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Hindawi Publishing Corporation

Journal of Applied Mathematics

Volume 2012, Article ID 308791, 18 pages

doi:10.1155/2012/308791

Research Article

Strong Convergence to Solutions of Generalized

Mixed Equilibrium Problems with Applications

Prasit Cholamjiak,1,2Suthep Suantai,2,3and Yeol Je Cho4

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, yjcho@gnu.ac.kr

Received 21 October 2011; Accepted 23 November 2011

Academic Editor: Yonghong Yao

Copyright q 2012 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

Let E be a real Banach space, C a nonempty, closed, and convex subset of E, and E∗the dual

space of E. Let T : C → C be a nonlinear mapping. The fixed points set of T is denoted by

F?T?, that is, F?T? ? {x ∈ C : x ? Tx}.

One classical way often used to approximate a fixed point of a nonlinear self-mapping

T on C was firstly introduced by Halpern ?1? which is defined by x1? x ∈ C and

xn?1? αnx ? ?1 − αn?Txn,

∀n ≥ 1,

?1.1?

where{αn}isarealsequencein?0,1?.Heproved,inarealHilbertspace,astrongconvergence

theorem for a nonexpansive mapping T when αn? n−afor any a ∈ ?0,1?.

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Subsequently, motivated by Halpern ?1?, many mathematicians devoted time to study

algorithm ?1.1? in different styles. Several strong convergence results for nonlinear mappings

were also continuously established in some certain Banach spaces ?see also ?2–9??.

Let f : C × C → R be a bifunction, A : C → E∗a mapping, and ϕ : C → R a

real-valued function. The generalized mixed equilibrium problem is to find ? x ∈ C such that

f?? x,y???A? x,y − ? x?? ϕ?y?≥ ϕ?? x?,

The solutions set of ?1.2? is denoted by GMEP?f,A,ϕ? ?see Peng and Yao ?10??.

If A ≡ 0, then the generalized mixed equilibrium problem ?1.2? reduces to the

following mixed equilibrium problem: finding ? x ∈ C such that

f?? x,y?? ϕ?y?≥ ϕ?? x?,

The solutions set of ?1.3? is denoted by MEP?f,ϕ? ?see Ceng and Yao ?11??.

Iff ≡ 0,thenthegeneralizedmixedequilibriumproblem?1.2?reducestothefollowing

mixed variational inequality problem: finding ? x ∈ C such that

?A? x,y − ? x?? ϕ?y?≥ ϕ?? x?,

The solutions set of ?1.4? is denoted by VI?C,A,ϕ? ?see Noor ?12??.

Ifϕ ≡ 0,thenthegeneralizedmixedequilibriumproblem?1.2?reducestothefollowing

generalized equilibrium problem: finding ? x ∈ C such that

f?? x,y???A? x,y − ? x?≥ 0,

The solutions set of ?1.5? is denoted by GEP?f,A? ?see Moudafi ?13??.

If ϕ ≡ 0, then the mixed equilibrium problem ?1.3? reduces to the following equilib-

rium problem: finding ? x ∈ C such that

f?? x,y?≥ 0,

The solutions set of ?1.6? is denoted by EP?f? ?see Combettes and Hirstoaga ?14??.

If f ≡ 0, then the mixed equilibrium problem ?1.3? reduces to the following convex

minimization problem: finding ? x ∈ C such that

ϕ?y?≥ ϕ?? x?,

The solutions set of ?1.7? is denoted by CMP?ϕ?.

∀y ∈ C.

?1.2?

∀y ∈ C.

?1.3?

∀y ∈ C.

?1.4?

∀y ∈ C.

?1.5?

∀y ∈ C.

?1.6?

∀y ∈ C.

?1.7?

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If ϕ ≡ 0, then the mixed variational inequality problem ?1.4? reduces to the following

variational inequality problem: finding ? x ∈ C such that

?A? x,y − ? x?≥ 0,

∀y ∈ C.

?1.8?

The solutions set of ?1.8? is denoted by VI?C,A? ?see Stampacchia ?7??.

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?:

?A1? f?x,x? ? 0 for all x ∈ C;

?A2? f is monotone, that is, f?x,y? ? f?y,x? ≤ 0 for all x,y ∈ C;

?A3? for all x,y,z ∈ C, limsupt↓0f?tz ? ?1 − t?x,y? ≤ f?x,y?;

?A4? for all x ∈ C, f?x,·? 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.

Let E be a real Banach space and let U ? {x ∈ E : ?x? ? 1} be the unit sphere of E. A

Banach space E is said to be strictly convex if, for any x,y ∈ U,

x/ ?y implies

????

x ? y

2

????< 1.

?2.1?

It is also said to be uniformly convex if, for any ε ∈ ?0,2?, there exists δ > 0 such that, for any

x,y ∈ U,

??x − y??≥ ε implies

????

x ? y

2

????< 1 − δ.

?2.2?

It is known that a uniformly convex Banach space is reflexive and strictly convex. Define a

function δ : ?0,2? → ?0,1? called the modulus of convexity of E as follows:

?

δ?ε? ? inf1 −

????

x ? y

2

????: x,y ∈ E, ?x? ???y??? 1,

??x − y??≥ ε

?

.

?2.3?

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Then E is uniformly convex if and only if δ?ε? > 0 for all ε ∈ ?0,2?. A Banach space E is said

to be smooth if the limit

lim

t→0

??x ? ty??− ?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 for x,y ∈ U. The normalized duality mapping J : E → 2E∗is defined by

?

J?x? ?

x∗∈ E∗: ?x,x∗? ? ?x?2? ?x∗?2?

?2.5?

for all x ∈ E. It is also known that if E is uniformly smooth, then J is uniformly norm-to-norm

continuous on each bounded subset of E ?see ?35??.

Let E be a smooth Banach space. The function φ : E × E → R is defined by

φ?x,y?? ?x?2− 2?x,Jy????y??2,

∀x,y ∈ E.

?2.6?

Remark 2.1. We know the following: for any x,y,z ∈ E,

?1? ??x? − ?y??2≤ φ?x,y? ≤ ??x? ? ?y??2;

?2? φ?x,y? ? φ?x,z? ? φ?z,y? ? 2?x − z,Jz − Jy?;

?3? φ?x,y? ? ?x − y?2in a real Hilbert space.

Lemma 2.2 ?see ?36??. Let E be a uniformly convex and smooth Banach space and let {xn} and {yn}

be sequences of E such that {xn} or {yn} is bounded and limn→∞φ?xn,yn? ? 0. Then limn→∞?xn−

yn? ? 0.

Let E be a reflexive, strictly convex, and smooth Banach space and let C be a nonempty

closed and convex subset of E. The generalized projection mapping, introduced by Alber ?37?, is

a mapping ΠC: E → C, that assigns to an arbitrary point x ∈ E the minimum point of the

functional φ?y,x?, that is, ΠCx ? x, where x is the solution to the minimization problem:

φ?x,x? ? min?φ?y,x?: y ∈ C?.

?2.7?

In fact, we have the following result.

Lemma 2.3 ?see ?37??. Let C be a nonempty, closed, and convex subset of a reflexive, strictly convex,

and smooth Banach space E and let x ∈ E. Then there exists a unique element x0 ∈ C such that

φ?x0,x? ? min{φ?z,x? : z ∈ C}.

Lemma 2.4 ?see ?36, 37??. Let C be a nonempty closed and convex subset of a reflexive, strictly

convex, and smooth Banach space E, x ∈ E, and z ∈ C. Then z ? ΠCx if and only if

?Jx − Jz,y − z?≤ 0,

∀y ∈ C.

?2.8?

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Lemma 2.5 ?see ?36, 37??. Let C be a nonempty closed and convex subset of a reflexive, strictly

convex, and smooth Banach space E and let x ∈ E. Then

φ?y,ΠCx?? φ?ΠCx,x? ≤ φ?y,x?,

∀y ∈ C.

?2.9?

Lemma 2.6 ?see ?38??. Let E be a uniformly convex and uniformly smooth Banach space and C a

nonempty, closed, and convex subset of E. Then ΠCis uniformly norm-to-norm continuous on every

bounded set.

We make use of the following mapping V studied in Alber ?37?:

V?x,x∗? ? ?x?2− 2?x,x∗? ? ?x∗?2

?2.10?

for all x ∈ E and x∗∈ E∗, that is, V?x,x∗? ? φ?x,J−1?x∗??.

Lemma 2.7 ?see ?39??. Let E be a reflexive, strictly convex, smooth Banach space. Then

V?x,x∗? ? 2

?

J−1x∗− x,y∗?

≤ V?x,x∗? y∗?

?2.11?

for all x ∈ E and x∗,y∗∈ E∗.

Lemma 2.8 ?see ?25??. Let C be a closed and convex subset of a smooth, strictly convex, and reflexive

Banach space E, let f be a bifunction from C × C to R which satisfies conditions ?A1?–?A4?, and let

r > 0 and x ∈ E. Then there exists z ∈ C such that

f?z,y??1

r

?Jz − Jx,y − z?≥ 0,

∀y ∈ C.

?2.12?

Following ?25, 40?, we know the following lemma.

Lemma 2.9 ?see ?41??. Let C be a nonempty closed and convex subset of a smooth, strictly convex,

and reflexive Banach space E. Let A : C → E∗be a continuous and monotone mapping, let f be a

bifunction from C × C to R satisfying ?A1?–?A4?, and let ϕ be a lower semicontinuous and convex

function from C to R. For all r > 0 and x ∈ E, there exists z ∈ C such that

f?z,y???Az,y − z?? ϕ?y??1

r

?Jz − Jx,y − z?≥ ϕ?z?,

∀y ∈ C.

?2.13?

Define the mapping Tr: E → 2Cas follows:

Tr?x? ?

?

z ∈ C : f?z,y???Az,y − z?? ϕ?y??1

r

?Jz − Jx,y − z?≥ ϕ?z?, ∀y ∈ C

?

.

?2.14?