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Volume 2010, Article ID 756492, 22 pagesdoi:10.1155/2010/756492 Research Article Strong Convergence for Mixed Equilibrium Problems of Infinitely Nonexpansive Mappings Jintana Joomwong Di

Volume 2010, Article ID 756492, 22 pages

doi:10.1155/2010/756492

Research Article

Strong Convergence for Mixed Equilibrium

Problems of Infinitely Nonexpansive Mappings

Jintana Joomwong

Division of Mathematics, Faculty of Science, Maejo University, Chiang Mai 50290, Thailand

Correspondence should be addressed to Jintana Joomwong,jintana@mju.ac.th

Received 29 March 2010; Accepted 24 May 2010

Academic Editor: Tomonari Suzuki

Copyrightq 2010 Jintana Joomwong This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited

We introduce a new iterative scheme for finding a common element of infinitely nonexpansivemappings, the set of solutions of a mixed equilibrium problems, and the set of solutions of thevariational inequality for anα-inverse-strongly monotone mapping in a Hilbert Space Then, the

strong converge theorem is proved under some parameter controlling conditions The results ofthis paper extend and improve the results of Jing Zhao and Songnian He2009 and many others.Using this theorem, we obtain some interesting corollaries

1 Introduction

LetH be a real Hilbert space with norm · and inner product ·, · And let C be a nonempty

closed convex subset ofH Let ϕ : C → R be a real-valued function and let Θ : C × C → R

be an equilibrium bifunction, that is,Θu, u  0 for each u ∈ C Ceng and Yao 1 consideredthe following mixed equilibrium problem

Findx∗∈ C such that

The set of solutions of1.1 is denoted by MEPΘ, ϕ It is easy to see that x∗is the solution

of problem 1.1 and x∗ ∈ dom ϕ  {x ∈ ϕx < ∞} In particular, if ϕ ≡ 0, the mixed

equilibrium problem1.1 reduced to the equilibrium problem

Findx∗∈ C such that

The set of solutions of1.2 is denoted by EPΘ If ϕ ≡ 0 and Θx, y  Ax, y − x for

allx, y ∈ C, where A is a mapping from C to H, then the mixed equilibrium problem 1.1becomes the following variational inequality

Findx∗∈ C such that



Ax∗, y − x∗

The set of solutions of1.3 is denoted by VIA, C.

The variational inequality and the mixed equilibrium problems which includefixed point problems, optimization problems, variational inequality problems have beenextensively studied in literature See, for example,2 8

In 1997, Combettes and Hirstoaga 9 introduced an iterative method for findingthe best approximation to the initial data and proved a strong convergence theorem.Subsequently, Takahashi and Takahashi7 introduced another iterative scheme for finding

a common element of EPΘ and the set of fixed points of nonexpansive mappings.Furthermore,Yao et al.8,10 introduced an iterative scheme for finding a common element

of EPΘ and the set of fixed points of finitely infinitely nonexpansive mappings

Very recently, Ceng and Yao 1 considered a new iterative scheme for finding

a common element of MEPΘ, ϕ and the set of common fixed points of finitely manynonexpansive mappings in a Hilbert space and obtained a strong convergence theorem.Now, we recall that a mappingA : C → H is said to be

i monotone if Au − Av, u − v ≥ 0, for all u, v ∈ C,

ii L-Lipschitz if there exists a constant L > 0 such that Au − Av ≤ Lu −

v, for all u, v ∈ C,

iii α-inverse strongly monotone if there exists a positive real number α such that Au−

Av, u − v ≥ αAu − Av2, for all u, v ∈ C.

It is obvious that anyα-inverse strongly monotone mapping A is monotone and Lipscitz A

mappingS : C → C is called nonexpansive if Su − Sv ≤ u − v, for all u, v ∈ C We denote

byFS : {x ∈ C : Sx  x} the set of fixed point of S.

In 2006, Yao and Yao11 introduced the following iterative scheme

LetC be a closed convex subset of a real Hilbert space Let A be an α-inverse strongly

monotone mapping ofC into H and let S be a nonexpansive mapping of C into itself such

thatFS ∩ VIA, C / ∅ Suppose that x1 u ∈ C and {x n } and {y n} are given by

y n  P C x n − λ n Ax n ,

x n1  α n u  β n x n  γ n SP C

y n − λ n Ay n

where{α n }, {β n }, and {γ n } are sequence in 0, 1 and {λ n } is a sequence in 0,2λ They proved

that the sequence{x n} defined by 1.4 converges strongly to a common element of FS ∩

VIA, C under some parameter controlling conditions

Moreover, Plubtieng and Punpaeng 12 introduced an iterative scheme 1.5 forfinding a common element of the set of fixed point of nonexpansive mappings, the set ofsolutions of an equilibrium problems, and the set of solutions of the variational of inequality

problem for anα-inverse strongly monotone mapping in a real Hilbert space Suppose that

x1 u ∈ C and {x n }, {y n }, and {u n} are given by

where{α n }, {β n }, and {γ n } are sequence in 0, 1, {λ n } is a sequence in 0,2λ, and {r n} ⊂

0, ∞ Under some parameter controlling conditions, they proved that the sequence {x n}defined by1.5 converges strongly to P FS∩VIA,C∩EPΘ u.

On the other hand, Yao et al 8 introduced an iterative scheme 1.7 for finding acommon element of the set of solutions of an equilibrium problem and the set of commonfixed point of infinitely many nonexpansive mappings inH Let {T n}∞

n1 be a sequence ofnonexpansive mappings ofC into itself and let {t n}∞n1be a sequence of real number in0, 1.

For eachn ≥ 1, define a mapping W nofC into itself as follows:

Such a mappingW nis called theW-mapping generated by T n , T n−1 , , T1andt n , t n−1 , , t1

In8, given x0∈ H arbitrarily, the sequences {x n } and {u n} are generated by

Subsequently, Ceng and Yao 13 introduced an iterative scheme by the viscosityapproximation method:

i1 FT i  ∩ VIA, C ∩ EPΘ, where z  P∩∞

i1 FT i∩VIA,C∩EPΘ u.

Motivated by the ongoing research in this field, in this paper we suggest and analyze

an iterative scheme for finding a common element of the set of fixed point of infinitelynonexpansive mappings, the set of solutions of an equilibrium problem and the set ofsolutions of the variational of inequality problem for an α-inverse strongly monotone

mapping in a real Hilbert space Under some appropriate conditions imposed on theparameters, we prove another strong convergence theorem and show that the approximatesolution converges to a unique solution of some variational inequality which is the optimalitycondition for the minimization problem The results of this paper extend and improve theresults of Zhao and He14 and many others For some related works, we refer the readers

to15–22 and the references therein

2 Preliminaries

LetH be a real Hilbert space and let C be a closed convex subset of H Then, for any x ∈ H,

there exists a unique nearest point inC, denoted by P C x such that

Moreover,P Cis characterized by the following properties:P c x ∈ C and

x − P C x, y − P C x ≤ 0,

x − y2

≥ x − P C x2y − P C x2, ∀x ∈ H, y ∈ C. 2.3

It is clear thatu ∈ VIA, C ⇔ u  P C u − λAu, λ > 0.

A spaceX is said to satisfy Opials condition if for each sequence {x n } in X which

converges weakly to a pointx ∈ X, we have

lim inf

n → ∞ x n − x < lim inf

n → ∞ x n − y, ∀y ∈ X, y / x. 2.4The following lemmas will be useful for proving the convergence result of this paper

Lemma 2.1 see 23 Let {x n } and {y n } be bounded sequences in a Banach space X and let {β n } be

a sequence in 0, 1 with 0 < lim inf n → ∞ β n lim supn → ∞ β n < 1 Suppose that x n1  1−β n y n

β n x n for all integer n ≥ 1 and lim sup n → ∞ y n1 −y n −x n1 −x n   0 Then lim n → ∞ y n −x n 

0.

Lemma 2.2 see 24 Let H be a real Hilbert space, let C be a closed convex subset of H, and let

T : C → C be a nonexpansive mapping with FT / ∅ If {x n } is a sequence in C weakly converging

to x and if I − Tx n converge strongly to y, then I − Tx  y.

Lemma 2.3 see 25 Assume that {a n } is a sequence of nonnegative real numbers such that

A1 Θx, x  0 for all x ∈ C;

A2 Θ is monotone, that is, Θx, y  Θy, x ≤ 0 for any x, y ∈ C;

A3 Θ is upper-hemicontinuous, that is, for each x, y, z ∈ C,

lim

t → 0supΘtz  1 − tx, y≤ Θx, y; 2.6

A4 Θx, · is convex and lower semicontinuous for each x ∈ C;

B1 for each x ∈ H and r > 0, there exists a bounded subset D x ⊂ C and y x ∈ C such

that for anyz ∈ C \ D x,

Lemma 2.4 Let C be a nonempty closed convex subset of a real Hilbert space H Let Θ be a bifunction

from C×C → R that satisfies (A1)–(A4) and let ϕ : C → R∪{∞} be a proper lower semicontinuous and convex function Assume that either (B1) or (B2) holds For r > 0 and x ∈ H, define a mapping

5 MEPΘ, ϕ is closed and convex.

Let {T n}∞n1 be a sequence of nonexpansive mappings of C into itself, where C is a

nonempty closed convex subset of a real Hilbert spaceH Given a sequence {t n}∞n1in0, 1,

Remark 2.7see 8 Using Lemma2.5, we define a mappingW : C → C as follows: Wx 

limn → ∞ W n x  lim n → ∞ U n,1 x, for all x ∈ C W is called the W-mapping generated by T1, T2,

andt1, t2,

SinceW nis nonexpansive,W : C → C is also nonexpansive.

Indeed, for allx, y ∈ C, W x − W y  limn → ∞ W n x − W n y ≤ x − y.

If{x n } is a bounded sequence in C, then we put D  {x n :n ≥ 0} Hence it is clear

from Remark 2.6that for any arbitrary > 0, there exists n0 ≥ 1 such that for all n > n0,

W n x n − Wx n   U n,1 x n − U1x n ≤ supx∈D U n,1 x − U1x < .

This implies that limn → ∞ W n x n − Wx n   0.

Lemma 2.8 see 27 Let C be a nonempty closed convex subset of a real Hilbert space H Let {T n}∞n1 be a sequence of nonexpansive self-mappings on C such that ∩∞

n1 FT n  / ∅ and let {t n } be a

sequence in 0, b for some b ∈ 0, 1 Then FW  ∩∞

n1 FT n .

3 Main Results

Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H Let ϕ : C →

R ∪ {∞} be a lower semicontinuous and convex function Let Θ be a bifunction from C × C → R

satisfying (A1)–(A4), let A be an α-inverse-strongly monotone mapping of C into H, and let {T n}∞n1

be a sequence of nonexpansive self-mapping on C such that ∩∞

n1 FT n  ∩ VIA, C ∩ MEPΘ, ϕ / ∅.

Suppose that {s n }, {α n }, {β n }, and {γ n } are sequences in 0, 1,{λ n } is a sequence in 0, 2α such that

λ n ∈ a, b for some a, b with 0 < a < b < 2α, and {r n } ⊂ 0, ∞ is a real sequence Suppose that the

following conditions are satisfied:

i α n  β n  γ n  1,

ii limn → ∞ α n  0 and∞

n1 α n  ∞,

iii 0 < lim inf n → ∞ β n≤ lim supn → ∞ β n < 1,

iv 0 < lim inf n → ∞ s n≤ lim supn → ∞ s n < 1/2 and lim n → ∞ |s n1 − s n |  0,

v limn → ∞ |λ n1 − λ n |  0,

vi lim infn → ∞ r n > 0 and lim n → ∞ |r n1 − r n |  0.

Let f be a contraction of C into itself with coefficient β ∈ 0, 1 Assume that either (B1) or (B2) holds Let the sequences {x n }, {u n }, and {y n } be generated by, x1∈ C and

for all n ∈ N, where W n is defined by1.6 and {t n } is a sequence in 0, b, for some b ∈ 0, 1 Then

the sequence {x n } converges strongly to a point x∗ ∈ ∩∞

n1 FT n  ∩ VIA, C ∩ MEPΘ, ϕ, where

x∗ P∩∞

n1 FT n∩VIA,C∩MEPΘ,ϕ fx∗.

Proof For any x, y ∈ C and λ n ∈ a, b ⊂ 0, 2α, we note that

I − λ n Ax − I − λ n Ay2x − y − λ n

Let{T r n} be a sequence of mappping defined as in Lemma2.4and letx∗∈ ∩∞

n1 FT n ∩VIA, C ∩ MEPΘ, ϕ Then x∗  W n x∗andx∗  P C x∗− λ n Ax∗  T r n x∗ Putv n  P C y n−

Therefore {x n } is bounded Consequently, {fx n }, {u n }, {y n }, {v n }, {W n v n }, {Au n}, and

{Ay n} are also bounded

Next, we claim that limn → ∞ x n1 − x n  0

Indeed, settingx n1  β n x n  1 − β n z n , for all n ≥ 1, it follows that

Now, we estimateW n1 v n − W n v n  and W n1 v n1 − W n1 v n

From the definition of{W n}, 1.6, and since T i,U n,iare nonexpansive, we deduce that,for eachn ≥ 1,

W n1 v n − W n v n   t1T1U n1,2 v n − t1T1U n,2 v n

≤ t1U n1,2 v n − U n,2 v n

 t1t2T2U n1,3 v n − t2T2U n,3 v n

≤ t1t2U n1,3 v n − U n,3 v n

whereQ  sup{u n , λ n Au n , x n  : n ≥ 1}.

Combining3.7 and 3.8, we obtain

Puttingy  u n1in3.10 and y  u nin3.11, we have

Θu n , u n1   ϕu n1  − ϕu n  r1

n u n1 − u n , u n − x n  ≥ 0, Θu n1 , u n   ϕu n  − ϕu n1 r1

n1 u n − u n1 , u n1 − x n1  ≥ 0.

3.12

So, fromA2 we get u n1 − u n , u n − x n /r n  − u n1 − x n1 /r n1 ≥ 0

Henceu n1 − u n , u n − u n1  u n1 − x n − r n /r n1 u n1 − x n1  ≥ 0.

Without loss of generality, we may assume that there exists a real numberc such that

r n > c > 0, for all n ≥ 1 Then we get

which imply that

From conditioniv and 3.20, 3.24, and 3.32, we have limn → ∞ W n v n −v n   0 Moreover,

from Remark2.7we get limn → ∞ Wv n − v n   0.

Next, we show that

lim sup

n → ∞ fx∗ − x∗, x n − x∗ ≤ 0, 3.37

Since{v n i } is bounded, there exists a subsequence {v n ij } of {v n i} which converges weakly to

z Without loss of generality, we can assume that v n i z.

Fromu n − x n  → 0, x n − Wv n  → 0 and Wv n − v n  → 0, we get u n i → z It follows from

A4 that u n i − x n i /r n i → 0 and from the lower semicontinuity of ϕ that

Fort with 0 < t ≤ 1 and y ∈ C, let y t  ty  1 − tz Since y ∈ C and z ∈ C, we have y t ∈ C

and henceΘy t , z  ϕz − ϕy t ≤ 0 So, from A1 and A4, we have

Lettingt → 0, it follows from the weakly semicontinuity of ϕ that

Hencez ∈ MEPΘ, ϕ.

Second, we show thatz ∈ FW  ∩∞

n1 FT n  Assume z /∈ FW Since u n i z and

z / Wz, by Opial’s condition, we have

which derives a contradiction Thus we havez ∈ FT.

Finally, by the same argument in the proof of 28, Theorem 3.1, we can show that

x n1 − x∗2≤ 1 − α n x n − x∗2 2α nfx n  − x∗, x n1 − x∗

By3.47 and Lemma2.3, we get that{x n } converges strongly to x∗

This completes the proof

Settingfx n  ≡ u and ϕ  0 in Theorem3.1., we have the following result

Corollary 3.2 see 14, Theorem 2.1 Let C be a nonempty closed convex subset of a real Hilbert space H Let Θ be a bifunction from C × C → R satisfying (A1)–(A4), let A be an α-inverse-strongly monotone mapping of C into H, and let {T n}∞

n1 be a sequence of nonexpansive self-mapping on C such that ∩∞

n1 FT n  ∩ VIA, C ∩ EPΘ / ∅ Suppose that x1  u ∈ C, {s n }, {α n }, {β n }, and {γ n } are sequences in 0, 1,{λ n } is a sequence in 0, 2α such that λ n ∈ a, b for some a, b with

0 < a < b < 2α and {r n } ⊂ 0, ∞ is a real sequence Suppose that the following conditions are

satisfied:

i α n  β n  γ n  1,

ii limn → ∞ α n  0 and∞

n1 α n  ∞,

iii 0 < lim inf n → ∞ β n≤ lim supn → ∞ β n < 1,

iv 0 < lim inf n → ∞ s n≤ lim supn → ∞ s n < 1/2 and lim n → ∞ |s n1 − s n |  0,

v limn → ∞ |λ n1 − λ n |  0,

vi lim infn → ∞ r n > 0 and lim n → ∞ |r n1 − r n |  0.

Let the sequence {x n } be generated by,

for all n ∈ N, where W n is defined by1.6 and {t n } is a sequence in 0, b, for some b ∈ 0, 1.

Then the sequence {x n } converges strongly to a point x∗ ∈ ∩∞

n1 FT n  ∩ VIA, C ∩ EPΘ, where

x∗ P∩ ∞

n1 FT n∩VIA,C∩EPΘ u.

Settingϕ  0 in Theorem3.1, we have the following result