Strong Convergence to Solutions of Generalized Mixed Equilibrium Problems with Applications

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Volume 2012, Article ID 308791,18pages doi:10.1155/2012/308791

Research Article

Strong Convergence to Solutions of Generalized Mixed Equilibrium Problems with Applications

Prasit Cholamjiak,

^{1, 2}

Suthep Suantai,

^{2, 3}

and Yeol Je Cho

⁴

1School of Science, University of Phayao, Phayao 56000, Thailand

2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

3Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

4Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Correspondence should be addressed to Yeol Je Cho,[email protected] Received 21 October 2011; Accepted 23 November 2011

Academic Editor: Yonghong Yao

Copyrightq2012 Prasit Cholamjiak 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 a Halpern-type iteration for a generalized mixed equilibrium problem in uniformly smooth and uniformly convex Banach spaces. Strong convergence theorems are also established in this paper. As applications, we apply our main result to mixed equilibrium, generalized equilibrium, and mixed variational inequality problems in Banach spaces. Finally, examples and numerical results are also given.

1. Introduction

LetEbe a real Banach space,Ca nonempty, closed, and convex subset ofE, andE^∗the dual space ofE. LetT :C → Cbe a nonlinear mapping. The fixed points set ofT is denoted by FT, that is,FT {x∈C: xTx}.

One classical way often used to approximate a fixed point of a nonlinear self-mapping TonCwas firstly introduced by Halpern1which is defined byx1x∈Cand

xn1αnx 1−αnTxn, ∀n≥1, 1.1

where{αn}is a real sequence in0,1. He proved, in a real Hilbert space, a strong convergence theorem for a nonexpansive mappingTwhenα_nn^−afor anya∈0,1.

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Subsequently, motivated by Halpern1, many mathematicians devoted time to study algorithm1.1in diﬀerent styles. Several strong convergence results for nonlinear mappings were also continuously established in some certain Banach spacessee also2–9.

Let f : C×C → R be a bifunction,A : C → E^∗ a mapping, and ϕ : C → Ra real-valued function. The generalized mixed equilibrium problem is to findx∈Csuch that

f x, y

Ax, y −x ϕ

y

≥ϕx, ∀y∈C. 1.2

The solutions set of1.2is denoted by GMEPf, A, ϕ see Peng and Yao10.

If A ≡ 0, then the generalized mixed equilibrium problem 1.2 reduces to the following mixed equilibrium problem: findingx∈Csuch that

f x, y

ϕ y

≥ϕx, ∀y∈C. 1.3

The solutions set of1.3is denoted by MEPf, ϕ see Ceng and Yao11.

Iff≡0, then the generalized mixed equilibrium problem1.2reduces to the following mixed variational inequality problem: findingx∈Csuch that

Ax, y −x ϕ

y

≥ϕx, ∀y∈C. 1.4

The solutions set of1.4is denoted by VIC, A, ϕ see Noor12.

Ifϕ≡0, then the generalized mixed equilibrium problem1.2reduces to the following generalized equilibrium problem: findingx∈Csuch that

f x, y

Ax, y −x

≥0, ∀y∈C. 1.5

The solutions set of1.5is denoted by GEPf, A see Moudafi13.

Ifϕ ≡ 0, then the mixed equilibrium problem1.3reduces to the following equilibrium problem: findingx∈Csuch that

f x, y

≥0, ∀y∈C. 1.6

The solutions set of1.6is denoted by EPf see Combettes and Hirstoaga14.

Iff ≡ 0, then the mixed equilibrium problem1.3reduces to the following convex minimization problem: findingx∈Csuch that

ϕ y

≥ϕx, ∀y∈C. 1.7

The solutions set of1.7is denoted by CMPϕ.

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Ifϕ≡0, then the mixed variational inequality problem1.4reduces to the following variational inequality problem: findingx∈Csuch that

Ax, y −x

≥0, ∀y∈C. 1.8

The solutions set of1.8is denoted by VIC, A see Stampacchia7.

The problem 1.2 is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others. For more details on these topics, see, for instance,14–34.

For solving the generalized mixed equilibrium problem, let us assume the following 25:

A1fx, x 0 for allx∈C;

A2fis monotone, that is,fx, y fy, x≤0 for allx, y∈C;

A3for allx, y, z∈C, lim sup_t↓0ftz 1−tx, y≤fx, y;

A4for allx∈C, fx,·is convex and lower semicontinuous.

The purpose of this paper is to investigate strong convergence of Halpern-type iteration for a generalized mixed equilibrium problem in uniformly smooth and uniformly convex Banach spaces. As applications, our main result can be deduced to mixed equilibrium, generalized equilibrium, mixed variational inequality problems, and so on. Examples and numerical results are also given in the last section.

2. Preliminaries and Lemmas

In this section, we need the following preliminaries and lemmas which will be used in our main theorem.

LetEbe a real Banach space and letU{x∈E:x1}be the unit sphere ofE. A Banach spaceEis said to be strictly convex if, for anyx, y∈U,

x /y implies xy

2

<1. 2.1

It is also said to be uniformly convex if, for anyε∈0,2, there existsδ >0 such that, for any x, y∈U,

x−y≥εimplies xy

2

<1−δ. 2.2

It is known that a uniformly convex Banach space is reflexive and strictly convex. Define a functionδ:0,2 → 0,1called the modulus of convexity ofEas follows:

δε inf

1− xy

2

: x, y∈E, xy1, x−y≥ε . 2.3

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ThenEis uniformly convex if and only ifδε>0 for allε∈0,2. A Banach spaceEis said to be smooth if the limit

limt→0

xty− x

t 2.4

exists for all x, y ∈ U. It is also said to be uniformly smooth if the limit 2.4 is attained uniformly forx, y∈U. The normalized duality mappingJ:E → 2^E^∗is defined by

Jx

x^∗∈E^∗:x, x^∗x²x^∗²

2.5 for allx∈E. It is also known that ifEis uniformly smooth, thenJis uniformly norm-to-norm continuous on each bounded subset ofEsee35.

LetEbe a smooth Banach space. The functionφ:E×E → Ris defined by φ

x, y

x²−2 x, Jy

y², ∀x, y∈E. 2.6 Remark 2.1. We know the following: for anyx, y, z∈E,

1 x − y²≤φx, y≤xy²;

2φx, y φx, z φz, y 2x−z, Jz−Jy;

3φx, y x−y²in a real Hilbert space.

Lemma 2.2see36. LetEbe a uniformly convex and smooth Banach space and let{x_n}and{y_n} be sequences ofEsuch that{xn}or{yn}is bounded and lim_n_{→ ∞}φxn, yn 0. Then lim_n_{→ ∞}xn− y_n0.

LetEbe a reflexive, strictly convex, and smooth Banach space and letCbe a nonempty closed and convex subset ofE. The generalized projection mapping, introduced by Alber37, is a mappingΠ_C : E → C, that assigns to an arbitrary pointx ∈Ethe minimum point of the functionalφy, x, that is,Π_Cxx, wherexis the solution to the minimization problem:

φx, x min φ

y, x

:y∈C

. 2.7

In fact, we have the following result.

Lemma 2.3see37. LetCbe a nonempty, closed, and convex subset of a reflexive, strictly convex, and smooth Banach spaceEand letx ∈ E. Then there exists a unique element x0 ∈ C such that φx0, x min{φz, x:z∈C}.

Lemma 2.4see 36,37. LetCbe a nonempty closed and convex subset of a reflexive, strictly convex, and smooth Banach spaceE,x∈E, andz∈C. Thenz Π_Cxif and only if

Jx−Jz, y−z

≤0, ∀y∈C. 2.8

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Lemma 2.5see 36,37. LetCbe a nonempty closed and convex subset of a reflexive, strictly convex, and smooth Banach spaceEand letx∈E. Then

φ

y,ΠCx

φΠCx, x≤φ y, x

, ∀y∈C. 2.9

Lemma 2.6see38. LetEbe a uniformly convex and uniformly smooth Banach space andCa nonempty, closed, and convex subset of E. ThenΠCis uniformly norm-to-norm continuous on every bounded set.

We make use of the following mappingV studied in Alber37:

Vx, x^∗ x²−2x, x^∗x^∗² 2.10

for allx∈Eandx^∗∈E^∗, that is,Vx, x^∗ φx, J⁻¹x^∗.

Lemma 2.7see39. LetEbe a reflexive, strictly convex, smooth Banach space. Then

Vx, x^∗ 2

J⁻¹x^∗−x, y^∗

≤V

x, x^∗y^∗

2.11

for allx∈Eandx^∗, y^∗∈E^∗.

Lemma 2.8see25. LetCbe a closed and convex subset of a smooth, strictly convex, and reflexive Banach spaceE, letf be a bifunction fromC×CtoRwhich satisfies conditionsA1–A4, and let r >0 andx∈E. Then there existsz∈Csuch that

f z, y

1 r

Jz−Jx, y−z

≥0, ∀y∈C. 2.12

Following25,40, we know the following lemma.

Lemma 2.9see41. LetCbe a nonempty closed and convex subset of a smooth, strictly convex, and reflexive Banach spaceE. Let A: C → E^∗be a continuous and monotone mapping, letf be a bifunction fromC×CtoRsatisfyingA1–A4, and letϕbe a lower semicontinuous and convex function fromCtoR. For allr >0 andx∈E, there existsz∈Csuch that

f z, y

Az, y−z ϕ

y 1

r

Jz−Jx, y−z

≥ϕz, ∀y∈C. 2.13

Define the mappingTr :E → 2^Cas follows:

Trx

z∈C:f z, y

Az, y−z ϕ

y 1

r

Jz−Jx, y−z

≥ϕz, ∀y∈C . 2.14

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Then, the followings hold:

1Tr is single-valued;

2Tr is firmly nonexpansive-type mapping [42], that is, for allx, y∈E, T_rx−T_ry, JT_rx−JT_ry

≤

T_rx−T_ry, Jx−Jy

; 2.15

3FTr GMEPf, A, ϕ;

4GMEPf, A, ϕis closed and convex.

Remark 2.10. It is known thatTis of firmly nonexpansive type if and only if φ

Tx, Ty φ

Ty, Tx

φTx, x φ Ty, y

≤φ Tx, y

φ Ty, x

2.16

for allx, y∈domTsee42.

The following lemmas give us some nice properties of real sequences.

Lemma 2.11see43. Assume that{an}is a sequence of nonnegative real numbers such that

an1≤1−αnanbn, ∀n≥1, 2.17 where{αn}is a sequence in0,1and{bn}is a sequence such that

a_∞

n1α_n ∞;

blim sup_n_{→ ∞}bn/αn≤0 or_∞

n1|bn|<∞.

Then lim_n_{→ ∞}a_n0.

Lemma 2.12see44. Let{γn}be a sequence of real numbers such that there exists a subsequence {γ_n_j}of{γ_n}such thatγ_n_j < γ_n_j₁for allj ≥1. Then there exists a nondecreasing sequence{m_k}ofN such that lim_k_{→ ∞}m_k∞and the following properties are satisfied by all (suﬃciently large) numbers k≥1:

γmk ≤γmk1, γk≤γmk1. 2.18 In fact,m_kis the largest numbernin the set{1,2, . . . , k}such that the conditionγ_n < γ_n1holds.

3. Main Results

In this section, we prove our main theorem in this paper. To this end, we need the following proposition.

Proposition 3.1. LetCbe a nonempty closed and convex subset of a reflexive, strictly convex, and uniformly smooth Banach spaceE. Let f be a bifunction from C×Cto RsatisfyingA1–A4, A:C → E^∗a continuous and monotone mapping, andϕa lower semicontinuous and convex function

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fromCtoRsuch that GMEPf, A, ϕ/∅. Let{rn} ⊂ 0,∞be such that lim inf_n_{→ ∞}rn > 0. For eachn≥1, letT_r_nbe defined as inLemma 2.9. Suppose thatx∈Cand{x_n}is a bounded sequence in Csuch that lim_n_{→ ∞}xn−Trnxn0. Then

lim sup

n→ ∞

Jx−Jp, x_n−p

≤0, 3.1

wherep Π_GMEPf,A,ϕx andΠ_GMEPf,A,ϕis the generalized projection ofConto GMEPf, A, ϕ.

Proof. Letx ∈ Cand putp ΠGMEPf,A,ϕx. SinceEis reflexive and{xn}is bounded, there exists a subsequence{xnk}of{xn}such thatxnk v∈Cand

lim sup

n→ ∞

Jx−Jp, xn−p

Jx−Jp, v−p

. 3.2

Puty_nT_r_nx_n. Since lim_k_{→ ∞}x_n_k−y_n_k0, we havey_n_k v. On the other hand, sinceEis uniformly smooth,Jis uniformly norm-to-norm continuous on bounded subsets ofE. So we have

klim→ ∞Jxnk −Jynk0. 3.3

Since lim inf_k_{→ ∞}rnk >0,

klim→ ∞

Jx_n_k −Jy_n_k rnk

0. 3.4

By the definition ofT_r_nk, for anyy∈C, we see that f

y_n_k, y

Ay_n_k, y−y_n_k ϕ

y 1

r_n_k

Jy_n_k−Jx_n_k, y−y_n_k

≥ϕ y_n_k

. 3.5

ByA2, for eachy∈C, we obtain f

y, y_n_k ϕ

y_n_k

≤ −f y_n_k, y

ϕ y_n_k

≤

Aynk, y−ynk

ϕ y

1 rnk

Jynk−Jxnk, y−ynk

. 3.6

For anyt ∈ 0,1andy ∈ C, we definey_t ty 1−tv. Theny_t ∈ C. It follows by the monotonicity ofAthat

f y_t, y_n_k

ϕ y_n_k

≤

Ay_n_k−Ay_t, y_t−y_n_k

Ay_t, y_t−y_n_k ϕ

yt 1

r_n_k

Jynk −Jxnk, yt−ynk

≤

Ay_t, y_t−y_n_k ϕ

y_t 1

r_n_k

Jy_n_k−Jx_n_k, y_t−y_n_k .

3.7

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ByA4,3.4, and the weakly lower semicontinuity ofϕ, lettingk → ∞, we obtain f

y_t, v

ϕv≤

Ay_t, y_t−v ϕ

y_t

. 3.8

ByA1,A4, and the convexity ofϕ, we have 0f

yt, yt ϕ

yt

−ϕ yt

≤tf y_t, y

1−tf y_t, v

tϕ y

1−tϕv−ϕ y_t t

f y_t, y

ϕ y

−ϕ y_t

1−t f

y_t, v

ϕv−ϕ y_t

≤t f

yt, y ϕ

y

−ϕ yt

1−tAyt, yt−v t

f y_t, y

ϕ y

−ϕ y_t

1−ttAy_t, y−v.

3.9

It follows that

f yt, y

ϕ y

−ϕ yt

1−t

Ayt, y−v

≥0. 3.10

ByA3, the weakly lower semicontinuity ofϕ, and the continuity ofA, lettingt → 0, we obtain

f v, y

ϕ y

−ϕv

Av, y−v

≥0, ∀y∈C. 3.11

This shows thatv∈GMEPf, A, ϕ. ByLemma 2.4, we have lim sup

n→ ∞

Jx−Jp, xn−p

Jx−Jp, v−p

≤0. 3.12

This completes the proof.

Theorem 3.2. LetCbe nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach spaceE. Letfbe a bifunction fromC×CtoRsatisfyingA1–A4,A:C → E^∗a continuous and monotone mapping, andϕa lower semicontinuous and convex function fromCtoR such that GMEPf, A, ϕ/∅. Define the sequence{x_n}as follows:x1x∈Cand

f yn, y

Ayn, y−yn ϕ

y 1

r_n

Jyn−Jxn, y−yn

≥ϕ yn

, ∀y∈C, x_n1 Π_CJ⁻¹

α_nJx 1−α_nJy_n

, ∀n≥1,

3.13

where{α_n} ⊂0,1and{r_n} ⊂0,∞satisfy the following conditions:

alim_n_{→ ∞}α_n0;

b_∞

n1αn∞;

clim inf_n_{→ ∞}r_n>0.

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Then{xn}converges strongly toΠGMEPf,A,ϕx, whereΠGMEPf,A,ϕis the generalized projection ofC onto GMEPf, A, ϕ.

Proof. From Lemma 2.94, we know that GMEPf, A, ϕ is closed and convex. Let p ΠGMEPf,A,ϕx. PutynTrnxnandznJ⁻¹αnJx1−αnJynfor alln∈N. So, byLemma 2.5, we have

φ p, x_n1

≤φ p, z_n

≤αnφ p, x

1−αnφ p, yn

≤α_nφ p, x

1−α_nφ p, x_n

.

3.14

By induction, we can show thatφp, xn≤φp, xfor eachn∈N. Hence{φp, xn}is bounded and thus{x_n}is also bounded.

We next show that if there exists a subsequence{x_n_k}of{x_n}such that

klim→ ∞

φ

p, xnk1

−φ p, xnk

0, 3.15

then

klim→ ∞

φ p, ynk

−φ p, xnk

0. 3.16

Sinceαnk → 0,

klim→ ∞Jz_n_k−Jy_n_k lim

k→ ∞α_n_kJx−Jy_n_k0. 3.17

SinceJis uniformly norm-to-norm continuous on bounded subsets ofE, so isJ⁻¹. It follows that

k→ ∞limznk−ynk0. 3.18

SinceEis uniformly smooth and uniformly convex, byLemma 2.6,Π_Cis uniformly norm-to- norm continuous on bounded sets. So we obtain

klim→ ∞x_n_k₁−y_n_k lim

k→ ∞Π_Cz_n_k−Π_Cy_n_k0, 3.19 and hence

klim→ ∞Jx_n_k₁−Jy_n_k0. 3.20

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Furthermore, lim_k_{→ ∞}φxnk1, ynk 0. Indeed, by the definition ofφ, we observe that φ

xnk1, ynk

xnk1²−2

xnk1, Jynk

ynk²

x_n_k₁, Jxnk1−Jy_n_k

y_n_k−x_n_k₁, Jy_n_k

. 3.21

It follows from 3.19 and 3.20 that lim_k_{→ ∞}φxnk1, ynk 0. On the other hand, from Remark 2.12, we have

φ p, y_n_k

−φ p, x_n_k

φ

p, x_n_k₁

−φ p, x_n_k

φ

p, y_n_k

−φ

p, x_n_k₁

φ

p, xnk1

−φ p, xnk

φ

xnk1, ynk

2

p−xnk1, Jxnk1−Jynk

.

3.22

It follows from3.20and3.21that_k^lim_{→ ∞}^φp,y^nk^−φp,x^nk^0.

We next consider the following two cases.

Case 1. φp, x_n1 ≤ φp, x_n for all suﬃciently large n. Hence the sequence {φp, x_n} is bounded and nonincreasing. So lim_n_{→ ∞}φp, xnexists. This shows that lim_n_{→ ∞}φp, xn1− φp, xn 0 and hence

n→ ∞lim φ

p, y_n

−φ p, x_n

0. 3.23

SinceTrn is of firmly nonexpansive type, byRemark 2.10, we have φ

yn, p φ

p, yn φ

yn, xn φ

Trnp, p

≤φ yn, p

φ p, xn

, 3.24

which implies

φ p, y_n

φ y_n, x_n

≤φ p, x_n

. 3.25

Hence

φ y_n, x_n

≤φ p, x_n

−φ p, y_n

−→0 3.26

asn → ∞. ByLemma 2.2, we obtain

nlim→ ∞x_n−y_n0. 3.27

Proposition 3.1yields that

lim sup

n→ ∞

Jx−Jp, x_n−p

≤0. 3.28

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It also follows that

lim sup

n→ ∞

Jx−Jp, yn−p

≤0. 3.29

Finally, we show thatxn → p. UsingLemma 2.7, we see that φ

p, x_n1

≤φ p, z_n V

p, α_nJx 1−α_nJy_n

≤V

p, αnJx 1−αnJyn−αn

Jx−Jp

αn

Jx−Jp

, zn−p V

p, α_nJp 1−α_nJy_n α_n

Jx−Jp, z_n−p

≤αnV p, Jp

1−αnV p, Jyn

αn

Jx−Jp, zn−p 1−αnφ

p, yn αn

Jx−Jp, zn−p

≤1−α_nφ p, x_n

α_n

Jx−Jp, z_n−p 1−αnφ

p, xn αn

Jx−Jp, zn−yn

Jx−Jp, yn−p .

3.30

Set a_n φp, x_n and b_n α_nJx − Jp, z_n − y_n Jx − Jp, y_n − p. We see that lim sup_n_{→ ∞}b_n/α_n ≤ 0. By Lemma 2.11, since _∞

n1α_n ∞, we conclude that lim_n_{→ ∞}φp, xn 0. Hencexn → pasn → ∞.

Case 2. There exists a subsequence{φp, x_n_j}of{φp, x_n}such thatφp, x_n_j< φp, x_n_j₁for allj ∈N. ByLemma 2.12, there exists a strictly increasing sequence{m_k}of positive integers such that the following properties are satisfied by all numbersk∈N:

φ p, xmk

≤φ

p, xmk1 , φ

p, xk

≤φ

p, xmk1

. 3.31

So we have

0≤ lim

k→ ∞

φ

p, xmk1

−φ p, xmk

≤lim sup

n→ ∞

φ p, x_n1

−φ p, x_n

≤lim sup

n→ ∞

φ p, z_n

−φ p, x_n

≤lim sup

n→ ∞

αnφ p, x

1−αnφ p, yn

−φ p, xn lim sup

n→ ∞

α_n φ

p, x

−φ p, y_n

φ

p, y_n

−φ

p, x_n

≤lim sup

n→ ∞ α_n φ

p, x

−φ p, y_n

0.

3.32

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This shows that

klim→ ∞

φ

p, xmk1

−φ p, xmk

0. 3.33

Following the proof line in Case1, we can show that lim sup

k→ ∞

Jx−Jp, y_m_k−p

≤0, φ

p, xmk1

≤1−αmkφ p, xmk

αmk

Jx−Jp, zmk−ymk

Jx−Jp, ymk−p .

3.34

This implies

αmkφ p, xmk

≤φ p, xmk

−φ

p, xmk1 α_m_k

Jx−Jp, z_m_k−y_m_k

Jx−Jp, y_m_k−p

≤αmk

Jx−Jp, zmk−ymk

Jx−Jp, ymk−p .

3.35

Hence lim_k_{→ ∞}φp, xmk 0. Using this and3.33together, we conclude that lim sup

k→ ∞ φ p, x_k

≤ lim

k→ ∞φ

p, x_m_k₁

0. 3.36

This completes the proof.

As a direct consequence ofTheorem 3.2, we obtain the following results.

Corollary 3.3. LetCbe nonempty closed and convex subset of a uniformly smooth and uniformly convex Banach spaceE. Letfbe a bifunction fromC×CtoRsatisfyingA1–A4andϕa lower semicontinuous and convex function fromCtoRsuch that MEPf, ϕ/∅. Define the sequence{xn} as follows:x1x∈Cand

f y_n, y

ϕ y

1 r_n

Jy_n−Jx_n, y−y_n

≥ϕ y_n

, ∀y∈C, xn1 ΠCJ⁻¹

αnJx 1−αnJyn

, ∀n≥1,

3.37

where{αn} ⊂0,1and{rn} ⊂0,∞satisfy the following conditions:

b_∞

n1α_n∞;

clim inf_n_{→ ∞}rn>0.

Then{x_n}converges strongly toΠ_MEPf,ϕx, whereΠ_MEPf,ϕis the generalized projection ofConto MEPf, ϕ.

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Corollary 3.4. LetCbe nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach spaceE. Letfbe a bifunction fromC×CtoRsatisfyingA1–A4, andA: C → E^∗ a continuous and monotone mapping such that GEPf, A/∅. Define the sequence{xn}as follows:

x1x∈Cand

f yn, y

Ayn, y−yn 1

rn

Jyn−Jxn, y−yn

≥0, ∀y∈C, x_n1 Π_CJ⁻¹

α_nJx 1−α_nJy_n

, ∀n≥1,

3.38

b_∞

n1αn∞;

Then{x_n}converges strongly toΠ_GEPf,Ax, whereΠ_GEPf,Ais the generalized projection ofConto GEPf, A.

Corollary 3.5. LetCbe nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach spaceE. LetA : C → E^∗ be a continuous and monotone mapping, and ϕa lower semicontinuous and convex function fromCtoRsuch that VIC, A, ϕ/∅. Define the sequence{x_n} as follows:x1x∈Cand

Ayn, y−yn ϕ

y 1

rn

Jyn−Jxn, y−yn

≥ϕ yn

, ∀y∈C, x_n1 Π_CJ⁻¹

α_nJx 1−α_nJy_n

, ∀n≥1,

3.39

b_∞

n1αn∞;

Then{x_n}converges strongly toΠ_VIC,A,ϕx, whereΠ_VIC,A,ϕis the generalized projection ofConto VIC, A, ϕ.

4. Examples and Numerical Results

In this section, we give examples and numerical results for our main theorem.

Example 4.1. LetERandC −1,1. Letfx, y −9x²xy8y²,ϕx 3x², andAx2x.

Findx∈−1,1such that f

x, y

Ax, y −x ϕ

y

≥ϕx, ∀y∈−1,1. 4.1

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Solution. It is easy to check thatf,ϕ, andAsatisfy all conditions inTheorem 3.2. For each r > 0 and x ∈ −1,1, Lemma 2.9ensures that there existsz ∈ −1,1 such that, for any y∈−1,1,

f z, y

Az, y−zϕ y

1

rz−x, y−z ≥ϕz

⇐⇒ −9z²yz8y²2z y−z

3y²1

rz−x y−z

≥3z²

⇐⇒11ry² 3rzz−xy−

14rz²z²−xz

≥0.

4.2

PutGy 11ry² 3rzz−xy−14rz²z²−xz. ThenGis a quadratic function ofy with coeﬃcienta11r,b 3rzz−x, andc −14rz²z²−xz. We next compute the discriminantΔofGas follows:

Δ b²−4ac

3r1z−x²44r

14rz²z²−xz

x²−23r1xz 3r1²z²616r²z²44rz²−44rxz x²−50rxz−2xz625r²z²50rz²z²

x²−225rzzx

625r²z²50rz²z² x−25rzz².

4.3

We know thatGy≥0 for ally∈−1,1if it has at most one solution in−1,1. SoΔ≤0 and hencex25rzz. Now we havezT_rxx/25r1.

Let{x_n}^∞_n1be the sequence generated byx1 x∈−1,1and f

y_n, y

Ay_n, y−y_n ϕ

y 1

rn

y_n−x_n, y−y_n

≥ϕ y_n

, ∀y∈−1,1, xn1αnx 1−αnyn, ∀n≥1,

4.4

and, equivalently,

xn1αnx 1−αnTrnxn, ∀n≥1. 4.5 We next give two numerical results for algorithm4.5.

Algorithm 4.2. Letαn 1/80nandrn n/n1. Choosex1 x 1. Then algorithm4.5 becomes

x_n1 1 80n

1− 1

80n

n1 26n1

x_n, ∀n≥1. 4.6

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Table 1

n xn

1 1.0000

2 0.0856

3 0.0111

4 0.0047

5 0.0033

... ...

261 0.0001

262 0.0000

Table 2

n xn

1 −1.0000

2 −0.0481

3 −0.0074

4 −0.0038

5 −0.0027

... ...

217 −0.0001

218 0.0000

Numerical Result I SeeTable 1.

Algorithm 4.3. Letα_n1/100nandr_n n1/2n. Choosex1x−1. Then algorithm4.5 becomes

x_n1− 1 100n

1− 1

100n

2n 27n25

x_n, ∀n≥1. 4.7

Numerical Result II SeeTable 2.

5. Conclusion

Tables 1 and 2 show that the sequence {x_n} converges to 0 which solves the generalized mixed equilibrium problem. On the other hand, using Lemma 2.93, we can check that GMEPf, A, ϕ FTr {0}.

Remark 5.1. In the view of computation, our algorithm is simple in order to get strong convergence for generalized mixed equilibrium problems.

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Acknowledgments

The first and the second authors wish to thank the Thailand Research Fund and the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. The third author was supported by Basic Science Research Program through the National Research Foundation of Korea NRF funded by the Ministry of Education, Science and TechnologyGrant no.

2011-0021821.

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