A New Iteration Method for Nonexpansive Mappings and Monotone Mappings in Hilbert Spaces

Volume 2010, Article ID 251761,16pages doi:10.1155/2010/251761

Research Article

A New Iteration Method for

Nonexpansive Mappings and Monotone Mappings

in Hilbert Spaces

Jong Soo Jung

Department of Mathematics, Dong-A University, Busan 604-714, South Korea

Correspondence should be addressed to Jong Soo Jung,[email protected]

Received 5 October 2009; Accepted 20 December 2009

Academic Editor: Jong Kim

Copyrightq2010 Jong Soo Jung. 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.

We introduce a new composite iterative scheme by the viscosity approximation method for nonexpansive mappings and monotone mappings in a Hilbert space. It is proved that the sequence generated by the iterative scheme converges strongly to a common point of set of fixed points of nonexpansive mapping and the set of solutions of variational inequality for an inverse-strongly monotone mappings, which is a solution of a certain variational inequality. Our results substantially develop and improve the corresponding results ofChen et al. 2007 and Iiduka and Takahashi 2005. Essentially a new approach for finding the fixed points of nonexpansive mappings and solutions of variational inequalities for monotone mappings is provided.

1. Introduction

LetH be a real Hilbert space and Ca nonempty closed convex subset ofH. Recall that a mapping f : C → C is a contractionon C if there exists a constant k ∈ 0,1such that

fx−fy ≤kx−y, x, y∈C.We useΣCto denote the collection of mappingsfverifying

the above inequality. That is ΣC {f : C → C | f is a contraction with constantk}. A

mappingS :C → Cis called nonexpansiveifSx−Sy ≤ x−y, x, y ∈ C; see1,2for the results of nonexpansive mappings. We denote byFSthe set of fixed points ofS; that is,

FS {x∈C:xSx}.

Let PC be the metric projection of H onto C. A mapping A of C into H is called

monotoneif forx, y ∈ C,x−y, Ax−Ay ≥ 0. The variational inequality problemis to find au∈Csuch that

for all v ∈ C; see 3–6. The set of solutions of the variational inequality is denoted by VIC, A. A mappingAofCintoHis called inverse-strongly monotoneif there exists a positive real numberαsuch that

x−y, Ax−Ay≥αAx−Ay2 1.2

for allx, y∈C; see7–9. For such a case,Ais calledα-inverse-strongly monotone.

In 2005, Iiduka and Takahashi 10 introduced an iterative scheme for finding a common point of the set of fixed points of a nonexapnsive mapping and the set of solutions of the variational inequality for an inverse-strong monotone mapping as follows. For anα -inverse-strongly monotone mappingAofCtoHand a nonexpansive mappingSofCinto itself such thatFS∩VIC, A/∅,x1x∈C,{αn} ⊂0,1, and{λn} ⊂0,2α,

xn1αnx 1−αnSPCxn−λnAxn 1.3

for everyn ≥ 1. They proved that the sequence generated by 1.3 converges strongly to

PFS∩VIC,Axunder the conditions on{αn}and{λn} :λn ∈c, dfor somec, dwith 0< c <

d <2α,

lim

n→ ∞αn0,

∞

n1

αn<∞,

∞

n1

|αn1−αn|<∞,

∞

n1

|λn1−λn|<∞. 1.4

On the other hand, the viscosity approximation method of selecting a particular fixed point of a given nonexpansive mapping was proposed by Moudafi11. In 2004, in order to extend Theorem 2.2 of Moudafi11to a Banach space setting, Xu12considered the the following explicit iterative process. ForS:C → Cnonexpansive mappings,f∈_Candαn∈0,1,

xn1αnfxn 1−αnSxn, n≥1. 1.5

Moreover, in12, he also studied the strong convergence of{xn}generated by1.5asn →

∞in either a Hilbert space or a uniformly smooth Banach space and showed that the strong limn→ ∞xnis a solution of a certain variational inequality.

In 2007, Chen et al. 13considered the following iterative scheme as the viscosity approximation method of1.3. For anα-inverse-strongly-monotone mappingAofCtoH

and a nonexpansive mappingSofCinto itself such thatFS∩VIC, A/∅,f ∈ΣC,x0 ∈C, {αn} ⊂0,1, and{λn} ⊂0,2α,

xn1αnfxn 1−αnSPCxn−λnAxn, n≥0, 1.6

and showed that the sequence{xn}generated by1.6converges strongly to a point inFS∩

VIC, Aunder condition1.4on{αn}and{λn}, which is a solution of a certain variational

inequality.

mapping A of C to H and a nonexpansive mapping S of C into itself such that FS ∩

VIC, A/∅,f ∈ΣC,x0∈C,{αn} ⊂0,1, and{λn} ⊂0,2α,

ynαnfxn 1−αnSPCxn−λnAxn,

xn1

1−βn

ynβnSPC

yn−λnAyn

, n≥0.

1.7

If βn 0, then the iterative scheme 1.7 reduces to the iterative scheme 1.6. Under

condition 1.4 on the sequences {αn} and {λn} and appropriate condition on sequence

{βn}, we show that the sequence {xn}generated by 1.7 converges strongly to a point in

FS∩VIC, A, which is a solution of a certain variational inequality. Using this result, we also obtain a strong convergence result for finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping. Moreover, we investigate the problem of finding a common point of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse-strongly monotone mapping. The main results develop and improve the corresponding results of Chen et al. 13 and Iiduka and Takahashi10. We point out that the iterative scheme1.7is a new approach for finding the fixed points of nonexpansive mappings and solutions of variational inequalities for monotone mappings.

2. Preliminaries and Lemmas

LetHbe a real Hilbert space with inner product·,·and norm · , andCa closed convex subset ofH. We write xn x to indicate that the sequence {xn} converges weakly tox.

xn → x implies that{xn} converges strongly to x. For every pointx ∈ H, there exists a

unique nearest point inC, denoted byPCx, such that

x−PCx≤x−y 2.1

for ally∈C.PCis called the metric projectionofHtoC. It is well known thatPC satisfies

x−y, PCx−PCy

≥PCx−PCy2 2.2

for everyx, y∈H. Moreover,PCxis characterized by the properties

x−PCx, PCx−y

≥0, 2.3

_x−y2 _≥

x−PCx2y−PCx2 2.4

for allx∈H, y∈C. In the context of the variational inequality problem, this implies that

We state some examples for inverse-strongly monotone mappings. IfA I−T, whereT is a nonexpansive mapping ofCinto itself andI is the identity mapping ofH, thenAis 1/ 2-inverse-strongly monotone and VIC, A FT. A mappingAofCintoHis called strongly monotoneif there exists a positive real numberηsuch that

x−y, Ax−Ay≥ηx−y2 2.6

for all x, y ∈ C. In such a case, we say that A isη-strongly monotone. IfA isη-strongly monotone andκ-Lipschitz continuous, that is,Ax−Ay ≤κx−yfor allx, y∈C, thenAis

η/κ2_{-inverse-strongly monotone.}

IfAis anα-inverse-strongly monotone mapping ofCintoH, then it is obvious thatA

is 1/α-Lipschitz continuous. We also have that for allx, y∈Candλ >0,

I−λAx−I−λAy2x−y−λAx−Ay2

x−y2−2λx−y, Ax−Ayλ2Ax−Ay2

≤x−y2λλ−2αAx−Ay2.

2.7

So, ifλ≤2α, thenI−λAis a nonexpansive mapping ofCintoH. The following result for the existence of solutions of the variational inequality problem for inverse-strongly monotone mappings was given in Takahashi and Toyoda14.

Proposition 2.1. LetCbe a bounded closed convex subset of a real Hilbert space and A anα -inverse-strongly monotone mapping ofCintoH. Then, VIC, Ais nonempty.

A set-valued mappingT :H → 2H_{is called monotoneif for allx, y∈H,f ∈Txand}

g ∈Tyimplyx−y, f−g ≥0. A monotone mappingT :H → 2H_{is maximalif the graph}

GTofTis not properly contained in the graph of any other monotone mapping. It is known that a monotone mappingT is maximal if and only if forx, f ∈H×H,x−y, f −g ≥0 for everyy, g ∈GTimpliesf ∈Tx. LetAbe an inverse-strongly monotone mapping of

CintoHand letNCvbe the normal conetoCatv, that is,NCv {w ∈H : v−u, w ≥

0, for all u∈C}, and define

Tv

⎧ ⎨ ⎩

AvNCv, v∈C,

∅, v /∈C.

2.8

ThenT is maximal monotone and 0∈Tvif and only ifv∈VIC, A; see15,16. We need the following lemmas for the proof of our main results.

Lemma 2.2see17. Let{sn}be a sequence of nonnegative real numbers satisfying

where{λn}and{βn}satisfy the following conditions:

i{λn} ⊂0,1and∞n0λn∞or, equivalently,∞n01−λn 0;

iilim sup_{n→ ∞}βn/λn≤0or∞n0|βn|<∞.

Thenlimn→ ∞sn0.

Lemma 2.3see 1, demiclosedness principle. Let H be a real Hilbert space,C a nonempty closed convex subset ofH, andT : C → Ea nonexpansive mapping. Then the mappingI −T is demiclosed onC, whereI is the identity mapping; that is,xn xinEandI−Txn → yimply

thatx∈CandI−Txy.

Lemma 2.4. In a real Hilbert spaceH, there holds the following inequality:

_xy2_≤

x22y, xy, 2.10

for allx, y∈H.

3. Main Results

In this section, we introduce a new composite iterative scheme for nonexpansive mappings and inverse-strongly monotone mappings and prove a strong convergence of this scheme.

Theorem 3.1. Let Cbe a closed convex subset of a real Hilbert space H. Let A be an α -inverse-strongly monotone mapping of Cto H and S a nonexpansive mapping of Cinto itself such that

FS∩VIC, A/∅, andf ∈ΣC. Let{xn}be a sequence generated by

x0∈C,

ynαnfxn 1−αnSPCxn−λnAxn,

xn1

1−βn

ynβnSPC

yn−λnAyn

, n≥0,

3.1

where{λn} ⊂0,2α,{αn} ⊂0,1, and{βn} ⊂0,1. If{αn},{λn}and{βn}satisfy the following

conditions:

ilimn→ ∞αn0;

_∞

n0αn∞;

iiβn⊂0, afor alln≥0and for somea∈0,1;

iiiλn∈c, dfor somec, dwith0< c < d <2α;

iv∞_n0|αn1−αn|<∞;∞n0|βn1−βn|<∞;∞n0|λn1−λn|<∞,

then{xn}converges strongly toq∈FS∩VIC, A, which is a solution of the following variational

inequality:

Proof. Letzn PCxn−λnAxnandwn PCyn−λnAynfor everyn ≥ 0. Letu ∈ FS∩

VIC, A. SinceI−λnAis nonexpansive anduPCu−λnAufrom2.5, we have

zn−uPCxn−λnAxn−PCu−λnAu

≤ xn−λnAxn−u−λnAu

≤ xn−u.

3.3

Similarly we havewn−u ≤ yn−u.

We divide the proof into several steps.

Step 1. We show that{xn}is bounded. In fact, since

yn−uαn

fxn−u

1−αnSzn−u

≤αnfxn−u 1−αnzn−u

≤αnfxn−fuαnfu−u 1−αnxn−u

≤αnkxn−u 1−αnxn−uαnfu−u

1−1−kαnxn−uαnfu−u

≤max

xn−u, 1

1−kfu−u

,

3.4

we have

xn1−u1−βn yn−u

βnSwn−u

≤1−βnyn−uβnwn−u

≤1−βnyn−uβnyn−u

≤max

xn−u, 1

1−kfu−u

.

3.5

By induction, we get

xn−u ≤max

x0−u,

1

1−kfu−u

, n≥0. 3.6

This implies that{xn}is bounded and so{yn},{zn},{wn},{Axn}, and{Ayn}are bounded.

Moreover, sinceSzn−u ≤ xn−uandSwn−u ≤ yn−u,{Szn}and{Swn}are also

bounded. By conditioni, we also obtain

Step 2. We show that limn→ ∞xn1−xn0. From3.1, we have

ynαnfxn 1−αnSzn,

yn−1αn−1fxn−1 1−αn−1Szn−1, n≥1.

3.8

Simple calculations show that

yn−yn−1 1−αnSzn−Szn−1 αn−αn−1fxn−1−Szn−1

αn

fxn−fxn−1. 3.9

Since

zn−zn−1 ≤ xn−λnAxn−xn−1−λn−1Axn−1

≤ xn−λnAxn−xn−1−λnAxn−1|λn−1−λn|Axn−1

≤ xn−xn−1|λn−1−λn|Axn−1

3.10

for everyn≥1,we have

yn−yn−1≤1−αnzn−zn−1|αn−αn−1|fxn−1−Szn−1αnkxn−xn−1

≤1−αnxn−xn−1|λn−1−λn|Axn−1 |αn−αn−1|fxn−1−Szn−1αnkxn−xn−1

≤1−1−kαnxn−xn−1L1|λn−1−λn|M1|αn−αn−1|

3.11

for everyn≥1, whereM1sup{fxn−Szn−1:n≥1}andL1sup{Axn:n≥0}.

On the other hand, from3.1we have

xn1

1−βn

ynβnSwn,

xn

1−βn−1

yn−1βn−1Swn−1.

3.12

Also, simple calculations show that

xn1−xn

1−βn

yn−yn−1

βnSwn−Swn−1 βn−βn−1

Swn−1−yn−1

. 3.13

Since

wn−wn−1 ≤yn−λnAyn

−yn−1−λn−1Ayn−1 ≤yn−λnAyn

−yn−1−λnAyn−1|λn−1−λn|Ayn−1 ≤yn−yn−1|λn−1−λn|Ayn−1

for everyn≥1,it follows that

xn1−xn ≤

1−βnyn−yn−1βnwn−wn−1βn−βn−1Swn−1−yn−1 ≤1−βnyn−yn−1βnyn−yn−1|λn−1−λn|Ayn−1

βn−βn−1Swn−1−yn−1

≤yn−yn−1|λn−1−λn|Ayn−1βn−βn−1Swn−1−yn−1.

3.15

Substituting3.11into3.15, we derive

xn1−xn ≤1−1−kαnxn−xn−1L1|λn−1−λn|M1|αn−αn−1| |λn−1−λn|Ayn−1βn−βn−1Swn−1−yn−1

≤1−1−kαnxn−xn−1L2|λn−1−λn|M1|αn−αn−1|M2βn−βn−1, 3.16

whereL2 sup{L1Ayn:n≥1}andM2 sup{Swn−yn:n≥0}. From conditionsi

andiv, it is easy to see that

lim

n→ ∞1−kαn 0,

∞

n0

1−kαn ∞,

∞

n0

M1|αn1−αn|M2βn1−βnL2|λn1−λn|

<∞.

3.17

ApplyingLemma 2.2to3.16, we have

xn1−xn −→0 as n−→ ∞. 3.18

By3.11, we also have thatyn1−yn → 0 asn → ∞.

Step 3. We show that limn→ ∞xn−yn0 and limn→ ∞xn−Szn0. Indeed, it follows that

xn1−ynβnSwn−yn

≤βn

Swn−SznSzn−yn

≤awn−znSzn−yn

≤ayn−xnSzn−yn

≤ayn−xn1xn1−xnSzn−yn,

3.19

which implies that

_xn1−yn≤ a

1−a

Obviously, by3.7andStep 2, we havexn1−yn → 0 asn → ∞. This implies that

xn−yn≤ xn−xn1xn1−yn−→0 as n−→ ∞. 3.21

By3.7and3.21, we also have

xn−Szn ≤xn−ynyn−Szn−→0 as n−→ ∞. 3.22

Step 4. We show that limn→ ∞xn−zn 0. To this end, letu ∈ FS∩VIC, A. Then, by

convexity of · 2_{, we have}

yn−u2αnfxn 1−αnSzn−u2

≤αnfxn−u2 1−αnSzn−u2

≤αnfxn−u2 1−αnzn−u2

≤αnfxn−u2 1−αn

xn−u2λnλn−2αAxn−Au2

≤αnfxn−u2xn−u2 1−αncd−2αAxn−Au2.

3.23

So we obtain

−1−αncd−2αAxn−Au2

≤αnfxn−u2

xn−uyn−uxn−u −yn−u

≤αnfxn−u2

xn−uyn−uxn−yn.

3.24

Sinceαn → 0 andxn−yn → 0 by conditioniand3.21, we haveAxn−Au → 0n →

∞. Moreover, from2.2we obtain

zn−u2PCxn−λnAxn−PCu−λnAu2

≤ xn−λnAxn−u−λnAu, zn−u

1

2

xn−λnAxn−u−λnAu2zn−u2

−xn−λnAxn−u−λnAu−zn−u2

≤ 1

2

xn−u2zn−u2− xn−zn2

2λnxn−zn, Axn−Au −λ2nAxn−Au2

,

and so

zn−u2≤ xn−u2− xn−zn22λnxn−zn, Axn−Au −λ2nAxn−Au2. 3.26

And hence

yn−u2≤αnfxn−u2 1−αnzn−u2

≤αnfxn−u2xn−u2−1−αnxn−zn2

21−αnλnxn−zn, Axn−Au −1−αnλn2Axn−Au2.

3.27

Then we have

1−αnxn−zn2≤αnfxn−u2

xn−uyn−uxn−u −yn−u

21−αnλnxn−zn, Axn−Au −1−αnλ2nAxn−Au2

≤αnfxn−u2

xn−uyn−uxn−yn

21−αnλnxn−zn, Axn−Au −1−αnλn2Axn−Au2.

3.28

Sinceαn → 0,xn−yn → 0 andAxn−Au → 0, we getxn−zn → 0. Also by3.21, we

have

_{yn−zn≤yn−xnxn−zn −→0 n−→ ∞. 3.29}

Step 5. We show that lim sup_{n→ ∞}fq−q, yn−q ≤ 0 forq∈FS∩VIC, A, whereqis a

solution of the variational inequality

I−fq, q−p≤0, p∈FS∩VIC, A. 3.30

To this end, choose a subsequence{zni}of{zn}such that

lim sup

n→ ∞

fq−q, Szn−q

lim

i→ ∞

fq−q, Szni−q

. _3.31

Since{zni}is bounded, there exists a subsequence{znij}of{zni}which converges weakly to

z. We may assume without loss of generality thatzni z. SinceSzni−zni ≤ Szni −xni

xni−zni → 0 by Steps4and5, we haveSzni z. Then we can obtainz∈FS∩VIC, A. Indeed, let us first show thatz∈VIC, A. Let

Tv

⎧ ⎨ ⎩

AvNCv, v∈C,

∅, v /∈C.

ThenT is maximal monotone. Letv, w∈GT. Sincew−Av∈NCvandzn∈C, we have

v−zn, w−Av ≥0. 3.33

On the other hand, fromzn PCxn−λnAxn, we havev−zn, zn−xn−λnAzn ≥ 0 and

hence

v−zn,

zn−xn

λn

Axn

≥0. 3.34

Therefore we have

v−zni, w ≥ v−zni, Av

≥ v−zni, Av −

v−zni,

zni −xni

λni

Axni

v−zni, Av−Axni−

zni−xni

λni

v−zni, Av−Azniv−zni, Azni−Axni

−

v−zni,

zni−xni

λni

≥ v−zni, Azni−Axni −

v−zni,

zni−xni

λni

.

3.35

Hence we havev−z, w ≥0 asi → ∞. SinceTis maximal monotone, we havez∈T−1_{0 and}

hencez∈VIC, A.

On the another hand, by Steps3and4,zn−Szn ≤ zn−xnxn−Szn → 0. So,

byLemma 2.3, we obtainz∈FSand hencez∈FS∩VIC, A. Then by3.30we have

lim sup

n→ ∞

fq−q, Szni−q

lim

i→ ∞

fq−q, Szni−q

fq−q, z−q

I−fq, q−z≤0.

3.36

Thus, from3.7we obtain

lim sup

n→ ∞ f

q−q, yn−q ≤lim sup

n→ ∞ f

q−q, yn−Sznlim sup

n→ ∞

fq−q, Szn−q

≤lim sup

n→ ∞

fq−qyn−Sznlim sup

n→ ∞

fq−q, Szn−q

≤0.

Step 6. We show that limn→ ∞xn−q0 forq∈FS∩VIC, A, whereqis a solution of the

variational inequality

I−fq, q−p≤0, p∈FS∩VIC, A. 3.38

Indeed, fromLemma 2.4, we have

xn1−q2≤yn−q2αn

fxn−q

1−αn

Szn−q2

≤1−αnSzn−q22αn

fxn−q, yn−q

≤1−αn2zn−q22αn

fxn−f

q, yn−q

2αn

fq−q, yn−q

≤1−αn2xn−q22αnkxn−qyn−q2αn

fq−q, yn−q

≤1−αn2xn−q22αnkxn−qyn−xnxn−q

2αn

fq−q, yn−q

1−21−kαnxn−q22αnkyn−xnxn−q2αn

fq−q, yn−q

≤1−αnxn−q2αnβn,

3.39

where

αn 21−kαn, βn

kB

1−kyn−xn

1 1−k

fq−q, yn−q

, 3.40

andBsup{xn−q:n≥0}. It is easily seen thatαn → 0,

_∞

n1αn∞, and lim supn→ ∞βn≤

0. Thus byLemma 2.2, we obtainxn → q. This completes the proof.

Remark 3.2. 1Theorem 3.1improves the corresponding results in Chen et al.13and Iiduka and Takahashi10. In particular, ifβn0 andfxn xis constant in3.1, thenTheorem 3.1

reduces to Theorem 3.1 of Iiduka and Takahashi10.

2We obtain a new composite iterative scheme for a nonexpansive mapping ifA0 inTheorem 3.1as followssee also Jung18:

x0∈C,

ynαnfxn 1−αnSxn,

xn1

1−βn

ynβnSyn, n≥0.

3.41

Corollary 3.3. Let Cbe a closed convex subset of a real Hilbert space H. Let A be an α -inverse-strongly monotone mapping ofCtoHsuch thatV IC, A/∅, andf ∈ΣC. Let{xn}be a sequence

generated by

x0∈C,

ynαnfxn 1−αnPCxn−λnAxn,

xn1

1−βn

ynβnPC

yn−λnAyn

, n≥0,

3.42

where{λn} ⊂0,2α,{αn} ⊂0,1, and{βn} ⊂0,1. If{αn},{λn}, and{βn}satisfy the following

conditions:

ilimn→ ∞αn0;

_∞

n0αn∞,

iiβn⊂0, afor alln≥0and for somea∈0,1,

iiiλn∈c, dfor somec, dwith0< c < d <2α,

iv∞_n0|αn1−αn|<∞;∞n0|βn1−βn|<∞;∞n0|λn1−λn|<∞,

then {xn} converges strongly to q ∈ VIC, A, which is a solution of the following variational

inequality:

I−fq, q−p≤0, p∈VIC, A. 3.43

4. Applications

In this section, as in10,13, we obtain two theorems in a Hilbert space by usingTheorem 3.1. A mappingT :C → Cis called strictly pseudocontractiveif there existsαwith 0≤α <1 such that

Tx−Ty2≤x−y2αI−Tx−I−Ty2 4.1

for everyx, y ∈ C. Ifα 0, thenT is nonexpansive. PutA I−T, whereT :C → Cis a strictly pseudocontractive mapping withα. ThenAis1−α/2-inverse-strongly monotone; see7. Actually, we have, for allx, y∈C,

I−Ax−I−Ay2≤x−y2αAx−Ay2. 4.2

On the other hand, sinceHis a real Hilbert space, we have

I−Ax−I−Ay2x−y2AAx−Ay2−2x−y, Ax−Ay. 4.3

Hence we have

x−y, Ax−Ay≥ 1−α

2 Ax−Ay

2

UsingTheorem 3.1, we first get a strong convergence theorem for finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping.

Theorem 4.1. Let C be a closed convex subset of a real Hilbert space H. LetT be an α-strictly pseudocontractive mapping ofCinto itself andSa nonexpansive mapping ofCinto itself such that

FS∩FT/∅, andf∈ΣC. Let{xn}be a sequence generated by

x0∈C,

ynαnfxn 1−αnS1−λnxnλnTxn,

xn1

1−βn

ynβnS

1−λnynλnTyn

, n≥0,

4.5

where {λn} ⊂ 0,1−α,{αn} ⊂ 0,1, and{βn} ⊂ 0,1. If{αn}, {λn}, and{βn} satisfy the

conditions:

ilimn→ ∞αn0;

_∞

n0αn∞,

iiβn⊂0, afor alln≥0and for somea∈0,1,

iiiλn∈c, dfor somec, dwith0< c < d <1−α,

iv∞_n0|αn1−αn|<∞;

_∞

n0|βn1−βn|<∞;

_∞

n0|λn1−λn|<∞,

then{xn} converges strongly toq ∈ FS∩FT, which is a solution of the following variational

inequality:

I−fq, q−p≤0, p∈FS∩FT. 4.6

Proof. PutAI−T. ThenAis1−α/2-inverse-strongly monotone. We haveFT VIC, A

andPCxn−λnAxn 1−λnxnλnTxn. Thus, the desired result follows fromTheorem 3.1.

UsingTheorem 3.1, we also have the following result.

Theorem 4.2. LetHbe a real Hilbert spaceH. LetAbe anα-inverse-strongly monotone mapping of

Hinto itself andSa nonexpansive mapping ofHinto itself such thatFS∩A−1_{0/∅, andf∈Σ}

C.

Let{xn}be a sequence generated by

x0∈H,

ynαnfxn 1−αnSxn−λnAxn,

xn1

1−βn

ynβnS

yn−λnAyn

, n≥0,

4.7

where{λn} ⊂0,2α,{αn} ⊂0,1, and{βn} ⊂0,1. If{αn},{λn}, and{βn}satisfy the conditions:

ilimn→ ∞αn0;

_∞

n0αn∞,

iiβn⊂0, afor alln≥0and for somea∈0,1,

iiiλn∈c, dfor somec, dwith0< c < d <2α,

then{xn} converges strongly toq ∈ FS∩A−10, which is a solution of the following variational

inequality:

I−fq, q−p≤0, p∈FS∩A−1_{0. 4.8}

Proof. We haveA−1_{0 V IH, A. So, puttingP}

H I, byTheorem 3.1, we obtain the desired

result.

Remark 4.3. Ifβn 0 in Theorems4.1and4.2, then Theorems4.1and4.2reduce to Chen et

al.13, Theorems 4.1 and 4.2. Theorems4.1and4.2also extend in Iiduka and Takahashi10, Theorems 4.1 and 4.2to the viscosity methods in composite iterative schemes.

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea NRF funded by the Ministry of Education, Science and Technology2009-0064444. The author thanks the referees for their valuable comments and suggestions, which improved the presentation of this paper.

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