Báo cáo hóa học: " Research Article Convergence Theorem for Equilibrium Problems and Fixed Point Problems of Infinite Family of Nonexpansive Mappings" docx

Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2007, Article ID 64363, 12 pages
doi:10.1155/2007/64363
Research Article
Convergence Theorem for Equilibrium Problems and Fixed Point
Problems of Infinite Family of Nonexpansive Mappings
Yonghong Yao, Yeong-Cheng Liou, and Jen-Chih Yao
Received 17 March 2007; Accepted 20 August 2007
Recommended by Billy E. Rhoades
We introduce an iterative scheme for finding a common element of the set of solutions
of an equilibrium problem and the set of common fixed points of infinite nonexpan-
sive mappings in a Hilbert space. We prove a strong-convergence theorem under mild
assumptions on parameters.
Copyright © 2007 Yong hong Yao 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.
1. Introduction
Let H be a real Hilbert space and let C be a nonempty closed convex subset of H.Let
h : C
× C→R be an equilibr ium bifunction, that is, h(u,u) = 0foreveryu ∈ C.Thenone
can define the equilibrium problem that is to find an element u
∈ C such that
EP(h):h(u,v)
≥ 0 ∀v ∈ C. (1.1)
Denote the set of solutions of EP(h)bySEP(h). This problem contains fixed point
problems, optimization problems, variational inequality problems, and Nash equilibrium
problems as special cases, see [1]. Some methods have been proposed to solve the equi-
librium problem, please consult [2–4].
Recently, Combettes and Hirstoaga [2] introduced an iterative scheme of finding the
best approximation to the initial data when SEP(h)

=∅ and proved a strong convergence
theorem. Motivated by the idea of Combettes and Hirstoaga, very recently, Takahashi and
Takahashi [4] introduced a new iterative scheme by the viscosity approximation method
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. Their results extend and
2 Fixed Point Theory and Applications
improve the corresponding results announced by Combettes and Hirstoaga [2], Moudafi
[5], Wittmann [6], and Tada and Takahashi [7].
In this paper, motivated and inspired by Combettes and Hirstoaga [2] and Takahashi
and Takahashi [4], we introduce an iterative scheme for finding a common element of the
set of solutions of EP(h) and the set of fixed points of infinite nonexpansive mappings in
a Hilbert space. We obtain a strong convergence theorem which improves and extends
the corresponding results of [2, 4].
2. Preliminaries
Let H be a real Hilbert space with inner product
·,· and norm ·.LetC be a nonempty
closed conv ex subset of H.Thenforanyx
∈ H, there exists a unique nearest point in C,
denoted by P
C
(x), such that x − P
C
(x)≤x − y for all y ∈ C.SuchaP
C
is called the
metric projection of H onto C.WeknowthatP
C
is nonexpansive. Further, for x ∈ H and
x
∗

∈ C,
x
∗
= P
C
(x) ⇐⇒

x − x
∗
,x
∗
− y

≥
0 ∀y ∈ C. (2.1)
Recall that a mapping T : C
→H is called nonexpansive if Tx− Ty≤x − y for
all x, y
∈ C. Denote the set of fixed p oints of T by F(T). It is well known that if C is a
bounded closed convex and T : C
→C is nonexpansive, then F(T)=∅; see, for instance,
[8]. We call a mapping f : H
→H contractive if there exists a constant α ∈ (0,1)suchthat
 f (x) − f (y)≤αx − y for all x, y ∈ H.
For an equilibrium bifunction h : C
× C → R,wecallh satisfying condition (A) if h
satisfies the following three conditions:
(i) h is monotone, that is, h(x, y)+h(y,x)
≤ 0forallx, y ∈ C;
(ii) for each x, y, z

∈ C,lim
t↓0
h(tz +(1− t)x, y) ≤ h(x, y);
(iii) for each x
∈ C, y → h(x, y) is convex and lower semicontinuous.
If an equilibr ium bifunction h : C
× C→R satisfies condition (A), then we have the fol-
lowing two important results. You can find the first lemma in [1] and the second one in
[2].
Lemma 2.1. Let C be a nonempty closed convex subset of H and let h be an equilibrium
bifunction of C
× C into R, satisfying condition (A). Let r>0 and x ∈ H. Then there exists
y
∈ C such that
h(y, z)+
1
r
z − y, y − x≥0 ∀z ∈ C. (2.2)
Lemma 2.2. Assume that h satisfies the same assumptions as Lemma 2.1.Forr>0 and
x
∈ H, define a mapping S
r
: H→C as follows:
S
r
(x) =

y ∈ C : h(y,z)+
1
r

z − y, y − x≥0, ∀z ∈ C

(2.3)
for all y
∈ H.Thenthefollowingholds:
Yonghong Yao et al. 3
(1) S
r
is single-valued and S
r
is fir mly nonexpansive, that is, for any x, y ∈ H,


S
r
x − S
r
y


2
≤

S
r
x − S
r
y,x − y

; (2.4)

(2) F(S
r
) = SEP(h) and SEP(h) is closed and c onvex.
We also need the following lemmas for proving our main results.
Lemma 2.3 (see [9]). Let
{x
n
} and {y
n
} be bounded sequences in a Banach space X and
let
{β
n
} beasequencein[0, 1] with 0 < liminf
n→∞
β
n
≤ limsup
n→∞
β
n
< 1.Supposex
n+1
=
(1 − β
n
)y
n
+ β
n

x
n
for all integers n ≥ 0 and limsup
n→∞
(y
n+1
− y
n
−x
n+1
− x
n
) ≤ 0.
Then lim
n→∞
y
n
− x
n
=0.
Lemma 2.4 (see [10]). Assume
{a
n
} is a sequence of nonnegative real numbers such that
a
n+1
≤ (1 − γ
n
)a
n

+ δ
n
,where{γ
n
} is a sequence in (0,1) and {δ
n
} is a sequence such that

∞
n=1
γ
n
=∞and limsup
n→∞
δ
n
/γ
n
≤ 0. Then lim
n→∞
a
n
= 0.
3. Iterative scheme and strong convergence theorems
In this section, we first introduce our iterative scheme. Consequently, we will establish
strong convergence theorems for this iteration scheme. To be more specific, let T
1
,T
2
,

be infinite mappings of C into C and let λ
1
,λ
2
, be real numbers such that 0 ≤ λ
i
≤ 1for
every i
∈ N.Foranyn ∈ N, define a mapping W
n
of C into C as foll ows:
U
n,n+1
= I,
U
n,n
= λ
n
T
n
U
n,n+1
+

1 − λ
n

I,
U
n,n−1

= λ
n−1
T
n−1
U
n,n
+

1 − λ
n−1

I,
.
.
.
U
n,k
= λ
k
T
k
U
n,k+1
+

1 − λ
k

I,
.

.
.
U
n,2
= λ
2
T
2
U
n,3
+

1 − λ
2

I,
W
n
= U
n,1
= λ
1
T
1
U
n,2
+

1 − λ
1


I.
(3.1)
SuchamappingW
n
is called the W-mapping generated by T
n
,T
n−1
, ,T
1
and λ
n
,
λ
n−1
, ,λ
1
;see[11].
Now we introduce the following iteration scheme: Let f be a contraction of H into it-
self with coefficient α
∈ (0,1) and given x
0
∈ H arbitrarily. Suppose the sequences {x
n
}
∞
n=1
4 Fixed Point Theory and Applications
and

{y
n
}
∞
n=1
are generated iteratively by
h

y
n
,x

+
1
r
n

x − y
n
, y
n
− x
n

≥
0, ∀x ∈ C,
x
n+1
= α
n

f

x
n

+ β
n
x
n
+ γ
n
W
n
y
n
,
(3.2)
where
{α
n
}, {β
n
},and{γ
n
} are three sequences in (0,1) such that α
n
+ β
n
+ γ
n

= 1, {r
n
}
is a real sequence in (0,∞), h is an equilibrium bifunction, and W
n
is the W-mapping
defined by (3.1).
We have the following cr ucial conclusions concerning W
n
. You can find them in [12,
13]. Now we only need the following similar version in Hilber t spaces.
Lemma 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H.LetT
1
,
T
2
, be nonexpansive mappings of C into C such that

∞
i=1
F(T
i
) is nonempty, and let
λ
1
,λ
2
, be real numbers such that 0 <λ
i
≤ b<1 for any i ∈ N. Then for every x ∈ C and

k
∈ N, the limit lim
n→∞
U
n,k
x exists.
Remark 3.2. From Lemma 3.1,wehavethatifC is bounded, then for all ε>0, there
exists a common positive integer number N
0
such that for n>N
0
, U
n,k
x − U
k
(x) <ε
for all x
∈ C. Indeed, by the similar argument to Lemma 3.2 in [13], let w ∈

∞
n=1
F(T
n
).
Since C is bounded, there exists a constant M>0suchthat
x − w≤M for all x ∈ C.
Fix k
∈ N.Thenforallx ∈ C and any n ∈ N with n ≥ k,wehaveU
n+1,k
x − U

n,k
x≤
2(

n+1
i
=k
λ
i
)x − w≤2M(

n+1
i
=k
λ
i
).
Let ε>0. Then there exists n
0
∈ N with n
0
≥ k such that for all x ∈ C, b
n
0
−k+2
<
ε(1
− b)/2M.Soforallx ∈ C and every m, n with m>n>n
0
,wehave



U
m,k
x − U
n,k
x


≤
m−1

j=n


U
j+1,k
x − U
j,k
x


≤
m−1

j=n

2

j+1


i=k
λ
i


x − w

≤
2M
m−1

j=n
b
j−k+2
≤
2Mb
n−k+2
1 − b
<ε.
(3.3)
Remark 3.3. Using Lemma 3.1, one can define a mapping W of C into C as Wx
=
lim
n→∞
W
n
x = lim
n→∞
U

n,1
x for every x ∈ C.SuchaW is called the W-mapping gen-
erated by T
1
,T
2
, and λ
1
,λ
2
, Weobservethatif{x
n
} is a bounded sequence in C,
then we have
lim
n→∞


Wx
n
− W
n
x
n


=
0. (3.4)
Indeed, from Remark 3.1,wehave:foranyε>0, there is n
0

such that Wx− W
n
x≤
ε for all x ∈{x
n
} and for all n ≥ n
0
.Inparticular,Wx
n
− W
n
x
n
≤ε for all n ≥ n
0
.
Consequently, lim
n→∞
Wx
n
− W
n
x
n
=0, as claimed.
Throughout this paper, we will assume that 0 <λ
i
≤ b<1foreveryi ∈ N.
Yonghong Yao et al. 5
Lemma 3.4. Let C be a nonempty closed convex subset of a real Hilbert space H.LetT

1
,
T
2
, be nonexpansive mappings of C into C such that

∞
i=1
F(T
i
) is nonempt y, and let
λ
1
,λ
2
, be real numbers such that 0 <λ
i
≤ b<1 for any i ∈ N. Then F(W) =

∞
i=1
F(T
i
).
Now we state and prove our main results.
Theorem 3.5. Let C beanonemptyclosedconvexsubsetofarealHilbertspaceH.Let
h : C
× C→R be an equilibrium bifunction satisfying condition (A) and let {T
i
}

∞
i=1
be an
infinite family of nonexpansive mappings of C into C such that

∞
i=1
F(T
i
)∩SEP(h)=∅.
Suppose
{α
n
}, {β
n
},and{γ
n
} are three seque nces in (0,1) such that α
n
+ β
n
+ γ
n
= 1 and
{r
n
}⊂(0,∞). Suppose the following conditions are satis fied:
(i) lim
n→∞
α

n
= 0 and

∞
n=0
α
n
=∞;
(ii) 0 < liminf
n→∞
β
n
≤ limsup
n→∞
β
n
< 1;
(iii) liminf
n→∞
r
n
> 0 and lim
n→∞
(r
n+1
− r
n
) = 0.
Let f be a contraction of H into itself and given x
0

∈ H arbitrarily. Then the sequences {x
n
}
and {y
n
} generated iteratively by (3.2) converge strongly to x
∗
∈

∞
i=1
F(T
i
) ∩ SEP(h),where
x
∗
= P

∞
i=1
F(T
i
)∩SEP(h)
f (x
∗
).
Proof. Let Q
= P

∞

i=1
F(T
i
)∩SEP(h)
.Notethat f is a contraction mapping with coefficient α ∈
(0,1). Then Qf(x) − Qf(y)≤f (x) − f (y)≤αx − y for all x, y ∈ H. Therefore,
Qf is a contraction of H into itself, which implies that there exists a unique element
x
∗
∈ H such that x
∗
= Qf(x
∗
). At the same time, we note that x
∗
∈ C.
Let p
∈

∞
i=1
F(T
i
)∩SEP(h). From the definition of S
r
, we note that y
n
= S
r
n

x
n
.Itfol-
lows that
y
n
− p=S
r
n
x
n
− S
r
n
p≤x
n
− p.Next,weprovethat{x
n
} and {y
n
} are
bounded. From ( 3.1)and(3.2), we obtain


x
n+1
− p


≤

α
n


f

x
n

−
p


+ β
n


x
n
− p


+ γ
n


W
n
y
n

− p


≤
α
n



f

x
n

−
f (p)


+


f (p) − p



+ β
n


x

n
− p


+ γ
n


y
n
− p


≤
α
n

α


x
n
− p


+


f (p) − p




+

1 − α
n



x
n
− p


≤
max



x
0
− p


,
1
1 − α


f (p) − p




.
(3.5)
Therefore,
{x
n
} is bounded. We also obtain that {y
n
}, {W
n
x
n
},and{ f (x
n
)} are all
bounded. We shall use M to denote the possible different constants appearing in the fol-
lowing reasoning.
Setting x
n+1
= β
n
x
n
+(1− β
n
)z
n
for all n ≥ 0, we have that

z
n+1
− z
n
=
x
n+2
− β
n+1
x
n+1
1 − β
n+1
−
x
n+1
− β
n
x
n
1 − β
n
=
α
n+1
1 − β
n+1

f


x
n+1

−
f

x
n

+

α
n+1
1 − β
n+1
−
α
n
1 − β
n

f

x
n

+
γ
n+1
1 − β

n+1

W
n+1
y
n+1
− W
n
y
n

+

γ
n+1
1 − β
n+1
−
γ
n
1 − β
n

W
n
y
n
.
(3.6)
6 Fixed Point Theory and Applications

So we have


z
n+1
− z
n


≤
αα
n+1
1 − β
n+1


x
n+1
− x
n


+




α
n+1
1 − β

n+1
−
α
n
1 − β
n







f

x
n



+


W
n
y
n




+
γ
n+1
1 − β
n+1


W
n+1
y
n+1
− W
n
y
n


.
(3.7)
Since T
i
and U
n,i
are nonexpansive from (3.1), we have


W
n+1
y
n

− W
n
y
n


=


λ
1
T
1
U
n+1,2
y
n
− λ
1
T
1
U
n,2
y
n


≤
λ
1



U
n+1,2
y
n
− U
n,2
y
n


≤
λ
1
λ
2


U
n+1,3
y
n
− U
n,3
y
n


≤···

≤
λ
1
λ
2
···λ
n


U
n+1,n+1
y
n
− U
n,n+1
y
n


≤
M
n

i=1
λ
i
,
(3.8)
and hence



W
n+1
y
n+1
− W
n
y
n


≤


W
n+1
y
n+1
− W
n+1
y
n


+


W
n+1
y

n
− W
n
y
n


≤


y
n+1
− y
n


+ M
n

i=1
λ
i
.
(3.9)
Substituting (3.9)into(3.7), we have


z
n+1
− z

n


≤
αα
n+1
1 − β
n+1


x
n+1
− x
n


+




α
n+1
1 − β
n+1
−
α
n
1 − β
n








f

x
n



+


W
n
y
n



+
γ
n+1
1 − β
n+1



y
n+1
− y
n


+
Mγ
n+1
1 − β
n+1
n

i=1
λ
i
.
(3.10)
On the other hand, from y
n
= S
r
n
x
n
and y
n+1
= S
r

n+1
x
n+1
,wehave
h

y
n
,x

+
1
r
n

x − y
n
, y
n
− x
n

≥
0 ∀x ∈ C, (3.11)
h

y
n+1
,x


+
1
r
n+1

x − y
n+1
, y
n+1
− x
n+1

≥
0 ∀x ∈ C. (3.12)
Yonghong Yao et al. 7
Putting x
= y
n+1
in (3.11)andx = y
n
in (3.12), we have
h

y
n
, y
n+1

+
1

r
n

y
n+1
− y
n
, y
n
− x
n

≥
0, (3.13)
h

y
n+1
, y
n

+
1
r
n+1

y
n
− y
n+1

, y
n+1
− x
n+1

≥
0. (3.14)
From the monotonicity of h,wehave
h

y
n
, y
n+1

+ h

y
n+1
, y
n

≤
0. (3.15)
So from (3.13), we can conclude that

y
n+1
− y
n

,
y
n
− x
n
r
n
−
y
n+1
− x
n+1
r
n+1

≥
0, (3.16)
and hence

y
n+1
− y
n
, y
n
− y
n+1
+ y
n+1
− x

n
−
r
n
r
n+1

y
n+1
− x
n+1


≥
0. (3.17)
Since liminf
n→∞
r
n
> 0, without loss of generality, we may assume that there exists a real
number τ such that r
n
>τ>0foralln ∈ N.Thenwehave


y
n+1
− y
n



2
≤

y
n+1
− y
n
,x
n+1
− x
n
+

1 −
r
n
r
n+1


y
n+1
− x
n+1


≤



y
n+1
− y
n





x
n+1
− x
n


+




1 −
r
n
r
n+1







y
n+1
− x
n+1



,
(3.18)
and hence


y
n+1
− y
n


≤


x
n+1
− x
n


+
M

τ


r
n+1
− r
n


. (3.19)
Substituting (3.19)into(3.10), we have


z
n+1
− z
n


≤
αα
n+1
1 − β
n+1


x
n+1
− x
n



+




α
n+1
1 − β
n+1
−
α
n
1 − β
n







f

x
n




+


W
n
y
n



+
γ
n+1
1 − β
n+1


x
n+1
− x
n


+
γ
n+1
1 − β
n+1
×
M

τ


r
n+1
− r
n


+
Mγ
n+1
1 − β
n+1
n

i=1
λ
i
≤


x
n+1
− x
n


+





α
n+1
1 − β
n+1
−
α
n
1 − β
n







f

x
n

+


W
n
y

n



+
γ
n+1
1 − β
n+1
×
M
τ


r
n+1
− r
n


+
Mγ
n+1
1 − β
n+1
n

i=1
λ
i

.
(3.20)
This together with α
n
→0andr
n+1
− r
n
→0 imply that limsup
n→∞
(z
n+1
− z
n
−x
n+1
−
x
n
) ≤ 0. Hence by Lemma 2.3,weobtainz
n
− x
n
→0asn→∞. Consequently,
lim
n→∞


x
n+1

− x
n


=
lim
n→∞

1 − β
n



z
n
− x
n


=
0. (3.21)
8 Fixed Point Theory and Applications
From (3.19)andlim
n→∞
(r
n+1
− r
n
) = 0, we have lim
n→∞

y
n+1
− y
n
=0. Since x
n+1
=
α
n
f (x
n
)+β
n
x
n
+ γ
n
W
n
y
n
,wehave


x
n
− W
n
y
n



≤


x
n
− x
n+1


+


x
n+1
− W
n
y
n


≤


x
n
− x
n+1



+ α
n


f

x
n

−
W
n
y
n


+ β
n


x
n
− W
n
y
n


,

(3.22)
that is,


x
n
− W
n
y
n


≤
1
1 − β
n


x
n
− x
n+1


+
α
n
1 − β
n



f

x
n

−
W
n
y
n


. (3.23)
Hencewehavelim
n→∞
x
n
− W
n
y
n
=0. For p ∈

∞
i=1
F(T
i
)∩SEP(h), note that S
r

is
firmly nonexpansive. Then we have


y
n
− p


2
=


S
r
n
x
n
− S
r
n
p


2
≤

S
r
n

x
n
− S
r
n
p,x
n
− p

=

y
n
− p,x
n
− p

=
1
2



y
n
− p


2
+



x
n
− p


2
−


x
n
− y
n


2

,
(3.24)
and hence


y
n
− p


2

≤


x
n
− p


2
−


x
n
− y
n


2
. (3.25)
Therefore, we have


x
n+1
− p


2
≤ α

n


f

x
n

−
p


2
+ β
n


x
n
− p


2
+ γ
n


W
n
y

n
− p


2
≤ α
n


f

x
n

−
p


2
+ β
n


x
n
− p


2
+ γ

n


y
n
− p


2
≤ α
n


f

x
n

−
p


2
+ β
n


x
n
− p



2
+ γ
n



x
n
− p


2
−


x
n
− y
n


2

≤
α
n



f

x
n

−
p


2
+


x
n
− p


2
− γ
n


x
n
− y
n


2

.
(3.26)
Then we have
γ
n


x
n
− y
n


2
≤ α
n


f

x
n

−
p


2
+




x
n
− p


+


x
n+1
− p



×



x
n
− p


−


x
n+1

− p



≤
α
n


f

x
n

−
p


2
+


x
n
− x
n+1






x
n
− p


+


x
n+1
− p



.
(3.27)
It is easily seen that lim inf
n→∞
γ
n
> 0. So we have
lim
n→∞


x
n
− y
n



=
0. (3.28)
From
W
n
y
n
− y
n
≤W
n
y
n
− x
n
 + x
n
− y
n
,wealsohaveW
n
y
n
− y
n
→0. At that
same time, we note that



Wy
n
− y
n


≤


Wy
n
− W
n
y
n


+


W
n
y
n
− y
n


. (3.29)

Yonghong Yao et al. 9
It follows from (3.29)andRemark 3.2 that lim
n→∞
Wy
n
− y
n
=0. Next, we show that
limsup
n→∞

f

x
∗

−
x
∗
,x
n
− x
∗

≤
0, (3.30)
where x
∗
= P
F(W)∩SEP(h)

f (x
∗
). First, we can choose a subsequence {y
n
j
} of {y
n
} such
that
lim
j→∞

f

x
∗

−
x
∗
, y
n
j
− x
∗

=
limsup
n→∞


f

x
∗

−
x
∗
, y
n
− x
∗

. (3.31)
Since
{y
n
j
} is bounded, there exists a subsequence {y
n
ji
} of {y
n
j
}, which converges weakly
to w. Without loss of generality, we can assume that y
n
j
→w weakly. From Wy
n

− y
n
→0,
we obtain Wy
n
j
→w weakly. Now we show w ∈ SEP(h).
By y
n
= S
r
n
x
n
,wehaveh(y
n
,x)+(1/r
n
)x − y
n
, y
n
− x
n
≥0forallx ∈ C.Fromthe
monotonicity of h,wehave(1/r
n
)x − y
n
, y

n
− x
n
≥−h(y
n
,x) ≥ h(x, y
n
), and h ence

x − y
n
j
,
y
n
j
− x
n
j
r
n
j

≥
h

x, y
n
j


. (3.32)
Since (y
n
j
− x
n
j
)/r
n
j
→0andy
n
j
→w weakly, from the lower semicontinuity of h(x, y)on
the second variable y,wehaveh(x,w)
≤ 0forallx ∈ C.Fort with 0 <t≤ 1andx ∈ C,
let x
t
= tx +(1− t)w.Sincex ∈ C and w ∈ C,wehavex
t
∈ C, and hence h(x
t
,w) ≤ 0. So
from the convexity of equilibrium bifunction h(x, y) on the second variable y,wehave
0
= h

x
t
,x

t

≤
th

x
t
,x

+(1− t)h

x
t
,w

≤
th

x
t
,x

. (3.33)
Hence h(x
t
,x) ≥ 0. Therefore, we have h(w,x) ≥ 0forallx ∈ C, and hence w ∈ SEP( h).
We will show w
∈ F(W). Assume w ∈ F(W). Since y
n
j

→w weakly and w=Ww,from
Opial’s condition, we have
liminf
j→∞


y
n
j
− w


< liminf
j→∞


y
n
j
− Ww


≤
liminf
j→∞



y
n

j
− Wy
n
j


+


Wy
n
j
− Ww



≤
liminf
j→∞


y
n
j
− w


.
(3.34)
This is a contradiction. So we get w

∈ F(W) =

∞
i=1
F(T
i
). Therefore, w ∈

∞
i=1
F(T
i
)∩
SEP(h). Since x
∗
= P

∞
i=1
F(T
i
)∩SEP(h)
f (x
∗
), we have
limsup
n→∞

f


x
∗

−
x
∗
,x
n
− x
∗

=
lim
j→∞

f

x
∗

−
x
∗
,x
n
j
− x
∗

=

lim
j→∞

f

x
∗

−
x
∗
, y
n
j
− x
∗

=

f

x
∗

−
x
∗
,w − x
∗


≤
0.
(3.35)
10 Fixed Point Theory and Applications
First, we prove that
{x
n
} converges strongly to x
∗
∈

∞
i=1
F(T
i
)∩SEP(h). From (3.2),
we have


x
n+1
− x
∗


2
≤


β

n

x
n
− x
∗

+ γ
n

W
n
y
n
− x
∗



2
+2α
n

f

x
n

−
x

∗
,x
n+1
− x
∗

≤

β
n


x
n
− x
∗


+ γ
n


W
n
y
n
− x
∗





2
+2α
n

f

x
∗

−
x
∗
,x
n+1
− x
∗

+2α
n

f

x
n

−
f


x
∗

,x
n+1
− x
∗

≤

β
n


x
n
− x
∗


+ γ
n


y
n
− x
∗




2
+2αα
n


x
n
− x
∗




x
n+1
− x
∗


+2α
n

f

x
∗

−
x

∗
,x
n+1
− x
∗

≤

1 − α
n

2


x
n
− x
∗


2
+ αα
n



x
n+1
− x
∗



2
+


x
n
− x
∗


2

+2α
n

f

x
∗

−
x
∗
,x
n+1
− x
∗


,
(3.36)
which implies that


x
n+1
− x
∗


2
≤

1 − α
n

2
+ αα
n
1 − αα
n


x
n
− x
∗



2
+
2α
n
1 − αα
n

f

x
∗

−
x
∗
,x
n+1
− x
∗

=
1 − 2α
n
+ αα
n
1 − αα
n


x

n
− x
∗


2
+
α
2
n
1 − αα
n


x
n
− x
∗


2
+
2α
n
1 − αα
n

f

x

∗

−
x
∗
,x
n+1
− x
∗

≤

1 −
2(1 − α)α
n
1 − αα
n



x
n
− x
∗


2
+
2(1
− α)α

n
1 − αα
n
×

Mα
n
2(1 − α)
+
1
1 − α

f

x
∗

−
x
∗
,x
n+1
− x
∗


=

1 − ϕ
n




x
n
− x
∗


2
+ φ
n
ϕ
n
,
(3.37)
where ϕ
n
= 2(1 − α)α
n
/(1 − αα
n
)andφ
n
= Mα
n
/2(1 − α)+1/(1 − α) f (x
∗
) − x
∗

,x
n+1
−
x
∗
. It is easily seen that

∞
n=0
ϕ
n
=∞and limsup
n→∞
φ
n
≤ 0. Now applying Lemma 2.4
and (3.35)to(3.37), we conclude that x
n
→x
∗
(n→∞). Consequently, from (3.28), we
have y
n
→x
∗
(n→∞). This completes the proof. 
Corollar y 3.6. Let C be a nonempty closed convex subset of a real Hilbert space H.Let
h : C
× C→R be an equilibrium bifunction satisfying condition (A) such that SEP(h)=∅.
Let

{α
n
}, {β
n
},and{γ
n
} be three sequences in (0,1) such that α
n
+ β
n
+ γ
n
= 1 and {r
n
}⊂
(0,∞) is a real sequence. Suppose the following conditions are sat isfied:
(i) lim
n→∞
α
n
= 0 and

∞
n=0
α
n
=∞;
(ii) 0 < liminf
n→∞
β

n
≤ limsup
n→∞
β
n
< 1;
(iii) liminf
n→∞
r
n
> 0 and lim
n→∞
(r
n+1
− r
n
) = 0.
Yonghong Yao et al. 11
Let f be a contraction of H into itself and given x
0
∈ H arbitrarily. Let {x
n
} and {y
n
} be
sequences generated iteratively by
h

y
n

,x

+
1
r
n

x − y
n
, y
n
− x
n

≥
0, ∀x ∈ C,
x
n+1
= α
n
f

x
n

+ β
n
x
n
+ γ

n
y
n
.
(3.38)
Then the sequences
{x
n
} and {y
n
} generated by (3.38) converge strongly to x
∗
∈ SEP(h),
where x
∗
= P
SEP(h)
f (x
∗
).
Proof. Take T
i
x = x for all i = 1,2, and for all x ∈ C in (3.1). Then W
n
x = x for all
x
∈ C. The conclusion follows from Theorem 3.1. This completes the proof. 
Corollar y 3.7. Let C be a nonempty closed convex subset of a real Hilbert space H.Let
{T
i

}
∞
i=1
be an infinite family of nonexpansive mappings of C into C such that

∞
i=1
F(T
i
)=∅.
Let
{α
n
}, {β
n
},and{γ
n
} be three sequences in (0,1) such that α
n
+ β
n
+ γ
n
= 1.Supposethe
following conditions are satisfied:
(i) lim
n→∞
α
n
= 0 and


∞
n=0
α
n
=∞;
(ii) 0 < liminf
n→∞
β
n
≤ limsup
n→∞
β
n
< 1.
Let f be a contraction of H into itself and given x
0
∈ H arbitrarily. Let {x
n
} beasequence
generated iteratively by
x
n+1
= α
n
f

x
n


+ β
n
x
n
+ γ
n
W
n
P
C
x
n
. (3.39)
Then the sequence
{x
n
} converges strongly to x
∗
= P

∞
i=1
F(T
i
)
f (x
∗
).
Proof. Set h(x, y)
= 0forallx, y ∈ C and r

n
= 1foralln ∈ N.Thenwehavey
n
= P
C
x
n
.
From (3.2), we have
x
n+1
= α
n
f

x
n

+ β
n
x
n
+ γ
n
W
n
P
C
x
n

. (3.40)
Then the conclusion follows from Theorem 3.5. This completes the proof.

Acknowledgments
The authors thank the referee for his/her helpful comments and suggestions, which im-
proved the presentation of this manuscript. The research was partially supported by the
Grant NSC 96-2221-E-230-003.
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Yonghong Yao: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
Email address:
Yeong-Cheng Liou: Department of Information Management, Cheng Shiu University,
Kaohsiung 833, Taiwan
Email address: simplex

Jen-Chih Yao: Department of Applied Mathematics, National Sun Yat-Sen University,
Kaohsiung 804, Taiwan
Email address: