Strong Convergence of an Iterative Method for Inverse Strongly Accretive Operators

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Volume 2008, Article ID 420989,9pages doi:10.1155/2008/420989

Research Article

Strong Convergence of an Iterative Method for Inverse Strongly Accretive Operators

Yan Hao

School of Mathematics, Physics and Information Science, Zhejiang Ocean University, Zhoushan 316004, China

Correspondence should be addressed to Yan Hao,[email protected] Received 12 May 2008; Accepted 10 July 2008

Recommended by Jong Kim

We study the strong convergence of an iterative method for inverse strongly accretive operators in the framework of Banach spaces. Our results improve and extend the corresponding results announced by many others.

Copyrightq2008 Yan Hao. 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

LetH be a real Hilbert space with norm·and inner product·,·,Ca nonempty closed convex subset of H, and A a monotone operator of C into H. The classical variational inequality problem is formulated as finding a pointx∈Csuch that

y−x, Ax ≥0 1.1

for ally∈C. Such a pointx∈Cis called a solution of the variational inequality1.1. Next, the set of solutions of the variational inequality1.1is denoted by VIC, A. In the case when CH, VIH, A A⁻¹0 holds, where

A⁻¹0{x∈H:Ax0}. 1.2

Recall that an operatorAofCintoHis said to be inverse strongly monotone if there exists a positive real numberαsuch that

x−y, Ax−Ay ≥αAx−Ay² 1.3

for allx, y∈Csee1–4. For such a case,Ais said to beα-inverse strongly monotone.

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Recall thatT :C → Cis nonexpansive if

Tx−Ty ≤ x−y, 1.4

for allx, y∈C.It is known that ifTis a nonexpansive mapping ofCinto itself, thenA I−T is 1/2-inverse strongly monotone andFT VIC, A, whereFTdenotes the set of fixed points ofT.

LetPC be the projection ofH onto the convex subsetC. It is known that projection operatorPCis nonexpansive. It is also known thatPCsatisfies

x−y, PCx−PCy

≥PCx−PCy², 1.5 forx, y∈H.Moreover,PCxis characterized by the propertiesPCx∈Candx−PCx, PCx−y ≥ 0 for ally∈C.

One can see that the variational inequality problem1.1is equivalent to some fixed- point problem. The elementx∈Cis a solution of the variational inequality1.1if and only ifx∈Csatisfies the relationxPCx−λAx,whereλ >0 is a constant.

To find a solution of the variational inequality for an inverse strongly monotone operator, Iiduka et al.2proved the following weak convergence theorem.

Theorem ITT. LetCbe a nonempty closed convex subset of a real Hilbert spaceHand letAbe an α-inverse strongly monotone operator ofCintoHwith VIC, A/∅. Let{xn}be a sequence defined as follows:

x1x∈C, xn1PC

αnxn 1−αn

PC

xn−λnAxn

1.6 for alln 1,2, . . . ,wherePC is the metric projection fromHontoC,{αn}is a sequence in−1,1, and{λn}is a sequence in0,2α. If{αn}and{λn}are chosen so thatαn∈a, bfor somea, bwith

−1 < a < b <1 andλn ∈c, dfor somec, dwith 0< c < d <21aα, then the sequence{xn} defined by1.6converges weakly to some element ofV IC, A.

Next, we assume thatCis a nonempty closed and convex subset of a Banach spaceE.

LetE^∗be the dual space ofEand let·,·denote the pairing betweenEandE^∗. Forq >1, the generalized duality mappingJq:E → 2^E^∗is defined by

Jqx f∈E^∗:x, fx^q,fx^q−1

1.7 for allx∈E. In particular,J J2is called the normalized duality mapping. It is known that Jqx q^q−2Jx for allx ∈ E. IfEis a Hilbert space, thenJ I. Further, we have the following properties of the generalized duality mappingJq:

1Jqx x^q−2J2xfor allx∈Ewithx /0;

2Jqtx t^q−1Jqxfor allx∈Eandt∈0,∞;

3Jq−x −Jqxfor allx∈E.

LetU{x∈X:x1}. A Banach spaceEis said to be uniformly convex if, for any ∈0,2, there existsδ >0 such that, for anyx, y∈U,

x−y ≥ implies xy

2

≤1−δ. 1.8

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It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach spaceEis said to be smooth if the limit

limt→0

xty − x

t 1.9

exists for all x, y ∈ U. It is also said to be uniformly smooth if the limit1.9is attained uniformly forx, y∈U. The norm ofEis said to be Fr´echet diﬀerentiable if, for anyx∈U, the limit1.9is attained uniformly for ally∈U. The modulus of smoothness ofEis defined by

ρτ sup 1

2

xyx−y

−1 :x, y∈X,x1,yτ

, 1.10

whereρ:0,∞ → 0,∞is a function. It is known thatEis uniformly smooth if and only if lim_τ→0ρτ/τ 0. Letqbe a fixed real number with 1< q ≤2. A Banach spaceEis said to beq-uniformly smooth if there exists a constantc >0 such thatρτ≤cτ^qfor allτ >0.

Note that

1 E is a uniformly smooth Banach space if and only if Jq is single-valued and uniformly continuous on any bounded subset ofE;

2all Hilbert spaces,Lporlpspacesp≥2, and the Sobolev spaces,Wm^p p≥2, are 2-uniformly smooth, whileLporlpandWm^p spaces1< p≤2arep-uniformly smooth.

Recall that an operatorAofCintoEis said to be accretive if there existsjx−y ∈ Jx−ysuch that

Ax−Ay, jx−y

≥0 1.11

for allx, y∈C.

Forα >0,recall that an operatorAofCintoEis said to beα-inverse strongly accretive if

Ax−Ay, Jx−y

≥αAx−Ay² 1.12

for allx, y∈C. Evidently, the definition of the inverse strongly accretive operator is based on that of the inverse strongly monotone operator.

LetDbe a subset ofCand letQbe a mapping ofCintoD. ThenQis said to be sunny if

Q

Qxtx−Qx

Qx, 1.13 wheneverQxtx−Qx ∈Cforx ∈Candt ≥0. A mappingQofCinto itself is called a retraction ifQ²Q. If a mappingQofCinto itself is a retraction, thenQzzfor allz∈RQ, whereRQis the range ofQ. A subsetDofCis called a sunny nonexpansive retract ofCif there exists a sunny nonexpansive retraction fromContoD. We know the following lemma concerning sunny nonexpansive retraction.

Lemma 1.1 see 5. LetC be a closed convex subset of a smooth Banach space E, letD be a nonempty subset ofC, and letQbe a retraction fromContoD. ThenQis sunny and nonexpansive if and only if

u−P u, Jy−P u

≤0 1.14

for allu∈Candy∈D.

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Recently, Aoyama et al. 6 first considered the following generalized variational inequality problem in a smooth Banach space. Let Abe an accretive operator ofC intoE.

Find a pointx∈Csuch that

Ax, Jy−x

≥0 1.15

for ally∈C. In order to find a solution of the variational inequality1.15, the authors proved the following theorem in the framework of Banach spaces.

Theorem AIT. LetEbe a uniformly convex and 2-uniformly smooth Banach space andCa nonempty closed convex subset ofE. LetQC be a sunny nonexpansive retraction fromEontoC,α >0,andA anα-inverse strongly accretive operator ofCintoEwithSC, A/∅, where

SC, A x^∗∈C: Ax^∗, J

x−x^∗

≥0, x∈C

. 1.16

If{λn}and{αn}are chosen such thatλn ∈a, α/K²for somea >0 andαn ∈b, cfor someb, c with 0< b < c <1, then the sequence{xn}defined by the following manners:

x1x∈C, x_n1αnxn

1−αn

QC

xn−λnAxn

, 1.17

converges weakly to some elementzofSC, A, whereKis the 2-uniformly smoothness constant ofE.

In this paper, motivated by Aoyama et al. 6, Iiduka et al. 2, Takahahsi and Toyoda 4, we introduce an iterative method to approximate a solution of variational inequality1.15for anα-inverse strongly accretive operators. Strong convergence theorems are obtained in the framework of Banach spaces under appropriate conditions on parameters.

We also need the following lemmas for proof of our main results.

Lemma 1.2see7. Letqbe a given real number with 1< q≤2 and letEbe aq-uniformly smooth Banach space. Then

xy^q≤ x^qq

y, Jqx

2Ky^q 1.18

for allx, y∈X, whereKis theq-uniformly smoothness constant ofE.

The following lemma is characterized by the set of solutions of variational inequality 1.15by using sunny nonexpansive retractions.

Lemma 1.3see6. LetCbe a nonempty closed convex subset of a smooth Banach spaceE. Let QCbe a sunny nonexpansive retraction fromEontoCand letAbe an accretive operator ofCintoE. Then, for allλ >0,

SC, A F

QI−λA

. 1.19

Lemma 1.4see8. LetCbe a nonempty bounded closed convex subset of a uniformly convex Banach spaceEand letT be nonexpansive mapping ofCinto itself. If{xn}is a sequence ofCsuch thatxn → xweakly andxn−Txn → 0, thenxis a fixed point ofT.

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Lemma 1.5see9. Let{xn},{ln}be bounded sequences in a Banach spaceEand let{αn}be a sequence in0,1which satisfies the following condition:

0<lim inf

n→∞ αn≤lim sup

n→∞ αn <1. 1.20

Suppose that

x_n1αnxn 1−αn

ln 1.21

for alln0,1,3, . . .and

lim sup

n→∞

l_n1−ln−x_n1−xn≤0. 1.22

Then lim_n→∞ln−xn0.

Lemma 1.6see10. Assume that{an}is a sequence of nonnegative real numbers such that

a_n1≤ 1−γn

anδn 1.23

for alln0,1,3, . . ., where{γn}is a sequence in0,1and{δn}is a sequence inRsuch that i_∞

n0 γ_n∞;

iilim sup_n→∞δn/γn≤0 or_∞

n0|δn|<∞.

Then lim_n→∞an 0.

2. Main results

Theorem 2.1. LetEbe a uniformly convex and 2-uniformly smooth Banach space andCa nonempty closed convex subset of E. Let QC be a sunny nonexpansive retraction from E onto C, u ∈ C an arbitrarily fixed point, and A an α-inverse strongly accretive operator of C into E such that SC, A/ ∅. Let {αn} and {βn} be two sequences in0,1 and let{λn} a real number sequence ina, α/K²for somea >0 satisfying the following conditions:

ilim_n→∞αn0 and_∞

n0αn∞;

ii0<lim inf_n→∞βn≤lim sup_n→∞βn<1;

iiilim_n→∞|λn1−λn|0.

Then the sequence{xn}defined by

x0∈C, yn βnxn

1−βn

QC

I−λnA xn, x_n1αnu

1−αn

yn, n≥0,

2.1

converges strongly toQu, whereQis a sunny nonexpansive retraction ofContoSC, A.

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Proof. First, we show thatI−λnAis nonexpansive for alln≥0. Indeed, for allx, y∈Cand λn∈a, α/K², fromLemma 1.2, one has

I−λnA x−

I−λnA

y²x−y−λnAx−Ay²

≤ x−y²−2λn

Ax−Ay, Jx−y

2K²λ²_nAx−Ay²

≤ x−y²−2λnαAx−Ay² 2K²λ²_nAx−Ay² x−y²2λn

K²λn−α

Ax−Ay²

≤ x−y².

2.2

Therefore, one obtains thatI−λnAis a nonexpansive mapping for alln≥0. For allp∈SC, A, it follows fromLemma 1.3thatpQCI−λnAp. PutρnQCI−λnAxn. Noticing that

ρn−pQC

I−λnA

xn−QC

I−λnA p

≤I−λnA xn−

I−λnA p

≤xn−p,

2.3

one has

yn−pβn

xn−p

1−βn

ρn−p

≤βnxn−p

1−βnρn−p

≤βnx−p

1−βnxn−p xn−p,

2.4

from which it follows that

x_n1−pαnu−p 1−αn

yn−p

≤αnu−p

1−αnyn−p

≤αnu−p

1−αnxn−p

≤max u−p,xn−p.

2.5

Now, an induction yields

xn−p≤max u−p,x0−p, n≥0. 2.6 Hence,{xn}is bounded, and so is{yn}. On the other hand, one has

ρ_n1−ρnQC

x_n1−λ_n1Ax_n1

−QC

xn−λnAxn

≤xn1−λn1Axn1

−

xn−λnAxn xn1−λn1Axn1

−

xn−λn1Axn

λn−λn1 Axn

≤xn1−xnλn1−λnAxn.

2.7

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Putln xn1−βnxn/1−βn, that is, xn1

1−βn

lnβnxn, n≥0. 2.8

Next, we computel_n1−ln.Observing that

ln1−ln α_n1u

1−α_n1

y_n1−β_n1x_n1

1−β_n1 −αnu

1−αn

yn−βnxn

1−βn

αn1

u−yn1 1−β_n1 −αn

u−yn

1−βn ρn1−ρn,

2.9

we have

ln1−ln≤ αn1

1−βn1

u−yn1 αn

1−βn

yn−uρn1−ρn. 2.10

Combining2.7with2.10, one obtains l_n1−ln−x_n1−xn ≤ αn1

1−βn1

u−yn1 αn

1−βn

yn−u λ_n1−λnAxn. 2.11

It follows that

lim sup

n→∞

ln1−ln−xn1−xn≤0. 2.12 Hence, fromLemma 1.5, we obtain lim_n→∞ln−xn0. From2.7and the conditionii, one arrives at

n→∞limxn1−xn0. 2.13

On the other hand, from2.1, one has xn1−xnαn

u−xn

1−αn

1−βn

ρn−xn

, 2.14

which combines with2.13, and from the conditionsi,ii, one sees that

n→∞limρn−xn0. 2.15

Next, we show that

lim sup

n→∞

u−Qu, J

xn−Qu

≤0. 2.16

To show2.16, we choose a sequence{xni}of{xn}that converges weakly toxsuch that lim sup

n→∞

u−Qu, J

xn−Qu lim

i→∞

u−Qu, J

xn,i−Qu

. 2.17

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Next, we prove thatx∈SC, A. Sinceλn ∈a, α/K²for somea > 0, it follows that{λni}is bounded and so there exists a subsequence{λn_ij}of{λni}which converges toλ0∈a, α/K². We may assume, without loss of generality, that λni → λ0. Since QC is nonexpansive, it follows fromyni QCxni−λniAxnithat

QC

xni−λ0Axni

−xni≤QC

xni−λ0Axni

−ρniρni−xni

≤xni−λ0Axni

−

xni−λniAxniρni −xni

≤λni−λ0Axniρni−xni.

2.18

It follows from2.15that

limi→∞QC

I−λ0A

xni −xni0. 2.19

FromLemma 1.4, we havex∈FQCI−λ0A. It follows fromLemma 1.3thatx∈SC, A.

Now, from2.17andLemma 1.1, we have

lim sup

n→∞

u−Qu, J

xn−Qu lim

i→∞

u−Qu, J

xni−Qu

u−Qu, J

x−Qu

≤0.

2.20

From2.1, we have x_n1−Qu²αn

u−Qu, J

x_n1−Qu

1−αn

yn−Qu, J

x_n1−Qu

≤αn

u−Qu, J

xn1−Qu

1−αn

2 yn−Qu²xn1−Qu²

≤αn

u−Qu, J

x_n1−Qu

1−αn

2 xn−Qu²x_n1−Qu² .

2.21

It follows that

xn1−Qu²≤

1−αnxn−Qu²2αn

u−Qu, J

xn1−Qu

. 2.22

ApplyingLemma 1.6to2.22, we can conclude the desired conclusion. This completes the proof.

As an application ofTheorem 2.1, we have the following results in the framework of Hilbert spaces.

Corollary 2.2. LetH be a Hilbert space andCa nonempty closed convex subset of H. Let PC be a metric projection fromH ontoC,u ∈ Can arbitrarily fixed point, andAanα-inverse strongly monotone operator ofCintoHsuch thatV IC, A/∅. Let{αn}and{βn}be two sequences in0,1

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and let{λn}be a real number sequence ina,2αfor somea >0 satisfying the following conditions:

ilim_n→∞αn0 and_∞

n0αn∞;

ii0<lim inf_n→∞βn≤lim sup_n→∞βn<1;

iiilim_n→∞|λn1−λn|0.

Then the sequence{xn}defined by

x0∈C, ynβnxn

1−βn

PC

I−λnA xn, xn1αnu

1−αn

yn, n≥0,

2.23

converges strongly toP u.

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