Báo cáo hóa học: " Research Article A New General Iterative Method for a Finite Family of Nonexpansive Mappings in Hilbert Spaces Urailuk Singthong1 and Suthep Suantai1, 2" potx

Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2010, Article ID 262691, 12 pages
doi:10.1155/2010/262691
Research Article
A New General Iterative Method for a Finite Family
of Nonexpansive Mappings in Hilbert Spaces
Urailuk Singthong
1
and Suthep Suantai
1, 2
1
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
2
PERDO National Centre of Excellence in Mathematics, Faculty of Science, Mahidol University,
Bangkok 10400, Thailand
Correspondence should be addressed to Suthep Suantai,
Received 10 February 2010; Revised 21 June 2010; Accepted 15 July 2010
Academic Editor: Massimo Furi
Copyright q 2010 U. Singthong and S. Suantai. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, d istribution, and
reproduction in any medium, provided the original work is properly cited.
We introduce a new general iterative method by using the K-mapping for finding a common fixed
point of a finite family of nonexpansive mappings in the framework of Hilbert spaces. A strong
convergence theorem of the purposed iterative method is established under some certain control
conditions. Our results improve and extend the results announced by many others.
1. Introduction
Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. A mapping
T of C into itself is called nonexpansive if Tx− Ty≤x − y for all x, y ∈ C. Apointx ∈ C is
called a fixed point of T provided that Tx  x. We denote by FT the set of fixed points of T
i.e., FT{x ∈ H : Tx  x}. Recall that a self-mapping f : C → C is a contraction on C,if

there exists a constant α ∈ 0, 1 such that fx − fy≤αx − y for all x, y ∈ C. A bounded
linear operator A on H is called strongly positive with coe
fficient γ if there is a constant γ>0
with the property

Ax, x

≥
γ

x

2
, ∀x ∈ H. 1.1
In 1953, Mann 1 introduced a well-known classical iteration to approximate a fixed point of
a nonexpansive mapping. This iteration is defined as
x
n1
 α
n
x
n


1 − α
n

T

x

n

,n≥ 0, 1.2
2 Fixed Point Theory and Applications
where the initial guess x
0
is taken in C arbitrarily, and the sequence {α
n
}
∞
n0
is in the interval
0, 1. But Mann’s iteration process has only weak convergence, even in a Hilbert space
setting. In general for example, Reich 2 showed that if E is a uniformly convex Banach space
and has a Frehet differentiable norm and if the sequence {α
n
} is such that Σ
∞
n1
α
n
1 − α
n
∞,
then the sequence {x
n
} generated by process 1.2 converges weakly to a point in FT.
Therefore, many authors try to modify Mann’s iteration process to have strong convergence.
In 2005, Kim and Xu 3 introduced the following iteration process:
x

0
 x ∈ C arbitrarily chosen,
y
n
 β
n
x
n


1 − β
n

Tx
n
,
x
n1
 α
n
u 

1 − α
n

y
n
.
1.3
They proved in a uniformly smooth Banach space that the sequence {x

n
} defined by 1.3
converges strongly to a fixed point of T under some appropriate conditions on {α
n
} and {β
n
}.
In 2008, Yao et al. 4 alsomodified Mann’s iterative scheme 1.2 to get a strong
convergence theorem.
Let {T
i
}
N
i1
be a finite family of nonexpansive mappings with F :

N
n1
FT
i

/
 ∅. There
are many authors introduced iterative method for finding an element of F which is an optimal
point for the minimization problem. For n>N, T
n
is understood as T
n mod N
with the mod
function taking values in {1, 2, ,N}.Letu be a fixed element of H.

In 2003, Xu 5 proved that the sequence {x
n
} generated by
x
n1


1 − 
n
A

T
n1
x
n
 
n1
u 1.4
converges strongly to the solution of the quadratic minimization problem
min
x∈F
1
2

Ax, x

−

x, u


, 1.5
under suitable hypotheses on 
n
and under the additional hypothesis
F  F

T
1
T
2
···T
N

 F

T
N
T
1
···T
N−1

 ··· F

T
2
T
3
···T
N

T
1

. 1.6
In 1999, Atsushiba and Takahashi 6 defined the mapping W
n
as follows:
U
n,0
 I,
U
n,1
 γ
n,1
T
1


1 − γ
n,1

I,
U
n,2
 γ
n,2
T
2
U
n,1



1 − γ
n,2

I,
U
n,3
 γ
n,3
T
3
U
n,2


1 − γ
n,3

I,
.
.
.
U
n,N−1
 γ
n,N−1
T
N
− 1U

n,N−2


1 − γ
n,N−1

I,
W
n
 U
n,N
 γ
n,N
T
N
U
n,N−1


1 − γ
n,N

I,
1.7
Fixed Point Theory and Applications 3
where {γ
n,i
}
N
i

⊆ 0, 1. This mapping is called the W-mapping generated by T
1
,T
2
, ,T
N
and
γ
n,1
,γ
n,2
, ,γ
n,N
.
In 2000, Takahashi and Shimoji 7 proved that if X is strictly convex Banach space,
then FW
n


N
i1
FT
i
, where 0 <λ
n,i
< 1,i 1, 2, ,N.
In 2007,Shang et al. 8 introduced a composite iteration scheme as follows:
x
0
 x ∈ C arbitrarily chosen,

y
n
 β
n
x
n


1 − β
n

W
n
x
n
,
x
n1
 α
n
γf

x
n



I − α
n
A


y
n
,
1.8
where f ∈

C
is a contraction, and A is a linear bounded operator.
Note that the iterative scheme 1.8 is not well-defined, because x
n
n ≥ 1 may not lie
in C,soW
n
x
n
is not defined. However, if C  H, the iterative scheme 1.8 is well-defined
and Theorem 2.1 8 is obtained. In the case C
/
 H, we have to modify the iterative scheme
1.8 in order to make it well-defined.
In 2009, Kangtunyakarn and Suantai 9 introduced a new mapping, called K-
mapping, for finding a common fixed point of a finite family of nonexpansive mappings. For
a finite family of nonexpansive mappings {T
i
}
N
i1
and sequence {γ
n,i

}
N
i
in 0, 1, the mapping
K
n
: C → C is defined as follows:
U
n,1
 γ
n,1
T
1


1 − γ
n,1

I,
U
n,2
 γ
n,2
T
2
U
n,1


1 − γ

n,2

U
n,1
,
U
n,3
 γ
n,3
T
3
U
n,2


1 − γ
n,3

U
n,2
,
.
.
.
U
n,N−1
 γ
n,N−1
T
N

− 1U
n,N−2


1 − γ
n,N−1

U
n,N−2
,
K
n
 U
n,N
 γ
n,N
T
N
U
n,N−1


1 − γ
n,N

U
n,N−1
.
1.9
The mapping K

n
is called the K-mapping generated by T
1
, ,T
N
and γ
n,1
,γ
n,2
, ,γ
n,N
.
In this paper, motivated by Kim and Xu 3,MarinoandXu10,Xu5, Yao et al. 4,
andShang et al. 8, we introduce a composite iterative scheme as follows:
x
0
 x ∈ C arbitrarily chosen,
y
n
 β
n
x
n


1 − β
n

K
n

x
n
,
x
n1
 P
C

α
n
γf

x
n



I − α
n
A

y
n

,
1.10
where f ∈

C
is a contraction, and A is a bounded linear operator. We prove, under certain

appropriate conditions on the sequences {α
n
} and {β
n
} that {x
n
} defined by 1.10 converges
strongly to a common fixed point of the finite family of nonexpansive mappings {T
i
}
N
i1
, which
solves a variational inequaility problem.
4 Fixed Point Theory and Applications
In order to prove our main results, we need the following lemmas.
Lemma 1.1. For all x, y ∈ H, there holds the inequality


x  y


2
≤

x

2
 2


y, x  y

,x,y∈ H.
1.11
Lemma 1.2 see 11. Let {x
n
} and {z
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 sup
n →∞
β
n
< 1. Suppose that
x
n1
 β
n
x
n


1 − β
n


z
n
1.12
for all integer n ≥ 0, and
lim sup
n →∞


z
n1
− z
n

−

x
n1
− x
n


≤ 0.
1.13
Then lim
n →∞
x
n
− z
n

  0.
Lemma 1.3 see 5. Assume that {a
n
} is a sequence of nonnegative real numbers such that a
n1
≤
1 − γ
n
a
n
 δ
n
n ≥ 0,where{γ
n
}⊂0, 1 and {δ
n
} is a sequence in R such that
i

∞
n1
γ
n
 ∞,
ii lim sup
n →∞
δ
n
/γ
n

≤ 0 or

∞
n1
|δ
n
| < ∞.
Then lim
n →∞
a
n
 0.
Lemma 1.4 see 10. Let A be a strongly positive linear bounded operator on a Hilbert space H
with coefficient
γ and 0 <ρ≤A
−1
.ThenI − ρA≤1 − ργ.
Lemma 1.5 see 10. Let H be a Hilbert space. Let A be a strongly positive linear bounded operator
with coefficient
γ>0. Assume that 0 <γ<γ/α.LetT : C → C be a nonexpansive mapping with a
fixed point x
t
∈ C of the contraction C  x → tγfx1 − tATx.Thenx
t
converges strongly as
t → 0 to a fixed point
x of T, which solves the variational inequality

A − γf


x, z − x

≥ 0,z∈ F

T

. 1.14
Lemma 1.6 see 1. Demiclosedness principle. Assume that T is nonexpansive self-mapping of
closed convex subset C of a Hilbert space H.IfT has a fixed point, then I − T is demiclosed. That is,
whenever {x
n
} is a sequence in C weakly converging to some x ∈ C and the sequence {I − Tx
n
}
strongly converges to some y, it follows that I − Tx  y. Here, I is identity mapping of H.
Lemma 1.7 see 9. Let C be a nonempty closed convex subset of a strictly convex Banach space.
Let {T
i
}
N
i1
be a finite family of nonexpansive mappings of C into itself with

N
i1
FT
i

/
 ∅, and let

λ
1
, ,λ
N
be real numbers such that 0 <λ
i
< 1 for every i  1, ,N− 1 and 0 <λ
N
≤ 1. Let K be
the K-mapping of C into itself generated by T
1
, ,T
N
and λ
1
, ,λ
N
.ThenFK

N
i1
FT
i
.
Fixed Point Theory and Applications 5
By using the same argument as in 9, Lemma 2.10, we obtain the following lemma.
Lemma 1.8. Let C be a nonempty closed convex subset of Banach space. Let {T
i
}
N

i1
be a finite family of
nonexpanxive mappings of C into itself and {λ
n,i
}
N
i1
sequences in 0, 1 such that λ
n,i
→ λ
i
, as n →
∞, i  1, 2, ,N. Moreover, for every n ∈ N,letK and K
n
be the K -mappings generated by
T
1
,T
2
, ,T
N
and λ
1
,λ
2
, ,λ
N
, and T
1
,T

2
, ,T
N
and λ
n,1
,λ
n,2
, ,λ
n,N
, respectively. Then, for
every bounded sequence x
n
∈ C, one has lim
n →∞
K
n
x
n
− Kx
n
  0.
Let H be real Hilbert space with inner product ·, ·, C a nonempty closed convex
subset of H. Recall that the metric nearest point projection P
C
from a real Hilbert space H
to a closed convex subset C of H is defined as follows. Given that x ∈ H, P
C
x is the only
point in C with the property x −P
C

x  inf{x − y : y ∈ C}.BelowLemma 1.9 can be found
in any standard functional analysis book.
Lemma 1.9. Let C be a closed convex subset of a real Hilbert space H. Given that x ∈ H and y ∈ C
then
i y  P
C
x if and only if the inequality x − y, y − z≥0 for all z ∈ C,
ii P
C
is nonexpansive,
iii x − y, P
C
x − P
C
y≥P
C
x − P
C
y
2
for all x, y ∈ H,
iv x − P
C
x, P
C
x − y≥0 for all x ∈ H and y ∈ C.
2. Main Result
In this section, we prove strong convergence of the sequences {x
n
} defined by the iteration

scheme 1.10.
Theorem 2.1. Let H be a Hilbert space, C a closed convex nonempty subset of H.LetA be a strongly
positive linear bounded operator with coefficient
γ>0, and let f ∈

c
·
Let {T
i
}
N
i1
be a finite family of
nonexpansive mappings of C into itself, and let K
n
be defined by 1.9. Assume that 0 <γ<γ/α and
F 

N
i1
FT
i

/
 ∅.Letx
0
∈ C, given that {α
n
}
∞

n0
and {β
n
}
∞
n0
are sequences in 0, 1, and suppose
that the following conditions are satisfied:
C1 α
n
→ 0;
C2

∞
n0
α
n
 ∞;
C3 0 < lim inf
n →∞
β
n
≤ lim sup
n →∞
β
n
< 1;
C4

∞

n1
|γ
n,i
− γ
n−1,i
| < ∞, for all i  1, 2, ,Nand {γ
n,i
}
N
i1
⊂ a, b,where0 <a≤ b<1;
C5

∞
n1
|α
n1
− α
n
| < ∞;
C6

∞
n1
|β
n1
− β
n
| < ∞.
If {x

n
}
∞
n1
is the composite process defined by 1.10,then{x
n
}
∞
n1
converges strongly to q ∈ F,which
also solves the following variational inequality:

γf

q

− Aq, p − q

≤ 0,p∈ F. 2.1
6 Fixed Point Theory and Applications
Proof. First, we observe that {x
n
}
∞
n0
is bounded. Indeed, take a point u ∈ F, and notice that
y
n
− u≤β
n

x
n
− u 

1 − β
n

K
n
x
n
− u≤x
n
− u. 2.2
Since α
n
→ 0, we may assume that α
n
≤A
−1
 for all n.ByLemma 1.4, we have I − α
n
A≤
1 − α
n
γ for all n.
It follows that
x
n1
− u 



P
C

α
n
γf

x
n



I − α
n
A

y
n

− P
C

u



≤



α
n

γf

x
n

− Au



I − α
n
A


y
n
− u



≤ α
n


γf


x
n

− Au




1 − α
n
γ



y
n
− u


≤ α
n


γf

x
n

− γf


u



 α
n


γf

u

− Au




1 − α
n
γ



y
n
− u


≤ αγα
n


x
n
− u

 α
n


γf

u

− Au




1 − α
n
γ


x
n
− u



1 −


γ − γα

α
n


x
n
− u

 α
n


γf

u

− Au




1 −

γ − γα

α
n



x
n
− u



γ − γα

α
n


γf

u

− Au


γ − γα
≤ max


x
n
− u

,



γf

u

− Au


γ − γα

.
2.3
By simple inductions, we have

x
n
− u

≤ max


x
0
− u

,


γf


u

− Au


γ − γα

,n≥ 0. 2.4
Therefore {x
n
} is bounded, so are {y
n
} and {fx
n
}. Since K
n
is nonexpansive and y
n

β
n
x
n
1 − β
n
K
n
x
n

, we also have
y
n1
− y
n
≤



β
n1
x
n1


1 − β
n1

K
n1
x
n1

−

β
n
x
n



1 − β
n

K
n
x
n






β
n1
x
n1
− β
n1
x
n
 β
n1
x
n
− β
n
x
n



1 − β
n1


K
n1
x
n1
− K
n1
x
n



1 − β
n1


K
n1
x
n
− K
n
x
n




1 − β
n1

K
n
x
n
−

1 − β
n

K
n
x
n


≤ β
n1

x
n1
− x
n





β
n1
− β
n



x
n



1 − β
n1


K
n1
x
n1
− K
n1
x
n



1 − β
n1



K
n1
x
n
− K
n
x
n




β
n
− β
n1



K
n
x
n

≤ β
n1

x

n1
− x
n




β
n1
− β
n



x
n



1 − β
n1


x
n1
− x
n




1 − β
n1


K
n1
x
n
− K
n
x
n




β
n
− β
n1



K
n
x
n

 x
n1

− x
n
 


β
n1
− β
n



x
n



1 − β
n1


K
n1
x
n
− K
n
x
n




β
n
− β
n1



K
n
x
n

.
2.5
Fixed Point Theory and Applications 7
By using the inequalities 2.6 and 2.11 of 9, L emma 2.11, we can conclude that

K
n
x
n−1
− K
n−1
x
n−1

≤ M
N


j1


γ
n,j
− γ
n−1,j


,
2.6
where M  sup{

N
j2
T
j
U
n,j−1
x
n
  U
n,j−1
x
n
T
1
x
n

  x
n
}.
By 2.5 and 2.6, we have

x
n1
− x
n





P
C

α
n
γf

x
n



I − α
n
A


y
n

−

P
C

α
n−1
γf

x
n−1



I − α
n−1
A

y
n−1



≤




I − α
n
A


y
n
− y
n−1

−

α
n
− α
n−1

Ay
n−1
γα
n

f

x
n

− f

x

n−1


 γ

α
n
− α
n−1

f

x
n−1



≤

1 − α
n
γ



y
n
− y
n−1




|
α
n
− α
n−1
|


Ay
n−1


 γαα
n

x
n
− x
n−1

 γ
|
α
n
− α
n−1
|



f

x
n−1



≤

1 − α
n
γ


x
n
− x
n−1




β
n
− β
n−1




x
n−1




1 − β
n



K
n
x
n−1
− K
n−1
x
n−1




β
n−1
− β
n




K
n−1
x
n−1



|
α
n
− α
n−1
|


Ay
n−1


 γαα
n

x
n
− x
n−1

 γ
|
α

n
− α
n−1
|


f

x
n−1



≤

1 − α
n
γ


x
n
− x
n−1




β
n

− β
n−1



x
n−1




1 − β
n


K
n
x
n−1
− K
n−1
x
n−1
 


β
n−1
− β
n



K
n−1
x
n−1


|
α
n
− α
n−1
|
Ay
n−1
  γαα
n
x
n
− x
n−1
  γ
|
α
n
− α
n−1
|
f


x
n−1




1 −

γ − γα

α
n

x
n
− x
n−1
  L


β
n−1
− β
n


 M

|

α
n
− α
n−1
|



1 − β
n


M
N

j1


γ
n,j
− γ
n−1,j


,
2.7
where L  sup{x
n−1
K
n−1

x
n−1
 : n ∈ N}, M

 max{Ay
n−1
γfx
n−1
}. Since

∞
n1
|α
n
−
α
n−1
| < ∞,

∞
n1
|β
n−1
−β
n
| < ∞,and

∞
n1
|γ

n,j
−γ
n−1,j
| < ∞, for all j  1, 2, ,N,byLemma 1.3,
we obtain x
n1
− x
n
→0. It follows that


x
n1
− y
n





P
C

α
n
γf

x
n




I − α
n
A

y
n

− P
C

y
n



≤


α
n
γf

x
n



I − α

n
A

y
n
− y
n


 α
n


γf

x
n

 Ay
n


.
2.8
8 Fixed Point Theory and Applications
Since α
n
→ 0and{fx
n
}, {Ay

n
} are bounded, we have x
n1
− y
n
→0asn →∞. Since


x
n
− y
n


≤

x
n
− x
n1




x
n1
− y
n



, 2.9
it implies that x
n
− y
n
→0asn →∞.
On the other hand, we have

K
n
x
n
− x
n

≤


x
n
− y
n





y
n
− K

n
x
n





x
n
− y
n


 β
n

x
n
− K
n
x
n

, 2.10
which implies that 1 − β
n
K
n
x

n
− x
n
≤x
n
− y
n
.
From condition C3 and x
n
− y
n
→0asn →∞,weobtain

K
n
x
n
− x
n

→ 0. 2.11
By C4, we have lim
n →∞
γ
n,i
 γ
i
∈ a, b for all i  1, 2, ,N.LetK be the K-mapping
generated by T

1
, ,T
N
and γ
1
, ,γ
N
. Next, we show that
lim sup
n →∞

γf

q

− Aq, x
n
− q

≤ 0,
2.12
where q  lim
t → 0
x
t
with x
t
being the fixed point of the contraction x → tγfxI − tAKx.
Thus, x
t

solves the fixed point equation x
t
 tγfx
t
I − tAKx
t
.ByLemma 1.5 and
Lemma 1.7, we have q ∈ F and γfq − Aq, p − q≥0 for all p ∈ F. It follows by 2.11
and Lemma 1.8 that Kx
n
− x
n
→0. Thus, we have x
t
− x
n
  I − tAKx
t
− x
n

tγfx
t
 − Ax
n
. It follows from Lemma 1.1 that for 0 <t<A
−1
,

x

t
− x
n

2




I − tA

Kx
t
− x
n

 t

γf

x
t

− Ax
n



2
≤


1 − γt

2

Kx
t
− x
n

2
 2t

γf

x
t

− Ax
n
,x
t
− x
n

≤

1 −
γt


2


Kx
t
− Kx
n

2
 2

Kx
t
− K
n
x
n

Kx
n
− x
n



Kx
n
− x
n


2

 2t

γf

x
t

− Ax
t
,x
t
− x
n



Ax
t
− Ax
n
,x
t
− x
n


≤


1 − 2
γt

γt

2


x
t
− x
n

2
 f
n

t

 2t

γf

x
t

− Ax
t
,x
t

− x
n

 2t

Ax
t
− Ax
n
,x
t
− x
n

,
2.13
where
f
n

t



2

x
t
− x
n




x
n
− Kx
n



x
n
− Kx
n

−→ 0, as n → 0. 2.14
Fixed Point Theory and Applications 9
It follows that

Ax
t
− γf

x
t

,x
t
− x
n


≤

−2
γt

γt

2
2t


x
t
− x
n

2

1
2t
f
n

t



Ax
t

− Ax
n
,x
t
− x
n

≤

−2 
γt
2

γ

x
t
− x
n

2

1
2t
f
n

t




Ax
t
− Ax
n
,x
t
− x
n

≤

−1 
γt
2


Ax
t
− Ax
n
,x
t
− x
n


1
2t
f

n

t



Ax
t
− Ax
n
,x
t
− x
n

≤
γt
2

Ax
t
− Ax
n
,x
t
− x
n


1

2t
f
n

t

.
2.15
Letting n →∞in 2.15 and 2.14,weget
lim sup
n →∞

Ax
t
− γf

x
t

,x
t
− x
n

≤
t
2
M
0
,

2.16
where M
0
> 0 is a constant such that M
0
≥ γAx
t
− Ax
n
,x
t
− x
n
 for all t ∈ 0, 1 and n ≥ 1.
Taking t → 0in2.16, we have
lim sup
t → 0
lim sup
n →∞

Ax
t
− γf

x
t

,x
t
− x

n

≤ 0.
2.17
On the other hand, one has

γf

q

− Aq, x
n
− q



γf

q

− Aq, x
n
− q

−

γf

q


− Aq, x
n
− x
t



γf

q

− Aq, x
n
− x
t

−

γf

q

− Ax
t
,x
n
− x
t




γf

q

− Ax
t
,x
n
− x
t

−

γf

x
t

− Ax
t
,x
n
− x
t



γf


x
t

− Ax
t
,x
n
− x
t

.


γf

q

− Aq, x
t
− q



Ax
t
− Aq, x
n
− x
t




γf

q

− γf

x
t

,x
n
− x
t



γf

x
t

− Ax
t
,x
n
− x
t


≤


γf

q

− Aq




x
t
− q





A



x
t
− q


 γα



x
t
− q




x
n
− x
t



γf

x
t

− Ax
t
,x
n
− x
t





γf

q

− Aq




x
t
− q





A

 γα



x
t
− q




x
n
− x
t



γf

x
t

− Ax
t
,x
n
− x
t

.
2.18
It follows that
lim sup
n →∞

γf

q

− Aq, x

n
− q

≤


γf

q

− Aq




x
t
− q





A

 γα



x

t
− q


lim sup
n →∞

x
n
− x
t

 lim sup
n →∞

γf

x
t

− Ax
t
,x
n
− x
t

.
2.19
10 Fixed Point Theory and Applications

Therefore, from 2.17 and lim
t → 0
x
t
− q  0, we have
lim sup
n →∞
γf

q

− Aq, x
n
− q≤lim sup
t → 0

lim sup
n →∞
γf

q

− Aq, x
n
− q

≤ lim sup
t → 0
lim sup
n →∞


γf

x
t

− Ax
t
,x
n
− x
t

≤ 0.
2.20
Hence 2.12 holds. Finally, we prove that x
n
→ q.Byusing2.2 and together with the
Schwarz inequality, we have


x
n1
− q


2




P
C

α
n
γf

x
n



I − α
n
A

y
n

− P
C

q



2
≤



α
n

γf

x
n

− Aq



I − α
n
A


y
n
− q



2




I − α
n

A


y
n
− q



2
 α
2
n


γf

x
n

− Aq


2
 2α
n


I − α
n

A


y
n
− q

,γf

x
n

− Aq

≤

1 − α
n
γ

2


y
n
− q


2
 α

2
n


γf

x
n

− Aq


2
 2α
n

y
n
− q, γf

x
n

− Aq

− 2α
2
n

A


y
n
− q

,γf

x
n

− Aq

≤

1 − α
n
γ

2


x
n
− q


2
 α
2
n



γf

x
n

− Aq


2
 2α
n

y
n
− q, γf

x
n

− γf

q

 2α
n

y
n

− q, γf

q

− Aq

− 2α
2
n

A

y
n
− q

,γf

x
n

− Aq

≤

1 − α
n
γ

2



x
n
− q


2
 α
2
n


γf

x
n

− Aq


2
 2α
n


y
n
− q





γf

x
n

− γf

q



 2α
n

y
n
− q, γf

q

− Aq

− 2α
2
n

A


y
n
− q

,γf

x
n

− Aq

≤

1 − α
n
γ

2


x
n
− q


2
 α
2
n



γf

x
n

− Aq


2
 2γαα
n


y
n
− q




x
n
− q


 2α
n


y
n
− q, γf

q

− Aq

− 2α
2
n

A

y
n
− q

,γf

x
n

− Aq

≤

1 − α
n
γ


2


x
n
− q


2
 α
2
n


γf

x
n

− Aq


2
 2γαα
n


x
n

− q


2
 2α
n

y
n
− q, γf

q

− Aq

− 2α
2
n

A

y
n
− q

,γf

x
n


− Aq

≤


1 − α
n
γ

2
 2γαα
n



x
n
− q


2
 2α
n

y
n
− q, γf

x
n


− Aq

 α
2
n


γf

x
n

− Aq


2
 2α
2
n


A

y
n
− q






γf

x
n

− Aq




1 − 2

γ − γα

α
n



x
n
− q


2
 α
n


2

y
n
− q, γf

q

− Aq

α
n



γf

x
n

− Aq


2
 2


A

y

n
− q





γf

x
n

− Aq



γ
2


x
n
− q


2

.
2.21
Fixed Point Theory and Applications 11

Since {x
n
}, {fx
n
},and{y
n
} are bounded, we can take a constant η>0 such that
η ≥


γf

x
n

− Aq


2
 2


A

y
n
− q






γf

x
n

− Aq



γ
2


x
n
− q


2
2.22
for all n ≥ 0. It then follows that


x
n1
− q



2
≤

1 − 2

γ
− γα

α
n



x
n
− q


2
 α
n
β
n
, 2.23
where β
n
 2y
n
− q, γfq − Aq  ηα
n

. By lim sup
n →∞
γf − Aq, y
n
− q≤0, we get
lim sup
n →∞
β
n
≤ 0. By applying Lemma 1.3 to 2.23, we can conclude that x
n
→ q.This
completes the proof.
If A  I and γ  1inTheorem 2.1, we obtain the following result.
Corollary 2.2. Let H be a Hilbert space, C a closed convex nonempty subset of H, and let f ∈

c
.
Let {T
i
}
N
i1
be a finite family of nonexpansive mappings of C into itself, and let K
n
be defined by 1.9.
Assume that F 

N
i1

FT
i

/
 ∅.Letx
0
∈ C, given that {α
n
}
∞
n0
and {β
n
}
∞
n0
are sequences in 0, 1,
and suppose that the following conditions are satisfied:
C1 α
n
→ 0;
C2

∞
n0
α
n
 ∞;
C3 0 < lim inf
n →∞

β
n
≤ lim sup
n →∞
β
n
< 1;
C4

∞
n1
|γ
n,i
− γ
n−1,i
| < ∞, for all i  1, 2, ,N and {γ
n,i
}
N
i1
⊂ a, b,where 0 <a≤
b<1;
C5

∞
n1
|α
n1
− α
n

| < ∞;
C6

∞
n1
|β
n1
− β
n
| < ∞.
If {x
n
}
∞
n1
is the composite process defined by
y
n
 β
n
x
n


1 − β
n

K
n
x

n
,
x
n1
 α
n
f

x
n



1 − α
n

y
n
,
2.24
then {x
n
}
∞
n1
converges strongly to q ∈ F, which also solves the following variational inequality:

f − I

q, p − q


≤ 0,p∈ F. 2.25
If N  1, A  I, γ  1, and f ≡ u ∈ C is a constant in Theorem 2.1,wegettheresultsof
Kim and Xu 3.
Corollary 2.3. Let H be a Hilbert space, C a closed convex nonempty subset of H, and let f ∈

c
.
Let T be a nonexpansive mapping of C into itself. FT
/
 ∅.Letx
0
∈ C, given that {α
n
}
∞
n0
and
{β
n
}
∞
n0
are sequences in 0, 1, and suppose that the following conditions are satisfied:
C1 α
n
→ 0;
C2

∞

n0
α
n
 ∞;
12 Fixed Point Theory and Applications
C3 0 < lim inf
n →∞
β
n
≤ lim sup
n →∞
β
n
< 1;
C4

∞
n1
|α
n1
− α
n
| < ∞;
C5

∞
n1
|β
n1
− β

n
| < ∞.
If {x
n
}
∞
n1
is the composite process defined by
y
n
 β
n
x
n


1 − β
n

Tx
n
,
x
n1
 α
n
u 

I − α
n


y
n
,
2.26
then {x
n
}
∞
n1
converges strongly to q ∈ F, which also solves the following variational inequality:

u − q, p − q

≤ 0,p∈ F. 2.27
Acknowledgments
The authors would like to thank the referees for valuable suggestions on the paper and thank
the Center of Excellence in Mathematics, the Thailand Research Fund, and the Graduate
School of Chiang Mai University f or financial support.
References
1 W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol.
4, pp. 506–510, 1953.
2 S. Reich, “Weak convergence theorems for nonexpansive mappings in Banach spaces,” Journal of
Mathematical Analysis and Applications, vol. 67, no. 2, pp. 274–276, 1979.
3 T H. Kim and H K. Xu, “Strong convergence of modified Mann iterations,” Nonlinear Analysis:
Theory, Methods & Applications, vol. 61, no. 1-2, pp. 51–60, 2005.
4 Y. Yao, R. Chen, and J C. Yao, “Strong convergence and certain control conditions for modified Mann
iteration,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 6, pp. 1687–1693, 2008.
5 H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and
Applications, vol. 116, no. 3, pp. 659–678, 2003.

6 S. Atsushiba and W. Takahashi, “Strong convergence theorems for a finite family of nonexpansive
mappings and applications,” Indian Journal of Mathematics, vol. 41, no. 3, pp. 435–453, 1999.
7 W. Takahashi and K. Shimoji, “Convergence theorems for nonexpansive mappings and feasibility
problems,” Mathematical and Computer Modelling, vol. 32, no. 11–13, pp. 1463–1471, 2000.
8 M. Shang, Y. Su, and X. Qin, “Strong convergence theorems for a finite family of nonexpansive
mappings,” Fixed Point Theory and Applications, vol. 2007, Article ID 76971, 9 pages, 2007.
9 A. Kangtunyakarn and S. Suantai, “A new mapping for finding common solutions of equilibrium
problems and fixed point problems of finite family of nonexpansive mappings,” Nonlinear Analysis:
Theory, Methods & Applications, vol. 71, no. 10, pp. 4448–4460, 2009.
10 G. Marino and H K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,”
Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006.
11 T. Suzuki, “Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter non-
expansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications,
vol. 305, no. 1, pp. 227–239, 2005.