Hybrid Steepest-Descent Methods for Solving Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators

Fixed Point Theory and Applications Volume 2010, Article ID 673932,16pages doi:10.1155/2010/673932

Research Article

Hybrid Steepest-Descent Methods for Solving Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators

Songnian He and Xiao-Lan Liang

College of Science, Civil Aviation University of China, Tianjin 300300, China

Correspondence should be addressed to Songnian He,[email protected] Received 30 September 2009; Accepted 13 January 2010

Academic Editor: Tomonari Suzuki

Copyrightq2010 S. He and X.-L. Liang. 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.

LetH be a real Hilbert space and letF : H → H be a boundedly Lipschitzian and strongly monotone operator. We design three hybrid steepest descent algorithms for solving variational inequality VIC, Fof finding a pointx^∗ ∈Csuch thatFx^∗, x−x^∗ ≥ 0, for allx∈C, whereC is the set of fixed points of a strict pseudocontraction, or the set of common fixed points of finite strict pseudocontractions. Strong convergence of the algorithms is proved.

1. Introduction

Let H be a real Hilbert space with the inner product ·,·and the norm · , let C be a nonempty closed convex subset ofH,and letF:C → Hbe a nonlinear operator. We consider the problem of finding a pointx^∗∈Csuch that

Fx^∗, x−x^∗ ≥0, ∀x∈C. 1.1

This is known as the variational inequality problemi.e., VIC, F,initially introduced and studied by Stampacchia1in 1964. In the recent years, variational inequality problems have been extended to study a large variety of problems arising in structural analysis, economics, optimization, operations research, and engineering sciences; see 1–6 and the references therein.

Yamada7proposed hybrid methods to solve VIC, F, whereCis composed of fixed points of a nonexpansive mapping; that is,Cis of the form

C≡FixT:{x∈H:Txx}, 1.2

whereT :H → His a nonexpansive mappingi.e.,Tx−Ty ≤ x−yfor allx, y ∈H, F:H → His Lipschitzian and strongly monotone.

He and Xu8proved that VIC, Fhas a unique solution and iterative algorithms can be devised to approximate this solution ifF is a boundedly Lipschitzian and strongly monotone operator and Cis a closed convex subset of H. In the case where C is the set of fixed points of a nonexpansive mapping, they invented a hybrid iterative algorithm to approximate the unique solution of VIC, Fand this extended the Yamada’s results.

The main purpose of this paper is to continue our research in8. We assume thatF is a boundedly Lipschitzian and strongly monotone operator as in8, but Cis the set of fixed points of a strict pseudo-contractionT : H → H, or the set of common fixed points of finite strict pseudo-contractions Ti : H → H i 1, . . . , N. For the two cases of C, we will design the hybrid iterative algorithms for solving VIC, F and prove their strong convergence, respectively. Relative definitions are stated as below.

LetCbe a nonempty closed and convex subset of a real Hilbert spaceH,F :C → H andT :C → C, then

1Fis called Lipschitzian onC, if there there exists a positive constantLsuch that Fx−Fy≤Lx−y, ∀x, y∈C; 1.3 2Fis called boundedly Lipschitzian onC, if for each nonempty bounded subsetBof

C, there exists a positive constantκ_Bdepending only on the setBsuch that

Fx−Fy≤κ_Bx−y, ∀x, y∈B; 1.4 3F is said to beη-strongly monotone onC, if there exists a positive constantη >0

such that

Fx−Fy, x−y ≥ηx−y², ∀x, y∈C; 1.5 4T is said to be aκ-strict pseudo-contraction if there exists a constantκ∈0,1such

that

Tx−Ty²≤x−y²κI−Tx−I−Ty², ∀x, y∈C. 1.6 Obviously, the nonexpansive mapping class is a proper subclass of the strict pseudo- contraction class and the Lipschitzian operator class is a proper subclass of the boundedly Lipschitzian operator class, respectively.

We will use the following notations:

ifor weak convergence and → for strong convergence, iiωwxn {x:∃xnj x}denotes the weakω-limit set of{xn},

iiiSu, r {x:x∈H,x−u ≤r}denotes a closed ball with centeruand radiusr.

2. Preliminaries

We need some facts and tools which are listed as lemmas below.

Lemma 2.1. LetHbe a real Hilbert space. The following expressions hold:

itx1−ty²tx²1−ty²−t1−tx−y², for allx, y∈H, for all t∈0,1.

iixy²≤ x²2y, xy, for all x, y∈H.

Lemma 2.2see9. Assume that{an} is a sequence of nonnegtive real numbers satisfying the property

a_n1 ≤ 1−γ_n

a_nγ_nσ_n, n0,1,2. . . . 2.1

If{γ_n}^∞_n0⊂0,1and{σ_n}^∞_n0satisfy the following conditions:

ilim_{n→ ∞}γ_n0, ii_∞

n1γn∞,

iiilim sup_{n→ ∞}σn≤0, or_∞

n1|γnσn|<∞, then lim_{n→ ∞}a_n0.

Lemma 2.3 see10. LetC be a nonempty closed convex subset of a real Hilbert space H and T : C → Cis a nonexpansive mapping. If a one has sequence{x_n}in Csuch thatx_n zand I−Tx_n → 0,thenzTz.

Lemma 2.4see11. LetCbe a nonempty closed convex subset of a real Hilbert spaceH, ifT : C → Cis aκ-strict pseudo-contraction, then the mappingI−T is demiclosed at 0. That is, if{x_n}is a sequence inCsuch thatx_nxandI−Tx_n → 0,thenI−Tx0.

Lemma 2.5see8. Assume thatCis a nonempty closed convex subset of a real Hilbert spaceH, F : C → H,if F is boundedly Lipschitzian andη-strongly monotone, then variational inequality 1.1has a unique solution.

Lemma 2.6. Assume thatT : H → His aκ-strict pseudo-contraction, and the constantαsatisfies κ≤α <1.Let

T_ααI 1−αT, 2.2

thenT_αis nonexpansive and FixT_α FixT.

Proof. UsingLemma 2.1iand the conception ofκ-strict pseudo-contraction, we get T_αx−T_αy²αx−y 1−αTx−Ty²

αx−y² 1−αTx−Ty²−α1−αI−Tx−I−Ty²

≤αx−y² 1−αx−y²κI−Tx−I−Ty²

−α1−αI−Tx−I−Ty²

x−y²−α−κ1−αI−Tx−I−Ty²

≤x−y², ∀x, y∈H,

2.3

soT_αis nonexpansive. FixT_α FixTis obvious.

Lemma 2.7. Assume thatH is a real Hilbert space,T : H → His aκ-strict pseudo-contraction such that FixT/∅,and F:H → His a boundedly Lipschitzian andη-strongly monotone operator.

Takex0 ∈FixTarbitrarily and setCSx0,2Fx0/η. Denote byLthe Lipschitz constant ofF onCand let

T^α,λ

I−μλF

Tα, 2.4

where the constantsμandλare such that 0< μ < η/L²and 0< λ <1, respectively, andTαis defined as inLemma 2.6above. ThenT^α,λrestricted toCis a contraction.

Proof. Ifx∈C, that is,x−x0 ≤2Fx0/η, byLemma 2.6, we have

Tαx−x0Tαx−Tαx0 ≤ x−x0 ≤ 2Fx0

η . 2.5

It suggests that Tαx ∈ C. Since F is Lipschitzian and η-strongly monotone on C, using Lemma 2.6, we obtain

T^α,λx−T^α,λy²I−μλF T_αx−

I−μλF T_αy² T_αx−T_αy

−μλ

FT_αx−FT_αy² Tαx−Tαy²μ²λ²FTαx−FTαy²

−2μλ

T_αx−T_αy, FT_αx−FT_αy

≤T_αx−T_αy²μ²λ²L²T_αx−T_αy²

−2μληT_αx−T_αy²

1μ²λ²L²−2μληT_αx−T_αy²

≤1−τλ²x−y², ∀x, y∈C.

2.6

Therefore,T^α,λrestricted to thatCis a contraction with coefficient 1−τλ, whereτ 1/2μ2η− μL².

Lemma 2.8see11. AssumeCis a closed convex subset of a Hilbert spaceH.

iGiven an integerN≥1, assume that for each 1≤i≤N,Ti:C → Cis aκi-strict pseudo- contraction for some 0≤κ_i<1. Assume{γ_i}^N_i1is a positive sequence such that_N

i1γ_i1.

ThenT _N

i1γiTiis aκ-strict pseudo-contraction, withκmax{κi: 1≤i≤N}.

iiLet{T_i}^N_i1,{γ_i}^N_i1,andTbe given as in (i) above. Suppose that_N

i1FixT_i/∅, then

FixT ^N

i1

FixT_i. 2.7

Lemma 2.9. Assume thatTi :H → His aκi-strict pseudo-contraction for some 0≤ κi <1 1 ≤ i≤N,letTαi αiI 1−αiTi, κi< αi<11≤i≤N,if_N

i1FixTi/∅, then

FixTα1Tα2· · ·TαN

N i1

FixTαi. 2.8

Proof. We prove it by induction. ForN 2, setTα1 α1I 1−α1T1, Tα2 α2I 1−α2T2, κ_i< α_i<1, i1,2. Obviously

FixT_α1

FixT_α2⊂FixT_α1T_α2. 2.9

Now we prove

FixT_α1T_α2⊂FixT_α1

FixT_α2. 2.10

for allq ∈ FixTα1Tα2, Tα1Tα2q q, if Tα2q q, then Tα1q q,the conclusion holds. In fact, we can claim thatT_α2q q. From Lemma 2.6, we know thatT_α2 is nonexpansive and FixTα1

FixTα2 FixT1

FixT2/∅.Takep∈FixTα1

FixTα2, then

p−q²p−Tα1Tα2q²p−α1Tα2q 1−α1T1Tα2q² α1p−T_α2q 1−α1p−T1T_α2q²

α1p−Tα2q² 1−α1p−T1Tα2q²

−α11−α1Tα2q−T1Tα2q²

≤α1p−Tα2q² 1−α1p−Tα2q²κ1Tα2q−T1Tα2q²

−α11−α1T_α2q−T1T_α2q²

≤p−Tα2q²−α1−κ11−α1Tα2q−T1Tα2q²

≤p−q²−α1−κ11−α1T_α2q−T1T_α2q².

2.11

Sinceκ1< α1<1, we get

Tα2q−T1Tα2q²≤0, 2.12

Namely,T_α2qT1T_α2q,that is,

T_α2q∈FixT1 FixT_α1, T_α2qT_α1T_α2qq. 2.13

Suppose that the conclusion holds forNk, we prove that

FixT_α1T_α2· · ·T_αk1 ^k1

i1

FixT_αi. 2.14

It suffices to verify

FixTα1Tα2· · ·Tαk1⊂^k1

i1

FixTαi 2.15

for allq ∈ FixTα1Tα2· · ·Tαk1, Tα1Tα2· · ·Tαk1q q. Using Lemma 2.6 again, take p ∈ _k1

i1 FixT_αi,

p−q² p−T_α1T_α2· · ·T_αk1q²

p−α1Tα2· · ·Tαk1q 1−α1T1Tα2· · ·Tαk1q² α1p−T_α2· · ·T_αk1q 1−α1p−T1T_α2· · ·T_αk1q² α1p−Tα2· · ·Tαk1q² 1−α1p−T1Tα2· · ·Tαk1q²

−α11−α1T_α2· · ·T_αk1q−T1T_α2· · ·T_αk1q²

≤α1p−Tα2· · ·Tαk1q²

1−α1p−T_α2· · ·T_αk1q²κ1T_α2· · ·T_αk1q−T1T_α2· · ·T_αk1q²

−α11−α1T_α2· · ·T_αk1q−T1T_α2· · ·T_αk1q²

≤p−q²−α1−κ11−α1T_α2· · ·T_αk1q−T1T_α2· · ·T_αk1q².

2.16

Sinceκ1< α1<1, we have

Tα2· · ·Tαk1q−T1Tα2· · ·Tαk1q²≤0, 2.17

this implies that

Tα2· · ·Tαk1q∈FixT1 FixTα1, 2.18

Namely,

Tα2· · ·Tαk1qTα1Tα2· · ·Tαk1qq. 2.19

From2.19and inductive assumption, we get q∈FixTα2· · ·Tαk1

k1

i2

FixTαi, 2.20

therefore

Tαiqq, i2,3, . . . , k1. 2.21 Substituting it into2.19, we obtainT_α1qq.Thus we assert that

q∈^k1

i1

FixT_αi. 2.22

3. Further Extension of Hybrid Iterative Algorithm

Yamada got the following result.

Theorem 3.1 see7. Assume thatH is a real Hilbert space, T : H → H is nonexpansive such that FixT/∅,andF : H → Hisη-strongly monotone andL-Lipschitzian. Fix a constant μ∈0,2η/L². Assume also that the sequence{λ_n} ⊂0,1satisfies the following conditions:

iλ_n → 0, n → ∞;

ii_∞

n0λn∞;

iii_∞

n0|λn1−λn|<∞, or limn→ ∞λn/λn1 1.

Takex0∈FixTarbitrarily and define{xn}by xn1T^λnxn

I−μλnF

Txn, 3.1

then{xn}converges strongly to the unique solution of VIFixT, F.

He and Xu 8 proved that VIC, F has a unique solution if F is a boundedly Lipschitzian and strongly monotone operator andCis a closed convex subset ofH. Using this result, they were able to relax the global Lipschitz condition on F in Theorem 3.1 to the weaker bounded Lipschitz condition and invented a hybrid iterative algorithm to approximate the unique solution of VIC, F. Their result extended the Yamada’s above theorem.

In this section, we mainly focus on further extension of our hybrid algorithm in8.

Consider VIC, F, whereCis composed of fixed points of aκ-strict pseudo-contractionT : H → H such that FixT/∅andF : H → H is stillη-strongly monotone and boundedly Lipschitzian. Fix a pointx0 ∈ FixTarbitrarily, setC Sx0,2Fx0/η. Denote byL the Lipschitz constant ofFonC. Fix the constantμsatisfying 0< μ < η/L². Assume also that the sequences{αn}and{λn}satisfyκ≤αn≤α <1 for a constantα∈0,1and 0< λn<1n≥0, respectively. LetTαn αnI 1−αnTandT^αn,λn I−μλnFTαn, define{xn}by the scheme:

xn1T^αn,λnxn

I−μλnF

Tαnxn, n≥0. 3.2

We have the following result.

Theorem 3.2. If the sequences{λn}and{αn}satisfy the following conditions:

iλ_n → 0 n → ∞;

ii _∞

n0λ_n∞;

iii_∞

n0|λ_n1−λ_n|< ∞,_∞

n0|α_n1−α_n| <∞, or lim_{n→ ∞}λ_n−1/λ_n 1, lim_{n→ ∞}|α_n− α_n−1|/λ_n 0,

then{x_n}generated by3.2converges strongly to the unique solutionx^∗of VIFixT, F.

Proof. We prove thatxn ∈Cfor alln≥ 0 by induction. It is trivial thatx0 ∈ C. Suppose we have provedxn ∈C, that is,

xn−x0 ≤ 2Fx0

η . 3.3

UsingLemma 2.7, We then derive from3.2and3.3that

x_n1−x0T^αn,λnx_n−x0

≤T^αn,λnx_n−T^αn,λnx0T^αn,λnx0−x0

≤1−τλnxn−x0μλnFx0 1−τλnxn−x0τλnμ

τFx0

≤max

x_n−x0,μ τFx0

≤max 2

η,μ τ

Fx0.

3.4

However, since 0< μ < η/L²andτ 1/2μ2η−μL²,we get

μ

τ μ

1/2μ

2η−μL² 2 η

η−μL² ≤ 2

η. 3.5

This together with3.4implies that

xn1−x0 ≤ 2Fx0

η . 3.6

It proves thatxn1∈C. Therefore,xn∈Cfor alln≥0. Thus{xn}is bounded. It is not difficult to verify that the sequences{Tx_n}and{FT_αnx_n}are all bounded.

By3.2andLemma 2.7, we have xn1−xnT^αn,λnxn−T^{αn−1,λn−1}xn−1

≤T^αn,λnxn−T^αn,λnxn−1T^αn,λnxn−1−T^αn−1,λnxn−1 T^αn−1,λnxn−1−T^{αn−1,λn−1}xn−1

≤1−τλ_nx_n−x_n−1 1−τλ_nT_αnx_n−1−T_αn−1x_n−1 μ|λ_n−λ_n−1|FT_αn−1x_n−1

≤1−τλnxn−xn−1 1−τλn|αn−αn−1|xn−1Txn−1 μ|λ_n−λ_n−1|FT_αn−1x_n−1

≤1−τλnxn−xn−1M|αn−αn−1||λn−λn−1|,

3.7

where M sup_nFTαnxn,xn,Txn < ∞. By Lemma 2.2and conditions i–iii, we conclude that

xn1−xn −→0 n−→ ∞. 3.8

Sinceλ_n → 0, it is straitforward from3.2that

xn1−TαnxnμλnFTαnxn −→0 n−→ ∞. 3.9

On the other hand

xn1−Tαnxnxn1−αnxn 1−αnTxn x_n1−x_n 1−α_nx_n−Tx_n

≥1−α_nx_n−Tx_n − x_n1−x_n.

3.10

By the conditionαn≤α <1 and3.8–3.10, we obtain xn−Txn ≤ 1

1−α_nxn1−Tαnxnxn1−xn

≤ 1

1−αxn1−Tαnxnxn1−xn−→0 n−→ ∞.

3.11

ByLemma 2.4and3.11, we obtain

ωwxn⊂FixT. 3.12

Lemma 2.5asserts that VIFixT, Fhas a unique solutionx^∗ ∈FixT. Now we prove that x_n−x^∗ → 0n → ∞. ByLemma 2.1ii,3.2, andLemma 2.7, we have

xn1−x^∗2T^αn,λnxn−x^∗2

T^αn,λnxn−T^αn,λnx^∗ T^αn,λnx^∗−x^∗2

≤T^αn,λnx_n−T^αn,λnx^∗22T^αn,λnx^∗−x^∗, x_n1−x^∗

≤1−τλnxn−x^∗22μλn−Fx^∗, xn1−x^∗.

3.13

Let us show that

lim sup

n→ ∞ −Fx^∗, xn−x^∗ ≤0. 3.14

In fact, there exists a subsequence{x_nj} ⊂ {x_n}such that lim sup

n→ ∞ −Fx^∗, x_n−x^∗ lim

j→ ∞−Fx^∗, x_nj−x^∗. 3.15

Without loss of generality, we may further assume that x_nj x ∈ FixT. Since x^∗ is the unique solution of VIFixT, F, we obtain

lim sup

n→ ∞ −Fx^∗, xn−x^∗ lim

j→ ∞−Fx^∗, xnj−x^∗−Fx^∗,x−x^∗ ≤0. 3.16 Finally conditionsi–iiiand 3.14allow us to applyLemma 2.2to the relation3.13to conclude that lim_{n→ ∞}x_n−x^∗0.

4. Parallel Algorithm and Cyclic Algorithm

In this section, we discuss the parallel algorithm and the cyclic algorithm, respectively, for solving the variational inequality over the set of the common fixed points of finite strict pseudo-contractions.

LetHbe a real Hilbert space andF :H → Haη-strongly monotone and boundedly Lipschitzian operator. Let N be a positive integer and T_i : H → H a κ_i-strict pseudo- contraction for some κ_i ∈ 0,1 i 1, . . . , N such that _N

i1FixT_i/∅. We consider the problem of findingx^∗∈_N

i1FixTisuch that

Fx^∗, x−x^∗ ≥0, ∀x∈^N

i1

FixT_i. 4.1

Since_N

i1FixT_iis a nonempty closed convex subset ofH, VI4.1has a unique solution.

Throughout this section,x0 ∈ _N

i1FixT_iis an arbitrary fixed point,C Sx0,2Fx0/η,

L is the Lipschitz constant ofF onC, the fixed constantμsatisfies 0 < μ < η/L², and the sequence{λ_n}belongs to0,1.

Firstly we consider the parallel algorithm. Take a positive sequence{γi}^N_i1 such that _N

i1γi1 and let

T ^N

i1

γiTi. 4.2

By usingLemma 2.8, we assert thatT is aκ-strict pseudo-contraction withκmax{κi :i 1, . . . , N}and FixT _N

i1FixT_iholds. Thus VI4.1is equivalent to VIFixT, Fand we can use scheme3.2to solve VI4.1. In fact, takingT _N

i1γ_iT_iin the scheme3.2, we get the so-called parallel algorithm

xn1T^αn,λnxn

I−μλnF

Tαnxn n≥0. 4.3

UsingLemma 2.8and Thorem 3.2, the following conclusion can be deduced directly.

Theorem 4.1. Suppose that{α_n}and{λ_n}satisfy the same conditions as inTheorem 3.2. Then the sequence{x_n}generated by the parallel algorithm4.3converges strongly to the unique solutionx^∗ of VI4.1.

For eachi1, . . . , N,let

T_αi α_iI 1−α_iT_i, 4.4

where the constantα_isuch thatκ_i < α_i<1. Then we turn to defining the cyclic algorithm as follows:

x1T_α1x0−μλ0FT_α1x0, x2Tα2x1−μλ1FTα2x1,

. . .

xNTαNxN−1−μλN−1FTαNxN−1, xN1Tα1xN−μλNFTα1xN,

· · ·.

4.5

Indeed, the algorithm above can be rewritten as xn1Tαn1xn−μλnF

Tαn1xn

, 4.6

whereTαn αnI1−αnTn, TnTnmodN, namely,Tnis one ofT1, T2, . . . , TNcircularly.

For convenience, we denote4.6as

xn1T^αn1,λnxn. 4.7

We get the following result

Theorem 4.2. If{λ_n} ⊂0,1satisfies the following conditions:

iλ_n → 0, n → ∞;

ii_∞

n0λ_n∞;

iii_∞

n0|λ_nN−λ_n|<∞, or lim_{n→ ∞}λ_n/λ_nN 1,

then the sequence{x_n}generated by4.6converges strongly to the unique solutionx^∗ofV I4.1.

Proof. We break the proof process into six steps.

1xn∈C. We prove it by induction. Definitelyx0∈C. Supposexn∈C, that is,

x_n−x0 ≤ 2Fx0

η . 4.8

We have fromx0∈_N

i1FixTi,4.8, andLemma 2.7that x_n1−x0T^αn1,λnx_n−x0

≤T^αn1,λnx_n−T^αn1,λnx0T^αn1,λnx0−x0

≤1−τλ_nx_n−x0μλ_nFx0 1−τλ_nx_n−x0τλ_nμ

τFx0

≤max

xn−x0,μ τFx0

≤max 2

η,μ τ

Fx0,

4.9

whereτ 1/2μ2η−μL².Observing 0< μ < η/L², we get μ

τ μ

1/2μ

2η−μL² 2 η

η−μL² ≤ 2

η. 4.10

This together with4.9implies that

xn1−x0 ≤ 2Fx0

η . 4.11

It suggests thatxn1∈C. Therefore,xn∈Cfor alln≥0. We can also prove that the sequences {x_n},{T_αnx_n},{FT_αnx_n}are all bounded.

2xnN−xn → 0 n → ∞.By4.6andLemma 2.7, we have x_nN−x_nT^{αnN,λnN−1}x_nN−1−T^αn,λn−1x_n−1

≤T^{αnN,λnN−1}x_nN−1−T^{αnN,λnN−1}x_n−1 T^{αnN,λnN−1}x_n−1−T^αn,λn−1x_n−1

≤1−τλnN−1xnN−1−xn−1 μ|λnN−1−λn−1|FTαnxn−1

≤1−τλ_nN−1x_nN−1−x_n−1M|λ_nN−1−λ_n−1|,

4.12

whereMsup_nFT_αnx_n−1<∞.Since{λ_n}satisfiesi–iii, usingLemma 2.2, we get xnN−xn −→0 n−→ ∞. 4.13 3x_n−T_αnNT_αnN−1· · ·T_αn1x_n → 0n → ∞.By4.3andλ_n → 0, we have

x_n1−T_αn1x_nμλ_nFT_αn1x_n−→0 n−→ ∞. 4.14

Recursively,

x_nN−T_αnNx_nN−1−→0 n−→ ∞,

x_nN−1−T_αnN−1x_nN−2−→0 n−→ ∞. 4.15

ByLemma 2.6,T_αnNis nonexpansive, we obtain

TαnNxnN−1−TαnNTαnN−1xnN−2−→0 n−→ ∞, T_αnNT_αnN−1x_nN−2−T_αnNT_αnN−1T_αnN−2x_nN−3−→0 n−→ ∞,

· · ·

TαnN· · ·Tαn2xn1−TαnN· · ·Tαn1xn−→0 n−→ ∞.

4.16

Adding all the expressions above, we get

xnN−TαnNTαnN−1· · ·Tαn1xn−→0 n−→ ∞. 4.17 Using this together with the conclusion of step2, we obtain

xn−TαnNTαnN−1· · ·Tαn1xn−→0 n−→ ∞. 4.18

4 ωwxn ⊂ _N

i1FixTi. Assume that{xnj} ⊂ {xn} such that xnj x, we prove x∈_N

i1FixT_i. By the conclusion of step3, we get

x_nj−T_αnjNT_αnjN−1· · ·T_αnj1x_nj−→0,

j −→ ∞

. 4.19

Observe that, for each nj, TαnjNTαnjN−1· · ·Tαnj1 is some permutation of the mappings Tα1, Tα2, . . . , TαN, since Tα1, Tα2, . . . , TαN are finite, all the full permutation are N!, there must be some permutation that appears infinite times. Without loss of generality, suppose that this permutation isTα1Tα2· · ·TαN, we can take a subsequence{xnjk} ⊂ {xnj}such that

xnjk−Tα1Tα2· · ·TαNxnjk−→0 k−→ ∞. 4.20

It is easy to prove thatT_α1T_α2· · ·T_αN is nonexpansive. ByLemma 2.3, we get

xTα1Tα2· · ·TαNx. 4.21

Using Lemmas2.6and2.9, we obtain

x∈FixTα1Tα2· · ·TαN

N i1

FixTαi

N i1

FixTi. 4.22

5lim sup_{n→ ∞}−Fx^∗, xn−x^∗ ≤ 0.In fact, there exists a subsequence{xnj} ⊂ {xn} such that

lim sup

n→ ∞ −Fx^∗, x_n−x^∗ lim

j→ ∞−Fx^∗, x_nj−x^∗. 4.23

Without loss of generality, we may further assume thatx_nj x∈_N

i1FixT_i.Sincex^∗is the solution of VI4.1, we obtain

lim sup

n→ ∞ −Fx^∗, x_n−x^∗ lim

j→ ∞−Fx^∗, x_nj−x^∗−Fx^∗,x−x^∗ ≤0. 4.24 6xn → x^∗.By4.6, Lemmas2.1ii, and2.7, we obtain

xn1−x^∗2T^αn1,λnxn−x^∗2

T^αn1,λnxn−T^αn1,λnx^∗ T^αn1,λnx^∗−x^∗2

≤T^αn1,λnx_n−T^αn1,λnx^∗22T^αn1,λnx^∗−x^∗, x_n1−x^∗

≤1−τλnxn−x^∗22μλn−Fx^∗, xn1−x^∗.

4.25

From the conclusion of step5andLemma 2.2, we get

xn → x^∗ n → ∞. 4.26

Acknowledgment

This research is supported by the Fundamental Research Funds for the Central Universities GRANT:ZXH2009D021.

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