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RESEA R C H Open Access
Convergence and weaker control conditions for
hybrid iterative algorithms
Shuang Wang
Correspondence:

School of Mathematical Sciences,
Yancheng Teachers University,
Yancheng, 224051 Jiangsu, PR
China,
Abstract
Very recently, Yao et al. (Appl. Math. Comput. 216, 822-829, 2010) have proposed a
hybrid iterative algorithm. Under the parameter sequences satisfying some quite
restrictive conditions, they derived a strong convergence theorem in a Hilbert space.
In this article, under the weaker conditions, we prove the strong convergence of the
sequence generated by their iterative algorithm to a common fixed point of an
infinite family of nonexpansive mappings, which solves a variational inequality. It is
worth pointing out that we use a new method to prove our results. An appropriate
example, such that all conditions of this result that are satisfied and that other
conditions are not satisfied, is provided. Furthermore, we also give a weak
convergence theorem for their iterative algorithm involving an infinite family of
nonexpansive mappings in a Hilbert space.
MSC: 47H05, 47H09, 47H10
Keywords: Strong convergence, Variational inequality, Fixed points, k-Lipschitzian,
η-strongly monotone, Hilbert space
1 Introduction
Let H be a real Hilbert space and C be a nonempty, closed, convex subset of H,letF :
H ® H be a nonlinear operator. The variational inequality problem is formulated as
finding a point x* Î C such that

Fx

∗
, v − x
∗

≥ 0, ∀v ∈ C
.
In 1964, Stampacchia [1] introduced and studied variational inequality initiall y. It is
now well known that variational inequalities cover as diverse disciplines as partial dif-
ferential equations, optimal control, op timization, mathematical programming,
mechanics, and finance [1-5].
It is known that a mapping T : H ® H is said to be nonexpansive if ||Tx - Ty|| ≤ ||x
- y||, ∀x, y Î H. We use F (T ) to denote the set of fixed points of T, that is F (T)={x
Î H : Tx = x}.
Yamada [2] introduced the fol lowing hybrid i terative method for solving the varia-
tional inequality:
x
n+1
= Tx
n
− μλ
n
F
(
Tx
n
)
, n ≥ 0
,
(1:1)
Wang Fixed Point Theory and Applications 2011, 2011:3

/>© 2011 Wang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution
License ( which p ermits unrestricte d use, distribution, and reproduction in any medium,
provided the original work is properly cited.
where F is a k-Lipschitzian and h-strongly monotone operator with k >0,h > 0 and
0<μ <2h /k
2
. Let a sequence {l
n
} of real numbers in (0,1) satisfy the conditions
below:
(C1) lim
n®∞
l
n
=0,
(C2)

∞
n
=
0
λ
n
=
∞
,
(C3)
lim
n→∞
(λ

n
− λ
n+1
)/λ
2
n
+1
=
0
.
He has proved that {x
n
} generated by (1.1) converges strongly to the unique solution
of the variational inequality:
F
˜
x, x −
˜
x≥0, ∀x ∈ F
(
T
).
An example of sequence {l
n
} which satisfies conditions (C1)-(C3) is given by l
n
=1/
n
s
, where 0 <s < 1. We note that the condition (C3) was first used by Lions [3]. It was

observed that Lion’s conditions on the sequence {l
n
} exclud ed the canonical choice l
n
= 1/n. This was overcome in 2003 by Xu and Kim [4], if {l
n
} satisfies conditions (C1),
(C2), and (C4)
(C4) : lim
n
→∞
λ
n
/λ
n+1
= 1, or equivalently, lim
n
→∞
(λ
n
− λ
n+1
)/λ
n+1
=
0
who proved th e strong convergenc e of {x
n
} to the unique solution u*ofthevaria-
tional inequality 〈Fu*, v - u*〉 ≥ 0, ∀v Î C. It is e asy to see that the condition (C4) is

strictly weaker than condition (C3), coupled with conditions (C1) an d (C2). Moreover ,
(C4) includes the important and natural choice {1/n} of {l
n
}.
Very recently, motivated by Xu and Kim [4], Yao et al. [5] considered the following
algorithms: for x
0
Î H arbitrarily,

y
n
= x
n
− λ
n
F(x
n
),
x
n+1
=(1− α
n
)y
n
+ α
n
W
n
y
n

, n ≥ 0
,
(1:2)
where F is a k-Lipschitzian and h-strongly monotone operator on H, and W
n
is a W-
mapping defined by (2.3) cited later. Take k Î [1, ∞), h Î (0, 1), and {l
n
} satisfying the
conditions (C1) and (C2). If a sequence {a
n
} satisfying (C5)
(C5) : α
n
∈

γ ,
1
2

for some γ>0
,
then they proved that the sequences {x
n
}and{y
n
} defined by (1.2) converge strongly
to
x
∗

∈∩
∞
n
=1
F(T
n
)
, which solves the following variational inequality:
Fx
∗
, x − x
∗
≥0, ∀x ∈∩
∞
n
=1
F(T
n
).
We remind the reader of the fact that in order to guarantee the strong convergence
of the iterative sequence {x
n
}, there is at least one pa rameter sequence converging to
zero (i.e., l
n
® 0) as a result of Yamada [2], Xu and Kim [4, Theorem 3.1, and Theo-
rem3.2]andYaoetal.[5,Theorem3.2].Inaddition,h Î (0, 1) and (C5) are quite
restrictive assumptions in Yao et al. [5].
In this article, under the convergence of no parameter sequences to zero and the
weaker conditions on a

n
and h,weprovethatthesequence{y
n
} generated by the
Wang Fixed Point Theory and Applications 2011, 2011:3
/>Page 2 of 14
iterative algorithm (1.2) converges to a common fixed point of an infinite family of
nonexpansive mappings, which solves the variational inequality 〈Fx*, u - x*〉 ≥ 0,
∀u ∈∩
∞
n
=1
F(T
n
)
. In the meantime, we illustrate that this result is more general than
Theorem 3.2 of Yao et a l. [5]. That is, we give an appropriate example such that all
conditions of this result are satisfied and the conditions h Î (0, 1), (C1), and (C5) in
Yao et al. [5, Theorem 3.2] are not satisfied. Furthermore, we also give a weak co nver-
gence theo rem for hybrid iterative algorithm (1.2) involving an infinite family of non-
expansive mappings in a Hilbert space H. It i s worth pointing out that we use a new
method to prove our main results. The results presented in this article can be viewed
as the improvement, supplement, and extension of the results obtained in [2-5].
2 Preliminaries
Let H be a real Hilbert space with inner product 〈·,·〉 and norm ||·||. For the sequence
{x
n
}inH,wewritex
n
⇀ x to indicate that the sequence {x

n
} converges weakly to x.
x
n
® x implies that {x
n
} converges strongly to x.Wedenotebyω
w
(x
n
) the weak
ω-limit set of {x
n
}, that is
ω
w
(x
n
)={x ∈ H : x
n
i
 x for some subsequence {x
n
i
} of {x
n
}}
.
AmappingF : H ⇀ H is called k-Lipschitzian if there exists a positive constant k
such that

|
|Fx − F
y
|| ≤ k||x −
y
||, ∀x,
y
∈ H
.
(2:1)
F is said to be h-strongly monotone if there exists a positive constant h such that
Fx − F
y
, x −
y
≥η||x,
y
||
2
, ∀x,
y
∈ H
.
(2:2)
It is known that X satisfies Opial’s property [6] provided, for each sequence {x
n
}inX,
the condition x
n
⇀ x implies

lim sup
n
→∞
||x
n
− x|| < lim sup
n
→∞
||x
n
− y||, ∀y ∈ X, y = x
.
It is known that each l
p
(1 ≤ p < ∞) enjoys this property, while L
p
does not unless
p =2.
Finally, it is known that in a Hilbert space, there holds the following equality
|
|λx +
(
1 − λ
)
y||
2
= λ||x||
2
+
(

1 − λ
)
||y||
2
− λ
(
1 − λ
)
||x − y||
2
for all x, y Î H and l Î [0,1] (see Takahashi [7]).
In order to prove our main results, we need the following lemmas:
Lemma 2.1.[8].Let H be a Hilbert space, C a closed, convex subset of H, a nd T :
C ® C a nonexpansive mapping with F (T ) ≠ ∅; if {x
n
} is a sequence in C weakly
converging to × and if {(I - T )x
n
} converges strongly to y, then (I - T )x = y.
Lemma 2.2.[9].Let {x
n
} and {z
n
} be bounded sequences in Banach space E and { g
n
}
be a sequence in [0,1] which satisfies the following condition:
0 < lim inf
n→∞
γ

n
≤ lim sup
n
→
∞
γ
n
< 1.
Suppose that x
n+1
= g
n
x
n
+(1-g
n
)z
n
, n ≥ 0, and lim sup
n®∞
(||z
n+1
- z
n
|| - ||x
n+1
- x
n
||)
≤ 0. Then, lim

n®∞
||z
n
- x
n
|| = 0.
Wang Fixed Point Theory and Applications 2011, 2011:3
/>Page 3 of 14
Lemma 2.3. [10,11]. Let {s
n
} be a sequence of non-negative real numbers satisfying
s
n+1
≤
(
1 − λ
n
)
s
n
+ λ
n
δ
n
+ γ
n
, n ≥ 0
,
where {l
n

} and {g
n
} satisfy the following conditions: (i) {l
n
} ⊂ [0,1] and

∞
n
=
0
λ
n
=
∞
, (ii) lim sup
n® ∞
or

∞
n
=
0
λ
n
δ
n
<
∞
, (iii) g
n

≥ 0(n ≥ 0),

∞
n
=
0
γ
n
<
∞
.
Then lim
n®∞
S
n
=0.
Lemma 2.4. [12]. Let {a
n
} and {b
n
} be sequences of nonnegative real numbers such
that

∞
n
=
0
b
n
<

∞
and a
n+1
≤ a
n
+ b
n
for all n ≥ 0. Then lim
n®∞
a
n
exists.
Lemma 2.5.[13].Let F be a k-Lipschitzian and h -strong ly monotone operator on a
Hilbert space H with 0<h ≤ k and 0<t <h/k
2
. Then S =(I - tF ):H ® H is a contrac-
tion with contraction coefficient
τ
t
=

1 − t(2η − tk
2
)
.
Let {T
i
: H ® H}.beafamilyofinfinitelynonexpansivemappings,{ξ
i
}beareal

sequence such that 0 <ξ
i
≤ b <1,∀i ≥ 1. For any n ≥ 1, define a mapping W
n
: H ® H
as follows:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪

⎪
⎪
⎩
U
n,n+1
= I,
U
n,n
= ξ
n
T
n
U
n,n+1
+(1− ξ
n
)I,
U
n,n−1
= ξ
n−1
T
n−1
U
n,n
+(1− ξ
n−1
)I
,
···

U
n,k
= ξ
k
T
k
U
n,k+1
+(1− ξ
k
)I
U
n,k−1
= ξ
k−1
T
k−1
U
n,k
+(1− ξ
k−1
)I
···
U
n,2
= ξ
2
T
2
U

n,3
+(1− ξ
2
)I,
W
n
= U
n,1
= ξ
1
T
1
U
n,2
+
(
1 − ξ
1
)
I.
(2:3)
Such a mapping W
n
: H ® H is called a W-mapping generated by T
n
, T
n-1
, , T
1
and

ξ
n
, ξ
n-1
, , ξ
1
.
We have the following crucial conclusion concerning W
n
. We can find them in
[14-17]. Now we only need the following similar version in Hilbert spaces:
Lemma 2.6. Let H be a real Hilbert space,{T
i
: H ® H} be a family of infinitely non-
expansive mappings with
∩
∞
i
=1
F(T
i
) =
∅
,{ξ
i
} be a real sequence such that 0<ξ
i
≤ b <1,
∀i ≥ 1. Then,
(1) W

n
is a nonexpansive and
F(W
n
)=∩
n
i
=1
F(T
i
)
for each n ≥ 1;
(2) For every x Î H and k Î N, the limit lim
n®∞
U
n, k
x exists;
(3) If we define a mapping W : H ® H as Wx =lim
n®1
W
n
x, and W
n
x =lim
n®∞
U
n,1
x, for every Î H, then,
F(W)=∩
∞

i
=1
F(T
i
)
;
(4) For any bounded sequence {x
n
} ⊂ H, we have lim
n®∞
||Wx
n
- W
n
x
n
|| = 0.
3 Main results
Let F be a k-Lipschitzian and h-strongly monotone operator on H with 0 <k, T : H ®
H be a nonexpansive mapping. Let t Î (0,h/k
2
)and
τ
t
=

1 − t(2η − tk
2
)
,andcon-

sider a mapping S
t
on H defined by
S
t
x = T [
(
I − tF
)
x], x ∈ H
.
Wang Fixed Point Theory and Applications 2011, 2011:3
/>Page 4 of 14
It is easy to see that S
t
is a contraction. Indeed, from Lemma 2.5, we have
|
|S
t
x − S
t
y|| ≤ ||T[(I − tF)x] − T[(I − tF)y]|
|
≤||(I − tF)x − (I − tF)y||
≤ τ
t
||x −
y
||,
for all x, y Î H. Hence, it has a unique fixed point, denoted as x

t
, which uniquely
solves the fixed point equation
x
t
= T[
(
I − tF
)
x
t
], x
t
∈ H
.
(3:1)
Theorem 3.1. Let H be a real Hilbert space. Let T : H ® H be a nonexpansive map-
ping such that F (T ) ≠ ∅,. Let F be a k-Lipschitzian and h-strongly monotone operator
on H with 0<h ≤ k . For each t Î (0, h/k
2
), let the net {x
t
} be generated by (3.1). Then,
as t ® 0, the net {x
t
} converges strongly to a fixed point x*ofT which solves the varia-
tional inequality:
Fx
∗
, x

∗
− u≤0, ∀u ∈ F
(
T
).
(3:2)
Proof. We first show the uniqueness of a solution of the variational inequality (3.2),
which is indeed a consequence of the strong monotonicity of F. Suppose x* Î F (T )
and
˜
x ∈ F
(
T
)
both are solutions to (3.2), then

Fx
∗
, x
∗
−
˜
x

≤ 0
,
(3:3)
and

F

˜
x,
˜
x − x
∗

≤ 0
.
(3:4)
Adding up (3.3) and (3.4) yields

Fx
∗
− F
˜
x, x
∗
−
˜
x

≤ 0
.
The strong monotonicity of F implies that
x
∗
=
˜
x
and the uniqueness is proved. Later,

we use x* Î F (T ) to denote the unique solution of (3.2).
Next, we prove that {x
t
} is bounded. Take u Î F (T ), from (3.1) and using Lemma
2.5, we have
||x
t
− u|| = ||T[(I − tF)x
t
] − Tu||
≤||(I − tF)x
t
− u||
≤||(I − tF)x
t
− (I − tF)u − tFu||
≤||(I − tF)x
t
− (I − tF)u|| + t||Fu|
|
≤ τ
t
||
x
t
− u
||
+ t
||
Fu

||
,
that is,
||x
t
− u|| ≤
t
1 − τ
t
||Fu||
.
(3:5)
Observe that
lim
t→0
+
t
1 − τ
t
=
1
η
.
Wang Fixed Point Theory and Applications 2011, 2011:3
/>Page 5 of 14
From t ® 0, we may assume, w ithout loss of generality, that
t ≤
η
k
2

−
ε
,whereisa
arbitrarily small positive number. Thus, we have
t
1 − τ
t
is continuous,
∀
t ∈ [0,
η
k
2
− ε
]
.
Therefore, we obtain
sup

t
1 − τ
t
: t ∈ (0,
η
k
2
− ε]

< +∞
.

(3:6)
From (3.5) and (3.6), we have that {x
t
} is bounded and so is {Fx
t
}.
On the other hand, from (3.1), we obtain
|
|x
t
− Tx
t
|| = ||T[
(
I − tF
)
x
t
] − Tx
t
|| ≤ ||
(
I − tF
)
x
t
− x
t
|| = t||Fx
t

|| → 0
(
t → 0
).
(3:7)
To prove that x
t
® x*. For a given u Î F (T ), using Lemma 2.5, we have
||x
t
− u||
2
= ||T[(I − tF)x
t
] − Tu||
2
≤||(I − tf )x
t
− (I − tF)u − tFu||
2
≤ τ
t
2
||x
t
− u||
2
+ t
2
||Fu||

2
+2t(I − tF)u − (I − tF)x
t
, Fu
≤ τ
t
||x
t
− u||
2
+ t
2
||Fu||
2
+2tu − x
t
, Fu +2t
2
Fx
t
− Fu, Fu
≤ τ
t
||
x
t
− u
||
2
+ t

2
||
Fu
||
2
+2t

u − x
t
, Fu

+2t
2
k
||
x
t
− u
|| ||
Fu
||.
Therefore,
|
|x
t
− u||
2
≤
t
2

1 − τ
t
||Fu
2
|| +
2t
1 − τ
t
u − x
t
, Fu +
2t
2
k
1 − τ
t
||x
t
− u|| ||Fu||
.
(3:8)
From
τ
t
=

1 − t(2η − tk
2
)
,wehave

lim
t→0
t
2
1 − τ
t
=
0
and
lim
t→0
2t
2
k
1 − τ
t
=
0
.
Observe that, if x
t
⇀ u, we have
lim
t→0
2t
1 − τ
t
u − x
t
, Fu =

0
.
Since {x
t
} is bounded, we see that if {t
n
} is a sequence in
(0,
η
k
2
− ε
]
such that t
n
® 0
and
x
t
n

˜
x
, then by (3.8), w e see
x
t
n
→
˜
x

. Moreover, by (3.7) and using Lemma 2.1,
we have
˜
x ∈ F
(
T
)
.Wenextprovethat
˜
x
solves the variational inequality (3.2). From
(3.1) and u Î F (T ), we have
||x
t
− u||
2
≤||(I − tF)x
t
− u
2
= ||x
t
− u
2
|| + t
2
||Fx
t
||
2

− 2tFx
t
, x
t
− u
,
that is,
Fx
t
, x
t
− u≤
t
2
||Fx
t
||
2
.
(3:9)
Now replacing t in (3.9) with t
n
and letting n ®
∞
, we have

F
˜
x,
˜

x − u

≤ 0
.
That is
˜
x ∈ F
(
T
)
is a solution of (3.2), hence
˜
x
=
x
∗
by uniqueness. In a nutshell, we
have shown that each cluster point of {x
t
}(att® 0) equals x*. Therefore, x
t
® x*as
t ® 0.
Theorem 3.2. Let H be a real Hilbert space. Let F be a k-Lipschitzian and h-strongly
monotone operator on H with 0<h ≤ k. Let
{T
n
}
∞
n

=1
: H →
H
be an infinite family of
Wang Fixed Point Theory and Applications 2011, 2011:3
/>Page 6 of 14
nonexpansive mappings such that
∩
∞
n=1
F(T
n
) =
∅
and W
n
be a W-mapping defined by
(2.3). Let {l
n
} be a sequence in [0, ∞) and {a
n
} be a sequence in [0,1], ε be a arbitrarily
small positive number. Assume that the control conditions (C2),(C1)’,and(C5)’ hold
for {l
n
} and {a
n
},
(C1)’:
0 <λ

n
≤
η
k
2
−
ε
, ∀n ≥ n
0
for some integer n
0
≥ 0, and
(C5) ‘:0<g ≤ lim inf
n®∞
a
n
lim sup
n®∞
a
n
<1for some gÎ(0, 1).
For x
0
Î H arbitrarily, let the sequence {y
n
} be generated by (1.2). Then,
y
n
→ x
∗

⇔ λ
n
F
(
x
n
)
→ 0
(
n →∞
),
where
x
∗
∈∩
∞
n
=1
F(T
n
)
solves the variational inequality
Fx
∗
, x
∗
− u≤0, u ∈∩
∞
n
=1

F(T
n
)
.
Proof. On the one hand, suppose that l
n
F(x
n
) ® 0(n ® ∞). We proceed with the fol-
lowing steps:
Step 1. We claim that {x
n
} is bounded. In fact, let
u ∈∩
∞
n
=1
F(T
n
)
, from (1.2), (C1)’ and
using Lemma 2.5, we have
|
|y
n
− u|| = ||x
n
− λ
n
F(x

n
) − u||
≤||(I − λ
n
F)x
n
− (I − λ
n
F)u − λ
n
Fu|
|
≤ τ
λ
n
||x
n
− u|| + λ
n
||Fu||,
(3:10)
∀n ≥ n
0
for some integer n
0
≥ 0, where
τ
λ
n
=


1 − λ
n
(2η − λ
n
k
2
) ∈ (0, 1
)
.Then,
from (1.2) and (3.10), we obtain
|
| x
n+1
− u|| = ||(1 − α
n
)(y
n
− u)+α
n
(W
n
y
n
− u)||
≤||y
n
− u||
≤ [1 − (1 − τ
λ

n
)] ||x
n
− u || + λ
n
||Fu||
≤ max

||x
n
− u|| ,
||λ
n
Fu||
1 − τ
λ
n

.
By induction, we have
||
x
n
− u
||
≤ max
{||
x
0
− u

||
, M
1
||
Fu
||}
,
∀n ≥ n
0
for some integer n
0
≥ 0, where
M
1
=sup{
λ
n
1 − τ
λ
n
:0<λ
n
≤
η
k
2
− ε} < +
∞
.
Therefore, {x

n
} is bounded. We also obtain that {y
n
}, {W
n
y
n
} and {Fx
n
} are bounded.
Step 2. We claim that lim
n®∞
||x
n+1
- x
n
|| = 0. To this end, define x
n+1
=(1-a
n
)x
n
+ a
n
z
n
. We observe that
||z
n+1
− z

n
|| =




x
n+2
− (1 − α
n+1
)x
n+1
α
n+1
−
x
n+1
− (1 − α
n
)x
n
α
n




≤





(1 − α
n+1
)y
n+1
+ α
n+1
W
n+1
y
n+1
− (1 − α
n+1
)x
n+1
α
n+1
−
(1 − α
n
)y
n
+ α
n
W
n
y
n
− (1 − α

n
)x
n
α
n




≤




α
n+1
W
n+1
y
n+1
− (1 − α
n+1
)λ
n+1
F(x
n+1
)
α
n+1
−

α
n
W
n
y
n
− (1 − α
n
)λ
n
F(x
n
)
α
n




≤
1 − α
n+1
α
n+1
||λ
n+1
F(x
n+1
)|| +
1 − α

n
α
n
||λ
n
F(x
n
)|| + ||W
n+1
y
n+1
− W
n
y
n
||
≤
1 − γ
γ
||λ
n+1
F(x
n+1
)|| +
1 − γ
γ
||λ
n
F(x
n

)|| + ||W
n+1
y
n+1
− W
n
y
n
||.
(3:11)
Wang Fixed Point Theory and Applications 2011, 2011:3
/>Page 7 of 14
From (2.3), for
u
∈∩
∞
n=1
F(T
n
)
, we have
|
|W
n+1
y
n
− W
n
y
n

|| = ||ξ
1
T
1
U
n+1,2
y
n
− ξ
1
T
1
U
n,2
y
n
n||
≤ ξ
1
||U
n+1,2
y
n
− U
n,2
y
n
n||
= ξ
1

||ξ
2
T
2
U
n+1,3
y
n
− ξ
2
T
2
U
n,3
y
n
||
≤ ξ
1
ξ
2
||U
n+1,3
y
n
− U
n,3
y
n
||

≤···
≤ ξ
1
ξ
2
···ξ
n
||U
n+1,n+1
y
n
− U
n,n+1
y
n
||
= ξ
1
ξ
2
···ξ
n
||ξ
n+1
T
n+1
y
n
+(1− ξ
n+1

)y
n
− y
n
||
≤

n+1

i=1
ξ
i

(||T
n+1
y
n
− u|| + ||u − y
n
||)
≤

n+1

i=1
ξ
i

(2||y
n

− u||)
≤ M
2
n+1

i
=1
ξ
i
,
(3:12)
where M
2
= sup{2 ||y
n
- u||, n ≥ 0}. By (1.2) and (3.12), we have
|| 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
|| + || W
n+1
y
n
− W
n
y
n
||
≤||x
n+1
− λ
n+1
F(x
n+1

) − x
n
+ λ
n
F(x
n
) || + M
2
n+1

i=1
ξ
i
≤||x
n+1
− x
n
|| + || λ
n+1
F(x
n+1
) || + || λ
n
F(x
n
) || + M
2
n+1

i

=1
ξ
i
.
(3:13)
Substituting (3.13) into (3.11), we have
|
|z
n+1
− z
n
|| ≤
1 − γ
γ
||λ
n+1
F(x
n+1
)|| +
1 − γ
γ
||λ
n
F(x
n
)|| + ||x
n+1
− x
n
|| + ||λ

n+1
F(x
n+1)
)|
|
+ ||λ
n
F(x
n
)|| + M
2
n+1

i=1
ξ
i
=
1
γ
||λ
n+1
F(x
n+1
)|| +
1
γ
||λ
n
F(x
n

)|| + ||x
n+1
− x
n
|| + M
2
n+1

i
=1
ξ
i
,
that is,
|
|z
n+1
− z
n
|| − || x
n+1
− x
n
|| ≤
1
γ
||λ
n+1
F(x
n+1

) || +
1
γ
||λ
n
F(x
n
) || + M
2
n+
1

i
=1
ξ
i
.
Observing l
n
F(x
n
) ® 0(n ® ∞) and 0 <ξ
i
≤ b < 1, it follows that
lim sup
n
→∞
(||z
n+1
− z

n
|| − || x
n+1
− x
n
||) ≤ 0
.
(3:14)
By (C5)’ and using Lemma 2.2, we have lim
n®∞
||z
n
- x
n
|| = 0. Therefore,
lim
n
→
∞
|| x
n+1
− x
n
|| = lim
n
→
∞
α
n
||z

n
− x
n
|| =0
.
Wang Fixed Point Theory and Applications 2011, 2011:3
/>Page 8 of 14
Step 3. We claim that lim
n®∞
||x
n
- W
n
x
n
|| = 0. Observe that
||
x
n
− W
n
x
n
||
≤
||
x
n
− x
n+1

||
+
||
x
n+1
− W
n
x
n
||
≤||x
n
− x
n+1
|| +(1− α
n
) ||y
n
− W
n
x
n
|| + α
n
|| W
n
y
n
− W
n

x
n
||
≤||x
n
− x
n+1
|| +(1− α
n
) ||y
n
− x
n
|| +(1− α
n
) ||x
n
− W
n
x
n
|| + α
n
|| y
n
− x
n
|
|
= || x

n
− x
n+1
|| + || y
n
− x
n
|| +
(
1 − α
n
)
|| x
n
− W
n
x
n
||,
that is,
|
|x
n
− W
n
x
n
|| ≤
1
α

n
(||x
n+1
− x
n
|| + || y
n
− x
n
||)
≤
1
γ
(|| x
n+1
− x
n
|| + || λ
n
F(x
n
) ||) → 0(n →∞)
.
(3:15)
Step 4. We claim that lim
n®∞
||x
n
- Wx
n

|| = 0. Indeed, we have
||
x
n
− Wx
n
||
≤
||
x
n
− W
n
x
n
||
+
||
W
n
x
n
− Wx
n
||.
(3:16)
By (3.15), (3.16) and using Lemma 2.6, we obtain
lim
n
→

∞
|| x
n
− Wx
n
|| =0
.
Step 5. We claim that lim sup
n®∞
〈Fx*, x*-x
n
〉 ≤ 0, where x*=lim
n®∞
x
t
and x
t
defined by x
t
= W[(1 - tF)x
t
]. Since x
n
is bounded, there exists a subsequence
{x
n
k
}
of
{x

n
} which converges wea kly to ω.FromStep4,weobtain
Wx
n
k

ω
. From Lemma
2.1, we have ω Î F (W). Hence, by Theorem 3.1, we have
lim sup
n
→∞
Fx
∗
, x
∗
− x
n
 = lim
k→∞
Fx
∗
, x
∗
− x
n
k
 = Fx
∗
, x

∗
− ω≤0
.
Step 6. We claim that {x
n
} converges strongly to
x
∗
∈∩
∞
n
=1
F(T
n
)
. From (1.2), we have
|| x
n+1
− x
∗
||
2
≤ (1 − α
n
)|| y
n
− x
∗
||
2

+ α
n
|| W
n
y
n
− x
∗
||
2
≤||y
n
− x
∗
||
2
= ||x
n
− λ
n
F(x
n
) − x
∗
||
2
≤||(I − λ
n
F)x
n

− (I − λ
n
F)x
∗
− λ
n
Fx
∗
||
2
≤ τ
2
λ
n
||x
n
− x
∗
||
2
+ λ
2
n
|| Fx
∗
||
2
+2λ
n
(I − λ

n
F)x
∗
− (I − λ
n
F)x
n
, Fx
∗

≤ τ
λ
n
|| x
n
− x
∗
||
2
+ λ
2
n
|| Fx
∗
||
2
+2λ
n
x
∗

− x
n
, Fx
∗
 +2λ
n
λ
n
Fx
n
, Fx
∗
−2λ
2
n
||Fx
∗
||
2
≤ [1 − (1 − τ
λ
n
)]|| x
n
− x
∗
||
2
+2λ
n

x
∗
− x
n
, Fx
∗
 +2λ
n
|| λ
n
Fx
n
|| ||Fx
∗
|| − λ
2
n
|| Fx
∗
||
2
≤ [1 − (1 − τ
λ
n
)]|| x
n
− x
∗
||
2

+(1− τ
λ
n
)

2λ
n
1 − τ
λ
n
x
∗
− x
n
, Fx
∗
 +
λ
n
M
3
1 − τ
λ
n
||λ
n
Fx
n
||


≤ [1 − (1 − τ
λ
n
)]|| x
n
− x
∗
||
2
+(1− τ
λ
n
)[2M
1
x
∗
− x
n
, Fx
∗
 + M
1
M
3
|| λ
n
Fx
n
||],
∀n ≥ n

0
for some integer n
0
≥ 0, where M
3
=2||Fx*||. For every n ≥ n
0
,put
μ
n
=1− τ
λ
n
and δ
n
=2M
1
〈x*-x
n
, Fx*〉 +M
1
M
3
||l
n
Fx
n
||. It follows that
|
|x

n+1
− x
∗
||
2
≤
(
1 − μ
n
)
||x
n
− x
∗
||
2
+ μ
n
δ
n
, ∀n ≥ n
0
.
It is easy to see that

∞
n
=1
μ
n

=
∞
and lim sup
n®∞
δ
n
≤ 0. Hence, by Lemma 2.3,
the sequence {x
n
} converges strongly to
x
∗
∈∩
∞
n
=1
F(T
n
)
.
Observe that
||y
n
− x
∗
|| ≤ ||y
n
− x
n
|| + || x

n
− x
∗
|| ≤ ||λ
n
F
(
x
n
)
+ || x
n
− x
∗
|| → 0
(
n →∞
).
It follows t hat the sequence {y
n
}convergesstronglyto
x
∗
∈∩
∞
n
=1
F(T
n
)

.Fromx*=
lim
t®0
x
t
and Theorem 3.1, we have x* is the unique solution of the variational
inequality: 〈Fx*, x*-u〉 ≤ 0,
∀
u ∈∩
∞
n
=1
F(T
n
)
.
Wang Fixed Point Theory and Applications 2011, 2011:3
/>Page 9 of 14
On the other hand, suppose that
y
n
→ x
∗
∈∩
∞
n=1
F(T
n
)
as n ® ∞,where

x
∗
∈∩
∞
n
=1
F(T
n
)
solves the variational inequality:
Fx
∗
, x
∗
− u≤0, u ∈∩
∞
n
=1
F(T
n
)
.
From (1.2), we have
|
|x
n+1
− x
∗
|| = ||(1 − α
n

)(y
n
− x
∗
)+α
n
(W
n
y
n
− x
∗
) |
|
≤ (1 − α
n
)|| y
n
− x
∗
|| + α
n
|| y
n
− x
∗
||
= ||y
n
− x

∗
→ 0
(
n →∞
)
,
(3:17)
that is,
x
n
→ x
∗
∈∩
∞
n
=1
F(T
n
)
. Again from (1.2), we obtain that
||λ
n
F
(
x
n
)
|| = ||y
n
− x

n
|| ≤ ||y
n
− x
∗
|| + || x
n
− x
∗
||
.
Since
y
n
→ x
∗
∈∩
∞
n
=1
F(T
n
)
and
x
n
→ x
∗
∈∩
∞

n
=1
F(T
n
)
,wegetl
n
F(x
n
) ® 0. This com-
pletes the proof.
Remark 3.3. It is clear that condition (C1)’ is strictly weaker than condition (C1). In
the meantime, condition (C5)’ is also strictly weaker than condition (C5).
Corollary 3.4. (Yao et al. [5, Theorem 3.2]). Let H be a real Hilbert space. Let F : H
® H be k-Lipschitzian and h-strongly monotone operator with k Î [1, ∞) and h Î (0,
1). Let
{T
n
}
∞
n
=1
: H →
H
be an infinite family of nonexpansive mappings such that
∩
∞
n
=1
F(T

n
) =
∅
and {W
n
} be W-mapping defined by (2.3). Let {l
n
} be a sequenc e in [0,
∞) and {a
n
} be a sequence in [0,1]. Assume that
(C1) lim
n®1
l
n
=0;
(C2)

∞
n
=
0
λ
n
=
∞
;
(C5)
α
n

∈

γ ,
1
2

for some g >0.
Then, the sequence {x
n
} and {y
n
} generated by (1.2) converge strongly to
x
∗
∈∩
∞
n
=1
F(T
n
)
, which solves the following va riational inequality 〈Fx*, x*-x〉 ≤ 0,
x
∗
∈∩
∞
n
=1
F(T
n

)
.
Proof. Since lim
n®∞
l
n
= 0, it i s easy to see that
λ
n
≤
η
k
2
−
ε
, ∀n ≥ n
0
for some inte-
ger n
0
≥ 0. Without loss of generality, we assume that
0 <λ
n
≤
η
k
2
−
ε
, ∀n ≥ n

0
for
some integer n
0
≥ 0. Repeating the same argument as in the proof of Theorem 3.2, we
know that {x
n
} is bounded, and so are the sequence {y
n
}and{F( x
n
)}. Therefore, w e
have l
n
F(x
n
) ® 0.
From
α
n
∈

γ ,
1
2

for some g >0,wehave0<g ≤ lim inf
n®∞
a
n

≤ lim sup
n®∞
a
n
<
1forsomeg Î (0, 1). Therefore, all conditions of Theorem 3.2 are satisfied. Henc e,
using Theorem 3.2, we have that {y
n
} converges strongly to
x
∗
∈∩
∞
n
=1
F(T
n
)
which
solves the following variational inequality 〈Fx*, x*-x〉 ≤ 0,
x
∗
∈∩
∞
n
=1
F(T
n
)
. It follows

from (3.17) that {x
n
} also converges strongly to
x
∗
∈∩
∞
n
=1
F(T
n
)
.Thiscompletesthe
proof.
Wang Fixed Point Theory and Applications 2011, 2011:3
/>Page 10 of 14
Remark 3.5. Theorem 3.2 is more general than Theorem 3.2 of Yao et al. [5]. The
following example shows that all conditions o f Theorem 3.2 are satisfied. However, the
conditions l
n
® 0, h Î (0, 1) and
α
n
∈

γ ,
1
2

for some g >0in [5, Theorem 3.2] are

not satisfied.
Example 3.6. Let H = R the set of r eal numbers and T
n
≡ T. Define a nonexpansive
mapping T : H ® H and an operator F : H ® H as follows:
Tx =0 and F
(
x
)
= x, ∀x ∈ R
.
It is easy to see that F(T) = {0},
∩
∞
n
=1
F(T
n
)={0
}
and W
n
x =(1-ξ
1
)x, ∀x Î R. Let
ξ
1
=
1
2

, we have
W
n
x =
1
2
x
, ∀x Î R. Given sequences {a
n
} and
{λ
n
} : α
n
=
2
3
,
λ
n
=
1
2
for
all n ≥ 0. For an arbitrary x
0
Î H, let {x
n
} defined as follows:


y
n
= x
n
− λ
n
F(x
n
),
x
n+1
=(1− α
n
)y
n
+ α
n
W
n
y
n
, n ≥ 0
,
that is,
y
n
= x
n
− λ
n

F(x
n
)=
1
2
x
n
,
x
n+1
=
1
3
y
n
+
2
3
W
n
y
n
=
2
3
y
n
=
1
3

x
n
, n ≥ 0
.
Observe that for all n ≥ 0,
|
|x
n+1
− 0|| =
1
3
|| x
n
− 0 ||
.
Hence, we have
||x
n+1
− 0 || =

1
3

n+1
|| x
0
− 0 |
|
for all n ≥ 0. Thi s implies that {x
n

}
converges strongly to
0 ∈∩
∞
n
=1
F(T
n
)
. Since
||y
n
− 0 || =
1
2
|| x
n
|| →
0
, we ha ve that {y
n
}
converges strongly to
0 ∈∩
∞
n
=1
F(T
n
)

.
Observe that 〈F(0), 0 - u〉 ≤ 0,
u
∈∩
∞
n
=1
F(T
n
)
, that is,0is the solution of the varia-
tional inequality 〈Fx*, x*-u〉 ≤ 0,
u
∈∩
∞
n
=1
F(T
n
)
.
Finally, we have
|
|λ
n
F(x
n
) || =
1
2

|| x
n
|| → 0(n →∞)
.
By F(x)=x, we have h = k =1.Furthermore, it is easy to see that the following hold
true:
(B1)
0 <λ
n
=
1
2
≤ 1 −
ε
, ∀n ≥ n
0
for some integer n
0
≥ 0;
(B2)

∞
n=0
λ
n
=

∞
n=0
1

2
=
∞
;
(B3)
0 <
1
2
≤ lim inf
n→∞
α
n
=
2
3
= lim sup
n→∞
α
n
<
1
for some constant
γ =
1
2
.
Wang Fixed Point Theory and Applications 2011, 2011:3
/>Page 11 of 14
Hence, there is no doubt that all conditions of Theorem 3.2 are satisfied. Since
λ

n
=
1
2
,
h =1and
α
n
∈ [γ ,
1
2
]
, the conditions that l
n
® 0,
α
n
∈

γ ,
1
2

for some g >0and h Î
(0, 1) of Yao et al. [5, Theorem 3.2] are not satisfied.
Next, we give a weak convergence theorem for hybrid iterative algorithm (1.2) invol-
ving an infinite family of nonexpansive mappings in a Hilbert space.
Theorem 3.7. Let H be a real Hilbert space. Let F : H ® H be k-Lipschitzian
and h-strongly monotone operator with 0<h ≤ k. Let
{T

n
}
∞
n
=1
: H →
H
be an infi-
nite family of nonexpansive mappings such that
∩
∞
n=1
F(T
n
) =
∅
, and {W
n
} be W-
mapping defined by (2.3). Let {l
n
} and {a
n
} be two sequences in (0, 1). Assume
that
(A1)

∞
n
=

0
λ
n
=
∞
;
(A2) 0 < lim inf
n®∞
a
n
≤ lim sup
n®∞
a
n
<1.
Then, the sequence {x
n
} and {y
n
} generated by (1.2) converge weakly to
x
∗
∈∩
∞
n
=1
F(T
n
)
.

Proof. From (A1), we have
0 <λ
n
≤
η
k
2
−
ε
, ∀n ≥ n
0
for some integer n
0
≥ 0. Repeat-
ing the same argument as in the proof of Theorem 3.2, we know that {x
n
} is bounded,
and so are the sequences {y
n
} and {F(x
n
)}. Assuming
p ∈∩
∞
n
=1
F(T
n
)
, we have

|
|x
n+1
− p ||
2
= || (1 − α
n
)(y
n
− p)+α
n
(W
n
y
n
− p) ||
2
=(1− α
n
)|| y
n
− p ||
2
+ α
n
|| W
n
y
n
− p ||

2
− (1 − α
n
)α
n
|| y
n
− W
n
y
n
||
2
≤||y
n
− p ||
2
− (1 − α
n
)α
n
|| y
n
− W
n
y
n
||
2
= || x

n
− p − λ
n
F(x
n
) ||
2
− (1 − α
n
)α
n
|| y
n
− W
n
y
n
||
2
≤ [|| x
n
− p|| + λ
n
|| F(x
n
) ||]
2
− (1 − α
n
)α

n
|| y
n
− W
n
y
n
||
2
= || x
n
− p ||
2
+ λ
2
n
|| F(x
n
) ||
2
+2λ
n
|| x
n
− p || || F(x
n
) || − (1 − α
n
)α
n

|| y
n
− W
n
y
n
||
2
≤||x
n
− p ||
2
+ M
4
(λ
2
n
+ λ
n
),
(3:17a)
where M
4
= sup{||F(x
n
)||
2
,2||x
n
- p|| ||F(x

n
)||, n ≥ 0}. Since

∞
n
=
0
λ
n
=
∞
, we have

∞
n
=
0
λ
2
n
<
∞
. Therefore ,

∞
n
=
0
M
4

(λ
2
n
+ λ
n
) <
∞
. Utilizing Lemma 2.4, we deduce that
lim
n®∞
||x
n
- p|| exists. Further-more, from(3.17), we have
(1 − α
n
)α
n
||y
n
− W
n
y
n
||
2
≤||x
n
− p ||
2
−||x

n+1
− p ||
2
+ M
4
(λ
2
n
+ λ
n
)
.
(3:18)
Since l
n
® 0,
λ
2
n
→
0
and (A2), it follows from (3.18) that
|
|y
n
− W
n
y
n
|| → 0

(
n →∞
)
.
Utilizing Lemma 2.6, we have
||y
n
− Wy
n
|| ≤ ||y
n
− W
n
y
n
|| + || W
n
y
n
− Wy
n
|| → 0
(
n →∞
).
Now, we show that ω
w
(y
n
) ⊂ F(T). Indeed, let x* Î ω

w
(y
n
). Then, there exists a sub-
sequence
{y
n
i
}
of {y
n
}suchthat
y
n
i
 x
∗
.Since||y
n
- Wy
n
|| ® 0, by Lemma 2.1, we
have
x
∗
∈ F(W)=∩
∞
n
=1
F(T

n
)
.
Wang Fixed Point Theory and Applications 2011, 2011:3
/>Page 12 of 14
Next, we show that ω
w
(y
n
) is a singleton. Indeed, let
{y
m
j
}
be another subsequence of
{y
n
}suchthat
y
m
j

˜
x
.Then,
˜
x ∈∩
∞
n
=1

F(T
n
)
.If
x
∗

=
˜
x
,then,byOpial’spropertyofH,
we have
lim
n→∞
||y
n
− x
∗
|| = lim
i→∞
||y
n
i
− x
∗
||
< lim
i→∞
||y
n

i
−
˜
x||
= lim
j→∞
||y
m
j
−
˜
x||
< lim
j→∞
||y
m
j
− x
∗
|
|
= lim
n
→
∞
||y
n
− x
∗
||.

This is a contradiction. Therefore, ω
w
(y
n
) is a singleton. Consequently, {y
n
} converges
weakly to
x
∗
∈∩
∞
n
=1
F(T
n
)
. From (1.2), we have that {x
n
} converges weakly to
x
∗
∈∩
∞
n
=1
F(T
n
)
. This completes the proof.

Remark 3.8. It is worth pointing out that the conditions (C1) and (C2) in [5, Theorem
3.2] are replaced by the one (A1) in Theorem 3.7. It is also worth pointing out that condi-
tion (A2) is strictly weaker than the condition (C5). The advantages of there results in this
study are that weaker and fewer restrictions are imposed on parameters a
n
, l
n
and h.
Acknowledgements
This study was supported by the Natural Science Foundation of Yancheng Teachers University under Grant
(10YCKL022).
Competing interests
The authors declare that they have no competing interests.
Received: 24 January 2011 Accepted: 20 June 2011 Published: 20 June 2011
References
1. Stampacchia, G: Formes bilineaires coercitives sur les ensembles convexes. C.R Acad Sci Paris. 258, 4413–4416 (1964)
2. Yamada, I: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed
point sets of nonexpansive mappings. In: Butnariu D, Censor Y, Reich S (eds.) Stud Comput Math, vol. 8, pp. 473–504.
Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), North-Holland,
Amsterdam (2001)
3. Lions, PL: Approximation de points fixes de contractions. C.R Acad Sci Paris. 284, 1357–1359 (1977)
4. Xu, HK, Kim, TH: Convergence of hybrid steepest-descent methods for variational inequalities. J Optim Theory Appl.
119, 185–201 (2003)
5. Yao, YH, Noor, MN, Liou, YC: A new hybrid iterative algorithm for variational inequalities. Appl Math Comput. 216,
822–829 (2010). doi:10.1016/j.amc.2010.01.087
6. Opial, Z: Weak convergence of successive approximations for nonexpansive mappings. Bull Am Math Soc. 73, 591–597
(1967). doi:10.1090/S0002-9904-1967-11761-0
7. Takahashi, W: Introduction to Nonlinear and Convex Analysis. Yokohama Publisher, Yokohama (2009)
8. Geobel, K, Kirk, WA: Topics in Metric Fixed point Theory. In Cambridge Stud Adv Math, vol. 28, pp. 473–504.Cambridge
University Press, Cambridge (1990)

9. Suzuki, T: Strong convergence of Krasnoselskii and Mann’s type sequences for one parameter nonexpansive semigroups
without Bochner integral. J Math Anal Appl. 35, 227–239 (2005)
10. Liu, LS: Iterative processes with errors for nonlinear strongly accretive mappings in Banach spaces. J Math Anal Appl.
194, 114–125 (1995). doi:10.1006/jmaa.1995.1289
11. Xu, H-K: Iterative algorithms for nonlinear operators. J Lond Math Soc. 66(1), 240–256 (2002). doi:10.1112/
S0024610702003332
12. Tan, KK, Xu, HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iterative process. J Math Anal
Appl. 178, 301–308 (1993). doi:10.1006/jmaa.1993.1309
13. Wang, S, Hu, CS: Two New Iterative Methods for a Countable Family of Nonexpansive Mappings in Hilbert Spaces.
Fixed Point Theory Appl 2010, 12 (2010). (Article ID 852030)
14. Takahashi, W, Shimoji, K: Convergence theorems for nonexpansive mappings and feasibility problems. Math Comput
Model. 32, 1463–1471 (2000). doi:10.1016/S0895-7177(00)00218-1
15. Shimoji, K, Takahashi, W: Strong convergence to common fixed points of infinite nonexpansive mappings and
applications. Taiwan J Math. 5, 387–404 (2001)
Wang Fixed Point Theory and Applications 2011, 2011:3
/>Page 13 of 14
16. Chang, S-S: A new method for solving equilibrium problem and variational inequality problem with application to
optimization. Nonlinear Anal. 70, 3307–3319 (2009). doi:10.1016/j.na.2008.04.035
17. Yao, YH, Liou, YC, Yao, JC: Convergence theorem for equilibrium problems and fixed point problems of infinite family
of nonexpansive mappings. Fixed Point Theory Appl 2007, 12 (2007). (Article ID 64363)
doi:10.1186/1687-1812-2011-3
Cite this article as: Wang: Convergence and weaker control conditions for hybrid iterative algorithms. Fixed Point
Theory and Applications 2011 2011:3.
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