3.21}

Corollary 3.5. LetH, f, Abe as inTheorem 3.2,T :H → Hbe a nonexpansive mapping such that the fixed points setFofTis nonempty. Let{αn} ⊂0,1be a real sequence such thatlimn→ ∞αn0.

Then for any0 < γ < γ, there exists a unique{xn}satisfying3.20. Moreover, the sequence {xn}

generated by3.20converges strongly asn → ∞to a common fixed pointx∗∈Fthat is the unique solution inFto VI3.21.

We now turn to discuss the convergence of general implicit viscosity iteration process

1.13constructed from a nonexpansive semigroupΓ:{Tt:t≥0}.

Theorem 3.6. LetΓ: {Tt :t≥ 0}be a nonexpansive semigroup onHandf :H → Hbe an

Lf-Lipschitzianφ-strongly pseudocontractive mapping withlimt→∞φt ∞. LetAbe a strongly

positive and linear bounded operator with coefficient γ. Then for any0 < γ ≤ γ, the sequence{xn}

generated by1.13is well defined. Suppose that for any bounded subsetK⊂H,

lim

s→0sup_x∈KTsx−x0, 3.22

lim

n→ ∞tnnlim→ ∞ αn

tn 0. 3.23

Then the sequence{xn}converges strongly asn → ∞to a common fixed pointx∗∈FΓthat is the unique solution inFΓto VI3.2.

Proof. Since limn→ ∞αn 0, we assume without loss of generality that αn < A−1, for any n≥1. Let

ByLemma 2.2, we know

Tnfx−Tnfy, x−y

I−αnATtnx−Ttny, x−yαnγfx−fy, x−y

≤ I−αnAx−y2αnγx−y2−φx−yx−y

≤x−y2−αnγφx−yx−y,

3.25

and thus Tnf is φ-strongly pseudocontractive and strongly continuous. It follows from

Lemma 2.2thatTnf has a unique fixed pointsayxn∈H, that is,{xn}generated by1.13is

well defined.

Takingp∈FΓ, we note

_xn−p2

αnγfxn−Ap, xn−pI−αnATtnxn−p, xn−p

≤αnγfxn−γfp, xn−pαnγfp−Ap, xn−pI−αnAxn−p2

≤1−αnγ−γxn−p2−αnγφxn−pxn−pαnγfp−Apxn−p, 3.26

and soxn−p ≤ φ−1γfp−Ap/γ, the sequence {xn}is bounded. It follows from the Lipschitzian conditions ofΓandf that{ATtnxn}and{fxn}are bounded.1.13implies that

xn−Ttnxnαnγfxn−ATtnxn−→0. 3.27

For any givent >0,

xn−Ttxn

t/tn−1

k0

Tk1tnxn−TktnxnTtxn−T

t tn

tn

xn

≤

t tn

xn−TtnxnT

t−

t tn

tn

xn−xn

≤tαn

tnATtnxn−γfxnmax{Tsxn−xn: 0≤s≤tn},

3.28

wheret/tnis the integral part of t/tn. Since limn→ ∞αn/tn 0 andT·x : R → H is

continuous for anyx∈H, it follows from3.22that

lim

n→ ∞xn−Ttxn0 ∀t≥0. 3.29

In addition, by1.13andLemma 2.1, we observe

xn−x∗2αnγfxn−Ax∗, xn−x∗I−αnATtnxn−x∗, xn−x∗

≤αnγfxn−γfx∗, xn−x∗

αnγfx∗−Ax∗, xn−x∗I−αnAxn−x∗2

≤1−αnγ−γxn−x∗2−αnγφxn−x∗xn−x∗ αnγfx∗−Ax∗, xn−x∗,

3.30

which implies that

γφxn−x∗xn−x∗ ≤γfx∗−Ax∗, xn−x∗. 3.31

For anyp∈FΓ, it follows from1.13that

ATtnxn−γfxn, xn−p 1 αn

Ttnxn−xn, xn−p

1 αn

_Ttnxn−

p, xn−p−xn−p2

≤0.

3.32

The rest of the proof is the same asTheorem 3.1. This completes the proof.

To illustrateTheorem 3.6, we give the following example concerned with a nonexpan-sive semigroupΓ:{Tt:t≥0}onH.

Example 3.7. LetHbe a Hilbert space. For each givent≥0, letTt:H → Hbe defined by

Ttxe−tx, ∀x∈H. 3.33

Then it is easy to check thatΓ:{Tt:t≥0}is a nonexpansive semigroup satisfying3.22

andFΓis a singleton{θ}, whereθis the zero point inH.

Combining the proofs of Theorems3.2and3.6, we can easily conclude the following result.

Theorem 3.8. Letf :H → Hbe anLf-Lipschitzian pseudocontractive mapping andΓ:{Tt:

t ≥ 0}be a nonexpansive semigroup onH such that3.22holds. LetAbe a strongly positive and linear bounded operator with coefficientγ. Then for any0 < γ < γ, the sequence{xn}generated by

1.13is well defined. Suppose that

lim

n→ ∞tnnlim→ ∞ αn

Then the sequence{xn}converges strongly asn → ∞to a common fixed pointx∗∈FΓthat is the unique solution inFΓto VI3.2.

Remark 3.9. 1 Theorems 3.6 and 3.8 improve and generalize Theorem 3.2 of 6 from nonexpansive mapping to nonexpansive semigroup, and from contraction mapping to φ -strongly pseudocontractive mapping and pseudocontractive mapping, respectively.2IfA

is the identity mappingI,f, andΓare restricted on a nonempty closed convex subset inH, then Theorem 3.6 of4follows by Theorems3.6and3.8. So, Theorems3.6and3.8generalize Theorem 3.6 of4.

Acknowledgments

The authors are grateful to Professor J. C. Yao and the referees for valuable comments and suggestions. This work was supported by The Key Program of NSFCGrant no. 70831005, the National Natural Science Foundation of China10671135, 11026063, and the Open Fund

PLN0904 of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation

Southwest Petroleum University.

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