Strong Convergence Theorems for Equilibrium Problems and Fixed Point Problems in Hilbert Spaces

Volume 2010, Article ID 491357,13pages doi:10.1155/2010/491357

Research Article

Strong Convergence Theorems for

Equilibrium Problems and Fixed Point Problems in

Hilbert Spaces

Jian-Wen Peng

_{and Yan Wang}

1_{College of Mathematics and Computer Science, Chongqing Normal University, Chongqing 400047, China}

2_{Division of Mathematics, Yibin No. 3 Middle School, Yibin, Sichuan 644000, China}

Correspondence should be addressed to Jian-Wen Peng,[email protected]

Received 21 August 2009; Revised 1 December 2009; Accepted 13 January 2010

Academic Editor: Petru Jebelean

Copyrightq2010 J.-W. Peng and Y. Wang. 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 an Ishikawa iterative scheme by the viscosity approximate method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in Hilbert space. Then, we prove some strong convergence theorems which extend and generalize S. Takahashi and W. Takahashi’s results2007.

1. Introduction

LetHbe a real Hilbert space and letCbe a nonempty closed convex subset ofH. LetFbe a bifunction fromC×CtoR, whereRis the set of real numbers. The equilibrium problem for

F:C×C → Ris to findx∈Csuch that

Fx, y≥0, ∀y∈C. 1.1

The set of solutions of1.1is denoted by EPF. Given a mappingT : C → H, let

Fx, y Tx, y−xfor allx, y ∈C. Then,z ∈ EPFif and only ifTz, y−z ≥ 0 for all

y∈C. Numerous problems in physics, optimization, and economics reduce to find a solution of1.1; for more details, see1,2.

Recall that a self-mappingSof a closed convex subsetCofH is nonexpansive3if there holds that

We denote the set of fixed points ofSbyFS. There are some methods for approximation of fixed points of a nonexpansive mapping. In 2000, Moudafi 4introduced the viscosity approximation method for nonexpansive mappingssee5for further developments in both Hilbert and Banach spaces. Some methods have been proposed to solve the equilibrium problem; see, for instance,1,2,6,7. Recently, Combettes and Hirstoaga6introduced an iterative scheme of finding the best approximation to the initial data when EPFis nonempty and proved a strong convergence theorem. S. Takahashi and W. Takahashi7introduced a Mann iterative scheme by the viscosity approximation method for finding a common element of the set of solution1.1and the set of fixed points of a nonexpansive mapping in a Hilbert space and proved a strong convergence theorem.

On the other hand, Ishikawa8 introduced the following iterative process defined recursively by

xn1αnxn 1−αnSyn,

ynβnxn1−βnSxn, ∀n∈N,

1.3

where the initial guessx0is taking inCarbitrarily,{αn}and{βn}are sequences in the interval 0,1.

In this paper, motivated by the ideas in 4–8, we introduce an Ishikawa iterative scheme by the viscosity approximation method for finding a common element of the set of solution1.1and the set of fixed points of a nonexpansive mapping in a Hilbert space.

Starting with an arbitraryx1∈H, define sequences{xn},{yn},and{un}by

Fun, y_r1 n

y−un, un−xn≥0, ∀y∈C,

xn1αnfxn 1−αnSyn,

ynβnxn1−βnSun, ∀n∈N,

1.4

where{αn},{βn} ⊂0,1and{rn} ⊂0,∞.

We will prove in Section 3that if the sequences {α_n},{β_n}, and{r_n} of parameters satisfy appropriate conditions, then the sequences{xn},{yn},and {un}generated by1.4

converge strongly to z ∈ FS∩EPF. The results in this paper extend and generalize S. Takahashi and W. Takahashi’s results7.

2. Preliminaries

LetHbe a real Hilbert space with inner product·,·, and norm · and letCbe a nonempty closed convex subset ofH.x_n → ximplies that{x_n}converges strongly toxandx_n x means that{xn}converges weakly tox. In a real Hilbert spaceH, we have

_λx 1−λy2λx2 1−λy2−λ1−λx−y2 2.1

For anyx∈H, there exists a unique nearest point inC, denoted byPCx, such that

x−PCx ≤ x−yfor ally∈C. Such aPCis called the metric projection ofHontoC. It is

also known thatyPCxis equivalent tox−y, y−z ≥0 for allz∈C.

For solving the equilibrium problem, let us assume that the bifunctionF satisfies the following conditions:

A1Fx, x 0 for allx∈C;

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

A3for eachx, y, z∈C,

lim

t↓0F

tz 1−tx, y≤Fx, y; 2.2

A4for eachx∈C, y→Fx, yis convex and lower semicontinuous.

We recall some lemmas needed later.

Lemma 2.1see2. LetCbe a nonempty closed convex subset ofHand letFbe a bifunction from

C×CtoRsatisfying (A1)–(A4). Letr >0 andx∈H. Then, there existsz∈Csuch that

Fz, y1_ry−z, z−x≥0, ∀y∈C. 2.3

Lemma 2.2see5. LetCbe a nonempty closed convex subset ofH, and letFbe a bifunction from

C×CtoRsatisfying (A1)–(A4). Forr >0 andx∈H,define a mappingTr :H → Cas follows:

Trx

z∈C:Fz, y1

r

y−z, z−x≥0, ∀y∈C

2.4

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

1Tr is single-valued;

2Tr is firmly nonexpansive, that is, for anyx, y∈H,

_Trx−Try2_≤T

rx−Try, x−y; 2.5

3FTr EPF;

4EPFis closed and convex.

Lemma 2.3see10. Let{an}be a sequence of nonnegative real numbers such that

where{bn}is a sequence of real numbers and{cn}is a sequence in0,1such that

i ∞_n1cn ∞,

iilim sup_{n→ ∞}bn/cn≤0 or ∞n1|bn|<∞.

Then, limn→ ∞an0.

3. Strong Convergence Theorem

In this section, we show a strong convergence theorem which solves the problem of finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space.

Theorem 3.1. LetCbe a nonempty closed convex subset ofH. LetFbe a bifunction fromC×CtoR satisfying (A1)–(A4) and letSbe a nonexpansive mapping ofCintoHsuch thatFS∩EPF/∅.

Letfbe a contraction ofHinto itself and let{x_n},{u_n},and{y_n}be sequences generated byx1∈H and1.4. If{αn},{βn} ⊂0,1and{rn} ⊂0,∞satisfy the following conditions:

lim

n→ ∞αn0, ∞

n1

αn∞,

∞

n1

|αn1−αn|<∞,

0<lim inf

n→ ∞ βn≤lim sup_{n→ ∞} βn<1, ∞

n1

_{βn1−βn<∞,}

lim inf

n→ ∞ rn>0, ∞

n1

|rn1−rn|<∞,

3.1

then,{x_n},{y_n},and{u_n}converge strongly toz∈FS∩EPF,wherezP_FS∩EPFfz.

Proof. LetQ PFS∩EPF.ThenQf is a contraction ofHinto itself. In fact, there existsa ∈

0,1such thatfx−fy ≤ax−yfor allx, y∈H. So, we have that

_Qfx−Qf

y≤fx−fy≤ax−y 3.2

for allx, y∈H.SinceHis complete, there exists a unique elementz∈Hsuch thatzQfz. Such az∈His an element ofC.

Letv∈FS∩EPF. Then fromu_nT_rnx_n,we have

un−vTrnxn−Trnv ≤ xn−v 3.3

Supposexn−v ≤M.Then, we have

xn1−v ≤αnfxn−v 1−αnSyn−v

≤αnfxn−fvαnfv−v 1−αnSyn−v

≤aαnxn−vαnfv−v 1−αnyn−v.

3.4

On the other hand

_{yn−v≤βnxn−v}

1−βnSun−v

≤βnxn−v1−βnun−v

≤βnxn−v1−βnxn−v

xn−v.

3.5

Putting3.5into3.4, we have

xn1−v ≤aαnxn−vαnfv−v 1−αnxn−v

1−αn1−axn−vαn1−a

_fv−v

1−a

≤1−αn1−aMαn1−aMM.

3.6

So, we have thatx_n1−v ≤Mfor anyn∈N. And hence{xn}is bounded. We also obtain

that{un},{yn},{Sun},{Syn},and{fxn}are bounded. Next, we show that limn→ ∞xn1−

xn0.In fact,

_{yn−yn−1βnxn}

1−βnSun−βn−1xn−1

1−βn−1

Sun−1

βnxn−xn−1

βn−βn−1

xn−1

1−βnSun−Sun−1

βn−1−βn

Sun−1

≤βn−βn−1xn−1βnxn−xn−1

1−βnun−un−1βn−βn−1Sun−1,

and hence

xn1−xnαnfxn 1−αnSyn−αn−1fxn−1−1−αn−1Syn−1

αnfxn−αnfxn−1 αnfxn−1−αn−1fxn−1

1−αnSyn−1−αnSyn−1 1−αnSyn−1−1−αn−1Syn−1

≤αnaxn−xn−1|αn−αn−1|fxn−1 1−αnyn−yn−1

|αn−αn−1|Syn−1

≤αnaxn−xn−1|αn−αn−1|fxn−1 1−αn

×βn−βn−1xn−1βnxn−xn−1

1−β_nu_n−u_n−1βn−βn−1Sun−1

|αn−αn−1|Syn−1

βn−αnβn−axn−xn−1|αn−αn−1|fxn−1Syn−1

1−α_nβ_n−β_n−1xn−1Sun−1 1−αn1−βnun−un−1

≤βn−αnβn−axn−xn−1|αn−αn−1|K1 1−αnβn−βn−1K2

1−αn1−βnun−un−1,

3.8

whereK1sup{fxnSyn:n∈N}andK2sup{xnSun:n∈N}.

On the other hand, fromunTrnxnandun1 Trn1xn1, we have

Fun, y_r1 n

y−un, un−xn≥0, ∀y∈C, 3.9

Fun1, y

1

rn1

y−un1, un1−xn1

≥0, ∀y∈C. 3.10

Puttingyu_n1in3.9andyunin3.10, we have

Fun, un1

1

rnun1−un, un−xn ≥0,

Fun1, un

1

rn1un−un1, un1−xn1 ≥

0.

3.11

So, from the monotonicity ofF, we get

un1−un,un_r−xn

n −

un1−xn1

rn1

and hence

un1−un, un−un1un1−xn−_rrn

n1un1−xn1

≥0. 3.13

Without loss of generality, let us assume that there exists a real numberbsuch thatrn> b >0

for alln∈N.Then, we have

un1−un2≤

un1−un, xn1−xn

1− rn

rn1

un1−xn1

≤ un1−un

xn1−xn1−_rrn n1

un1−xn1

,

3.14

and hence

un1−un ≤ xn1−xn_r1

n1|rn1−rn|un1−xn1

≤ xn1−xn1_b|rn1−rn|L,

3.15

whereLsup{un−xn:n∈N}.So from3.8, we have

xn1−xn ≤βn−αnβn−axn−xn−1|αn−αn−1|K1

1−α_nβ_n−β_n−1K2 1−αn1−βnxn−xn−1

1

b|rn−rn−1|L

1−α_n1−ax_n−x_n−1|αn−αn−1|K1

1−αnβn−βn−1K2 1−αn1−βn_b1|rn−rn−1|L.

3.16

Using Lemma 2.1 in10, we obtain

lim

n→ ∞xn1−xn0. 3.17

From3.15and|rn1−rn| → 0,we have

lim

n→ ∞un1−un0. 3.18

It follows from3.7that

lim

Sincexnαn−1fxn−1 1−αn−1Syn−1,we have

_{xn−Syn≤xn−Syn−1Syn−1−Syn}

≤αn−1fxn−1−Syn−1yn−1−yn.

3.20

Fromαn → 0,we havexn−Syn → 0.Forv∈FS∩EPF,we have

un−v2Trnxn−Trnv2

≤ Trnxn−Trnv, xn−v

un−v, xn−v

1

2

un−v2xn−v2− xn−un2

,

3.21

and hence

un−v2≤ xn−v2− xn−un2. 3.22

Therefore, from the convexity of · 2_{, we have}

_yn−v2_≤β

nxn−v21−βnSun−v2

≤βnxn−v21−βnun−v2

≤βnxn−v21−βnxn−v2− xn−un2

xn−v2−1−βnxn−un2,

3.23

and hence

xn1−v2αnfxn 1−αnSyn−v2

≤αnfxn−v2 1−αnSyn−v2

≤αnfxn−v2 1−αnyn−v2

≤αnfxn−v2 1−αn

xn−v2−1−βnxn−un2

≤αnfxn−v2xn−v2−1−αn1−βnxn−un2.

3.24

So, we have

1−α_n1−β_nx_n−u_n2≤α_nfx_n−v2x_n−v2− x_n1−v2

≤αnfxn−v2xn−xn1xn−vxn1−v.

Without loss of generality, let us assume that there exists two real numbersβ∗andβsuch that 1> β≥βn≥β∗>0 for alln∈N.Hence,

1−αn

1−βxn−un2≤1−αn1−βnxn−un2

≤αnfxn−v2xn−xn1xn−vxn1−v.

3.26

It follows thatxn−un → 0.We also have

Sun−xn ≤Syn−xnSun−Syn

≤Syn−xnun−yn

≤Syn−xnun−xnxn−yn

Syn−xnun−xn1−βnxn−Sun.

3.27

It follows that

β∗_Su

n−xn ≤βnSun−xn ≤Syn−xnun−xn. 3.28

Hence,Su_n−x_n → 0.Since

Sun−un ≤ Sun−xnxn−un, 3.29

we also have lim_{n→ ∞}Su_n−u_n0. Next, we show that

lim sup

n→ ∞

fz−z, xn−z≤0, 3.30

wherez PFS∩EPFfz. To show this inequality, we choose a subsequence{xni}of {xn}

such that

lim

n→ ∞fz−z, xni−zlim sup n→ ∞

fz−z, xn−z. _3.31

Since{uni}is bounded, there exists a subsequence {unij}of{uni}which converges weakly

tow. Without loss of generality, we can assume that{uni} w. FromSun−un → 0,we

obtainSu_ni w.Let us showw∈EPF.Byu_nT_rnx_n,we have

Fun, y_r1 n

y−un, un−xn≥0, ∀y∈C. 3.32

FromA2, we also have

1

rn

and hence,

y−uni,

uni−xni

rni

≥Fy, uni

. 3.34

Sinceu_ni−x_ni/r_ni → 0 andu_ni w, fromA4, we have

fy, w≤0, ∀y∈C. 3.35

Fortwith 0< t≤1 andy∈C, letytty 1−tw.Sincey∈Candw∈C, we obtainyt∈C

and henceFyt, w≤0. So we have

0Fy_t, y_t≤tFy_t, y 1−tFy_t, w≤tFy_t, y. 3.36

Dividing byt, we get

Fyt, y≥0. 3.37

Lettingt → 0 and fromA3, we get

Fw, y≥0 3.38

for ally ∈Cand hencew ∈EPF. We shall show thatw ∈FS.Assumew /∈FS.Since

uni wandw /Sw, from the Opial theorem11we have

lim inf

i→ ∞ uni−w<lim infi→ ∞ uni −Sw ≤lim inf

i→ ∞ {uni−SuniSuni −Sw} ≤lim inf

i→ ∞ uni −w.

3.39

This is a contradiction. So, we get w ∈ FS. Therefore, w ∈ FS∩EPF. Since z

PFS∩EPFfz, we have

lim sup

n→ ∞

fz−z, xn−z_ilim_{→ ∞}fz−z, xni−z

lim

i→ ∞

fz−z, uni−z

fz−z, w−z≤0.

Fromxn1−zαnfxn−z 1−αnSyn−z, we have

1−α_n2Sy_n−z2≥ x_n1−z2−2αnfxn−z, xn1−z

, 3.41

_yn−z2_β

nxn 1−βnSun−z2

≤βnxn−z21−βnSun−z2

≤βnxn−z21−βnun−z2

≤ xn−z2.

3.42

It follows that

xn1−z2≤1−αn2Syn−z22αnfxn−z, xn1−z

≤1−αn2yn−z22αnfxn−fz, xn1−z

2α_nfz−z, x_n1−z

≤1−α_n2x_n−z22α_nax_n−zx_n1−z

2αnfz−z, xn1−z

≤1−α_n2x_n−z2α_nax_n−z2x_n1−z2

2αnfz−z, xn1−z

.

3.43

Hence

xn1−z2≤ 1−αn 2_α

na

1−α_na xn−z

2 2αn

1−α_na

fz−z, xn1−z

. 3.44

Fromαn → 0, we know that there exists a positive integern0, such that 1>1−αna >1/2 for

alln≥n0. Then

1−αn2αna

1−α_na

1−2αnαna

1−α_na

α2 n

1−α_na

1−21−aαn 1−αna

α2 n

1−αna

≤1−21−aαn2α2n, ∀n≥n0.

Putting above inequality into3.44, we get

xn1−z2≤1−21−aαnxn−z22Mα2n₁₋2α_αn

naσn, ∀n≥n0, 3.46

whereMsup{xn−z2:n∈N}, andσnfz−z, xn1−z.

It follows fromLemma 2.3that

xn−→z∈FS∩EPF. 3.47

It follows fromx_n−u_n → 0 and3.42thatu_n → zandy_n → z.

ByTheorem 3.1, we can obtain the following new result.

Corollary 3.2. LetCbe a nonempty closed convex subset ofH. LetSbe a nonexpansive mapping ofC intoHsuch thatFS/∅.Letfbe a contraction ofHinto itself and let{xn}and{yn}be sequences

generated byx1∈Hand

xn1αnfxn 1−αnSyn,

ynβnxn1−βnSPCxn, ∀n∈N.

3.48

If{αn},{βn} ⊂0,1satisfy the following conditions:

lim

n→ ∞αn0, ∞

n1

αn∞,

∞

n1

|αn1−αn|<∞,

0<lim inf

n→ ∞ βn≤lim sup_{n→ ∞} βn<1, ∞

n1

_{βn1−βn<∞,} 3.49

then,{xn}and{yn}converge strongly toz∈FS,wherezPFSfz.

Proof. PutFx, y 0 for allx, y ∈Candr_n 1 for alln∈NinTheorem 3.1. Then, we get

unPCxn. So fromTheorem 3.1, the sequences{xn}and{yn}converge strongly toz∈FS,

wherezPFSfz.

Remark 3.3. Theorem 3.1andCorollary 3.2, respectively, extend and generalize Theorem 3.2

and Corollary 3.3 in7from the Mann iterative form to the Ishikawa iterative form.

Acknowledgments

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