An Application of Hybrid Steepest Descent Methods for Equilibrium Problems and Strict Pseudocontractions in Hilbert Spaces

Volume 2011, Article ID 173430,15pages doi:10.1155/2011/173430

Research Article

An Application of Hybrid Steepest Descent Methods for Equilibrium Problems and Strict Pseudocontractions in Hilbert Spaces

Ming Tian

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

Correspondence should be addressed to Ming Tian,[email protected] Received 9 December 2010; Accepted 13 February 2011

Academic Editor: Shusen Ding

Copyrightq2011 Ming Tian. 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 use the hybrid steepest descent methods for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a strict pseudocontraction mapping in the setting of real Hilbert spaces. We proved strong convergence theorems of the sequence generated by our proposed schemes.

1. Introduction

LetHbe a real Hilbert space andCa closed convex subset ofH, and letφbe a bifunction of C×CintoR, whereRis the set of real numbers. The equilibrium problem forφ:C×C → R is to findx∈Csuch that

EP :φ x, y

≥0 ∀y∈C 1.1

denoted the set of solution by EPφ. Given a mappingT :C → H, letφx, y Tx, y−x for allx, y ∈C, thenz ∈ EPφif and only ifTz, y−z ≥ 0 for all y ∈ C, that is, z is a solution of the variational inequality. Numerous problems in physics, optimizations, and economics reduce to find a solution of1.1. Some methods have been proposed to solve the equilibrium problem, see, for instance,1,2.

A mappingTofCinto itself is nonexpansive ifTx−Ty ≤ x−y, for allx, y∈C. The set of fixed points ofTis denoted byFT. In 2007, Plubtieng and Punpaeng3, S. Takahashi and W. Takahashi4, and Tada and W. Takahashi5considered iterative methods for finding an element of EPφ∩FT.

Recall that an operatorAis strongly positive if there exists a constantγ >0 with the property

Ax, x ≥γx², ∀x∈H. 1.2

In 2006, Marino and Xu6introduced the general iterative method and proved that for a givenx0 ∈H, the sequence{xn}is generated by the algorithm

xn1αnγfxn I−αnATxn, n≥0, 1.3 whereTis a self-nonexpansive mapping onH,fis a contraction ofHinto itself withβ∈0,1 and {α_n} ⊂ 0,1 satisfies certain conditions, andA is a strongly positive bounded linear operator onHand converges strongly to a fixed-pointx^∗ofT which is the unique solution to the following variational inequality:

γf−Ax^∗, x−x^∗ ≤ 0, forx ∈FT, and is also the optimality condition for some minimization problem. A mappingS : C → H is said to bek-strictly pseudocontractive if there exists a constantk∈0,1such that

Sx−Sy² ≤x−y²kI−Sx−I−Sy², ∀x, y∈C. 1.4 Note that the class ofk-strict pseudo-contraction strictly includes the class of nonex- pansive mapping, that is,Sis nonexpansive if and only ifSis 0-srictly pseudocontractive; it is also said to be pseudocontractive ifk1. Clearly, the class ofk-strict pseudo-contractions falls into the one between classes of nonexpansive mappings and pseudo-contractions.

The set of fixed points ofS is denoted by FS. Very recently, by using the general approximation method, Qin et al.7obtained a strong convergence theorem for finding an element ofFS. On the other hand, Ceng et al.8proposed an iterative scheme for finding an element of EPφ∩FSand then obtained some weak and strong convergence theorems.

Based on the above work, Y. Liu9introduced two iteration schemes by the general iterative method for finding an element of EPφ∩FS.

In 2001, Yamada10introduced the following hybrid iterative method for solving the variational inequality:

xn1Txn−μλnFTxn, n≥0, 1.5 whereFisk-Lipschitzian andη-strongly monotone operator withk >0,η >0, 0< μ <2η/k², then he proved that if {λn} satisfyies appropriate conditions, the {xn} generated by 1.5 converges strongly to the unique solution of variational inequality

Fx, x −x ≥ 0, ∀x∈FixT, x∈FixT. 1.6 Motivated and inspired by these facts, in this paper, we introduced two iteration methods by the hybrid iterative method for finding an element of EPφ∩FS, whereS : C → H is ak-strictly pseudocontractive non-self mapping, and then obtained two strong convergence theorems.

2. Preliminaries

Throughout this paper, we always assume thatCis a nonempty closed convex subset of a Hilbert spaceH. We writexn xto indicate that the sequence{xn}converges weakly to x.xn → ximplies that{xn}converges strongly tox. For anyx∈ H, there exists a unique nearest point inC, denoted byP_Cx, such that

x−PCx ≤x−y, ∀y∈C. 2.1 Such aPCxis called the metric projection ofHontoC. It is known thatPCis nonexpansive.

Furthermore, forx∈Handu∈C,up_cx,⇔ x−u, u−y ≥0, for ally∈C.

It is widely known thatH satisfies Opial’s condition11, that is, for any sequence {x_n}withx_n x, the inequality

lim inf

n→ ∞ xn−x<lim inf

n→ ∞ xn−y, 2.2

holds for everyy∈Hwithy /x. In order to solve the equilibrium problem for a bifunction φ:C×C → R, let us assume thatφsatisfies the following conditions:

A1φx, x 0, for allx∈C,

A2φis monotone, that is,φx, y φy, x≤0, for allx, y∈C, A3For allx, y, z∈C.

limt↓0φ

tz 1−tx, y

≤φ x, y

; 2.3

A4For each fixedx∈C, the functiony→φx, yis convex and lower semicontinuous.

Let us recall the following lemmas which will be useful for our paper.

Lemma 2.1see12. Letφbe a bifunction fromC×CintoRsatisfying (A1), (A2),(A3) and (A4) then, for anyr >0 andx∈H, there existsz∈Csuch that

φ z, y

1

ry−z, z−x ≥0, ∀y∈C. 2.4 Further, ifTrx{z∈C:φz, y 1/ry−z, z−x ≥0,∀y∈C}, then the following hold:

1T_r is single-valued,

2T_r is firmly nonexpansive, that is, T_rx−T_ry² ≤

T_rx−T_ry, x−y

, ∀x, y∈H; 2.5

3FTr EPφ,

4EPφis nonempty, closed and convex.

Lemma 2.2see13. IfS:C → His ak-strict pseudo-contraction, then the fixed-point setFS is closed convex, so that the projectionP_FSis well difened.

Lemma 2.3see14. LetS : C → H be ak-strict pseudo-contraction. DefineT :C → Hby Tx λx 1−λSx for eachx ∈ C, then, asλ ∈ k,1, T is nonexpansive mapping such that FT FS.

Lemma 2.4see15. In a Hilbert spaceH, there holds the inequality xy² ≤ x²2

y, xy

, ∀x, y∈H. 2.6

Lemma 2.5see16. Assume that{an}is a sequence of nonnegative real numbers such that

a_n1≤ 1−γ_n

a_nγ_nδ_n, n≥0, 2.7

where{γ_n}is a sequence in (0,1) and{δ_n}is a sequence in^Ê, such that i_∞

n1γ_n∞,

iilim sup_{n→ ∞}δ_n≤0 or_∞

n1|δ_nγ_n|<∞.

Then lim_{n→ ∞}an0.

3. Main Results

Throughout the rest of this paper, we always assume thatFis aL-lipschitzian continuous and η-strongly monotone operator withL, η >0 and assume that 0< μ <2η/L².τ μη−μL²/2.

Let {Tλn} be mappings defined asLemma 2.1. Define a mappingSn : C → H bySnx β_nx 1−β_nSx, for allx∈C, whereβ_n∈k,1, then, byLemma 2.3,S_nis nonexpansive. We consider the mappingGnonHdefined by

G_nx

I−α_nμF

S_nT_λnx, x∈H, n∈N, 3.1

whereα_n∈0,1. By Lemmas2.1and 2.3, we have

Gnx−Gny≤1−αnτTλnx−Tλny

≤1−α_nτx−y. 3.2 It is easy to see thatGnis a contraction. Therefore, by the Banach contraction principle, G_nhas a unique fixed-pointx^F_n ∈Hsuch that

x^F_n

I−αnμF

SnTλnx^F_n. 3.3

For simplicity, we will writexnforx^F_n provided no confusion occurs. Next, we prove that the sequence{x_n}converges strongly to aq∈FS∩EPφwhich solves the variational inequality

Fq, p−q

≥0, ∀p∈FS∩EP φ

. 3.4

Equivalently,qPFS∩EPφI−μFq.

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H and φ a bifunction from C×C into R satisfying (A1), (A2), (A3), and (A4). Let S : C → H be a k- strictly pseudocontractive nonself mapping such thatFS∩EPφ/φ. LetF : H → H be an L-Lipschitzian continuous andη-strongly monotone operator onHwithL, η >0 and 0< μ <2η/L², τ μη−μL²/2. Let{x_n}be asequence generated by

φ un, y

1

λny−un, un−xn ≥0, ∀y∈C, y_nβ_nu_n

1−β_n Su_n, xn

I−αnμF

yn, ∀n∈N,

3.5

whereun Tλnxn,yn Snun, and{λn} ⊂ 0,∞satisfy lim inf_{n→ ∞}λn > 0 if{αn} and{βn} satisfy the following conditions:

i{αn} ⊂0,1, limn→ ∞αn0,

ii0≤k≤βn≤λ <1 and lim_{n→ ∞}βnλ,

then{x_n} converges strongly to a pointq ∈ FS∩EPφwhich solves the variational inequality 3.4.

Proof. First, takep ∈FS∩EPφ. Sinceun Tλnxnandp Tλnp, fromLemma 2.1, for any n∈N, we have

un−pTλnxn−Tλnp≤xn−p. 3.6

Then, sinceS_npp, we obtain that

yn−pSnun−Snp≤un−p≤xn−p. 3.7

Further, we have

xn−p−αnμFp

I−μαnF yn−

I−μαnF p

≤αn−μF

p 1−αnτyn−p. 3.8 It follows thatx_n−p ≤ μFp/τ.

Hence,{xn}is bounded, and we also obtain that{un}and{yn}are bounded. Notice that

u_n−y_n≤ un−x_nx_n−y_n

un−xnαn−μFyn. 3.9 ByLemma 2.1, we have

u_n−p²T_λnx_n−T_λnp²≤

x_n−p, u_n−p 1

2 xn−p²un−p²− un−xn²

. 3.10

It follows that

un−p² ≤xn−p²− xn−un². 3.11 Thus, fromLemma 2.4,3.7, and3.11, we obtain that

xn−p²αn

−μFp

I−μαnF yn−

I−μαnF p²

≤1−α_nτ²y_n−p²2α_n

−μFp, x_n−p

≤1−αnτ²un−p²2αn

−μFp, xn−p

≤1−α_nτ² x_n−p²− xn−u_n²

2α_n−μFpx_n−p 1−2αnτ αnτ²xn−p²

−1−αnτ²xn−un²2αn −μFpxn−p

≤x_n−p² αnτ²x_n−p²−1−α_nτ²x_n−u_n²2α_n−μFpx_n−p. 3.12

It follows that

1−αnτ²xn−un² ≤αnτ²xn−p²2αnμFpxn−p. 3.13 Sinceαn → 0, therefore

nlim→ ∞xn−un0. 3.14

From3.9, we derive that

nlim→ ∞un−yn0. 3.15

DefineT :C → HbyTx λx 1−λSx, thenT is nonexpansive withFT FS byLemma 2.3. We note that

Tun−un ≤Tun−ynyn−un≤λ−βnun−Sunyn−un. 3.16

So by3.15andβn → λ, we obtain that

nlim→ ∞Tun−un0. 3.17

Since{un}is bounded, so there exists a subsequence{uni}which converges weakly to q. Next, we show thatq∈FS∩EPφ. SinceCis closed and convex,Cis weakly closed. So we haveq∈C. Let us show thatq∈FS. Assume thatq∈FT, Sinceu_ni qandq /Tq, it follows from the Opial’s condition that

lim inf

n→ ∞ uni−q<lim inf

n→ ∞ uni−Tq

≤lim inf

n→ ∞

uni−TuniTuni−Tq

≤lim inf

n→ ∞ u_ni−q.

3.18

This is a contradiction. So, we getq∈FTandq∈FS.

Next, we show thatq∈EPφ. Sinceu_nT_λnx_n, for anyy∈C, we obtain φ

u_n, y 1

λn

y−u_n, u_n−x_n

≥0. 3.19

FromA2, we have

1 λ_n

y−u_n, u_n−x_n

≥φ y, u_n

. 3.20

Replacingnbyn_i, we have

y−uni,u_ni−xni

λ_ni

≥φ y, uni

. 3.21

Sinceuni −xni/λni → 0 anduni q, it follows fromA4that 0 ≥ φy, q, for all y ∈ C. Letzt ty 1−tqfor all t ∈ 0,1andy ∈ C, then we havezt ∈ Cand hence φz_t, q≤0. Thus, fromA1andA4, we have

0φzt, zt≤tφ zt, y

1−tφ zt, q

≤tφ zt, y

, 3.22

and hence 0 ≤ φzt, y. FromA3, we have 0 ≤ φq, yfor ally ∈ Cand henceq ∈EPφ.

Therefore,q∈FS∩EPφ. On the other hand, we note that x_n−q−α_nμFq

I−μα_nF y_n−

I−μα_nF

q. 3.23

Hence, we obtain xn−q²

−αnμFq, xn−q

I−μαnF yn−

I−μαnF

q, xn−q

≤αn

−μFq, xn−q

1−αnτxn−q². 3.24

It follows that

xn−q²≤ 1 τ

−μFq, xn−q

. 3.25

This implies that

x_n−q²≤

−μFq, xn−q

τ . 3.26

In particular,

x_ni−q²≤

−μFq, xni −q

τ . 3.27

Sincexni q, it follows from3.27thatxni → qasi → ∞. Next, we show thatq solves the variational inequality3.4.

As a matter of fact, we have xn

I−αnμF yn

I−αnμF

SnTλnxn, 3.28

and we have

μFxn− 1 αn

I−SnTλnxn−μαnFxn−FSnTλnxn

. 3.29

Hence, forp∈FS∩EPφ, μF

xn, xn−p − 1

α_n

I−SnTλnxn−μαnFxn−FSnTλnxn

, xn−p

− 1 αn

I−S_nT_λnx_n−I−S_nT_λnp, x_n−p μ

Fx_n−FS_nT_λnx_n, x_n−p . 3.30

SinceI−SnTλn is monotonei.e.,x−y,I−SnTλnx−I−SnTλny ≥0, for allx, y∈H. This is due to the nonexpansivity ofS_nT_λn.

Now replacingnin3.30withniand lettingi → ∞, we obtain μF

q, q−p lim

i→ ∞

μFxni, xni−p

≤ lim

i→ ∞μ

Fxni −FSnTλnxni, xni−p

0. 3.31

That is,q∈FS∩EPφis a solution of3.4. To show that the sequence{xn}converges strongly toq, we assume thatx_nk → x. Similiary to the proof above, we derive x ∈FS∩ EPφ. Moreover, it follows from the inequality3.31that

μF

q, q−x

≤0. 3.32

Interchangeqandxto obtain μF

x,x−q

≤0. 3.33

Adding up3.32and3.33yields μηq−x²≤

q−x, μF

q− μF

x

≤0. 3.34

Hence,qx, and therefore x_n → qasn → ∞, I−μF

q−q, q−p

≥0,∀p∈FS∩EP φ

. 3.35

This is equivalent to the fixed-point equation PFS∩EPφ

I−μF

qq. 3.36

Theorem 3.2. LetCbe a nonempty closed convex subset of a real Hilbert spaceHandφa bifunction from C × C into R satisfying (A1), (A2), (A3) and (A4). Let S : C → H be a k-strictly pseudocontractive nonself mapping such thatFS∩EPφ/φ. LetF:H → Hbe anL-Lipschitzian continuous andη-strongly monotone operator onH withL, η > 0. Suppose that 0 < μ < 2η/L², τ μη−μL²/2. Let{xn}and{un}be sequences generated byx1∈Hand

φ un, y

1 λ_n

y−un, un−xn

≥0, ∀y∈C, y_nβ_nu_n

1−β_n Su_n, xn1

I−αnμF

yn, ∀n∈N,

3.37

whereunTλnxn,ynSnunif{αn},{βn}, and{λn}satisfy the following conditions:

i{α_n} ⊂0,1, lim_{n→ ∞}α_n0,_∞

n1α_n∞,_∞

n1|α_n1−α_n|<∞, ii0≤k≤β_n≤λ <1 and lim_{n→ ∞}β_nλ,_∞

n1|β_n1−β_n|<∞, iii{λn} ∈0,∞, limn→ ∞λn>0 and_∞

n1|λn1−λn|<∞,

then{xn}and{un}converge strongly to a pointq∈FS∩EPφwhich solves the variational inequality3.4.

Proof. We first show that{x_n}is bounded. Indeed, pick anyp∈FS∩EPφto derive that x_n1−p−α_nμFp

I−μα_nF y_n−

I−μα_nF p

≤α_n−μF

p 1−α_nτx_n−p

≤1−αnτxn−pαn−μF p.

3.38

By induction, we have

xn−p ≤max

x1−p,1

τ −μF p

, ∀n∈N, 3.39

and hence {xn} is bounded. From3.6 and 3.7, we also derive that {un} and {yn} are bounded. Next, we show thatxn1−xn → 0. We have

xn1−xnI−αnμF yn−

I−αn−1μF yn−1 I−αnμF

yn−

I−αnμF

yn−1

I−αnμF

yn−1−

I−αn−1μF yn−1

≤1−α_nτy_n−y_n−1|α_n−α_n−1|μFy_n−1

≤1−αnτyn−yn−1K|αn−αn−1|,

3.40

where

KsupμFy_n:n∈N

<∞. 3.41

On the other hand, we have

y_n−y_n−1S_nu_n−S_n−1u_n−1

≤ Snun−Snun−1Snun−1−Sn−1un−1

≤ un−un−1Snun−1−Sn−1un−1.

3.42

Fromun1Tλn1xn1andun Tλnxn, we note that φ

u_n1, y 1

λn1

y−u_n1, u_n1−x_n1

≥0, ∀y∈C, 3.43 φ

un, y 1

λ_n

y−un, un−xn

≥0, ∀y∈C. 3.44

Puttingyu_nin3.43andyu_n1in3.44, we have φu_n1, u_n 1

λ_n1u_n−u_n1, u_n1−x_n1 ≥0, φu_n, u_n1 1

λnu_n1−u_n, u_n−x_n ≥0. 3.45 So, fromA2, we have

un1−un,un−xn

λ_n −un1−xn1

λ_n1

≥0 , 3.46

and hence

u_n1−u_n, u_n−u_n1u_n1−x_n− λn

λ_n1u_n1−x_n1

≥0. 3.47

Since lim_{n→ ∞}λ_n > 0, without loss of generality, let us assume that there exists a real number a such thatλn> a >0 for alln∈N. Thus, we have

u_n1−u_n²≤

u_n1−u_n, x_n1−x_n

1− λ_n λn1

u_n1−x_n1

≤ un1−un

xn1−xn 1− λn

λ_n1

un1−xn1

u_n1−u_n ≤ x_n1−x_n 1

a|λ_n1−λ_n|M0,

3.48

whereM0sup{un−xn:n∈N}. Next, we estimateSnun−1−Sn−1un−1. Notice that Snun−1−Sn−1un−1βnun−1

1−βn Sun−1

−

βn−1un−1

1−βn−1

Sun−1

≤βn−βn−1un−1−Sun−1. 3.49 From3.48,3.49, and3.42, we obtain that

yn−yn−1≤ xn−xn−1M0

a |λn−λn−1|βn−βn−1un−1−Sun−1

≤ xn−xn−1|λn−λn−1|M1βn−βn−1M1,

3.50

whereM1is an appropriate constant such that M1≥ M0

a u_n−1−Su_n−1, ∀n∈N. 3.51 From3.41and3.50, we obtain

xn1−xn ≤K|αn−αn−1| 1−αnτ

xn−xn−1|λn−λn−1|M1βn−βn−1M1

≤1−αnτxn−xn−1M

|αn−αn−1||λn−λn−1|βn−βn−1, 3.52

whereMmaxK, M1. Hence, few byLemma 2.5, we have

nlim→ ∞xn1−xn0. 3.53

From3.48and3.50,|λ_n−λ_n−1| → 0 and|β_n−β_n−1| → 0, we have

nlim→ ∞un1−un0, lim

n→ ∞yn1−yn0. 3.54

Since

x_n1

I−α_nμF

y_n, 3.55

it follows that

xn−yn≤ xn−xn1xn1−yn

x_n−x_n1α_n−μFy_n. 3.56

Fromα_n → 0 and3.53, we have

nlim→ ∞x_n−y_n0. 3.57

Forp∈FS∩EPφ, we have

u_n−p²T_λnx_n−T_λnp²≤

x_n−p, u_n−p 1

2 xn−p²un−p²− un−xn²

. 3.58

This implies that

un−p²≤xn−p²− un−xn². 3.59

Then, from3.7and3.59, we derive that xn1−p²−μαnFp

I−μαnF yn−

I−μαnF p²

≤1−αnτ²yn−p²α²_n−μFp²2αn−μFpyn−p

≤un−p²α²_n−μFp²2αn−μFpyn−p

≤x_n−p²− xn−u_n²α²_n−μFp²2α_n−μFpy_n−p.

3.60

Sinceα_n → 0,x_n−x_n1 → 0, we have

nlim→ ∞x_n−u_n0. 3.61

From3.57and3.61, we obtain that

un−yn ≤ un−xnxn−yn → 0, asn → ∞. 3.62

DefineT :C → HbyTx λx 1−λSx, thenT is nonexpansive withFT FS byLemma 2.3. Notice that

Tun−un ≤Tun−ynyn−un

≤λ−βnun−Sunyn−un. 3.63

By3.62andβn → λ, we obtain that

nlim→ ∞Tun−un0. 3.64

Next, we show that lim sup_{n→ ∞}μFq, q−xn ≤0, whereq PFS∩EPφI−μFqis a unique solution of the variational inequality3.4. Indeed, take a subsequence{x_ni}of{x_n} such that

ilim→ ∞

μFq, q−xni

lim sup

n→ ∞

μFq, q−xn

. 3.65

Since{xni} is bounded, there exists a subsequence {xn_ij} of {uni} which converges weakly tow.

Without loss of generality, we can assume thatuni w. From3.61and3.64, we obtainxni wandTuni w. By the same argument as in the proof ofTheorem 3.1, we have w∈FS∩EPφ. SinceqP_FS∩EPφI−μFq, it follows that

lim sup

n→ ∞

μFq, q−x_n

μFq, q−w

≤0. 3.66

Fromxn1−q−αnμFq I−μαnFyn−I−μαnFq, we have xn1−q² ≤I−μαnF

yn−

I−μαnF

q²2αn

−μFq, xn1−q

≤1−α_nτ²x_n−q²2α_n

−μFq, x_n1−q .

3.67

This implies that x_n1−q²≤

1−2α_nτ αnτ²x_n−q²2α_n

−μFq, x_n1−q 1−2α_nτx_n−q² αnτ²x_n−q²2α_n

−μFq, x_n1−q

1−2α_nτx_n−q²2α_nτ αnτ²

2τ M^∗1 τ

−μFq, x_n1−q

1−γnxn−q²γnδn,

3.68

whereM^∗sup{xn−q²:n∈N},γn2αnτ, andδn αnτ²/2τM^∗1/τ−μFq, xn1−q.

It is easy to see thatγn → 0,_∞

n1γn∞, and lim sup_{n→ ∞}δn ≤0 by3.66. Hence by Lemma 2.5, the sequence{x_n}converges strongly toq.

Acknowledgments

M. Tian was supported in part by the Science Research Foundation of Civil Aviation University of Chinano. 2010kys02. He was also Supported in part by The Fundamental Research Funds for the Central UniversitiesGrant no. ZXH2009D021.

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