Báo cáo hóa học: "Strong convergence theorems for variational inequalities and fixed points of a countable family of nonexpansive mappings" pdf

RESEARC H Open Access
Strong convergence theorems for variational
inequalities and fixed points of a countable
family of nonexpansive mappings
Aunyarat Bunyawat
1
and Suthep Suantai
2*
* Correspondence:

2
Centre of Excellence in
Mathematics, CHE, Si Ayutthaya
Road, Bangkok 10400, Thailand
Full list of author information is
available at the end of the article
Abstract
A new general iterative method for finding a common element of the set of
solutions of variational inequality and the set of common fixed points of a countable
family of nonexpansive mappings is introduced and studied. A strong convergence
theorem of the proposed iterative scheme to a common fixed point of a countable
family of nonexpansive mappings and a solution of variational inequality of an
inverse strongly monotone mapping are established. Moreover, we apply our main
result to obtain strong convergence theorems for a countable family of
nonexpansive mappings and a strictly pseudocontractive mapping, and a countable
family of uniformly k-strictly pseudocontractive mappings and an inverse strongly
monotone mapping. Our main results improve and extend the corresponding result
obtained by Klin-eam and Suantai (J Inequal Appl 520301, 16 pp, 2009).
Mathematics Subject Classification (2000): 47H09, 47H10
Keywords: countable family of nonexpansive mappings, variational inequality,
inverse strongly monotone mapping, strictly pseudocontractive mapping, countable

family of uniformly k-strictly pseudocontractive mappings
1 Introduction
Let H be a real Hilbert space and C be a nonempty closed convex subset of H.Inthis
paper, we always assume that a bounded linear operator A on H is strongly positive,
that is, there is a constant
¯
γ
>
0
such that
Ax, x≥ ¯
γ
||x||
2
for all x Î H. Recall that a
mapping T of H into itself is called nonexpansive if ||Tx - Ty|| ≤ ||x-y|| for all x, y Î
H. The set of all fixe d points of T is denoted by F(T), that is, F(T)={x Î C : x = Tx}.
A self-mapping f : H ® H is a contraction on H if there is a constant a Î [0, 1) such
that ||f(x)-f(y)||≤ a ||x-y|| for all x, y Î H.
Iterative methods for nonexpansive mappings have recently been applied to solve
conv ex minimization problems. A typical problem is to minimize a quadratic function
over the set of the fixed points of a nonexpansive mapping on H:
min
x∈F
1
2
Ax, x−x, b
,
(1:1)
where F is the fixed point set of a nonexpansive mapping T on H and b is a given

point in H. A mapping B of C into H is called monotone if 〈Bx - By, x-y〉 ≥ 0forall
x, y Î C. The variational inequality problem is to find x Î C such that 〈Bx, y-x〉 ≥ 0
Bunyawat and Suantai Fixed Point Theory and Applications 2011, 2011:47
/>© 2011 Bunyawat and Suantai ; licensee Springer. This is an Open Access article distributed under the terms of the Creativ e Commons
Attribution License (http://creati vecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
for all y Î C. The set of solutions of the variational inequality is denoted by VI(C, B).
A mapping B of C to H is called inverse strong ly monotone if there exists a positive
real number b such that 〈x-y, Bx - By〉 ≥ b ||Bx - By||
2
for all x, y Î C.
Starting with an arbitrary initial x
0
Î H, define a sequence {x
n
} recursively by
x
n+1
=
(
I − α
n
A
)
Tx
n
+ α
n
bn≥ 0
.

(1:2)
It is proved by Xu [1] that the sequence {x
n
} generated by (1.2) converges strongly to
the unique solution of the minimization problem (1.1) provided the sequence {a
n
}
satisfies certain conditions.
On the other hand, Moudafi [2] introduced the viscosity approximation method for
nonexpansive mappings. Let f beacontractiononH. Starting with an arbitrary initial
x
0
Î H, define a sequence {x
n
} recursively by
x
n+1
=
(
1 − σ
n
)
Tx
n
+ σ
n
f
(
x
n

)
n ≥ 0
,
(1:3)
where {s
n
} is a sequence in (0, 1). It is proved by Moudafi [2] and Xu [3] that under
certain appropriate conditions imposed on {s
n
}, the sequence {x
n
} generated by (1.3)
strongly converges to the unique solution x* in C of the variational inequality

(
I − f
)
x
∗
, x − x
∗
≥0 x ∈ C
.
Recently, Marino and Xu [4] combined the iterative method (1.2) with the viscosity
approximation method (1.3) and considered the following general iteration process:
x
n+1
=
(
I − α

n
A
)
Tx
n
+ α
n
γ f
(
x
n
)
n ≥
0
(1:4)
and proved that if the sequence {a
n
} satisfies appropriate conditions, the sequence {x
n
}
generated by (1.4) converges strongly to the unique solution of the variational inequality

(
A − γ f
)
x
∗
, x − x
∗
≥0 x ∈

C
which is the optimality condition for the minimization problem
min
x∈C
1
2
Ax, x−h(x)
,
where h is a potential function for g f (i.e., h’(x)=g f(x) for x Î H).
Chen, Zhang and Fan [5] introduced the following iterative process: x
0
Î C,
x
n+1
= α
n
f
(
x
n
)
+
(
1 − α
n
)
TP
C
(
x

n
− λ
n
Bx
n
)
, n ≥ 0
,
(1:5)
where {a
n
} ⊂ (0, 1) and {l
n
} ⊂ [a, b] for some a, b with 0 <a<b<2b.
They proved that under certain appropriate conditions imposed on {a
n
} and {l
n
}, the
sequence {x
n
} generated by (1.5) converges strongly to a common element of the set of
fixed points of nonexpansive mapping and the set of solutions of the variational
inequality for an inverse strongly monotone mapping (say
¯
x
∈ C
), which solves the var-
iational inequality


(
I − f
)
¯
x, x −
¯
x≥0 ∀x ∈ F
(
T
)
∩ VI
(
C, B
).
Klin-eam and Suantai [6] modify the iterative methods (1.4) and (1.5) by proposing
the following general iterative method: x
0
Î C,
x
n+1
= P
C
(
α
n
γ
f (
x
n
)

+
(
I − α
n
A
)
TP
C
(
x
n
− λ
n
Bx
n
))
, n ≥ 0
,
(1:6)
Bunyawat and Suantai Fixed Point Theory and Applications 2011, 2011:47
/>Page 2 of 13
where P
C
is the projection of H onto C, f is a contraction, A is a strongly positive lin-
ear bounded operator, B is a b-inverse strongly monotone mapp ing, {a
n
} ⊂ (0, 1) and
{l
n
} ⊂ [a, b] for some a, b with 0 <a<b<2b. They noted that when A = I and g =1,

the iterative scheme (1.6) reduced to the iterative scheme (1.5).
Wangkeeree, Petrot and Wangkeeree [7] introduced the following iterative process:
⎧
⎪
⎨
⎪
⎩
x
0
= x ∈ H,
y
n
= β
n
x
n
+(1− β
n
)T
n
x
n
,
x
n+1
= α
n
γ f (x
n
)+(I − α

n
A)y
n
, n ≥
0
(1:7)
where {a
n
}and{b
n
} ⊂ (0, 1) and T
n
is a countable family of nonexpansive mappings, f
is a contraction, and A is a strongly positive linear bounded operator. They proved
that under certain appropriate conditions imposed on {a
n
}, {b
n
} and {T
n
}, the sequence
{x
n
} converges strongly to
˜
x
, which solves the variational inequality:

(
A − γ f

)
˜
x,
˜
x − z≤0 z ∈ F
(
T
).
In this paper, motivated and inspired by Klin-eam and Suantai [6], we introduced the
following iteration to find some solutions of variational inequality and fixed points of
countable family of nonexpansive mappings in a Hilbert spaces H: x
0
Î C,
x
n+1
= P
C
(
α
n
γ f
(
x
n
)
+
(
I − α
n
A

)
T
n
P
C
(
x
n
− λ
n
Bx
n
))
, n ≥ 0
,
(1:8)
where P
C
is the projection of H onto C, f is a contraction, A is a strongly positive lin-
ear bounded operator, T
n
is a countable family of nonexpansive mappings of C into
itself, B is a b-inverse strongly monotone mapping, {a
n
} ⊂ (0, 1), and {l
n
} ⊂ [a, b]for
some a, b with 0 <a<b<2b.
2 Preliminaries
Let H be a real Hilbert space with inner product 〈·,·〉 andnorm||·||,andletC be a

closed convex subset of H.Wewritex
n
⇀ x to indicate that the sequence {x
n
}con-
verges weakly to x,andx
n
® x implies that {x
n
} converges strongly to x. For every
point x Î H, there exists a unique nearest poi nt in C, denoted by P
C
x,suchthat||x-
P
C
x|| ≤ ||x-y|| for all y Î C and P
C
x is called the metric projection of H onto C.We
know that P
C
is a nonexpansive mapping of H onto C. It is also known that P
C
satisfies
〈x-y, P
C
x-P
C
y〉 ≥ ||P
C
x - P

C
y||
2
for every x, y Î H. Moreover, P
C
x is characterized by
the properties: P
C
x Î C and 〈x - P
C
x, P
C
x - y 〉 ≥ 0 fo r all y Î C. In the context of the
variational inequality problem, this implies that
u
∈ VI
(
C, A
)
⇔ u = P
C
(
u − λAu
)
, ∀λ>0
.
A set-valued mapping T : H ® 2
H
is called monotone if for all x, y Î H, f Î Tx and g
Î Ty imply 〈x-y, f-g〉 ≥ 0. A monotone mapping T : H ® 2

H
is maximal if the graph
G(T)ofT 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 ) Î H × H, 〈x-
y, f-g〉 ≥ 0 for every (y, g) Î G(T) implies f Î Tx.LetA be an inverse strongly mono-
tone mapping of C into H,andletN
C
v be t he normal cone to C at v Î C,i.e.,N
C
v =
{w Î H : 〈v-u, w〉 ≥ 0, ∀u Î C}, and define
Tv =

Av + N
C
v, v ∈ C,
∅, v ∈ C
.
Bunyawat and Suantai Fixed Point Theory and Applications 2011, 2011:47
/>Page 3 of 13
Then, T is maximal monotone and 0 Î Tv if and only if v Î VI(C, A).
Lemma 2.1 Let C be a closed convex subset of a real Hilbert space H. Given x Î H
and y Î C, then
(i) y = P
C
x if and only if the inequality 〈x-y, y-z〉 ≥ 0 for all z Î 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 Î H,
(iv) 〈x-P
C
x, P
C
x-y〉 ≥ 0 for all x Î H and y Î C.
Lemma 2.2 [4]Assume A is a strongly positive linear bounded operator on a Hilbert
space H with coefficient
¯
γ
>
0
and 0 < r ≤ ||A||
-1
, then
|
|I − ρA|| ≤ 1 − ρ ¯
γ
.
Lemma 2.3 [8]Assume {a
n

} is a sequence of nonnegative real numbers such that
a
n+1
≤
(
1 − γ
n
)
a
n
+ δ
n
, n ≥
0
where {g
n
} ⊂ (0, 1) and {δ
n
} is a sequence in ℝ such that
(i)

∞
n
=1
γ
n
=
∞
,
(ii) lim sup

n®∞
δ
n
/g
n
≤ 0 or

∞
n
=1
|δ
n
| <
∞
.
Then, lim
n®∞
a
n
=0.
Lemma 2.4 [9]Let C be a closed convex subset of a real Hilbert space H, and let T :
C ® C be a nonexpansive mapping such that F(T) ≠ ∅. If a sequence {x
n
} in C such
that x
n
⇀ z and x
n
- Tx
n

® 0, then z = Tz.
To deal with a family of mappings, the following conditions are introduced: Let C be
a subset of a real Banach space E,andlet
{T
n
}
∞
n
=
1
be a family of mappings of C such
that
∩
∞
n
=1
F( T
n
) =
∅
.Then,{T
n
}issaidtosatisfytheAKTT-condition [10] if for each
bounded subset B of C,
∞

n
=1
sup{||T
n+1

z − T
n
z|| : z ∈ B} < ∞
.
Lemm a 2.5 [10]Let C be a nonempty and closed subset of a Banach space E and let
{T
n
} be a f amily of mappings of C into itself which satisfies the AKTT-condition. Then,
for each x Î C,{T
n
x} converges strongly to a point in C. Moreover, let the mapping T
be defined by
Tx = lim
n
→∞
T
n
x ∀x ∈ C
.
Then, for each bounded subset B of C,
lim sup
n
→∞
{||Tz − T
n
z|| : z ∈ B} =0
.
In the sequel, we will write ({T
n
}, T ) satisfies the AKTT-condition if {T

n
} satisfies
the AKTT-condition, and T is defined by Lemma 2.5 with
F( T)=∩
∞
n
=1
F( T
n
)
.
3 Main results
In this section, we prove a strong convergence theorem for a countab le family of no n-
expansive mappings.
Bunyawat and Suantai Fixed Point Theory and Applications 2011, 2011:47
/>Page 4 of 13
Theorem 3.1 LetCbeaclosedconvexsubsetofarealHilbertspaceH,andletB:
C ® Hbeab-inverse strongly monotone mapping, also let A be a strongly positive
linear bounded operator of H into itself with coefficient
¯
γ
>
0
such that ||A|| = 1 and
let f : C ® C be a contraction with coefficient a(0 < a <1). Assume that
0 <γ < ¯γ
/α
.
Let {T
n

} be a countable family of nonexpansive mappings from a subset C into itself
with
F = ∩
∞
n
=1
F( T
n
) ∩ VI(C, B) =
∅
. Suppose {x
n
} is the sequence generated by the
following algorithm: x
0
Î C,
x
n+1
= P
C
(
α
n
γ f
(
x
n
)
+
(

I − α
n
A
)
T
n
P
C
(
x
n
− λ
n
Bx
n
))
for all n =0,1,2, ,where {a
n
} ⊂ (0, 1) and {l
n
} ⊂ (0, 2b ). If {a
n
} and {l
n
} are
chosen so that l
n
Î [ a, b] for some a, b with 0 <a<b<2b ,
(C1) lim
n→0

α
n
=0; (C2)
∞

n=1
α
n
= ∞;
(C3)
∞

n
=1
|
α
n+1
− α
n
|
< ∞;(C4)
∞

n
=1
|
λ
n+1
− λ
n

|
< ∞
.
Suppose that ({T
n
}, T ) satisfies the AKTT-condition. Then,{x
n
} converges strongly to
q Î F, where q = P
F
(g f + I-A)(q) which solves the following variational inequality:

(
γ f − A
)
q, p − q≤0 ∀p ∈ F
.
Proof.First,weshowthatthesequence{x
n
} is bounded. Consid er the mapping I
-l
n
B. Since B is a b-inverse strongly monotone mapping, we have that for all x, y Î C,
|
|(I − λ
n
B)x − (I − λ
n
B)y||
2

= ||(x − y) − λ
n
(Bx − By)||
2
= ||x − y||
2
− 2λ
n
x − y, Bx − By + λ
2
n
||Bx − By||
2
≤||x − y||
2
+ λ
n
(
λ
n
− 2β
)
||Bx − By||
2
.
For 0 < l
n
<2b, implies that ||k(I-l
n
B)x-(I- l

n
B)y||
2
≤ ||x-y||
2
.
So, the mapping I-l
n
B is nonexpansive.
Put y
n
= P
C
(x
n
- l
n
Bx
n
) for all n ≥ 0. Let u Î F. Then u = P
C
(u-l
n
Bu).
From P
C
is nonexpansive implies that
|
|y
n

− u|| = ||P
C
(x
n
− λ
n
Bx
n
) − P
C
(u − λ
n
Bu)|
|
≤||(x
n
− λ
n
Bx
n
) − (u − λ
n
Bu)||
= ||
(
I − λ
n
B
)
x

n
−
(
I − λ
n
B
)
u||.
Since I-l
n
B is nonexpansive, we have that ||y
n
-u|| ≤ ||x
n
-u||. Then
||x
n+1
− u|| = ||P
C
(α
n
γ f (x
n
)+(I − α
n
A)T
n
y
n
) − u||

≤||α
n
γ f (x
n
)+(I − α
n
A)T
n
y
n
− u||
= ||α
n
(
γ f
(
x
n
)
− Au
)
+
(
I − α
n
A
)(
T
n
y

n
− u
)
||
.
Since A is strongly positive linear bounded operator, we have
|
|x
n+1
− u|| ≤ α
n
||γ f (x
n
) − Au|| +(1− α
n
¯γ )||T
n
y
n
− u||
≤ α
n
||γ f
(
x
n
)
− γ f
(
u

)
|| + α
n
||γ f
(
u
)
− Au|| +
(
1 − α
n
¯γ
)
||T
n
y
n
− u||
.
Bunyawat and Suantai Fixed Point Theory and Applications 2011, 2011:47
/>Page 5 of 13
By contraction of f, we have
|
|x
n+1
− u|| ≤ αγ α
n
||x
n
− u|| + α

n
||γ f (u) − Au|| +(1− α
n
¯γ )||T
n
y
n
− u||
= αγ α
n
||x
n
− u|| + α
n
||γ f (u) − Au|| +(1− α
n
¯γ )||T
n
y
n
− T
n
u|
|
≤ αγ α
n
||x
n
− u|| + α
n

||γ f (u) − Au|| +(1− α
n
¯γ )||y
n
− u||
≤ αγ α
n
||x
n
− u|| + α
n
||γ f (u) − Au|| +(1− α
n
¯γ )||x
n
− u||
≤ (αγ α
n
+1− α
n
¯γ )||x
n
− u|| + α
n
||γ f (u) − Au||
≤ (1 − α
n
( ¯γ − αγ ))||x
n
− u|| + α

n
( ¯γ − αγ )
||γ f (u) − Au||
¯γ − αγ
≤ max

||x
n
− u||,
||γ f (u) − Au||
¯γ − αγ

.
It follows from induction that
|
|x
n
− u|| ≤ max

||x
0
− u||,
||γ f(u)−Au||
¯γ −αγ

, n ≥ 0.
Therefore, {x
n
} is bounded, so are {y
n

}, { T
n
y
n
}, {Bx
n
}, and {f (x
n
)}.
Next, we show that ||x
n+1
- x
n
|| ® 0 and ||y
n
-T
n
y
n
|| ® 0asn ® ∞.
Since P
C
is nonexpansive, we also have
|
|y
n+1
− y
n
|| = ||P
C

(x
n+1
− λ
n+1
Bx
n+1
) − P
C
(x
n
− λ
n
Bx
n
)||
≤||x
n+1
− λ
n+1
Bx
n+1
− (x
n
− λ
n
Bx
n
)||
≤||x
n+1

− λ
n+1
Bx
n+1
− (x
n
− λ
n+1
Bx
n
)|| + |λ
n
− λ
n+1
|||Bx
n
|
|
= ||
(
I − λ
n+1
B
)
x
n+1
−
(
I − λ
n+1

B
)
x
n
|| + |λ
n
− λ
n+1
|||Bx
n
||.
Since I-l
n
B is nonexpansive, we have
||y
n+1
−
y
n
||
≤
||
x
n+1
− x
n
||
+
|
λ

n
− λ
n+1
|||
Bx
n
||.
So we obtain
||x
n+1
− x
n
|| = ||P
C
(α
n
γ f (x
n
)+(I − α
n
A)T
n
y
n
) − P
C
(α
n−1
γ f (x
n−1

)+(I − α
n−1
A)T
n−1
y
n−1
)|
|
≤||α
n
γ (f (x
n
) − f(x
n−1
)) + γ (α
n
− α
n−1
)f (x
n−1
)+(I − α
n
A)(T
n
y
n
− T
n−1
y
n−1

)
+(α
n
− α
n−1
)AT
n−1
y
n−1
||
≤ α
n
αγ ||x
n
− x
n−1
|| + γ |α
n
− α
n−1
|||f (x
n−1
)|| +(1− α
n
¯γ )||T
n
y
n
− T
n−1

y
n−1
||
+ |α
n
− α
n−1
|||AT
n−1
y
n−1
||
≤ α
n
αγ ||x
n
− x
n−1
|| + γ |α
n
− α
n−1
|||f (x
n−1
)|| +(1− α
n
¯γ )(||T
n
y
n

− T
n
y
n−1
||
+ ||T
n
y
n−1
− T
n−1
y
n−1
||)+|α
n
− α
n−1
|||AT
n−1
y
n−1
||
≤ α
n
αγ ||x
n
− x
n−1
|| + γ |α
n

− α
n−1
|||f (x
n−1
)|| +(1− α
n
¯γ )(||y
n
− y
n−1
||
+ ||T
n
y
n−1
− T
n−1
y
n−1
||)+|α
n
− α
n−1
|||AT
n−1
y
n−1
||
= α
n

αγ ||x
n
− x
n−1
|| + γ |α
n
− α
n−1
|||f (x
n−1
)|| +(1− α
n
¯γ )||y
n
− y
n−1
||
+(1− α
n
¯γ )||T
n
y
n−1
− T
n−1
y
n−1
|| + |α
n
− α

n−1
|||AT
n−1
y
n−1
||
≤ α
n
αγ ||x
n
− x
n−1
|| + γ |α
n
− α
n−1
|||f (x
n−1
)|| +(1− α
n
¯γ )||x
n
− x
n−1
||
+(1− α
n
¯γ )|λ
n−1
− λ

n
|||Bx
n−1
|| +(1− α
n
¯γ )||T
n
y
n−1
− T
n−1
y
n−1
||
+ |α
n
− α
n−1
|||AT
n−1
y
n−1
||
=(1− ( ¯γ − αγ )α
n
)||x
n
− x
n−1
|| + γ |α

n
− α
n−1
|||f(x
n−1
)||
+(1− α
n
¯γ )|λ
n−1
− λ
n
|||Bx
n−1
|| +(1− α
n
¯γ )||T
n
y
n−1
− T
n−1
y
n−1
||
+ |α
n
− α
n−1
|||AT

n−1
y
n−1
||
≤ (1 − ( ¯γ − αγ )α
n
)||x
n
− x
n−1
|| +2L|α
n
− α
n−1
| + M|λ
n−1
− λ
n
|
+ sup
y
∈{
y
n
}
||T
n
y − T
n−1
y||,

where L =max{sup
nÎN
||AT
n-1
y
n-1
||, sup
nÎN
g ||f (x
n-1
)||} and M = sup{||Bx
n-1
|| :
nÎN}.
Bunyawat and Suantai Fixed Point Theory and Applications 2011, 2011:47
/>Page 6 of 13
Since {T
n
} satisfies the AKTT-condition, we get that
∞

n
=1
sup
y∈{y
n
}
||T
n
y − T

n−1
y|| < ∞
.
From condition (C3), (C4) and by Lemma 2.3, we have ||x
n+1
- x
n
|| ® 0.
For u Î F and u = P
C
(u-l
n
Bu), we have
|
|x
n+1
− u||
2
= ||P
C
(α
n
γ f (x
n
)+(I − α
n
A)T
n
y
n

) − P
C
(u)||
2
≤||α
n
(γ f (x
n
) − Au)+(I − α
n
A)(T
n
y
n
− u)||
2
≤ (α
n
||γ f (x
n
) − Au|| + ||I − α
n
A|| ||T
n
y
n
− u||)
2
≤ (α
n

||γ f (x
n
) − Au|| +(1− α
n
¯γ )||y
n
− u||)
2
≤ α
n
||γ f (x
n
) − Au||
2
+(1− α
n
¯γ )||y
n
− u||
2
+2α
n
(1 − α
n
γ )||γ f (x
n
) − Au|| ||y
n
− u||
≤ α

n
||γ f (x
n
) − Au||
2
+(1− α
n
γ )||(I − λ
n
B)x
n
− (I − λ
n
B)u||
2
+2α
n
(1 − α
n
γ )||γ f (x
n
) − Au|| ||y
n
− u||
≤ α
n
||γ f (x
n
) − Au||
2

+(1− α
n
¯γ )(||x
n
− u||
2
− 2λ
n
x
n
− u, Bx
n
− Bu
+ λ
2
n
||Bx
n
− Bu||
2
)+2α
n
(1 − α
n
¯γ )||γ f(x
n
) − Au|| ||y
n
− u||
≤ α

n
||γ f (x
n
) − Au||
2
+(1− α
n
¯γ )(||x
n
− u||
2
− 2λ
n
β||Bx
n
− Bu||
2
+ λ
2
n
||Bx
n
− Bu||
2
)+2α
n
(1 − α
n
¯γ )||γ f(x
n

) − Au|| ||y
n
− u||
= α
n
||γ f (x
n
) − Au||
2
+(1− α
n
¯γ )

||x
n
− u||
2
+ λ
n
(λ
n
− 2β)||Bx
n
− Bu||
2

+2α
n
(1 − α
n

¯γ )||γ f (x
n
) − Au|| ||y
n
− u||
≤ α
n
||γ f (x
n
) − Au||
2
+ ||x
n
− u||
2
+(1− α
n
¯γ )b(b − 2β)||Bx
n
− Bu||
2
+2α
n
(
1 − α
n
¯γ
)
||γ f
(

x
n
)
− Au|| ||y
n
− u||.
So, we obtain
−(1 − α
n
¯γ )b(b − 2β)||Bx
n
− Bu||
2
≤ α
n
||γ f (x
n
) − Au||
2
+(||x
n
− u|| + ||x
n+1
− u||)(||x
n
− u|| − ||x
n+1
− u||)+ε
n
≤ α

n
||γ f
(
x
n
)
− Au||
2
+ ε
n
+ ||x
n
− x
n+1
||
(
||x
n
− u|| + ||x
n+1
− u||
)
,
where
ε
n
=2α
n
(
1 − α

n
¯γ
)
||γ f
(
x
n
)
− Au|| ||y
n
− u|
|
.
Since a
n
® 0 and ||x
n+1
- x
n
|| ® 0, we obtain ||Bx
n
-Bu|| ® 0asn ® ∞.
Further, by Lemma 2.1, we have
|
|y
n
− u||
2
= ||P
C

(x
n
− λ
n
Bx
n
) − P
C
(u − λ
n
Bu)||
2
≤(x
n
− λ
n
Bx
n
) − (u − λ
n
Bu), y
n
− u
=
1
2
(||(x
n
− λ
n

Bx
n
) − (u − λ
n
Bu)||
2
+ ||y
n
− u||
2
−||(x
n
− λ
n
Bx
n
) − (u − λ
n
Bu) − (y
n
− u)||
2
)
≤
1
2
(||x
n
− u||
2

+ ||y
n
− u||
2
−||(x
n
− y
n
) − λ
n
(Bx
n
− Bu)||
2
)
≤||x
n
− u||
2
−||x
n
− y
n
||
2
+2λ
n
x
n
− y

n
, Bx
n
− Bu−λ
2
n
||Bx
n
− Bu||
2
.
Bunyawat and Suantai Fixed Point Theory and Applications 2011, 2011:47
/>Page 7 of 13
So, we have
|
|x
n+1
− u||
2
= ||P
C
(α
n
γ f (x
n
)+(I − α
n
A)T
n
y

n
) − P
C
(u)||
2
≤||α
n
(γ f (x
n
) − Au)+(I − α
n
A)(T
n
y
n
− u)||
2
≤ (α
n
||γ f (x
n
) − Au|| + ||I − α
n
A|| ||T
n
y
n
− u||)
2
≤ (α

n
||γ f (x
n
) − Au|| +(1− α
n
¯γ )||y
n
− u||)
2
≤ α
n
||γ f (x
n
) − Au||
2
+(1− α
n
¯γ )||y
n
− u||
2
+2α
n
(1 − α
n
¯γ )||γ f(x
n
) − Au|| ||y
n
− u||

≤ α
n
||γ f (x
n
) − Au||
2
+(1− α
n
γ )||x
n
− u||
2
− (1 − α
n
γ )||x
n
− y
n
||
2
+2(1− α
n
¯γ )λ
n
x
n
− y
n
, Bx
n

− Bu−(1 − α
n
¯γ )λ
2
n
||Bx
n
− Bu||
2
+2α
n
(
1 − α
n
¯γ
)
||γ f
(
x
n
)
− Au|| ||y
n
− u||,
which implies
(1 − α
n
¯γ )||x
n
− y

n
||
2
≤ α
n
||γ f (x
n
) − Au||
2
+(||x
n
− u|| + ||x
n+1
− u||)||x
n
− x
n+1
||
+2(1− α
n
¯γ )λ
n
x
n
− y
n
, Bx
n
− Bu−(1 − α
n

¯γ )λ
2
n
||Bx
n
− Bu||
2
+2α
n
(
1 − α
n
¯γ
)
||γ f
(
x
n
)
− Au|| ||y
n
− u||.
Since a
n
® 0, ||x
n+1
- x
n
|| ® 0, and ||Bx
n

-Bu|| ® 0, we obtain ||x
n
-y
n
|| ® 0as
n ® ∞.
Next, we have
||x
n+1
− T
n
y
n
|| = ||P
C
(α
n
γ f (x
n
)+(I − α
n
A)T
n
y
n
) − P
C
(T
n
y

n
)|
|
≤||α
n
γ f (x
n
)+(I − α
n
A)T
n
y
n
− T
n
y
n
||
= α
n
||γ f
(
x
n
)
− AT
n
y
n
||.

Since a
n
® 0and{f (x
n
)}, {AT
n
y
n
} are bounded, we have ||x
n+1
- T
n
y
n
|| ® 0as
n ® ∞.Since
||x
n
− T
n
y
n
|| ≤ ||x
n
− x
n+1
|| + ||x
n+1
− T
n

y
n
||
,
it implies that ||x
n
-T
n
y
n
|| ® 0asn ® ∞. Since
||
x
n
− T
n
x
n
||
≤
||
x
n
− T
n
y
n
||
+
||

T
n
y
n
− T
n
x
n
||
≤||x
n
− T
n
y
n
|| + ||
y
n
− x
n
||,
we obtain ||x
n
-T
n
x
n
|| ® 0asn ® ∞. Moreover, from
|
|

y
n
− T
n
y
n
|| ≤ ||
y
n
− x
n
|| + ||x
n
− T
n
y
n
||
,
it follows that ||y
n
-T
n
y
n
|| ® 0asn ® ∞.
By ||y
n
-x
n

|| ® 0, ||T
n
y
n
-x
n
|| ® 0 and Lemma 2.5, we have
||
Tx
n
− x
n
||
≤
||
Tx
n
− Ty
n
||
+
||
Ty
n
− T
n
y
n
||
+

||
T
n
y
n
− x
n
||
≤||x
n
−
y
n
|| +sup{||T
n
z − Tz|| : z ∈{
y
n
}} + ||T
n
y
n
− x
n
||
.
Hence, lim
n®∞
||Tx
n

-x
n
|| = 0. Observe that P
F
(g f +I-A) is a contraction.
Bunyawat and Suantai Fixed Point Theory and Applications 2011, 2011:47
/>Page 8 of 13
By Lemma 2.2, we have that
||I −
A
|| ≤ 1 −¯
γ
, and since
0 <γ < ¯γ
/α
, we get
||P
F
(γ f + I − A)x − P
F
(γ f + I − A)y|| ≤ ||(γ f + I − A)x − (γ f + I − A)y||
≤ γ ||f (x) − f (y)|| + ||I − A|| ||x − y|
|
≤ γα||x − y|| +(1−¯γ )||x − y||
=
(
1 −
(
¯γ − γα
))

||x − y||.
Then, Banach’s contraction mapping principle guarantees that P
F
(g f +I-A)hasa
unique fixed point, say q Î H. That is, q = P
F
(g f + I-A)q. By Lemma 2.1, we obtain

(
γ f − A
)
q, x − q≤0forallx ∈ F
.
(3:1)
Choose a subsequence
{y
n
k
}
of {y
n
} such that
lim sup
n
→∞
(γ f − A)q, T
n
y
n
− q = lim

k→∞
(γ f − A)q, T
n
k
y
n
k
− q
.
As
{y
n
k
}
is bounded, there exists a subsequence
{y
n
k
i
}
of
{y
n
k
}
which converges weakly
to p. Without loss of generality, we may assume that
y
n
k


p
.
Since ||y
n
-T
n
y
n
|| ® 0, we obtain
T
n
k
y
n
k

p
. Since ||x
n
-Tx
n
|| ® 0, ||x
n
-y
n
|| ® 0
and by Lemma 2.4-2.5, we have
p ∈∩
∞

n
=1
F( T
n
)
. Let
Sv =

Bv + N
C
v, v ∈ C
,
∅, v ∈ C
.
where N
C
v is normal cone to C at v Î C, that is N
C
v ={w Î H : 〈v-u, w〉 ≥ 0, ∀u Î
C}. Then S is a maximal monotone. Let (v, w) Î G(S). Since w-BvÎ N
C
v and y
n
Î C,
we have 〈v-y
n
, w-Bv〉 ≥ 0. On the othe r hand, by Lemma 2.1 and from y
n
= P
C

(x
n
-
l
n
Bx
n
), we have
v − y
n
, y
n
− (x
n
− λ
n
Bx
n
)≥0
v − y
n
,
(
y
n
− x
n
)
/λ
n

+ Bx
n
≥0
.
Hence,
v − y
n
k
, w≥v − y
n
k
, Bv
≥v − y
n
k
, Bv−

v − y
n
k
,
y
n
k
− x
n
k
λ
n
+ Bx

n
k

=

v − y
n
k
, Bv − Bx
n
k
−
y
n
k
− x
n
k
λ
n

= v − y
n
k
, Bv − By
n
k
 + v − y
n
k

, By
n
k
− Bx
n
k
−

v − y
n
k
,
y
n
k
− x
n
k
λ
n

≥v − y
n
k
, By
n
k
− Bx
n
k

−

v − y
n
k
,
y
n
k
− x
n
k
λ
n

.
This implies 〈v-p, w〉 ≥ 0. Since S is maximal monotone, we have p Î S
-1
0and
hence p Î VI(C , B). We obtain that p Î F. By (3.1), we have 〈(g f-A)q, p-q〉 ≤ 0. It
follows that
lim sup
n
→
∞
(γ f − A)q , T
n
y
n
− q = lim

k→∞
(γ f − A)q , T
n
k
y
n
k
− q = (γ f − A)q, p − q≤0
.
Bunyawat and Suantai Fixed Point Theory and Applications 2011, 2011:47
/>Page 9 of 13
Finally, we prove x
n
® q.By||y
n
- u|| ≤ ||x
n
-u|| and Schwarz inequality, we have
||x
n+1
− q||
2
= ||P
C
(α
n
γ f (x
n
)+(I − α
n

A)T
n
y
n
) − P
C
(q)||
2
≤||α
n
(γ f (x
n
) − Aq)+(I − α
n
A)(T
n
y
n
− q)||
2
≤||( I − α
n
A)(T
n
y
n
− q)||
2
+ α
2

n
||γ f (x
n
) − Aq||
2
+2α
n
(I − α
n
A)(T
n
y
n
− q), γ f (x
n
) − Aq
≤ (1 − α
n
¯γ )
2
||y
n
− q||
2
+ α
2
n
||γ f (x
n
) − Aq||

2
+2α
n
T
n
y
n
− q, γ f(x
n
) − Aq−2α
2
n
A(T
n
y
n
− q), γ f(x
n
) − Aq
≤ (1 − α
n
¯γ )
2
||x
n
− q||
2
+ α
2
n

||γ f (x
n
) − Aq||
2
+2α
n
T
n
y
n
− q, γ f(x
n
) − γ f (q) +2α
n
T
n
y
n
− q, γ f (q) − Aq
− 2α
2
n
A(T
n
y
n
− q), γ f (x
n
) − Aq
≤ (1 − α

n
¯γ )
2
||x
n
− q||
2
+ α
2
n
||γ f (x
n
) − Aq||
2
+2α
n
||T
n
y
n
− q|| ||γ f (x
n
) − γ f (q)|| +2α
n
T
n
y
n
− q, γ f (q) − Aq


− 2α
2
n
A(T
n
y
n
− q), γ f (x
n
) − Aq
≤ (1 − α
n
¯γ )
2
||x
n
− q||
2
+ α
2
n
||γ f (x
n
) − Aq||
2
+2γαα
n
||y
n
− q|| ||x

n
− q|| +2α
n
T
n
y
n
− q, γ f(q) − Aq
− 2α
2
n
A(T
n
y
n
− q), γ f (x
n
) − Aq
≤ (1 − α
n
¯γ )
2
||x
n
− q||
2
+ α
2
n
||γ f (x

n
) − Aq||
2
+2γαα
n
||x
n
− q||
2
+2α
n
T
n
y
n
− q, γ f(q) − Aq
− 2α
2
n
A(T
n
y
n
− q), γ f (x
n
) − Aq
≤ ((1 − α
n
¯γ )
2

+2γαα
n
)||x
n
− q||
2
+ α
n
(2T
n
y
n
− q, γ f (q) − Aq
+ α
n
||γ f (x
n
) − Aq||
2
+2α
n
||A(T
n
y
n
− q)|| ||γ f (x
n
) − Aq||)
=(1− 2( ¯γ − γα)α
n

)||x
n
− q||
2
+ α
n
(2T
n
y
n
− q, γ f (q) − Aq
+ α
n
||γ f (x
n
) − Aq||
2
+2α
n
||A(T
n
y
n
− q)|| ||γ f (x
n
) − Aq||
+ α
n
γ
2

||x
n
− q||
2
)
.
Since {x
n
}, {f (x
n
)} and {T
n
y
n
} are bounded, we can take a constant h > 0 such that
η ≥||γ f
(
x
n
)
− Aq||
2
+2||A
(
T
n
y
n
− q
)

|| ||γ f
(
x
n
)
− Aq|| + ¯γ
2
||x
n
− q||
2
for all n ≥ 0. It follows that
|
|x
n+1
− q||
2
≤
(
1 − 2
(
¯γ − γα
)
α
n
)
||x
n
− q||
2

+ α
n
β
n
,
(3:2)
where b
n
=2〈T
n
y
n
-q, g f(q) -Aq〉 +ha
n
. By lim sup
n®∞
〈(g f-A)q, T
n
y
n
-q〉 ≤ 0, we
get lim sup
n®∞
b
n
≤ 0. By Lemma 2.3 and (3.2), we can conclude that x
n
® q.This
completes the proof. ■
Corollary 3.2 Let C be a closed convex subset of a real Hilbert space H, and let B : C

® Hbeab-inverse strongly monotone mapping, also let f : C ® C be a contraction
with coefficient a(0 < a <1). Let {T
n
} be a countable family of nonexpansive mappings
from a subset C into itself with
F = ∩
∞
n
=1
F( T
n
) ∩ VI(C, B) =
∅
. Suppose {x
n
} is the
sequence generated by the following algorithm: x
0
Î C,
x
n+1
= α
n
f
(
x
n
)
+
(

1 − α
n
)
T
n
P
C
(
x
n
− λ
n
Bx
n
)
for all n = 0, 1, 2, , where {a
n
} ⊂ (0, 1) and {l
n
} ⊂ (0, 2b ). If {a
n
} and {l
n
} are cho-
sen so that l
n
Î [ a, b] for some a, b with 0 <a<b<2b,
Bunyawat and Suantai Fixed Point Theory and Applications 2011, 2011:47
/>Page 10 of 13
(C1) lim

n→0
α
n
=0; (C2)
∞

n=1
α
n
= ∞;
(C3)
∞

n
=1
|
α
n+1
− α
n
|
< ∞;(C4)
∞

n
=1
|
λ
n+1
− λ

n
|
< ∞
.
Suppose that ({T
n
}, T ) satisfies the AKTT-condition. Then {x
n
} converges strongly to q
Î F, where q = P
F
(g f + I-A)(q) which solves the following variational inequality:

(
γ f − I
)
q, p − q≤0 ∀p ∈ F
.
Proof. Taking A = I and g = 1 in Theorem 3.1, we get the results. ■
4 Applications
In this section, we apply the iterative scheme (1.8) and Theorem 3.1 for finding a com-
mon fixed point of countable family of nonexpansive mappings and strictly pseudocon-
tractive mapping and inverse strongly monotone mapping.
A mapping T : C ® C is called strictly pseudocontractive if there exists k with 0 ≤ k
<1 such that
||Tx − Ty||
2
≤||x − y||
2
+ k||

(
I − T
)
x −
(
I − T
)
y||
2
∀x, y ∈ C
.
If k = 0, then T is nonexpansive. Put B = I-T, where T : C ® C is a strictly pseudo-
contractive mapping with k.Then,B is ((1 - k)/2)-inverse strongly monotone and B
-1
(0) = F(T). Hence, for all x, y Î C,
|
|
(
I − B
)
x −
(
I − B
)
y||
2
≤||x − y||
2
+ k||Bx − By||
2

.
Conversely, since H is a real Hilbert space, we have
||
(
I − B
)
x −
(
I − B
)
y||
2
≤||x − y||
2
+ ||Bx − By||
2
− 2x − y, Bx − By
.
Thus, we have
x − y, Bx − By≥
1 − k
2
||Bx − By||
2
.
Theorem 4.1 Let C be a closed convex subset of a real Hilbert space H, and let A be
a strongly positive linear bounded operator of H into itself with coefficient
¯
γ
>

0
such
that ||A|| = 1 and let f : C ® C be a contraction with coefficient a(0 < a <1). Assume
that
0 <γ < ¯γ
/α
. Let {T
n
} be a family of nonexpansive mappings of C into itself and
let S be a strictly pseudocontractive mapping of C into itself with b such that
F = ∩
∞
n
=1
F( T
n
) ∩ F(S) =
∅
. Suppose {x
n
} is a se quence generated by the following algo-
rithm: x
0
Î C,
x
n+1
= P
C
(
α

n
γ f
(
x
n
)
+
(
I − α
n
A
)
T
n
((
1 − λ
n
)
x
n
− λ
n
Sx
n
))
for all n =0,1,2, ,where {a
n
} ⊂ [0, 1) and {l
n
} ⊂ [0, 1 - b). If {a

n
} and {l
n
} are
chosen so that l
n
Î [ a, b] for some a, b with 0 <a<b<1 - b,
(C1) lim
n→0
α
n
=0; (C2)
∞

n=1
α
n
= ∞;
(C3)
∞

n
=1
|
α
n+1
− α
n
|
< ∞;(C4)

∞

n
=1
|
λ
n+1
− λ
n
|
< ∞
.
Bunyawat and Suantai Fixed Point Theory and Applications 2011, 2011:47
/>Page 11 of 13
Suppose that ({T
n
}, T) satisfies the AKTT-condition. Then,{x
n
} converges strongly to q
Î F, such that

(
γ f − A
)
q, p − q≤0 ∀p ∈ F
.
Proof.PutB = I-S,thenB is ((1 - k)/2)-inverse strongly monotone and F(S)=VI
(C, B)andP
C
(x

n
- l
n
Bx
n
)=(1- l
n
)x
n
+l
n
Sx
n
. Therefore, by Theorem 3.1, the conclu-
sion follows. ■
Lemma 4.2 [9]Let T : C ® H be a k-strictly pseudocontractive, then
(i) the fixed point set F(T) of T is closed convex so that the projection P
F(T)
is well
defined;
(ii) define a mapping S : C ® Hby
Sx = μx +
(
1 − μ
)
Tx, x ∈ C
.
(4:1)
If μ Î [k, 1), then S is a nonexpansive mapping such that F(T)=F(S).
A family of mappings

{T
n
: C → H}
∞
n
=
1
is called a family of uniformly k-strictly pseu-
docontractions, if there exists a constant k Î [0, 1) such that
||T
n
x − T
n
y||
2
≤||x − y||
2
+ k||
(
I − T
n
)
x −
(
I − T
n
)
y||
2
∀x, y ∈ C, ∀n ≥ 1

.
Let {T
n
: C ® C}beacountablefamilyofuniformlyk-strictly pseudocontractions.
Let
{S
n
: C → C}
∞
n
=
1
be the sequence of mappings defined by (4.1), i.e.,
S
n
x = μx +
(
1 − μ
)
T
n
x, x ∈ C, ∀n ≥ 1withμ ∈ [k,1
).
Corollary 4.3 Let C be a closed convex subset of a real Hilbert space H, and let B : C
® Hbeab-inverse strongly monotone mapping, also let A be a strongly positi ve linear
bounded operator of H into itself with coefficient
¯
γ
>
0

such that ||A|| = 1 and let f : C
® C be a contrac tion with coefficient a(0 < a <1). Assume that
0 <γ < ¯γ
/α
. Let {T
n
}
be a countable family o f uniformly k-strictly pseudocontractions from a subset C into
itself with
F = ∩
∞
n
=1
F( T
n
) ∩ VI(C, B) =
∅
. Suppose {x
n
} is the sequence generated by the
following algorithm: x
0
Î C,
x
n+1
= P
C
(
α
n

γ f
(
x
n
)
+
(
I − α
n
A
)
S
n
P
C
(
x
n
− λ
n
Bx
n
))
for all n = 0, 1, 2, , where { a
n
} ⊂ (0, 1) and {l
n
} ⊂ (0, 2b). If {a
n
} and {l

n
} are cho-
sen so that l
n
Î [ a, b] for some a, b with 0 <a<b<2b,
(C1) lim
n→0
α
n
=0; (C2)
∞

n=1
α
n
= ∞;
(C3)
∞

n
=1
|
α
n+1
− α
n
|
< ∞;(C4)
∞


n
=1
|
λ
n+1
− λ
n
|
< ∞
.
Then,{x
n
} converges strongly to q Î F, where q = P
F
(g f + I-A)(q) which solves the
following variational inequality:

(
γ f − A
)
q, p − q≤0 ∀p ∈ F
.
Proof.Let{T
n
} be a countable family of uniformly k-strictly pseudo-contractions
from a subset C into itself. Set S
n
= μI +(1-μ)T
n
where μ Î [k, 1). By Lemma 4.2, we

have S
n
is nonexpansive and F (S
n
)=F (T
n
). Therefore, by Theorem 3.1, the conclu-
sion follows. ■
Bunyawat and Suantai Fixed Point Theory and Applications 2011, 2011:47
/>Page 12 of 13
Acknowledgements
The authors would like to thank the Centre of Excellence in Mathematics for financial support under the project RG-1-
53-02-2. The first author is also supported by the Graduate School, Chiang Mai University, Thailand.
Author details
1
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
2
Centre of
Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand
Authors’ contributions
AB study and researched nonlinear analysis and also wrote this article. SS participated in the process of the study and
helped to draft the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Received: 25 January 2011 Accepted: 5 September 2011 Publi shed: 5 September 2011
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Cite this article as: Bunyawat and Suantai: Strong convergence theorems for variational inequalities and fixed
points of a countable family of nonexpansive mappings. Fixed Point Theory and Applications 2011 2011:47.
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