CUBO A Mathematical Journal

Vol.10, N
o
¯ 04, (15–26). December 2008

A Strong Convergence Theorem by a New Hybrid Method

for an Equilibrium Problem with Nonlinear

Mappings in a Hilbert Space

Rinko Shinzato and Wataru Takahashi

Department of Mathematical and Computing Sciences,

Tokyo Institute of Technology,

Ohokayama, Meguro-ku, Tokyo 152-8552, Japan

emails: shinzato.l.aa@m.is.titech.ac.jp, wataru@is.titech.ac.jp

ABSTRACT

In this paper, we prove a strong convergence theorem for finding a common element

of the set of solutions of an equilibrium problem, the set of solutions of the variational

inequality for a monotone mapping and the set of fixed points of a nonexpansive map-

ping in a Hilbert space by using a new hybrid method. Using this theorem, we obtain

three new results for finding a solution of an equilibrium problem, a solution of the

variational inequality for a monotone mapping and a fixed point of a nonexpansive

mapping in a Hilbert space.

RESUMEN

En este art́ıculo, probamos un teorema de convergencia fuerte para encontrar un ele-

mento común del conjunto de soluciones de un problema de equilibrio; del conjunto de

soluciones de una desigualdad variacional para una aplicación monótona y del conjunto

de punto fijos de una aplicación no expansiva en un espacio de Hilbert mediante el uso



16 Rinko Shinzato and Wataru Takahashi CUBO
10, 4 (2008)

de un nuevo método h́ıbrido. Usando nuestro teorema obtenemos tres nuevos resultados

para encontrar una solución de un problema de equiĺıbrio; una solución de la desigual-

dad variacional para una aplicación monótona y un punto fijo para una aplicación no

expansiva en un espacio de Hilbert.

Key words and phrases: Hilbert space, equilibrium problem, nonexpansive mapping, inverse-

strongly monotone mapping, iteration, strong convergence theorem.

Math. Subj. Class.: 47H05, 47H09, 47J25.

1 Introduction

Let H be a real Hilbert space with inner product 〈·, ·〉 and norm ‖ · ‖ and let C be a nonempty

closed convex subset of H. Let f be a bifunction from C × C to R, where R is the set of real

numbers. The equilibrium problem for f : C × C → R is to find x̂ ∈ C such that

f (x̂, y) ≥ 0 (1.1)

for all y ∈ C. The set of such solutions x̂ is denoted by EP (f ). The problem (1.1) is very general in

the sense that it includes, as special cases, optimization problems, variational inequalities, minimax

problems, Nash equilibrium problem in noncoopetative games and others; see, for instance, [1] and

[6]. A mapping S of C into H is called nonexpansive if

‖Sx − Sy‖ ≤ ‖x − y‖

for all x, y ∈ C. We denote by F (S) the set of fixed points of S. A mapping A : C → H is called

inverse-strongly monotone if there exists α > 0 such that

〈x − y, Ax − Ay〉 ≥ α‖Ax − Ay‖2

for all x, y ∈ C. The variational inequality problem is to find a u ∈ C such that

〈v − u, Au〉 ≥ 0 (1.2)

for all v ∈ C. The set of such solutions u is denoted by V I(C, A). Setting A = I−S, where S : C →H

is nonexpansive, we have from [14] that A : C → H is a 1
2
-inverse-strongly monotone mapping.

Recently, Tada and Takahashi [9, 10] and Takahashi and Takahashi [11] obtained weak and strong

convergence theorems for finding a common element of the set of solutions of an equilibrium

problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. In particular,

Tada and Takahashi [10] established a strong convergence theorem for finding a common element

of such two sets by using the hybrid method introduced in Nakajo and Takahashi [7]. On the other

hand, Takahashi and Toyoda [16] introduced an iterative method for finding a common element

of the set of solutions of the variational inequality for an inverse-strongly monotone mapping and



CUBO
10, 4 (2008)

A Strong Convergence Theorem ... 17

the set of fixed points of a nonexpansive mapping. Very recently, Takahashi, Takeuchi and Kubota

[15] proved the following theorem by a new hybrid method which is different from Nakajo and

Takahashi’s hybrid method. We call such a method the shrinking projection method.

Theorem 1.1 (Takahashi, Takeuchi and Kubota [15]). Let H be a Hilbert space and let C be a

nonempty closed convex subset of H. Let T be a nonexpansive mapping of C into H such that

F (T ) 6= ∅ and let x0 ∈ H. For C1 = C and u1 = PC1 x0, define a sequence {un} of C as follows:











yn = αnun + (1 − αn)T un,

Cn+1 = {z ∈ Cn : ‖yn − z‖ ≤ ‖un − z‖},

un+1 = PCn+1 x0, n ∈ N,

where 0 ≤ αn ≤ a < 1. Then, {un} converges strongly to z0 = PF (T )x0, where PF (T ) is the metric

projection of H onto F (T ).

In this paper, motivated by Tada and Takahashi [10], Takahashi and Toyoda [16], and Taka-

hashi, Takeuchi and Kubota [15], we prove a strong convergence theorem for finding a common

element of the set of solutions of an equilibrium problem, the set of solutions of the variational

inequality for an inverse-strongly monotone mapping and the set of fixed points of a nonexpansive

mapping in a Hilbert space by using the shrinking projection method. Using this theorem, we ob-

tain three new results for finding a solution of an equilibrium problem, a solution of the variational

inequality for an inverse-strongly monotone mapping and a fixed point of a nonexpansive mapping

in a Hilbert space.

2 Preliminaries

Let H be a real Hilbert space with inner product 〈·, ·〉 and norm ‖ · ‖. We denote by “→” strong

convergence and by “⇀” weak convergence. We know from [14] that, for all x, y ∈ H and λ ∈ [0, 1],

there holds

‖λx + (1 − λ)y‖2 = λ‖x‖2 + (1 − λ)‖y‖2 − λ(1 − λ)‖x − y‖2.

Let C be a nonempty closed convex subset of H. For any x ∈ H, there exists a unique nearest

point in C, denoted by PC x, such that

‖x − PC x‖ ≤ ‖x − y‖

for all y ∈ C. PC is called the metric projection of H onto C. We know that PC satisfies

‖PC x − PC y‖
2 ≤ 〈PC x − PC y, x − y〉 (2.1)

for all x, y ∈ H. Further, we have that

〈x − PC x, PC x − y〉 ≥ 0 (2.2)



18 Rinko Shinzato and Wataru Takahashi CUBO
10, 4 (2008)

for all x ∈ H and y ∈ C. A mapping A : C → H is called inverse-strongly monotone if there exists

α > 0 such that

〈x − y, Ax − Ay〉 ≥ α‖Ax − Ay‖2

for all x, y ∈ C. The set of solutions of the variational inequality for A is denoted by V I(C, A). We

know that, for all λ > 0,

u ∈ V I(C, A) ⇐⇒ u = PC (u − λAu).

We also know that, for any λ with 0 < λ ≤ 2α, a mapping I − λA : C → H is nonexpansive; see

[16, 14] for more details. It is also known that H satisfies Opial’s condition, i.e., for any sequence

{xn} with xn ⇀ x, the inequality

lim inf
n→∞

‖xn − x‖ < lim inf
n→∞

‖xn − y‖

holds for every y ∈ H with y 6= x. A Hilbert space H also has the Kadec-Klee property, i.e., if

{xn} is a sequence of H with xn ⇀ x and ‖xn‖ → ‖x‖, then there holds xn → x.

A set-valued mapping T : H → 2H is called monotone if for all x, y ∈ H, f ∈ T x and g ∈ T y

imply 〈x − y, f − g〉 ≥ 0. A monotone mapping T : H → 2H is maximal if the graph G(T ) of T is

not properly contained in the graph of any other monotone mapping. It is known that a monotone

mapping T is maximal if and only if for (x, f ) ∈ H × H, 〈x − y, f − g〉 ≥ 0 for every (y, g) ∈ G(T )

implies f ∈ T x. Let A be an inverse-strongly monotone mapping of C into H and let NC v be the

normal cone to C at v ∈ C, i.e., NC v = {w ∈ H : 〈v − u, w〉 ≥ 0, ∀u ∈ C}, and define

T v =

{

Av + NC v, v ∈ C,

∅, v /∈ C.

Then T is maximal monotone and 0 ∈ T v if and only if v ∈ V I(C, A); see [8].

For solving an equilibrium problem for a bifunction f : C × C → R, let us assume that f

satisfies the following conditions:

(A1) f (x, x) = 0 for all x ∈ C;

(A2) f is monotone, i.e. f (x, y) + f (y, x) ≤ 0 for all x, y ∈ C;

(A3) for all x, y, z ∈ C,

lim sup
t↓0

f (tz + (1 − t)x, y) ≤ f (x, y);

(A4) for all x ∈ C, f (x, ·) is convex and lower semicontinuous.

The following lemma appears implicitly in Blum and Oettlli [1].

Lemma 2.1 (Blum and Oettli). Let C be a nonempty closed convex subset of H and let f be a

bifunction of C × C into R satisfying (A1) − (A4). Let r > 0 and x ∈ H. Then, there exists z ∈ C

such that

f (z, y) +
1

r
〈y − z, z − x〉 ≥ 0 f or all y ∈ C.



CUBO
10, 4 (2008)

A Strong Convergence Theorem ... 19

The following lemma was also given in [2].

Lemma 2.2. Assume that f : C × C → R satisfies (A1) − (A4). For r > 0 and x ∈ H, define a

mapping Tr : H → C as follows:

Tr(x) =

{

z ∈ C : f (z, y) +
1

r
〈y − z, z − x〉 ≥ 0 for all y ∈ C

}

for all x ∈ H. Then, the following hold:

(1) Tr is single-valued;

(2) Tr is a firmly nonexpansive mapping, i.e., for all x, y ∈ H,

‖Trx − Try‖
2 ≤ 〈Trx − Try, x − y〉;

(3) F (Tr) = EP (f );

(4) EP (f ) is closed and convex.

3 Strong convergence theorem

In this section, using the shrinking projection method, we prove a strong convergence theorem for

finding a common element of the set of solutions of an equilibrium problem, the set of solutions of

the variational inequality for an inverse-strongly monotone mapping and the set of fixed points of

a nonexpansive mapping in a Hilbert space.

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let f be a

bifunction from C × C to R satisfying (A1) − (A4) and let S be a nonexpansive mapping from

C into H and let A be an α-inverse-strongly monotone mapping of C into H such that F (S) ∩

V I(C, A) ∩ EP (f ) 6= ∅. Let {xn} be a sequence in C generated by x0 = x ∈ C, C0 = C and



















un = Trn (xn),

yn = αnxn + (1 − αn)SPC (un − λnAun),

Cn+1 = {z ∈ Cn : ‖yn − z‖ ≤ ‖xn − z‖},

xn+1 = PCn+1 x, n ∈ N ∪ {0},

where 0 ≤ αn ≤ c < 1, 0 < d ≤ rn < ∞ and 0 < a ≤ λn ≤ b < 2α. Then, {xn} converges

strongly to PF (S)∩V I(C,A)∩EP (f )x.

Proof. From [7], we know that

‖yn − z‖ ≤ ‖xn − z‖

⇐⇒‖yn − xn‖
2

+ 2〈yn − xn, xn − z〉 ≤ 0.



20 Rinko Shinzato and Wataru Takahashi CUBO
10, 4 (2008)

So, Cn is a closed convex subset of H for all n ∈ N ∪{0}. Next we show by mathematical induction

that F (S) ∩ V I(C, A) ∩ EP (f ) ⊂ Cn for all n ∈ N ∪ {0}. Put zn = PC (un − λnAun) for all

n ∈ N ∪ {0}. From C0 = C, we have

F (S) ∩ V I(C, A) ∩ EP (f ) ⊂ C0.

Suppose that F (S)∩V I(C, A)∩EP (f ) ⊂ Ck for some k ∈ N∪{0}. Let u ∈ F (S)∩V I(C, A)∩EP (f ).

Since I − λkA and Tr
k

are nonexpansive and u = PC (u − λkAu), we have

‖zk − u‖ = ‖PC (uk − λkAuk) − PC (u − λkAu)‖

≤ ‖(I − λkA)uk − (I − λkA)u‖

≤ ‖uk − u‖

= ‖Tr
k
xk − Tr

k
u‖

≤ ‖xk − u‖.

So, we have

‖yk − u‖ = ‖αkxk + (1 − αk)Szk − u‖

≤ αk‖xk − u‖ + (1 − αk)‖Szk − u‖

≤ αk‖xk − u‖ + (1 − αk)‖zk − u‖

≤ αk‖xk − u‖ + (1 − αk)‖xk − u‖

= ‖xk − u‖.

Since u ∈ Ck, we have u ∈ Ck+1. This implies that

F (S) ∩ V I(C, A) ∩ EP (f ) ⊂ Cn

for all n ∈ N ∪ {0}. So, {xn} is well-defined.

From the definition of xn+1, we have

‖xn+1 − x‖ ≤ ‖u − x‖

for all u ∈ F (S) ∩ V I(C, A) ∩ EP (f ) ⊂ Cn+1. Then, {xn} is bounded. Therefore, {yn}, {zn},

{un} and {Szn} are also bounded.

Let us show that ‖xn+1 − xn‖ → 0. From xn+1 ∈ Cn+1 ⊂ Cn and xn = PCn x, we have

‖xn − x‖ ≤ ‖xn+1 − x‖

for all n ∈ N ∪ {0}. Thus {‖xn − x‖} is nondecreasing. Thus limn→∞ ‖xn − x‖ exists. Since

‖xn+1 − xn‖
2

= ‖xn+1 − x‖
2
+ ‖xn − x‖

2
+ 2〈xn+1 − x, x − xn〉

= ‖xn+1 − x‖
2 − ‖xn − x‖

2 − 2〈xn − xn+1, x − xn〉

≤ ‖xn+1 − x‖
2 − ‖xn − x‖

2



CUBO
10, 4 (2008)

A Strong Convergence Theorem ... 21

for all n ∈ N ∪ {0}, we have limn→∞ ‖xn+1 − xn‖ = 0.

Since xn+1 ∈ Cn+1, we have

‖xn − yn‖ ≤ ‖xn − xn+1‖ + ‖xn+1 − yn‖ ≤ 2‖xn − xn+1‖.

This together with ‖xn+1 − xn‖ → 0 implies that

‖xn − yn‖ → 0.

We also show that ‖Aun − Au‖ → 0. For all u ∈ F (S) ∩ V I(C, A) ∩ EP (f ), we have

‖zn − u‖
2

= ‖PC (un − λnAun) − PC (u − λnAu)‖
2

≤ ‖(un − λnAun) − (u − λnAu)‖
2

= ‖un − u − λn(Aun − Au)‖
2

= ‖un − u‖
2 − 2λn〈un − u, Aun − Au〉 + λ

2
n‖Aun − Au‖

2

≤ ‖un − u‖
2 − 2λnα‖Aun − Au‖

2
+ λ2n‖Aun − Au‖

2

= ‖un − u‖
2
+ λn(λn − 2α)‖Aun − Au‖

2

≤ ‖un − u‖
2
+ a(b − 2α)‖Aun − Au‖

2
.

Since ‖ · ‖2 is convex and ‖un − u‖ ≤ ‖xn − u‖, we have

‖yn − u‖
2 ≤ αn‖xn − u‖

2
+ (1 − αn)‖Szn − u‖

2

≤ αn‖xn − u‖
2
+ (1 − αn){‖un − u‖

2
+ a(b − 2α)‖Aun − Au‖

2}

≤ ‖xn − u‖
2
+ a(b − 2α)‖Aun − Au‖

2.

Therefore, we have

−a(b − 2α)‖Aun − Au‖
2 ≤ ‖xn − u‖

2 − ‖yn − u‖
2

= (‖xn − u‖ + ‖yn − u‖)(‖xn − u‖ − ‖yn − u‖)

≤ (‖xn − u‖ + ‖yn − u‖)‖xn − yn‖.

Since {xn} and {yn} are bounded and ‖xn − yn‖ → 0, we obtain ‖Aun − Au‖ → 0. Further we

show that ‖zn − un‖ → 0. For all u ∈ F (S) ∩ V I(C, A) ∩ EP (f ), we have from (2.1) that

‖zn − u‖
2

= ‖PC (un − λnAun) − PC (u − λnAu)‖
2

≤ 〈(un − λnAun) − (u − λnAu), zn − u〉

=
1

2
{‖(un − λnAun) − (u − λnAu)‖

2
+ ‖zn − u‖

2

− ‖(un − λnAun) − (u − λnAu) − (zn − u)‖
2}

≤
1

2
{‖un − u‖

2
+ ‖zn − u‖

2 − ‖(un − zn) − λn(Aun − Au)‖
2}

=
1

2
{‖un − u‖

2
+ ‖zn − u‖

2 − ‖un − zn‖
2

+ 2λn〈un − zn, Aun − Au〉 − λ
2
n‖Aun − Au‖

2},



22 Rinko Shinzato and Wataru Takahashi CUBO
10, 4 (2008)

and hence

‖zn − u‖
2 ≤ ‖un − u‖

2 − ‖un − zn‖
2
+ 2λn〈un − zn, Aun − Au〉.

From this inequality and ‖un − u‖ ≤ ‖xn − u‖, we have

‖yn − u‖
2 ≤ αn‖xn − u‖

2
+ (1 − αn)‖zn − u‖

2

≤ αn‖xn − u‖
2
+ (1 − αn){‖un − u‖

2 − ‖un − zn‖
2

+ 2λn〈un − zn, Aun − Au〉}

≤ ‖xn − u‖
2 − (1 − αn)‖un − zn‖

2

+ 2λn(1 − αn)〈un − zn, Aun − Au〉,

and hence

(1 − αn)‖un − zn‖
2 ≤ ‖xn − u‖

2 − ‖yn − u‖
2

+ 2λn(1 − αn)〈un − zn, Aun − Au〉

≤ (‖xn − u‖ + ‖yn − u‖)‖xn − yn‖

+ 2λn(1 − αn)〈un − zn, Aun − Au〉.

Since 0 ≤ αn ≤ c < 1, ‖xn − yn‖ → 0 and ‖Aun − Au‖ → 0, we have that

‖un − zn‖ → 0.

Let us show ‖xn − un‖ → 0. For all u ∈ F (S) ∩ V I(C, A) ∩ EP (f ), we have from Lemma 2.2 and

F (Trn ) = EP (f ) that

‖un − u‖
2

= ‖Trn xn − Trn u‖
2 ≤ 〈Trn xn − Trn u, xn − u〉

= 〈un − u, xn − u〉

=
1

2
{‖un − u‖

2
+ ‖xn − u‖

2 − ‖un − xn‖
2},

and hence

‖un − u‖
2 ≤ ‖xn − u‖

2 − ‖un − xn‖
2.

From this inequality and ‖zn − u‖ ≤ ‖un − u‖, we have

‖yn − u‖
2 ≤ αn‖xn − u‖

2
+ (1 − αn)‖zn − u‖

2

≤ αn‖xn − u‖
2
+ (1 − αn){‖xn − u‖

2 − ‖un − xn‖
2},

and hence

(1 − αn)‖un − xn‖
2 ≤ ‖xn − u‖

2 − ‖yn − u‖
2 ≤ (‖xn − u‖ + ‖yn − u‖)‖xn − yn‖.

Therefore, we obtain

‖un − xn‖ → 0.



CUBO
10, 4 (2008)

A Strong Convergence Theorem ... 23

Since (1 − αn)(Szn − zn) = αn(zn − xn) + (yn − zn), we have

(1 − αn)‖Szn − zn‖ ≤ ‖zn − xn‖ + ‖yn − zn‖

≤ ‖zn − xn‖ + ‖yn − xn‖ + ‖xn − zn‖ = 2‖zn − xn‖ + ‖yn − xn‖

≤ 2(‖zn − un‖ + ‖un − xn‖) + ‖yn − xn‖.

Therefore, we also obtain ‖Szn − zn‖ → 0.

Since {zn} is bounded, there exists a subsequence {zni} of {zn} such that zni ⇀ z0. Then,

we can obtain that z0 ∈ F (S) ∩ V I(C, A) ∩ EP (f ). In fact, let us first show z0 ∈ F (S). Assume

that z0 /∈ F (S). By Opial’s condition,

lim inf
i→∞

‖zni − z0‖ < lim inf
i→∞

‖zni − Sz0‖ = lim inf
i→∞

‖zni − Szni + Szni − Sz0‖

= lim inf
i→∞

‖Szni − Sz0‖

≤ lim inf
i→∞

‖zni − z0‖.

This is a contradiction. Therefore, we have z0 ∈ F (S). Let us show z0 ∈ V I(C, A). Define

T v =

{

Av + NC v, v ∈ C,

∅, v /∈ C.

Then T is maximal monotone and T −10 = V I(C, A); see [8]. Let (v, u) ∈ G(T ). Since u−Av ∈ NC v

and zn = PC (un − λnAun) ∈ C, we have 〈v − zn, u − Av〉 ≥ 0. By the definition of zn, we also

have

〈v − zn, zn − (un − λnAun)〉 ≥ 0,

and hence

〈v − zn,
zn − un

λn
+ Aun〉 ≥ 0.

Therefore,

〈v − zni , u〉 ≥ 〈v − zni , Av〉

≥ 〈v − zni , Av − {
zni − uni

λni
+ Auni }〉

= 〈v − zni , Av − Azni〉 + 〈v − zni , Azni − Auni〉 − 〈v − zni ,
zni − uni

λni
〉

≥ −‖v − zni‖‖Azni − Auni‖ − ‖v − zni‖‖
zni − uni

λni
‖.

Since ‖zn − un‖ → 0 and A is Lipschits continuous, we have 〈v − z0, u〉 ≥ 0. Since T is maximal

monotone, we have z0 ∈ T
−1

0 and hence z0 ∈ V I(C, A).

Finally, we show that z0 ∈ EP (f ). By un = Trn xn, we have

f (un, y) +
1

rn
〈y − un, un − xn〉 ≥ 0



24 Rinko Shinzato and Wataru Takahashi CUBO
10, 4 (2008)

for all y ∈ C. From (A2) we also have

1

rn
〈y − un, un − xn〉 ≥ f (y, un)

and hence

〈y − uni ,
uni − xni

rni
〉 ≥ f (y, uni ).

Since ‖un − zn‖ → 0 and zni ⇀ z0, we have uni ⇀ z0. Since 0 < d ≤ rn < ∞ and ‖un − xn‖ → 0,

we have from (A4) that 0 ≥ f (y, z0) for all y ∈ C. For t ∈ (0, 1] and y ∈ C, let yt = ty + (1 − t)z0.

Since y ∈ C and z0 ∈ C, we have yt ∈ C and hence f (yt, z0) ≤ 0. So, from (A1) and (A4) we have

0 = f (yt, yt) ≤ tf (yt, y) + (1 − t)f (yt, z0) ≤ tf (yt, y)

and hence 0 ≤ f (yt, y). From (A3), we have 0 ≤ f (z0, y) for all y ∈ C and hence z0 ∈ EP (f ).

Therefore z0 ∈ F (S) ∩ V I(C, A) ∩ EP (f ).

From z′ = PF (S)∩V I(C,A)∩EP (f )x, z0 ∈ F (S) ∩ V I(C, A) ∩ EP (f ) and ‖xn − x‖ ≤ ‖z
′ − x‖, we

have

‖z′ − x‖ ≤ ‖z0 − x‖ ≤ lim inf
i→∞

‖zni − x‖

≤ lim sup
i→∞

‖zni − x‖

≤ lim sup
i→∞

{‖zni − uni‖ + ‖uni − xni‖ + ‖xni − x‖}

≤ ‖z′ − x‖.

Thus, we have

lim
i→∞

‖zni − x‖ = ‖z0 − x‖ = ‖z
′ − x‖.

This implies z0 = z
′
. Further, since a Hilbert space has the Kadec-Klee property, we have that

zni → z
′
. From ‖zn − xn‖ → 0, we also have xni → z

′
. Therefore, xn → z

′
. This completes the

proof.

4 Applications

In this section, using Theorem 3.1, we prove three new results for finding a solution of an equilibrium

problem, a solution of the variational inequality for an inverse-strongly monotone mapping and a

fixed point of a nonexpansive mapping in a Hilbert space. First, we obtain a result for finding a

common element of the set of solutions of an equilibrium problem and the set of fixed points of a

nonexpansive mapping in a Hilbert space.

Theorem 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let f be a

bifunction from C × C to R satisfying (A1) − (A4) and let S be a nonexpansive mapping from C



CUBO
10, 4 (2008)

A Strong Convergence Theorem ... 25

into H such that F (S)∩EP (f ) 6= ∅. Let {xn} be a sequence in C generated by x0 = x ∈ C, C0 = C

and


















un = Trn (xn),

yn = αnxn + (1 − αn)S(un),

Cn+1 = {z ∈ Cn : ‖yn − z‖ ≤ ‖xn − z‖},

xn+1 = PCn+1 x, n ∈ N ∪ {0},

where 0 ≤ αn ≤ c < 1 and 0 < d ≤ rn < ∞. Then, {xn} converges strongly to PF (S)∩EP (f )x.

Proof. Putting A = 0 in Theorem 3.1, we obtain the desired result.

Next, we obtain a result for finding a common element of the set of solutions of an equilibrium

problem and the set of solutions of the variational inequality for an inverse-strongly monotone

mapping in a Hilbert space.

Theorem 4.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let f be a

bifunction from C × C to R satisfying (A1) − (A4) and let A be an α-inverse-strongly monotone

mapping of C into H such that V I(C, A) ∩ EP (f ) 6= ∅. Let {xn} be a sequence in C generated by

x0 = x ∈ C, C0 = C and


















un = Trn (xn),

yn = αnxn + (1 − αn)PC (un − λnAun),

Cn+1 = {z ∈ Cn : ‖yn − z‖ ≤ ‖xn − z‖},

xn+1 = PCn+1 x, n ∈ N ∪ {0},

where 0 ≤ αn ≤ c < 1, 0 < d ≤ rn < ∞ and 0 < a ≤ λn ≤ b < 2α. Then, {xn} converges

strongly to PV I(C,A)∩EP (f )x.

Proof. Putting S = I in Theorem 3.1, we obtain the desired result.

Finally, we obtain a result for finding a common element of the set of solutions of the variational

inequality for an inverse-strongly monotone mapping and the set of fixed points of a nonexpansive

mapping in a Hilbert space.

Theorem 4.3. Let C be a nonempty closed convex subset of a real Hilbert space H. Let S be a

nonexpansive mapping from C into H and let A be an α-inverse-strongly monotone mapping of C

into H such that F (S) ∩ V I(C, A) 6= ∅. Let {xn} be a sequence in C generated by x0 = x ∈ C,

C0 = C and










yn = αnxn + (1 − αn)SPC (xn − λnAxn),

Cn+1 = {z ∈ Cn : ‖yn − z‖ ≤ ‖xn − z‖},

xn+1 = PCn+1 x, n ∈ N ∪ {0},

where 0 ≤ αn ≤ c < 1 and 0 < a ≤ λn ≤ b < 2α. Then, {xn} converges strongly to PF (S)∩V I(C,A)x.

Proof. Putting f = 0 in Theorem 3.1, we obtain the desired result.

Received: January 2008. Revised: February 2008.



26 Rinko Shinzato and Wataru Takahashi CUBO
10, 4 (2008)

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