A Hybrid Projection Algorithm for Finding Solutions of Mixed Equilibrium Problem and Variational Inequality Problem

Volume 2010, Article ID 383740,19pages doi:10.1155/2010/383740

Research Article

A Hybrid Projection Algorithm for Finding

Solutions of Mixed Equilibrium Problem and

Variational Inequality Problem

Filomena Cianciaruso,

_{Giuseppe Marino,}

Luigi Muglia,

_{and Yonghong Yao}

1_{Dipartimento di Matematica, Universit´a della Calabria, 87036 Arcavacata di Rende (CS), Italy}

2_{Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China}

Correspondence should be addressed to Giuseppe Marino,[email protected]

Received 3 June 2009; Accepted 16 September 2009

Academic Editor: Mohamed A. Khamsi

Copyrightq2010 Filomena Cianciaruso 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 propose a modified hybrid projection algorithm to approximate a common fixed point of a

k-strict pseudocontraction and of two sequences of nonexpansive mappings. We prove a strong convergence theorem of the proposed method and we obtain, as a particular case, approximation of solutions of systems of two equilibrium problems.

1. Introduction

In this paper, we define an iterative method to approximate a common fixed point of ak -strict pseudocontraction and of two sequences of nonexpansive mappings generated by two sequences of firmly nonexpansive mappings and two nonlinear mappings. Let us recall from

1that thek-strict pseudocontractions in Hilbert spaces were introduced by Browder and Petryshyn in2.

Definition 1.1. S :C → Cis said to bek-strict pseudocontractiveif there existsk ∈0,1such that

_Sx−Sy2_≤

x−y2kI−Sx−I−Sy2, ∀x, y∈C. 1.1

in the setting of Hilbert spaces or uniformly convex Banach spaces. Although nonexpansive mappings are 0-strict pseudocontractions, iterative methods fork-strict pseudocontractions are far less developed than those for nonexpansive mappings. The reason, probably, is that the second term appearing in the previous definition impedes the convergence analysis for iterative algorithms used to find a fixed point of thek-strict pseudocontractionS. However,

k-strict pseudocontractions have more powerful applications than nonexpansive mappings do in solving inverse problems. In the recent years the study of iterative methods like Mann’s like methods and CQ-methods has been extensively studied by many authors1,10–13and the references therein.

If Cis a closed and convex subset of a Hilbert space H and F : C×C → R is a bi-function we call equilibrium problem

Find x∈C s.t.Fx, y≥0, ∀y∈C, 1.2

and we will indicate the set of solutions withEPF.

IfA : C → His a nonlinear mapping, we can chooseFx, y Ax, y−x, so an equilibrium pointi.e., a point of the setEPFis a solution of variational inequality problem

VIP

Findx∈C s.t.Ax, y−x≥0, ∀y∈C. 1.3

We will indicate withV IC, Athe set of solutions of VIP.

The equilibrium problems, in its various forms, found application in optimization problems, fixed point problems, convex minimization problems; in other words, equilibrium problems are a unified model for problems arising in physics, engineering, economics, and so onsee10.

As in the case of nonexpansive mappings, also in the case ofk-strict pseudocontraction mappings, in the recent years many papers concern the convergence of iterative methods to a solutions of variational inequality problems or equilibrium problems; see example for,

10,14–18.

Here we prove a strong convergence theorem of the proposed method and we obtain, as a particular case, approximation of solutions of systems of two equilibrium problems.

2. Preliminaries

LetHbe a real Hilbert space and letCbe a nonempty closed convex subset ofH. We denote byPC the metric projection ofHontoC. It is well known19that

x−PCx, PCx−y

≥0, ∀x∈Hand y∈C. 2.1

Lemma 2.1. (see [20]) LetXbe a Banach space with weakly sequentially continuous duality mapping

J, and suppose thatxn_n∈Nconverges weakly tox0 ∈X, then for anyx∈X,

lim inf

n→ ∞ xn−x0 ≤lim infn→ ∞ xn−x. 2.2

Recall that a pointu∈Cis a solution of a VIP if and only if

uPCI−λAu ∀λ >0, that is,u∈V IC, A⇐⇒u∈FixPCI−λA, ∀λ >0. 2.3

Definition 2.2. An operator A : C → His said to be α-inverse strongly monotone operatorif there exists a constantα >0 such that

Ax−Ay, x−y≥αAx−Ay2 ∀x, y∈C. 2.4

If α 1 we say that A is firmly nonexpansive. Note that every α-inverse strongly monotone operator is also 1/αLipschitz continuoussee21.

Lemma 2.3. (see [2]). LetCbe a nonempty closed convex subset of a real Hilbert spaceHand let

S : C → Cbe ak-strict pseudocontractive mapping. ThenSt : tI 1−tSwitht ∈ k,1is a

nonexpansive mapping withFixSt FixS.

3. Main Theorem

Theorem 3.1. LetCbe a closed convex subset of a real Hilbert spaceH. Let

iAbe anα-inverse strongly monotone mapping ofCintoH,

iiBaβ-inverse strongly monotone mapping ofCintoH,

iii Tn_n∈NandVn_n∈Ntwo sequences of firlmy nonexpansive mappings fromCtoH.

LetS:C → Cbe ak-strict pseudocontractionFixS/∅.

SetSkkI 1−kSand let us define the sequencexnn∈Nas follows:

x1∈C,

C1C,

unTnI−rnAxn

znVnI−λnBun,

ynαnxn 1−αnSkzn,

Cn1w∈Cn: yn−w≤ xn−w

,

xn1PCn1x1, ∀n∈N,

3.1

where

i αn_n∈N⊂0, awitha <1;

ii λn_n∈N⊂b, c⊂0,2β;

Moreover suppose that

iF:FixS ∩nFixVnI−λnB ∩nFixTnI−rnA/∅;

ii TnI−rnAn∈N pointwise converges in C to an operator R and VnI−λnBn∈N pointwise converges inCto an operatorW;

iiiFixW ∩nFixVnI−λnBandFixR ∩nFixTnI−rnA.

Thenxn_n∈Nstrongly converges tox∗PFx1.

Proof. We begin to observe that the mappingsTnI−rnAandVnI−λnBare nonexpansive

for alln∈Nsince they are compositions of nonexpansive mappingssee22, page 419. As a rule, ifp∈F

_un−p2_≤

xn−p2, _zn−p2_≤

un−p2≤xn−p2.

3.2

Now we divide the proof in more steps.

Step 1. Cnis closed and convex for eachn∈N.

IndeedCn1is the intersection ofCnwith the half space

w∈H:w, xn−yn

≤L, 3.3

whereL xn2− yn2/2.

Step 2. F⊆Cnfor eachn∈N.

For eachw∈Fwe have

yn−wαnxn 1−αnSkzn−w

≤αnxn−w 1−αnzn−w

αnxn−w 1−αnVnI−λnBun−w

≤αnxn−w 1−αnun−w

αnxn−w 1−αnTnI−rnAxn−w

≤αnxn−w 1−αnxn−w

xn−w.

3.4

Step 3. limn→∞xn−x1exists andxnn∈Nis asymptotically regular, that is, limn→∞xn1−

xn0.

Sincexn PCnx1, xn1 PCn1x1, andCn1⊆Cn, by2.1choosingyxn1, xx1and

CCn, we have

0≤ x1−xn, xn−xn1

x1−xn, xn−x1x1−xn1

≤ −x1−xn2x1−xnx1−xn1,

3.5

that is,xn−x1 ≤ xn1−x1.

ByxnPCnx1andF⊆Cn, we have

x1−xn ≤ x1−PFx1. 3.6

Then limn→∞xn−x1exists andxnn∈Nis bounded. Moreover

xn1−xn2 xn1−x1x1−xn2

xn1−x12xn−x122xn1−x1, x1−xn

xn1−x12xn−x122xn1−xn, x1−xn −2xn−x12

≤ xn1−x12− xn−x12 by3.5,

3.7

and consequently limn→∞xn1−xn0.

Step 4. limn→∞xn−yn0 and limn→∞xn−Skzn0.

Byxn1∈Cn1, it follows

yn−xn1≤ xn−xn1,

yn−xn≤yn−xn1xn1−xn ≤2xn1−xn −→0.

3.8

Moreover

_{yn−xn 1−αnxn−Skzn, 3.9}

Step 5. limn→∞Bun−Bw0, for eachw∈F.

Forw∈F, we have

_yn−w2 _≤

αnxn−w2 1−αnSkzn−w2

≤αnxn−w2 1−αnzn−w2

≤αnxn−w2 1−αnVnI−λnBun−VnI−λnBw2

≤αnxn−w2 1−αnI−λnBun−I−λnBw2

αnxn−w2 1−αn

un−w2λ2nBun−Bw2−2λnBun−Bw, un−w

≤αnxn−w2 1−αn

un−w2−λn

2β−λn

Bun−Bw2

≤ xn−w2 1−αnλn

λn−2β

Bun−Bw2.

3.10

Consequently

1−αnλn

2β−λn

Bun−Bw2≤ xn−w2−yn−w2

xn−w −yn−wxn−wyn−w

≤xn−ynxn−wyn−w,

3.11

and byStep 4, the assumptions onαn_n∈Nandλn_n∈N, we obtain the claim ofStep 5. Step 6. limn→∞un−zn0.

SinceVnis firmly nonexpansive, for anyw∈F, we have

zn−w2≤ I−λnBun−I−λnBw, zn−w

1

4

I−λnBun−I−λnBw zn−w2

−I−λnBun−I−λnBw−zn−w2

≤ 1

4

un−w2−λn

2β−λn

Bun−Bw2zn−w2

−un−zn−λnBun−Bw2

≤ 1

4

un−w2zn−w2− un−zn−λnBun−Bw2

1

4

un−w2zn−w2− un−zn2

2λnun−zn, Bun−Bw −λ2nBun−Bw2

which implies

3zn−w2≤ un−w2− un−zn22λnun−zn, Bun−Bw

≤ xn−w2− un−zn22λnun−zn Bun−Bw.

3.13

Consequently

_yn−w2_≤

αnxn−w2 1−αnzn−w2

≤ xn−w2−1−αnun−zn221−αnλnun−znBun−Bw

3.14

which implies

1−αnun−zn2≤ xn−w2−yn−w221−αnλnun−zn Bun−Bw

≤xn−w −yn−wxn−wyn−w

21−αnλnun−zn Bun−Bw

≤xn−ynxn−wyn−w21−αnλnun−zn Bun−Bw.

3.15

By the assumptions onαn_n∈N, Steps4and6, and the boundedness ofxn_n∈N yn_n∈Nand

un_n∈Nthe claim follows.

Step 7. limn→∞xn−un0 and limn→∞xn−Skxn0.

SinceTnis firmly nonexpansive, for eachp∈ ∩nFixTnI−rnA, we have

un−p2TnI−rnAxn−TnI−rnAp2

≤un−p,I−rnAxn−I−rnAp

1

2

I−rnAxn−I−rnAp2un−p2

−I−rnAxn−I−rnAp−

un−p2

1

2

xn−p2−rn2α−rnAxn−Ap2un−p2

−xn−un−rn

Axn−Ap2

≤ 1

2

xn−p2un−p2− xn−un2

−r2

nAxn−Ap 2

2rn

xn−un, Axn−Ap

,

and consequently

_un−p2 _≤

xn−p2− xn−un22rnxn−unAxn−Ap. 3.17

Then, for eachw∈F, we have

_yn−w2_≤

αnxn−w2 1−αnun−w2

≤ xn−w2−1−αnxn−un2

21−αnrnxn−un Axn−Aw by3.17,

3.18

consequently

1−αnxn−un2≤ xn−w2−yn−w221−αnrnxn−unAxn−Aw

≤xn−ynxn−wyn−w21−αnrnxn−unAxn−Aw,

3.19

and by the assumptions onαn_n∈N,Step 4and the boundedness ofxn_n∈N and yn_n∈Nit follows thatxn−un → 0 asn → ∞. ByStep 6we note that alsoxn−zn → 0.

Finally

xn−Skxn ≤ xn−SkznSkzn−Skxn

≤ xn−Skznzn−xn

≤ xn−Skznzn−unun−xn,

3.20

and by previous steps, it follows thatxn−Skxn → 0 asn → ∞.

Step 8. The set of weak cluster points ofxn_n∈Nis contained inF. We will use three times the Opial’sLemma 2.1.

Letpbe a weak cluster point ofxn_n∈Nand letxnj_j∈Nbe a subsequence of xnn∈N such thatxnj p.

We prove thatp ∈ FixS FixSk. We suppose for absurd that p /Skp. By Opial’s Lemma 2.1andxn−Skxn → 0 asn → ∞, we obtain

lim inf

j→∞ xnj−p

<lim inf

j→∞

xnj−Skp

lim inf

j→∞

xnj−Skxnj−Skxnj−Skp

≤lim inf

j→∞ _x

nj−Skxnj

Skxnj−Skp

lim inf

j→∞ xnj−p

3.21

Since FixR ∩nFixTnI−rnAit is enough to prove thatp∈FixR. Now ifp /Rp

we note that

lim inf

j→∞ xnj−p

<lim inf

j→∞

xnj−Rp

≤lim inf

j→∞ _x

nj−Tnj

I−rnjA

xnj

Tnj

I−rnjA

xnj−Tnj

I−rnjA

pTnj

I−rnjA

p−Rp

≤lim inf

j→∞ _x

nj−unj

xnj−p

Tnj

I−rnjA

p−Rp

lim inf

j→∞ xnj−p

.

3.22

This leads to a contraddiction again. By the hypotheses andStep 7the claim follows. By the same idea and usingStep 6, we prove thatp∈FixW ∩nFixVnI−λnB.

Step 9. xn → x∗PFx1.

Sincex∗PFx1∈CnandxnPCnx1, we have

x1−xn ≤ x1−x∗. 3.23

Letxnj_j∈Nbe a subsequence ofxnn∈Nsuch thatxnj p. ByStep 8,p∈F. Thus

x1−x∗ ≤x1−p≤lim inf j→∞

x1−xnj

≤lim sup

j→∞ x1−xnj

≤ x1−x∗.

3.24

Therefore we have

x1−x∗x1−p lim

j→∞ x1−xnj

. 3.25

SinceHhas the Kadec-Klee property, thenxnj → pasj → ∞.

Moreover, byx1 −x∗ x1 −pand by the uniqueness of the projectionPFx1, it

follows thatpx∗PFx1.

Thence every subsequencexnj_j∈N converges tox∗ as j → ∞ and consequently

xn → x∗, asn → ∞.

Remark 3.2. Let us observe that one can choose Tn_n∈N and Vn_n∈N as sequences of γn

-inverse strongly monotone operators andηn-inverse strongly monotone operators provided

The hypotheses iiand iii in the main Theorem 3.1seem very strong but, in the sequel, we furnish two cases in whichiiandiiiare satisfied.

Let us remember that the metric projection on a convex closed set PC is a firmly

nonexpansive mappingsee19so we claim that have the following proposition.

Proposition 3.3. If rn_n∈N ⊂ 0,∞ is such thatlimnrn r > 0 and A anα-inverse strongly

monotone, thenPCI−rnArealizes conditions (ii) and (iii) withRPCI−rA.

Proof. To proveiiwe note that for eachx∈C,

PCI−rnAx−PCI−rAx ≤ I−rnAx−I−rAx ≤ |rn−r|Ax. 3.26

Moreover,iiifollows directly by2.2.

Now we consider the mixed equilibrium problem

Findx∈C:fx, yhx, yAx, y−x≥0, ∀y∈C. 3.27

In the sequel we will indicate withMEPf, h, Athe set of solution of our mixed equilibrium problem. IfA0 we denoteMEPf, h,0withMEPf, h.

We notice that forh0 andA0 the problem is the well-known equilibrium problem

23–25. Ifh0 andAis anα-inverse strongly monotone operator we have the equilibrium problems studied firstly in26and then in18,22,27. Ifhx, y ϕy−ϕxandA0 we refound the mixed equilibrium problem studied in16,28,29.

Definition 3.4. A bi-functiong:C×C → Ris monotone ifgx, ygy, x≤0 for allx, y∈C. A functionG:C → Ris upper hemicontinuous if

lim sup

t→0

Gtx 1−ty≤Gy. _3.28

Next lemma examines the case in whichA0.

Lemma 3.5. LetCbe a convex closed subset of a Hilbert spaceH. Letf:C×C → Rbe a bi-function such that

f1fx, x 0for allx∈C;

f2fis monotone and upper hemicontinuous in the first variable;

f3fis lower semicontinuous and convex in the second variable.

Leth:C×C → Rbe a bi-function such that

h1hx, x 0for allx∈C;

h2his monotone and weakly upper semicontinuous in the first variable;

Moreover let us suppose that

Hfor fixedr > 0andx ∈C, there exists a bounded setK ⊂Canda∈ Ksuch that for all

z∈C\K,−fa, z hz, a 1/ra−z, z−x<0,

forr >0andx∈HletTr :H → Cbe a mapping defined by

Trx

z∈C:fz, yhz, y 1 r

y−z, z−x≥0, ∀y∈C

, 3.29

called resolvent offandh. Then

1Trx /∅;

2Trxis a single value; 3Tr is firmly nonexpansive;

4MEPf, h FixTrand it is closed and convex.

Proof. Letx0 ∈H. For anyy∈Cdefine

Gr,x0y

z∈C:−fy, zhz, y1 r

y−z, z−x≥0

. 3.30

We will prove that, by KKM’s lemma,∩y∈CGr,x0yis nonempty.

First of all we claim thatGr,x0is a KKM’s map. In fact if there exists{y1, . . . , yN} ⊂ C

such thaty_iαiyiwithiαi1does not appartiene toGr,x0yifor anyi1, . . . , Nthen

−fyi, y

hy, yi

1

r

yi−y, y−x0

<0, ∀i. 3.31

By the convexity offandhand the monotonicity off, we obtain that

0fy, yhy, y1 r

y−y, y−x0

≤

i

αif

y, yi

i

αih

y, yi 1 r i αi

yi−y, y−x0

≤ −

i

αif

yi, y

i

αih

y, yi 1 r i αi

yi−y, y−x0

i

αi

−fyi, y

hy, yi

1

r

yi−y, y−x0

<0,

3.32

Now we prove thatGr,x0

w

Gr,x0. We recall that, by the weak lower semicontinuity of · 2_{, the relation}

lim sup

m

y−zm, zm−x0

≤y−z, z−x0

3.33

holds. Letz∈Gr,x0y

w

and letzm_mbe a sequence inGr,x0ysuch thatzm z.

We want to prove that

−fy, zhz, y1 r

y−z, z−x0

≥0. 3.34

Sincef is lower semicontinuous and convex in the second variable andhis weakly upper semicontinuous in the first variable, then

0≤lim sup

m

−fy, zm

hzm, y

1

r

y−z, z−x0

≤lim sup

m

−fy, zm

lim sup

m

hzm, y

1

r lim supm

y−z, z−x0

≤ −lim inf

m f

y, zm

lim sup

m

hzm, y

1

r lim sup_m

y−z, z−x0

≤ −fy, zhz, y1

r

y−z, z−x0

.

3.35

Now we observe thatGr,x0y

w

Gr,x0yis weakly compact for at least a pointy∈C. In

fact by hypothesisHthere exist a boundedK ⊂Canda∈K, such that for allz ∈ C\K

it resultsz /∈Gr,x0a. ThenGr,x0a ⊂ K, that is, it is bounded. It follows thatGr,x0ais weakly

compact. Then by KKM’s lemma∩y∈CGr,x0yis nonempty. However ifz∈ ∩y∈CGr,x0then

−fy, zhz, y1 r

y−z, z−x0

≥0, ∀y∈C. 3.36

As in24, Lemma 3, sincef is upper hemicontinuous and convex in the first variable and monotone, we obtain that3.36is equivalent to claim thatzis such that

fz, yhz, y1 r

y−z, z−x0

≥0, ∀y∈C, 3.37

that is,z∈Trx0. This prove1. To prove2and3we considerz1 ∈Trx1andz2 ∈Trx2.

They satisfy the relations

fz1, z2 hz1, z2 1

rz2−z1, z1−x1 ≥0, fz2, z1 hz2, z1 1

rz1−z2, z2−x2 ≥0.

By the monotonicity offandh, summing up both the terms,

0≤ 1

rz2−z1, z1−x1 − z2−z1, z2−x2

1

rz2−z1, z1−x1−z2x2

1

r

−z2−z12z2−z1, x2−x1

3.39

so we conclude

z2−z12≤ z2−z1, x2−x1 3.40

that means simultaneously thatz1z2ifx1x2andTr is firmly nonexpansive.

To prove4, it is enough to followiiiandivin25, Lemma 2.12.

Remark 3.6. We note that ifh 0, our lemma reduces to25, Lemma 2.12. The coercivity conditionHis fulfilled.

Moreover our lemma is more general than16, Lemma 2.2. In fact

iour hypotheses onf are weakerf weak upper semicontinuous impliesf upper hemicontinuous;

iiifϕsatisfies the condition in Lemma 2.2 , choosinghx, y ϕy−ϕxone has thathis concave and upper semicontinuous in the first variable and convex and lower semicontinous in the second variable;

iiithe coercivity conditionHby the equivalence of3.36and3.37is the same.

Lemma 3.7. Let us suppose that (f1)–(f3), (h1)–(h3) and (H) hold. Letx, y∈H,r1, r2>0. Then

_Tr

2y−Tr1x≤y−x

r2−r1

r2

Tr2y−y. 3.41

Proof. ByLemma 3.5, definingu1Tr1xandu2:Tr2y, we know that

fu2, z hu2, z 1

r2

z−u2, u2−y ≥0, ∀z∈C,

fu1, z hu1, z 1

r1

z−u1, u1−x ≥0, ∀z∈C.

3.42

In particular,

fu2, u1 hu2, u1 1

r2

u1−u2, u2−y

≥0,

fu1, u2 hu1, u2

1

r1

u2−u1, u1−x ≥0.

Hence, summing up this two inequalities and using the monotonicity offandh,

u2−u1,

u1−x

r1 −

u2−y

r2

≥0. 3.44

We derive from3.44that

u2−u1, u1−u2−xu2−

r1

r2

u2−y

≥0, 3.45

and so

−u2−u12

u2−u1,

u2−y

1−r1

r2

y−x≥0. 3.46

Then,

u2−u12≤ u2−u1

y−x1− r1

r2

u2−y

, 3.47

and thus the claim holds.

Proposition 3.8. Let us suppose that f and h are two bi-functions satisfying the hypotheses of Lemma 3.5. Let Tr be the resolvent off and h. LetAbe anα-inverse strongly monotone operator.

Let us suppose thatrn_n∈N⊂0,∞is such thatlimnrn r >0. ThenTrnI−rnArealize (ii) and (iii) inTheorem 3.1.

Proof. Letxbe in a bounded closed convex subsetKofC. To proveiit is enough to observe that byLemma 3.7

TrnI−rnAx−TrI−rAx ≤ |rn−r|Ax

|rn−r|

r TrI−rAx−I−rAx. 3.48

Whenn → ∞, by boundedness of the terms that do not depend onn, we obtainii.

To proveiii letW TrI−rAthe pointwise limit ofTrnI−rnA. It is necessary to prove only that FixW ⊂ ∩nFixTrnI −rnA. Letx ∈ FixW. We want to prove that

x∈MEPf, h, A. Letwn TrnI−rnAx. Thus, by definition ofTrn,wnis the unique point such that

fwn, y

hwn, y

1

rn

y−wn, wn−I−rnAx

≥0, ∀y. 3.49

By monotonicity offandhthis implies

hwn, y

1

rn

y−wn, wn−I−rnAx

≥fy, wn

Passing to the limit onn, byf3andh2we obtain

hx, yy−x, Ax≥fy, x, ∀y. 3.51

Let nowuty 1−txwitht∈0,1. Then by the convexity offandh

0fu, u hu, u≤tfu, yhu, y 1−tfu, x hu, x

≤tfu, yhu, yu−x, Ax

tfu, yhu, yy−x, Ax.

3.52

Passingt → 0we obtain byf1andh1

fx, yhx, yAx, y−x≥0. 3.53

That is,x∈MEPf, h, A. At this point we observe that from the definitions ofMEPf, h, A

andTrn, one hasMEPf, h, A FixTrnI−rnA.

By Propositions 3.3 and 3.8 we can exhibit iterative methods to approximate fixed points of thek-strict pseudo contraction that are also

1solution of a system of two variational inequalities VIC,Aand VIC,B VnTn

PC;

2solution of a system of two mixed equilibrium problemsTnTrn andVnTλn;

3solution of a mixed equilibrium problem and a variational inequalityTnTrnand

VnPC.

However when the properties of the mapping Tn and Vn are well known, one can

prove convergence theorems likeTheorem 3.1without use of Opial’s lemma.

In next theorem our purpose is to prove a strong convergence theorem to approximate a fixed point ofSthat is also a solution of a mixed equilibrium problem and a solution of a variational inequalityV IC, B. One can note that we relax the hypotheses on the convergence of the sequencesrn_n∈Nandλn_n∈N.

Theorem 3.9. LetCbe a closed convex subset of a real Hilbert spaceH, letf, h:C×C → Rbe two bi-functions satisfying (f1)–(f3),(h1)–(h3), and (H). LetS:C → Cbe ak-strict pseudocontraction.

LetAbe anα-inverse strongly monotone mapping ofCintoHand letBbe aβ-inverse strongly monotone mapping ofCintoH.

SetSkkI 1−kS, one defines the sequencexnn∈Nas follows:

x1∈C,

C1C,

fun, y

hun, y

1

rn

y−un, un−xn

Axn, y−un

≥0,

znPCI−λnBun,

ynαnxn 1−αnSkzn,

Cn1w∈Cn: yn−w≤ xn−w

, xn1PCn1x1, ∀n∈N,

3.54

where

i αn_n∈N⊂0, awitha <1;

ii λn_n∈N⊂b, c⊂0,2β;

iii rn_n∈N⊂d, e⊂0,2α.

Thenxn_n∈Nstrongly converges tox∗PFx1.

Proof. First of all we observe that byLemma 3.5we have thatun TrnI−rnAxn. We can follow the proof ofTheorem 3.1from Steps1–7. We prove only the following.

Step 10. The set of weak cluster points ofxn_n∈Nis contained inF.

Letpbe a cluster point ofxn; we begin to prove thatp∈MEPf, h, A. We know that

fun, y

hun, y

Axn, y−un

1

rn

y−un, un−xn

≥0, ∀y∈C, 3.55

and byf2

hun, y

Axn, y−un

1

rn

y−un, un−xn

≥fy, un

, ∀y∈C. 3.56

Letxnj_j∈Nbe a subsequence ofxnn∈Nweakly convergent top, then byStep 7unj pas

j → ∞. Letρt:ty 1−tp, t∈0,1. Then by3.56

ρt−unj, Aρt

ρt−unj, Aρt−Axnj

Axnj, ρt−unj

≥ρt−unj, Aρt−Axnj

fy, unj

−hunj, y

− 1

rnj

y−unj, unj−xnj

ρt−unj, Aρt−Aunj

ρt−unj, Aunj−Axnj

fy, unj

−hunj, y

− 1

rnj

y−unj, unj−xnj

≥ρt−unj, Aunj−Axnj

fy, unj

−hunj, y

− 1

rnj

y−unj, unj−xnj

.

SinceAis Lipschitz continuous andunj−xnj → 0 asj → ∞, we haveAunj−Axnj → 0 asj → ∞.

By conditionf3, forx ∈ H fixed, the functionfx,·is lower semicontinuos and convex, and thus weakly lower semicontinuous30.

Sincexn−un → 0, asn → ∞and by the assumption onrnwe obtainunj−xnj/rnj → 0. Then we obtain byh2

ρt−p, Aρt

≥fy, p−hp, y. 3.58

Usingf1,f3,h1,h3we obtain

0fρt, ρt

hρt, ρt

≤tfρt, y

1−tfρt, p

thρt, y

1−thρt, p

≤tfρt, y

thρt, y

1−tfρt, p

−hp, ρt

≤tfρt, y

thρt, y

1−tρt−p, Aρt

tfρt, y

hρt, y

1−ty−p, Aρt

.

3.59

Consequently

fρt, y

hρt, y

1−ty−p, Aρt

≥0 3.60

byf2andh2, ast → 0, we obtainp∈MEPf, h, A. Now we prove thatp∈V IC, B.

We define the maximal monotone operator

Tx

BxNCx, ifx∈C,

∅, se x /∈C, 3.61

whereNCxis the normal cone toCatx, that is,

NCx{w∈H:x−u, w ≥0,∀u∈C}. 3.62

Sincezn∈C, by the definition ofNCwe have

x−zn, y−Bx

≥0. 3.63

ButznPCI−λnBun, then

x−zn, zn−I−λnBun ≥0, 3.64

and hence

x−zn,

zn−un

λn

Bun

By3.63,3.65, and by theβ-inverse monotonicity ofB, we obtain

x−znj, y

≥x−znj, Bx

≥x−znj, Bx

− x−znj,

znj−unj

λnj

Bunj

!

x−znj, Bx−Bznj

x−znj, Bznj−Bunj

− x−znj,

znj−unj

λnj

!

.

3.66

Byxn−zn → 0 asn → ∞immediately consequence of Steps6and7, it follows that

znj pasj → ∞. Then

x−p, y≥0, 3.67

moreover, sinceTis a maximal operator, 0∈Tp, that is,p∈V IC, B.

Finally, to prove thatp∈FixS FixSkwe followStep 8as inTheorem 3.1. Since alsoStep 9can be followed as inTheorem 3.1, we obtain the claim.

References

1 G. L. Acedo and H.-K. Xu, “Iterative methods for strict pseudo-contractions in Hilbert spaces,”

Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 7, pp. 2258–2271, 2007.

2 F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,”Journal of Mathematical Analysis and Applications, vol. 20, pp. 197–228, 1967.

3 F. E. Browder, “Nonexpansive nonlinear operators in a Banach space,”Proceedings of the National Academy of Sciences of the United States of America, vol. 54, pp. 1041–1044, 1965.

4 K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,”

Proceedings of the American Mathematical Society, vol. 35, pp. 171–174, 1972.

5 W. A. Kirk, “A fixed point theorem for mappings which do not increase distances,”The American Mathematical Monthly, vol. 72, pp. 1004–1006, 1965.

6 Q. Liu, “Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings,”Nonlinear Analysis: Theory, Methods & Applications, vol. 26, no. 11, pp. 1835–1842, 1996.

7 J. Schu, “Iterative construction of fixed points of asymptotically nonexpansive mappings,”Journal of Mathematical Analysis and Applications, vol. 158, no. 2, pp. 407–413, 1991.

8 H. K. Xu, “Inequalities in Banach spaces with applications,”Nonlinear Analysis: Theory, Methods & Applications, vol. 16, no. 12, pp. 1127–1138, 1991.

9 H. K. Xu, “Existence and convergence for fixed points of mappings of asymptotically nonexpansive type,”Nonlinear Analysis: Theory, Methods & Applications, vol. 16, no. 12, pp. 1139–1146, 1991. 10 L.-C. Ceng, S. Al-Homidan, Q. H. Ansari, and J.-C. Yao, “An iterative scheme for equilibrium

problems and fixed point problems of strict pseudo-contraction mappings,”Journal of Computational and Applied Mathematics, vol. 223, no. 2, pp. 967–974, 2009.

11 G. Marino and H.-K. Xu, “Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces,”Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 336–346, 2007. 12 G. Marino, V. Colao, X. Qin, and S. M. Kang, “Strong convergence of the modified Mann iterative

method for strict pseudo-contractions,”Computers and Mathematics with Applications, vol. 57, no. 3, pp. 455–465, 2009.

13 H. Zhou, “Convergence theorems of fixed points forκ-strict pseudo-contractions in Hilbert spaces,”

14 V. Colao, G. Marino, and H.-K. Xu, “An iterative method for finding common solutions of equilibrium and fixed point problems,”Journal of Mathematical Analysis and Applications, vol. 344, no. 1, pp. 340– 352, 2008.

15 A. Moudafi, “Viscosity approximation methods for fixed-points problems,”Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.

16 J. W. Peng, “Iterative algorithms for mixed equilibrium problems, strictly pseudocontractions and monotone mappings,”Journal of Optimization Theory and Applications. In press.

17 J. W. Peng, Y. C. Liou, and J. C. Yao, “An iterative algorithm combining viscosity method with parallel method for a generalized equilibrium problem and strict pseudocontractions,”Fixed Point Theory and Applications, vol. 2009, Article ID 794178, 21 pages, 2009.

18 Y. J. Cho, X. Qin, and J. I. Kang, “Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems,”Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 9, pp. 4203–4214, 2009.

19 F. E. Browder, “Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces,”Archive for Rational Mechanics and Analysis, vol. 24, pp. 82–90, 1967.

20 Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,”Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967.

21 W. Takahashi and M. Toyoda, “Weak convergence theorems for nonexpansive mappings and monotone mappings,”Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 417–428, 2003.

22 S. Takahashi and W. Takahashi, “Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space,”Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 3, pp. 1025–1033, 2008.

23 M. Bianchi and S. Schaible, “Generalized monotone bifunctions and equilibrium problems,”Journal of Optimization Theory and Applications, vol. 90, no. 1, pp. 31–43, 1996.

24 E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,”The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.

25 P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 117–136, 2005.

26 A. Moudafi, “Weak convergence theorems for nonexpansive mappings and equilibrium problems,”

Journal of Nonlinear and Convex Analysis, vol. 9, no. 1, pp. 37–43, 2008.

27 S. Wang, H. Zhou, and J. Song, “Viscosity approximation methods for equilibrium problems and fixed point problems of nonexpansive mappings and inverse-strongly monotone mappings,”Methods and Applications of Analysis, vol. 14, no. 4, pp. 405–419, 2007.

28 L.-C. Ceng and J.-C. Yao, “A hybrid iterative scheme for mixed equilibrium problems and fixed point problems,”Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 186–201, 2008. 29 O. Chadli, Z. Chbani, and H. Riahi, “Equilibrium problems with generalized monotone bifunctions

and applications to variational inequalities,”Journal of Optimization Theory and Applications, vol. 105, no. 2, pp. 299–323, 2000.