Weak and Strong Convergence Theorems for Equilibrium Problems and Countable Strict Pseudocontractions Mappings in Hilbert Space

Volume 2010, Article ID 474813,11pages doi:10.1155/2010/474813

Research Article

Weak and Strong Convergence Theorems for Equilibrium Problems and Countable Strict Pseudocontractions Mappings in Hilbert Space

Rudong Chen,

Xilin Shen,

and Shujun Cui

1Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China

2Department of Mathematics, Xinxiang College, Xinxiang, Henan 453002, China

Correspondence should be addressed to Rudong Chen,[email protected] Received 27 August 2009; Accepted 10 January 2010

Academic Editor: Jong Kim

Copyrightq2010 Rudong Chen 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.

We introduce two iterative sequence for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a countable family of strict pseudocontractions in Hilbert Space. Then we study the weak and strong convergence of the sequences.

1. Introduction

LetCbe a nonempty closed convex subset of a Hilbert spaceHand letT be a self-mapping ofC. ThenT is said to be a strict pseudocontraction mappings if for allx, y∈C, there exists a constant 0≤κ <1 such that

Tx−Ty²≤x−y²κI−Tx−I−Ty² 1.1 if1.1holds, we also say thatT is aκ-strict pseudocontraction. We useFTto denote the set of fixed points of T, → to denote weakstrongconvergence, andWwxn {x :

∃x_nk x}to denote theW-limit set of{x_n}.

Letf:C×C → Rbe a bifunction whereRis the set of real numbers. Then, we consider the following equilibrium problem:

Find z∈Csuch thatf z, y

≥0, ∀y∈C. 1.2

The set of suchz ∈ Cis denoted by EPf. Numerous problems in physics, optimization, and economics can be reduced to find a solution of1.2. Some methods have been proposed to solve the equilibrium problemsee1–3. Recently, S. Takahashi and W. Takahashi4 introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of the equilibrium problem and the set of fixed points of a nonexpansive mapping in Hilbert spaces. They also studied the strong convergence of the sequences generated by their algorithm for a solution of the EP which is also a fixed point of a nonexpansive mapping defined on a closed convex subset of a Hilbert space.

In this paper, thanks to the condition introduced by Aoyama et al.5, We introduce two iterative sequence for finding a common element of the set of solutions of an equilibrium problems and the set of fixed points of a countable family of strict pseudocontractions mappings in Hilbert Space. Then we study the weak and strong convergence of the sequences. The additional condition is inspired by Marino and Xu6and Kim and Xu7.

2. Preliminaries

For solving the equilibrium problem, let us assume that the bifunction f satisfies the following conditionssee3:

A1fx, x 0 for allx∈C;

A2f is monotone, that is, fx, y fy, x≤0 for anyx, y∈C;

A3f is upper-hemicontinuous, that is, for each x,y, z ∈ C,lim sup_t→0ftz 1 − tx, y≤fx, y;

A4fx,·is convex and lower semicontinuous for eachx∈C.

LetHbe a real Hilbert space. Then there hold the following well-known results:

tx 1−ty²tx² 1−ty²−t1−tx−y², ∀x, y∈H, ∀t∈0,1;

xy²x²−y²−2

x−y, y

, ∀x, y∈H. 2.1

If{x_n}is a sequence inHweakly convergent toz, then lim sup

n→ ∞

x_n−y²lim sup

n→ ∞ x_n−z²z−y² ∀y∈H. 2.2 Recall that the nearest point projectionP_C fromHontoCassigns to eachx ∈Hits nearest point denoted byPCxinC; that is,PCxis the unique point inCwith the property

x−P_Cx ≤x−y, ∀y∈C. 2.3 Givenx∈Handz∈C, thenzP_Cxif and only if there holds the following relation:

x−z, y−z ≤0, ∀y∈C. 2.4

Lemma 2.1see6. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetT: C → Cbe aκ-strict pseudocontraction such thatFT/∅.

1(Demi-closed principle)Tis demi-closed onC, that is, ifxn x∈Candxn−Txn → 0, thenxTx.

2T satisfies the Lipschitz condition

Tx−Ty≤Lx−y 1κ

1−κx−y ∀x, y∈C. 2.5 3The fixed point setFTofTis closed and convex so that the projectionP_FTis well defined.

Lemma 2.2see5. LetCbe a nonempty closed convex subset of a Banach space and let{Tn}be a sequence of mapping ofCinto itself. Suppose_∞

n1sup_x∈CT_n1x−T_nx<∞.Then, for eachy∈C, {T_ny}converges strongly to some point ofC. Moreover, letT be a mapping ofCinto itself defined by

Ty lim

n→ ∞Tny ∀y∈C. 2.6

Then lim_{n→ ∞}sup_x∈CTnx−Tx0.

Lemma 2.3see8. LetCbe a closed convex subset ofH. Let{x_n}be a sequence inHandu∈H.

LetqP_Cu. If{x_n}is such thatW_wx_n⊂Cand satisfies the condition

xn−u ≤u−q ∀n, 2.7

thenx_n → q.

Lemma 2.4see9. LetCbe a nonempty closed convex subset ofH. Letf be a bifunction from C×C into Rsatisfying (A1), (A2), (A3), and(A4). Then, for anyλ >0 andx∈H, there existsz∈C such that

f z, y

1 λ

y−z, z−x

≥0, ∀y∈C. 2.8

Further, ifTλx{z∈C:fz, y 1/λy−z, z−x ≥0,∀y∈C}, then the following holds:

1T_λxis single-valued;

2T_λxis firmly nonexpansive, that is,

T_λx−T_λy²≤ T_λx−T_λy, x−y, ∀x, y∈H; 2.9 3FT_λ EPf;

4EPfis closed and convex.

3. Weak Convergence Theorems

Theorem 3.1. Let Cbe a nonempty closed convex subset of a Hilbert space H and let {T_n} be a sequence ofκ_n-strict pseudocontractions mappings onCinto itself with 0 ≤ κ_n < 1. Assume that κ max{κn : n ≥ 1}.Letf : C×C → Rbe a bifunction satisfying (A1), (A2), (A3), (A4), and EPf∩_∞

n1FT_n/∅. Let{x_n}and{z_n}be sequence generated byx1∈Cand f

z_n, y 1

r_n

y−z_n, z_n−x_n

≥0, ∀y∈C, x_n1α_nz_n 1−α_nT_nz_n, ∀n≥1.

3.1

Assume that{α_n} ⊂ 0,1with κδ < α_n < 1−δfor all n, whereδ ∈ 0,1is a small enough constant, and{r_n}is a sequence in0,∞with lim inf_{n→ ∞}r_n > 0 and _∞

n1|r_n1−r_n| < ∞.Let _∞

n1sup_x∈BTn1x−Tnx<∞for any bounded subsetBofCand letTbe a mapping ofCinto itself defined byTxlim_{n→ ∞}T_nxfor allx∈Cand suppose thatFT _∞

n1FT_n.Then the sequences {x_n}and{z_n}converge weakly to an element ofFT∩EPf.

Proof. Pickp∈FT∩EPf. Then from the definition ofTrinLemma 2.4, we haveznTrnxn, and thereforez_n−pT_rnx_n−T_rnp ≤ x_n−p. It follows from3.1that

xn1−p²1−αnTnzn−p αnzn−p²

α_nz_n−p² 1−α_nT_nz_n−p²−α_n1−α_nz_n−T_nz_n²

≤αnzn−p² 1−αn zn−p²κzn−Tnzn²

−α_n1−α_nz_n−T_nz_n²

zn−p²−αn−κ1−αnzn−Tnzn²

≤x_n−p²−α_n−κ1−α_nz_n−T_nz_n².

3.2

Sinceκδ < α_n<1−δfor alln, we getx_n1−p ≤ x_n−p; that is, the sequence{x_n−p}

is decreasing. Hence lim_{n→ ∞}x_n−pexists. In particular,{x_n}is bounded. SinceT_ris firmly nonexpensive,{zn}is also bounded. Also3.2implies that

z_n−T_nz_n²≤ 1

δ² x_n−p²−x_n1−p²

. 3.3

Taking the limit asn → ∞yields that

nlim→ ∞z_n−T_nz_n0. 3.4

Since{z_n}is bounded, it follows that ∞ n1

sup

x∈{zn}T_n1x−T_nx<∞. 3.5

We applyLemma 2.2to get

z_n−Tz_n ≤ z_n−T_nz_nT_nz_n−Tz_n

≤ zn−Tnznsup{Tnz−Tz:z∈ {zn}} −→0. 3.6

Next, we claim that lim_{n→ ∞}z_n−x_n0. Indeed, letpbe an arbitrary element ofFT∩EPf.

Then as above

zn−p²Trxn−Trp²

≤ T_rx_n−T_rp, x_n−p z_n−T_rp, x_n−p 1

2 zn−p²xn−p²− xn−zn² ,

3.7

and hence

z_n−p²≤x_n−p²− xn−z_n². 3.8

Therefore, from3.2, we have

xn1−p² ≤zn−p²−αn−κ1−αnzn−Tnzn²

≤z_n−p²

≤xn−p²− xn−zn²,

3.9

and hence

xn−zn² ≤xn−p²−xn1−p². 3.10

So, from the existence of lim_{n→ ∞}xn−p, we have

n→ ∞limx_n−z_n0. 3.11

Next, we claim thatWwxn⊂FT∩EPf. since{xn}is bounded andHis reflexive, W_wx_nis nonempty. Letw ∈ W_wx_nbe an arbitrary element. Then a subsequencex_ni of {xn}converges weakly tow. Hence, from3.11we know thatzni w.Aszn−Tzn → 0, we obtain thatTzni w. Let us showWwxn⊂EPf. SinceznTrnxn, we have

f zn, y

1 rn

y−zn, zn−xn

≥0, ∀y∈C. 3.12

ByA2, we have

1 rn

y−z_n, z_n−x_n

≥f y, z_n

, 3.13

and hence

y−z_ni,zni−xni

rni

≥f y, z_ni

. 3.14

FromA4, we have

0≥f y, w

∀y∈C. 3.15

Then, fort∈0,1andy∈C, fromA1, andA4, we also have 0f

ty 1−tw, ty 1−tw

≤tf

ty 1−tw, y

1−tf

ty 1−tw, w

≤tf

ty 1−tw, y ,

3.16

Takingt → 0and usingA3, we get f

w, y

≥0 ∀y∈C, 3.17

and hencew ∈ EPf. SinceT is a strict pseudocontraction mapping, byLemma 2.11we know that the mappingT is demiclosed at zero. Note thatz_n −Tz_n → 0 and z_ni w.

Thus,w ∈ FT. Consequently, we deduce that w ⊂ FT∩EPf. Since wis an arbitrary element, we conclude thatWwxn⊂FT∩EPf.

To see that {x_n} and {z_n} are actually weakly convergent, we take x,x ∈ W_wx_n x_ni x, x_mj x. Since lim_{n→ ∞}x_n−p exist for everyp ∈ FT, by2.2, we have

n→ ∞limx_n−x ² lim

i→ ∞x_ni−x ² lim

i→ ∞x_ni−x²x−x ² lim

j→ ∞

xmj−x²x−x ² lim

j→ ∞

x_mj−x²2x−x ² lim

n→ ∞x_n−x ²2x−x ².

3.18

Hencexxand proof is completed.

4. Strong Convergence Theorems

Theorem 4.1. LetCbe a closed convex subset of a real Hilbert spaceH. Let{T_n}be a sequence ofκ_n- strict pseudocontractions mappings onCinto itself with 0≤κ_n<1. Assume thatκmax{κ_n:n≥ 1}.Letf :C×C → Rbe a bifunction satisfying (A1), (A2), (A3), (A4) and EPf∩_∞

n1FTn/∅.

ForC1Candx1P_C1x0, let{x_n}and{z_n}be sequence generated byx0 ∈Cand

ynαnxn 1−αnTnxn, z_n∈C such that f

z_n, y 1

r_ny−z_n, z_n−y_n ≥0, ∀y∈C, C_n1{v∈C:z_n−v ≤ x_n−v},

xn1PCn1x0, n≥1.

4.1

Assume that{αn} ⊂ 0,1with κδ < αn < 1−δfor all n, whereδ ∈ 0,1is a small enough constant, and{rn}is a sequence in0,∞with lim inf_{n→ ∞}rn > 0 and_∞

n1|rn1−rn| < ∞. Let _∞

n1sup_x∈BT_n1x−T_nx<∞for any bounded subsetBofCand letTbe a mapping ofCinto itself defined byTx lim_{n→ ∞}T_nxfor allx∈CSuppose thatFT _∞

n1FT_n.Then,{x_n}converges strongly toPFT∩EPfx0.

Proof. First, we show thatC_n is closed and convex. It is obvious thatC1 Cis closed and convex. Suppose thatCk is closed and convex for somek ≥ 1. Forz ∈ Ck, we know that zk−z ≤ xk−zis equivalent to

z_k−x_k²2z_k−x_k, x_k−z ≤0. 4.2

SoC_k1is closed and convex. Then,C_nis closed and convex.

Next, we show by induction thatFT∩EPf⊂C_nfor alln≥1.FT∩EPf⊂C1is obvious. Suppose thatFT∩EPf⊂Ckfor somek≥1. Letp∈FT∩EPf⊂Ck. Putting z_nT_rny_nfor alln, we know from4.1that

zn−p²Trnyn−p²

≤y_n−p²

1−αnTnxn−p αnxn−p²

α_nx_n−p² 1−α_nT_nx_n−p²−α_n1−α_nx_n−T_nx_n²

≤αnxn−p² 1−αn xn−p²κxn−Tnxn²

−α_n1−α_nx_n−T_nx_n²

x_n−p²−α_n−κ1−α_nx_n−T_nx_n²

≤x_n−p²−δ²x_n−T_nx_n²

≤xn−p²,

4.3 and hencep∈C_k1. This implies thatFT∩EPf⊂C_nfor alln≥1.

This implied that{xn}is well defined.

FromxnPCnx0, we have

x0−x_n ≤x0−y ∀y∈C_n. 4.4

UsingFT∩EPf⊂Cn, we have

x0−x_n ≤ x0−u ∀u∈FT∩EP f

, n≥1. 4.5

Then,{xn}is bounded. So are{yn}and{zn}. In particular,

x0−x_n ≤x0−p wherepP_FT∩EPfx0. 4.6

Fromx_nP_Cnx0andx_n1P_Cn1x0∈C_n1⊂C_n,we have

x0−x_n ≤ x0−x_n1. 4.7

Since{x_n−x0}is bounded, lim_{n→ ∞}x_n−x0exists. Fromx_nP_Cnx0andx_n1P_Cn1x0 ∈ C_n1⊂C_n.we also have

x0−xn, xn−xn1 ≥0. 4.8

In fact, from4.8, we have

xn−xn1xn−x0x0−xn1²

x0−xn1²− x0−xn²−2x0−xn, xn−xn1

≤ x0−xn1²− x0−xn².

4.9

Since lim_{n→ ∞}x_n−x0exists, we have thatx_n−x_n1 → 0. On the other handx_n1∈C_n1⊂ Cnimplies that

zn−xn1 ≤ xn−xn1 −→0. 4.10

Further, we have

n→ ∞limz_n−x_n0. 4.11

From4.3, we have

xn−Tnxn² ≤ 1

δ² xn−p²−zn−p²

. 4.12

On the other hand, we have

xn−p²−zn−p² xn²− zn²2zn−xn, p

≤ xn−z_nx_nz_n 2px_n−z_n. 4.13

Then, we have

nlim→ ∞x_n−p²−z_n−p²0. 4.14

Therefore, we have

nlim→ ∞xn−Tnxn0. 4.15

We applyLemma 2.2to get

xn−Txn ≤ xn−TnxnTnxn−Txn

≤ xn−Tnxnsup{Tnx−Tx:x∈ {xn}} −→0. 4.16

Lastly, we show that the sequence {xn} converges to PFT∩EPfx0. Since {xn} is bounded andHis reflexive,W_wx_nis nonempty. Letw ∈W_wx_nbe an arbitrary element.

Then a subsequence xni of {xn} converges weakly to w. From Lemma 2.1 and 4.16, we obtain thatωwxn⊂FT. Next, we showWwxn⊂EPf. Letpbe an arbitrary element of FT∩EPf. Fromz_nT_rny_nandy_n−p ≤ x_n−p, we have

zn−p²≤Tryn−Trp²

≤ T_ry_n−T_rp, y_n−p z_n−T_rp, y_n−p 1

2 zn−p²yn−p²−yn−zn² 1

2 zn−p²xn−p²−yn−zn² ,

4.17

and hence

yn−zn² ≤xn−p²−zn−p². 4.18

Therefore, we have

nlim→ ∞yn−zn0. 4.19

As in the proof ofTheorem 3.1, we have f

z_n, y 1

rny−z_n, z_n−y_n ≥0, ∀y∈C. 4.20 ByA2, we have

1 r_n

y−z_n, z_n−y_n

≥f y, z_n

, 4.21

and hence

y−z_ni,z_ni−y_ni rni

≥f y, z_ni

. 4.22

FromA4, we have

0≥f y, w

∀y∈C. 4.23

Then, fort∈0,1andy∈C, fromA1andA4, we also have 0f

ty 1−tw, ty 1−tw

≤tf

ty 1−tw, y

1−tf

ty 1−tw, w

≤tf

ty 1−tw, y .

4.24

Takingt → 0and usingA3, we get f

w, y

≥0 ∀y∈C, 4.25

and hence w ∈ EPf. Lemma 2.3 and 4.6 ensure the strong convergence of {x_n} to P_FT∩EPfx0. This completes the proof.

Acknowledgment

This work is supported by the National Science Foundation of China, Grant 10771050.

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