Convergence Theorems Concerning Hybrid Methods for Strict Pseudocontractions and Systems of Equilibrium Problems

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Volume 2010, Article ID 396080,14pages doi:10.1155/2010/396080

Research Article

Convergence Theorems Concerning Hybrid

Methods for Strict Pseudocontractions and

Systems of Equilibrium Problems

Peichao Duan

College of Science, Civil Aviation University of China, Tianjin 300300, China Correspondence should be addressed to Peichao Duan,[email protected]

Received 23 May 2010; Revised 23 August 2010; Accepted 26 August 2010

Academic Editor: S. Reich

Copyrightq2010 Peichao Duan. 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.

Let{Si}Ni1be N strict pseudo-contractions defined on a closed and convex subset C of a real

Hilbert space H. We consider the problem of finding a common element of fixed point set of these mappings and the solution set of a system of equilibrium problems by parallel and cyclic algorithms. In this paper, new iterative schemes are proposed for solving this problem. Furthermore, we prove that these schemes converge strongly by hybrid methods. The results presented in this paper improve and extend some well-known results in the literature.

1. Introduction

LetHbe a real Hilbert space with inner product·,·and norm · . LetCbe a nonempty, closed, and convex subset ofH.

Let{Fk}be a countable family of bifunctions fromC×CtoR, whereRis the set of

real numbers. Combettes and Hirstoaga1considered the following system of equilibrium problems:

findingx∈Csuch thatF_kx, y≥0, ∀k∈Γ, ∀y∈C, 1.1

whereΓis an arbitrary index set. IfΓis a singleton, then problem1.1becomes the following equilibrium problem:

findingx∈Csuch thatFx, y≥0, ∀y∈C. 1.2

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A mappingSofCis said to be aκ-strict pseudocontraction if there exists a constant

κ∈0,1such that

_Sx₋_Sy2_≤_x₋_y2_κ_I₋_S_x₋_I₋_S_y2 _1.3

for allx, y∈C; see2. We denote the fixed point set ofSbyFS, that is,FS {x∈

C:Sxx}.

Note that the class of strict pseudocontractions properly includes the class of nonexpansive mappings which are mappingSonCsuch that

_Sx₋_Sy_≤_x₋_y 1.4

for all x, y ∈ C. That is, S is nonexpansive if and only if S is a 0-strict pseudocontraction.

The problem 1.1 is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games, and others; see, for instance,1,3,4and the references therein. Some methods have been proposed to solve the equilibrium problem1.1; related work can also be found in5–8.

Recently, Acedo and Xu9considered the problem of finding a common fixed point of a finite family of strict pseudo-contractive mappings by the parallel and cyclic algorithms. Very recently, Duan and Zhao10considered new hybrid methods for equilibrium problems and strict pseudocontractions. In this paper, motivated by5, 8–12, applying parallel and cyclic algorithms, we obtain strong convergence theorems for finding a common element of the fixed point set of a finite family of strict pseudocontractions and the solution set of the system of equilibrium problems1.1by the hybrid methods.

We will use the following notations:

1for the weak convergence and → for the strong convergence,

2ωwxn {x:∃xnj x}denotes the weakω-limit set of{xn}.

2. Preliminaries

We will use the facts and tools in a real Hilbert spaceHwhich are listed below.

Lemma 2.1. LetHbe a real Hilbert space. Then the following identities hold: ix−y2x2− y2−2x−y, y, for allx, y∈H,

iitx1−ty2tx21−ty2−t1−tx−y2, for allt∈0,1, for allx, y∈H.

Lemma 2.2see6. LetHbe a real Hilbert space. Given a nonempty, closed, and convex subset

C⊂H, pointsx, y, z∈H, and a real numbera∈R, then the set

v∈C:y−v2≤ x−v2z, va 2.1

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Recall that given a nonempty, closed, and convex subsetCof a real Hilbert spaceH, for anyx∈, there exists the unique nearest point inC, denoted byP_Cx, such that

x−PCx ≤x−y 2.2

for ally∈C. Such aP_C is called the metricor the nearest pointprojection ofHontoC. As we all knowyPCxif and only if there holds the relation

x−y, y−z≥0 ∀z∈C. 2.3

Lemma 2.3see13. LetCbe a nonempty, closed, and convex subset ofH. Let{x_n}be a sequence inHandu∈H. LetqP_Cu. Suppose that{x_n}is such thatω_wx_n⊂Cand satisfies the following condition:

xn−u ≤u−q ∀n. 2.4

Thenxn → q.

Proposition 2.4see9. LetCbe a nonempty, closed, and convex subset of a real Hilbert spaceH.

iIfT :C → Cis aκ-strict pseudocontraction, thenT satisfies the Lipschitz condition

_Tx₋_Ty_≤ 1κ

1−κx−y, ∀x, y∈C. 2.5

iiIfT :C → Cis aκ-strict pseudocontraction, then the mappingI−T is demiclosed (at 0). That is, if{xn}is a sequence inCsuch thatxn xandI−Txn → 0, thenI−Tx0. iiiIfT :C → Cis aκ-strict pseudocontraction, then the fixed point setFTofT is closed

and convex. Therefore the projectionP_F_Tis well defined.

ivGiven an integer N ≥ 1, assume that, for each1 ≤ i ≤ N,T_i : C → Cis aκ_i-strict pseudocontraction for some0≤κi<1. Assume that{λi}Ni1is a positive sequence such that

N

i1λi 1. Then Ni1λiTiis aκ-strict pseudocontraction, withκ max{κi : 1 ≤ i ≤

N}.

vLet{Ti}Ni1 and{λi}Ni1 be given as in item (iv). Suppose that{Ti}Ni1 has a common fixed point. Then

F _N

i1

λiTi

N i1

FTi. 2.6

Lemma 2.5 see2. LetS : C → Hbe aκ-strict pseudocontraction. DefineT : C → Hby

Tx λx 1−λSxfor anyx ∈C. Then, for anyλ ∈ κ,1,T is a nonexpansive mapping with

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For solving the equilibrium problem, let one assume that the bifunctionFsatisfies the following conditions:

A1Fx, x 0for allx∈C;

A2Fis monotone, that is,Fx, y Fy, x≤0for anyx, y∈C;

A3for eachx, y, z∈C, lim sup_t_→₀ Ftz 1−tx, y≤ Fx, y;

A4Fx,·is convex and lower semicontinuous for eachx∈C.

Lemma 2.6see3. LetCbe a nonempty, closed, and convex subset ofH, letFbe bifunction from

C×CtoRwhich satisfies conditions (A1)–(A4), and letr > 0andx∈H. Then there existsz∈C such that

Fz, y1_ry−z, z−x≥0, ∀y∈C. 2.7

Lemma 2.7see1. Forr >0, x∈H, define the mappingTr :H → Cas follows:

Trx z∈C|Fz, y 1/ry−z, z−x≥0, ∀y∈C 2.8

for allx∈H. Then, the following statements hold:

iTris single valued;

iiTr is firmly nonexpansive, that is, for anyx, y∈H,

_T_r_x₋_T_r_y2_≤_T

rx−Try, x−y; 2.9

iiiFT_r EPF;

ivEPFis closed and convex.

3. Parallel Algorithm

In this section, we apply the hybrid methods to the parallel algorithm for finding a common element of the fixed point set of strict pseudocontractions and the solution set of the problem

1.1in Hilbert spaces.

Theorem 3.1. Let Cbe a nonempty, closed, and convex subset of a real Hilbert space H, and let

Fk, k ∈ {1,2, . . . , M},be bifunctions from C×CtoR which satisfies conditions (A1)–(A4). Let, for each1 ≤ i ≤ N,S_i : C → Cbe a κ_i-strict pseudocontraction for some0 ≤ κi < 1. Letκ

max{κi: 1≤i≤N}.Assume thatΩ ∩N_i1FSi∩∩Mk1EPFk/∅. Assume also that{η

n i }Ni1is a finite sequence of positive numbers such that N_i₁η_in 1for alln∈Nandinf_n≥1η_in >0for all

1≤i≤N. Let the mappingAnbe defined by

An N

i1

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Givenx1∈C, let{xn},{un},and{yn}, be sequences which are generated by the following algorithm:

unTrM,nFMTrMFM−1−1,n · · ·TrF22,nT F1

r1,nxn,

Aλnn λnI 1−λnAn,

ynαnxn 1−αnAλnn un,

Cnz∈C:yn−z≤ xn−z,

Qn{z∈C:xn−z, x1−xn ≥0},

xn1PCn∩Qnx1,

3.2

where {αn} ⊂ 0, afor somea ∈ 0,1,{λn} ⊂ κ, bfor someb ∈ κ,1, and{rk,n} ⊂ 0,∞ satisfieslim infn→ ∞rk,n>0for allk∈ {1,2, . . . , M}. Then,{xn}converge strongly toPΩx1.

Proof. DenoteΘk_n T_rk,nFk · · ·TF2

r2,nTrF11,n for every k ∈ {1,2, . . . , M} and Θ0n I for all n ∈ N.

Thereforeun ΘMn xn. The proof is divided into six steps.

Step 1. The sequence{x_n}is well defined.

It is obvious thatCn is closed and Qn is closed and convex for every n ∈ N. From

Lemma2.2, we also get thatCnis convex.

Takep ∈ Ω, since for each k ∈ {1,2, . . . , M},T_rk,nFk is nonexpansive, p T_rk,nFkp, and

un ΘMn xn, we have

_u_n₋_p_ΘM

n xn−ΘMn p≤ xn−p 3.3

for alln∈N. From Proposition2.4, Lemma2.5, and3.3, we get

_y_n₋_p_α_n_x_n 1−α_nAλn_n u_n−p≤αnxn−p 1−αnAλnn un−p≤xn−p. 3.4

Sop ∈ Cn for alln ∈ N. Thus Ω ⊂ Cn.Next we will show by induction that Ω ⊂ Qn for

all n ∈ N. Forn 1, we have Ω ⊂ C Q1. Assume that Ω ⊂ Qn for some n ≥ 1. Since

xn1PCn∩Qnx1, we obtain

xn1−z, x1−xn1 ≥0, ∀z∈Cn∩Qn. 3.5

AsΩ⊂ Cn∩Qn by induction assumption, the inequality holds, in particular, for allz ∈ Ω.

This together with the definition ofQn1 implies thatΩ⊂ Qn1. HenceΩ⊂ Qnholds for all

n≥1.ThusΩ⊂C_n∩Q_n, and therefore the sequence{x_n}is well defined.

Step 2.

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From the definition ofQn we imply thatxn PQnx1. This together with the fact that Ω⊂Qnfurther implies that

xn−x1 ≤x1−p ∀p∈Ω. 3.7

Then{xn}is bounded and3.6holds. From3.3,3.4, and Proposition2.4i, we also obtain

that{u_n},{y_n},and{S_ix_n}are bounded.

Step 3. The following limit holds:

lim

n→ ∞xn1−xn0. 3.8

Fromxn PQnx1andxn1∈Qn, we getxn1−xn, xn−x1 ≥0. This together with Lemma2.1 iimplies that

xn1−xn2 xn1−x1−xn−x12

xn1−x12− xn−x12−2xn1−xn, xn−x1

≤ xn1−x12− xn−x12.

3.9

Thenx_n−x1 ≤ xn1−x1, that is, the sequence{xn−x1}is nondecreasing. Since{xn−x1}

is bounded, limn→ ∞xn−x1exists. Then3.8holds.

Step 4. The following limit holds:

lim

n→ ∞Anxn−xn0. 3.10

Fromxn1∈Cn, we have

_y_n₋_x_n_≤_x_n₁₋_x_n_y_n₋_x_n₁_≤2x_n1−xn. 3.11

By3.6, we obtain

lim

n→ ∞yn−xn0. 3.12

Next we will show that

lim

n→ ∞

Θk

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Indeed, forp ∈ Ω, it follows from the firm nonexpansivity ofT_rk,nFk that for each k ∈

{1,2, . . . , M},we have

Θk

nxn−p 2

Trk,nFkΘk−n 1xn−Trk,nFkp 2

≤Θk

nxn−p,Θk−n 1xn−p

1

2

Θk

nxn−p 2

Θk−1 n xn−p

2

−Θk

nxn−Θk−n 1xn 2

.

3.14

Thus we get

Θk

nxn−p 2

≤Θk−1 n xn−p

2

−Θk

, k1,2, . . . , M, 3.15

which implies that, for eachk∈ {1,2, . . . , M},

Θk

nxn−p 2

≤Θ0

nxn−p 2

−Θk

−Θk−1

n xn−Θk−n 2xn 2

− · · · −Θ2

nxn−Θ1nxn 2

−Θ1

nxn−Θ0nxn 2

≤xn−p2−Θknxn−Θk−n 1xn 2

.

3.16

Therefore, by the convexity of · 2_{and Lemma}_2.5_{, we get}

_y_n₋_p2_≤_α

nxn−p2 1−αnAλnn un−p 2

≤αnxn−p2 1−αnun−p2

≤αnxn−p2 1−αnΘknxn−p 2

≤αnxn−p2 1−αnxn−p2−Θknxn−Θk−n 1xn 2

xn−p2−1−αnΘknxn−Θk−n 1xn 2

.

3.17

It follows that

1−αnΘknxn−Θk−n 1xn 2

≤xn−p2−yn−p2 ≤xn−ynxn−pyn−p.

3.18

Since{αn} ⊂0, a, we get from3.12that3.13holds; then we have

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Observe thatyn−un ≤ yn−xnxn−un; we also haveyn−un → 0 asn → ∞. On

the other hand, fromy_nα_nx_n 1−α_nAλn_n u_n, we observe that

1−αnAλnn un−un1−αn

Aλnn un−un

yn−un−αnxn−un

≤yn−unαnxn−un.

3.20

From{αn} ⊂0, a,3.19, andyn−un → 0, we obtainAλnnun−un → 0 asn → ∞. It is

easy to see that

Aλnn xn−xn≤Aλnnxn−AλnnunAλn

nun−unun−xn ≤2un−xnAλnnun−un. 3.21

Combining the above arguments and3.2, we have

Aλnn xn−xnλnxn 1−λnAnxn−xn 1−λnAnxn−xn. 3.22

Now, it follows from{λn} ⊂κ, bthatAnxn−xn → 0 asn → ∞.

Step 5. The following implication holds:

ωwxn⊂Ω. 3.23

We first show thatωwxn⊂ ∩N_i1FSi. To this end, we takeω∈wxnand assume thatxnj ω

asj → ∞for some subsequence{x_nj}ofx_n.Without loss of generality, we may assume that

η_inj → ηi as j → ∞, ∀1≤i≤N. 3.24

It is easily seen that eachηi>0 and Ni1ηi1. We also have

Anjx → Ax as j → ∞ ∀x∈C, 3.25

where A N_i₁ηiSi.Note that, by Proposition 2.4,A is aκ-strict pseudocontraction and

FA ∩N

i1FSi. Since

Axnj−xnj≤Anjxnj−AxnjAnjxnj−xnj

≤N

i1

η_inj−ηiSixnjAnjxnj−xnj,

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we obtain by virtue of3.10and3.24that

lim

n→ ∞

Axnj−xnj0. 3.27

So by the demiclosedness principleProposition2.4ii, it follows thatω∈FA ∩N_i₁FSi,

and henceω_wx_n⊂ ∩_iN₁FS_i.

Next we will show thatω∈ ∩M_k₁EPFk. Indeed, by Lemma2.6, we have that, for each

k1,2, . . . , M,

Fk

Θk nxn, y

_r1 n

y−Θk

nxn,Θknxn−Θk−n 1xn

≥0, ∀y∈C. 3.28

FromA2, we get

1

rn

y−Θk

nxn,Θknxn−Θk−n 1xn

≥Fk

y,Θk

nxn

, ∀y∈C. 3.29

Hence,

y−Θk

njxnj, Θk

njxnj−Θk−nj1xnj

rnj

≥Fk

y,Θk

njxnj

, ∀y∈C. 3.30

From 3.13, we obtain that Θk_njxnj ωas j → ∞ for each k 1,2, . . . , M especially,

unj ΘMnjxnj. Together with3.13andA4we have, for eachk1,2, . . . , M,that

0≥Fky, ω, ∀y∈C. 3.31

For any 0< t ≤ 1 andy ∈C, letyt ty 1−tω. Sincey ∈ Candω ∈ C, we obtain that

yt∈C, and henceFkyt, ω≤0. So, we have

0Fkyt, yt≤tFkyt, y 1−tFkyt, ω≤tFkyt, y. 3.32

Dividing by t, we get, for eachk1,2, . . . , M,that

Fkyt, y≥0, ∀y∈C. 3.33

Lettingt → 0 and fromA3, we get

Fkω, y≥0 3.34

for ally ∈Candω ∈EPFkfor eachk 1,2, . . . , M,that is,ω ∈ ∩M_k1EPFk.Hence3.23

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Step 6. From3.6and Lemma2.3, we conclude thatxn → q, whereqPΩx1.

Remark 3.2. In 2007, Acedo and Xu studied the following CQ method9:

x0∈C chosen arbitrarily,

ynαnxn 1−αnAnxn,

Cn

z∈C:y_n−z2≤ x_n−z2−1−α_nα_n−κx_n−A_nx_n2,

xn1PCn∩Qnx0.

3.35

In this paper, we first turn the strict pseudocontractionAninto nonexpansive mapping

Aλnn then replaceCnwith a more simple form in the iterative algorithm.

Remark 3.3. IfF_kx, y 0,N1, andλ_nδ, we can obtain14, Theorem 1.

Remark 3.4. IfM 1,N 1,κ 0, andλn 0 and we use 1−αn to replaceαn, we can get

the result that has been studied by Tada and Takahashi in8for nonexpansive mappings. If

Fkx, y 0,N1,κ0, andλn0, we can get7, Theorem 3.1.

Fk, k∈ {1,2, . . .},be bifunctions fromC×CtoRwhich satisfies conditions (A1)–(A4). Let, for each

1 ≤ i≤ N,S_i : C → Cbe aκ_i-strict pseudocontraction for some0 ≤ κ_i < 1. Letκ max{κ_i : 1 ≤i≤ N}.Assume thatΩ ∩_iN₁FS_i∩∩_kM₁EPF_k_/∅. Assume also that{η_in}N_i₁is a finite sequence of positive numbers such that N_i₁η_in 1for allnandinfn≥1η_in >0for all1 ≤i≤N. Let the mappingA_nbe defined by

An N

i1

η_inSi. 3.36

Givenx1∈CC1, letxn,un, andynbe sequences which are generated by the following algorithm:

unTrM,nFMTrMFM−1−1,n · · ·TrF22,nT F1

r1,nxn,

Aλnn λnI 1−λnAn,

ynαnxn 1−αnAλnn un,

Cn1

z∈Cn:yn−z ≤ xn−z,

xn1PCn1x1,

3.37

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Proof. The proof of this theorem is similar to that of Theorem3.1.

Step 1. The sequence{x_n}is well defined. We will show by induction thatC_n is closed and convex for alln. For n 1, we haveC C1 which is closed and convex. Assume thatCn

for somen≥ 1 is closed and convex; from Lemma2.2, we have thatCn1is also closed and

convex; The proof ofΩ⊂C_nis similar to the one in Step1of Theorem3.1.

Step 2. xn−x1 ≤ q−x1for alln, whereqPΩx1.

Step 3. xn1−xn → 0 asn → ∞.

Step 4. A_nx_n−x_n → 0 asn → ∞.

Step 5. ωwxn⊂Ω.

Step 6. x_n → q.

The proof of Step2–Step6is similar to that of Theorem3.1.

Remark 3.6. IfM1, we can obtain the two corresponding theorems in10.

4. Cyclic Algorithm

Let C be a closed, and convex subset of a Hilbert spaceH, and let {S_i}_iN−₀1 be Nκ_i-strict pseudocontractions onCsuch that the common fixed point set

N−1

i0

FSi/∅. 4.1

Letx0 ∈C,and let{αn}∞n0be a sequence in0,1. The cyclic algorithm generates a sequence

{xn}∞n1in the following way:

x1α0x0 1−α0S0x0,

x2α1x1 1−α1S1x1,

. . .

xNαN−1xN−1 1−αN−1SN−1xN−1,

xN1αNxN 1−αNS0xN,

· · ·.

4.2

In general,xn1is defined by

xn1αnxn 1−αnSnxn, 4.3

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Theorem 4.1. Let Cbe a nonempty, closed, and convex subset of a real Hilbert space H, and let

Fk, k ∈ {1,2, . . . , M},be bifunctions from C×CtoR which satisfies conditions (A1)–(A4). Let, for each0 ≤ i ≤ N−1,Si : C → Cbe aκi-strict pseudocontraction for some0 ≤ κi < 1. Let

κmax{κi : 0≤i≤N−1}.Assume thatΩ ∩N−_i01FSi∩∩Mk1EPFk/∅. Givenx0 ∈C, let

xn,un, andynbe sequences which are generated by the following algorithm:

unTrM,nFMTrMFM−1−1,n · · ·TrF22,nTrF11,nxn,

Sλn_nλnI 1−λnSn,

ynαnxn 1−αnSλn_nun,

Cnz∈C:yn−z≤ xn−z,

xn1PCn∩Qnx0,

4.4

where {α_n} ⊂ 0, afor somea ∈ 0,1,{λ_n} ⊂ κ, bfor someb ∈ κ,1, and{r_k,n} ⊂ 0,∞

satisfieslim infn→ ∞rk,n>0for allk∈ {1,2, . . . , M}. Then,{xn}converge strongly toPFx0.

Proof. The proof of this theorem is similar to that of Theorem3.1. The main points are the following.

Step 1. The sequence{xn}is well defined.

Step 2. x_n−x0 ≤ q−x0for alln, whereqPΩx0.

Step 3. xn1−xn → 0.

Step 4. Snxn−xn → 0.To prove the above steps, one simply replacesAnwithSnin the

proof of Theorem3.1.

Step 5. ωwxn ⊂ Ω.Indeed, let ω ∈ ωwxnandxnm ωfor some subsequence{xnm} of

{xn}. We may assume thatl nm mod Nfor allm. Since, byxn1−xn → 0, we also

havex_nm_j ωfor allj ≥0, we deduce that

_x_nm_j₋_S_l_j_x_nm_j_x_nm_j₋_S_nm_j_x_nm_j _→ 0. 4.5

Then the demiclosedness principle implies thatω ∈FSljfor allj. This ensures thatω ∈

∩N

i1FSi.The Proof ofω∈ ∩Mk1EPFkis similar to that of Theorem3.1.

Step 6. The sequencex_nconverges strongly toq.

The strong convergence toqof{xn}is a consequence of Step2, Step5, and Lemma2.3.

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κmax{κi: 0≤i≤N−1}.Assume thatΩ ∩N−_i01FSi∩∩Mk1EPFk/∅. Givenx0∈CC0, letx_n,u_n, andy_nbe sequences whic are generated by the following algorithm:

un TrM,nFMTrMFM−1−1,n · · ·TrF22,nTrF11,nxn,

Sλn_n λnI 1−λnSn,

yn αnxn 1−αnSλn_nun,

Cn1

z∈Cn:yn−z≤ xn−z,

xn1 PCn1x0,

4.6

where {αn} ⊂ 0, afor somea ∈ 0,1,{λn} ⊂ κ, bfor someb ∈ κ,1, and{rk,n} ⊂ 0,∞ satisfieslim infn→ ∞rk,n>0for allk∈ {1,2, . . .}. Then,{xn}converge strongly toPΩx0.

Proof. The proof of this theorem is similar to that of Step1in Theorem3.5and Step2–Step6

in Theorem4.1.

Remark 4.3. IfM1, we can obtain the two corresponding theorems in10.

Acknowledgment

This research is supported by Fundamental Research Funds for the Central Universities

GRANT:ZXH2009D021.

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