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Fixed point theorems for mappings with condition (B)
Fixed Point Theory and Applications 2011, 2011:92 doi:10.1186/1687-1812-2011-92
Lai-Jiu Lin ()
Chih Sheng Chuang ()
Zenn Tsun Yu ()
ISSN 1687-1812
Article type Research
Submission date 19 July 2011
Acceptance date 2 December 2011
Publication date 2 December 2011
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Fixed point theorems for mappings with
condition (B)
Lai-Jiu Lin
∗1
, Chih-Sheng Chuang
1
and Zenn-Tsun Yu
2
1
Department of Mathematics, National Changhua University of Education,

Changhua 50058, Taiwan
2
Department of Electronic Engineering, Nan Kai University of Technology,
Nantour 54243, Taiwan
∗
Corresponding author:
Email address:
C-SC:
Z-TY:
Abstract
In this article, a new type of mappings that satisfies condition (B) is in-
troduced. We study Pazy’s type fixed point theorems, demiclosed principles,
and ergodic theorem for mappings with condition (B). Next, we consider the
weak convergence theorems for equilibrium problems and the fixed points of
mappings with condition (B).
Keywords: fixed point; equilibrium problem; Banach limit; generalized hy-
brid mapping; projection.
1
1 Introduction
Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → H
be a mapping, and let F(T ) denote the set of fixed points of T . A mapping
T : C → H is said to be nonexpansive if ||T x − T y|| ≤ ||x − y|| for all x, y ∈ C.
A mapping T : C → H is said to be quasi-nonexpansive mapping if F (T ) = ∅ and
||T x − y|| ≤ ||x − y|| for all x ∈ C and y ∈ F (T ).
In 2008, Kohsaka and Takahashi [1] introduced nonspreading mapping, and
obtained a fixed point theorem for a single nonspreading mapping, and a common
fixed point theorem for a commutative family of nonspreading mappings in Banach
spaces. A mapping T : C → C is called nonspreading [1] if
2||T x − T y||
2

≤ ||T x − y||
2
+ ||T y − x||
2
for all x, y ∈ C. Indeed, T : C → C is a nonspreading mapping if and only if
||T x − T y||
2
≤ ||x − y||
2
+ 2x − Tx, y − T y
for all x, y ∈ C [2].
Recently, Takahashi and Yao [3] introduced two nonlinear mappings in Hilbert
spaces. A mapping T : C → C is called a T J-1 mapping [3] if
2||T x − T y||
2
≤ ||x − y||
2
+ T x − y
2
for all x, y ∈ C. A mapping T : C → C is called a T J-2 [3] mapping if
3||T x − T y||
2
≤ 2||T x − y||
2
+ T y − x
2
for all x, y ∈ C.
In 2010, Takahashi [4] introduced the hybrid mappings. A mapping T : C → C
is hybrid [4] if
||T x − T y||

2
≤ ||x − y||
2
+ x − Tx, y − T y
for each x, y ∈ C. Indeed, T : C → C is a hybrid mapping if and only if
3||T x − T y||
2
≤ ||x − y||
2
+ ||T x − y||
2
+ ||T y − x||
2
2
for all x, y ∈ C [4].
In 2010, Aoyoma et al. [5] introduced λ-hybrid mappings in a Hilbert space.
Note that the class of λ-hybrid mappings contain the classes of nonexpansive map-
pings, nonspreading mappings, and hybrid mappings. Let λ be a real number. A
mapping T : C → C is called λ-hybrid [5] if
||T x − T y||
2
≤ ||x − y||
2
+ 2λx − Tx, y − T y
for all x, y ∈ C.
In 2010, Kocourek et al. [6] introduced (α, β)-generalized hybrid mappings, and
studied fixed point theorems and weak convergence theorems for such nonlinear
mappings in Hilbert spaces. Let α, β ∈ R. A mapping T : C → H is (α, β)-
generalized hybrid [6] if
α||T x − T y||

2
+ (1 − α)||T y − x||
2
≤ β||T x − y||
2
+ (1 − β)||x − y||
2
for all x, y ∈ C.
In 2011, Aoyama and Kohsaka [7] introduced α-nonexpansive mapping on Ba-
nach spaces. Let C be a nonempty closed convex subset of a Banach space E, and
let α be a real number such that α < 1. A mapping T : C → E is said to be
α-nonexpansive if
||T x − T y||
2
≤ α||T x − y||
2
+ α||T y − x||
2
+ (1 − 2α)||x − y||
2
for all x, y ∈ C.
Furthermore, we observed that Suzuki [8] introduced a new class of nonlinear
mappings which satisfy condition (C) in Banach spaces. Let C be a nonempty
subset of a Banach space E. Then, T : C → E is said to satisfy condition (C) if
for all x, y ∈ C,
1
2
||x − T x|| ≤ ||x − y|| ⇒ ||T x − T y|| ≤ ||x − y||.
In fact, every nonexpansive mapping satisfies condition (C), but the converse may
be false [8, Example 1]. Besides, if T : C → E satisfies condition (C) and F (T ) = ∅ ,

3
then T is a quasi-nonexpansive mapping. However, the converse may be false [8,
Example 2].
Motivated by the above studies, we introduced Takahashi’s (
1
2
,
1
2
)-generalized
hybrid mappings with Suzuki’s sense on Hilbert spaces.
Definition 1.1. Let C be a nonempty closed convex subset of a real Hilbert space
H, and let T : C → H be a mapping. Then, we say T satisfies condition (B) if for
all x, y ∈ C,
1
2
||x − T x|| ≤ ||x − y|| ⇒ ||T x − T y||
2
+ ||x − T y||
2
≤ ||T x − y||
2
+ ||x − y||
2
.
Remark 1.1.
(i) In fact, if T is the identity mapping, then T satisfies condition (B).
(ii) Every (
1
2

,
1
2
)-generalized hybrid mapping satisfies condition (B). But the
converse may be false.
(iii) If T : C → C satisfies condition (B) and F (T ) = ∅, then T is a quasi-
nonexpansive mapping, and this implies that F (T ) is a closed convex subset
of C [9].
Remark
1.2
. Let
H
=
R
, let
C
be nonempty closed convex subset of
H
, and let
T : C → H be a function. In fact, we have
1
2
||x − T x|| ≤ ||x − y||
⇔ (T x)
2
+ x
2
− 2xT x ≤ 4x
2
+ 4y

2
− 8xy
⇔ (T x)
2
− 2xT x ≤ 3x
2
+ 4y
2
− 8xy
⇔ T x(T x − 2x) ≤ (3x − 2y)(x − 2y),
and
|T x − T y|
2
+ |x − Ty|
2
≤ |T x − y|
2
+ |x − y|
2
⇔ (T x)
2
+(T y)
2
−2T xT y+x
2
+(T y)
2
−2xT y ≤ (T x)
2
+y

2
−2yT x+x
2
+y
2
−2xy
4
⇔ 2(T y)
2
− 2T y(T x + x) ≤ 2y
2
− 2y(T x + x)
⇔ 2(T y)
2
− 2y
2
≤ 2(T y − y)(T x + x)
⇔ (T y − y)(T y + y) ≤ (T y − y)(T x + x)
⇔ (T y − y)[(T y + y) − (T x + x)] ≤ 0.
Example 1.1. Let H = C = R, and let T : C → H be defined by T x := −x for
each x ∈ C. Hence, we have the following conditions:
(1) T is (
1
2
,
1
2
)-generalized hybrid mapping, and T satisfies condition (B).
(2) T is not a nonspreading mapping. Indeed, if x = 1 and y = −1, then
2||T x − T y||

2
= 8 > 0 = ||T x − y||
2
+ ||T y − x||
2
.
(3) T is not a T J-1 mapping. Indeed, if x = 1 and y = −1, then
2||T x − T y||
2
= 8 > 4 = 4 + 0 = ||x − y||
2
+ T x − y
2
.
(4) T is not a T J-2 mapping. Indeed, if x = 1 and y = −1, then
3||T x − T y||
2
= 12 > 0 = 2||T x − y||
2
+ T y − x
2
.
(5) T is not a hybrid mapping. Indeed, if x = 1 and y = −1, then
3||T x − T y||
2
= 12 > 4 = ||x − y||
2
+ ||T x − y||
2
+ ||T y − x||

2
.
(6) Now, we want to show that if α = 0, then T is not a α-nonexpansive mapping.
For α > 0, let x = 1 and y = −1,
||T x − Ty||
2
= 4 > 4 − 8α = α||T x − y||
2
+ α||T y − x||
2
+ (1 − 2α)||x − y||
2
.
For α < 0, let x = y = 1,
||T x − T y||
2
= 0 > 8α = α||T x − y||
2
+ α||T y − x||
2
+ (1 − 2α)||x − y||
2
.
(7) Similar to (6), if α+β = 1, then T is not a (α, β)-generalized hybrid mapping.
5
Example 1.2. Let H = R, C = [−1, 1], and let T : C → C be defined by
T (x) :=




x if x ∈ [−1, 0],
−x if x ∈ (0, 1],
for each x ∈ C. First, we consider the following conditions:
(a) For x ∈ [−1, 0] and
1
2
||x − T x|| ≤ ||x − y||, we know that
(a)
1
if y ∈ [−1, 0], then T y = y and (T y − y)[(T y + y) − (T x + x)] = 0;
(a)
2
if y ∈ (0, 1], then T y = −y and (T y −y)[(T y +y)−(T x +x)] = 4xy ≤ 0.
(b) For x ∈ (0, 1] and
1
2
||x − T x|| ≤ ||x − y||, we know that
(b)
1
if y ≥ x, then x ≤ y − x, T x = −x, and T y = −y. So, (T y − y)[(T y +
y) − (T x + x)] = 0;
(b)
2
if y < x, then x ≤ x − y and this implies that y ≤ 0. So, (T y − y)[(T y +
y) − (T x + x)] = 0.
By these conditions and Remark 1.2, we know that T satisfies condition (B). In fact,
T is (
1
2
,

1
2
)-generalized hybrid mapping. Furthermore, we know that the following
conditions:
(1) T is a nonspreading mapping. Indeed, we know that the following conditions
hold.
(1)
1
If x > 0 and y > 0, then
2||T x − T y||
2
= 2||x − y||
2
≤ 2||x + y||
2
= ||T x − y||
2
+ ||T y − x||
2
;
(1)
2
If x ≤ 0 and y ≤ 0, then
2||T x − T y||
2
= 2||x − y||
2
= ||T x − y||
2
+ ||T y − x||

2
;
(1)
3
If x > 0 and y ≤ 0, then ||T x − T y||
2
= ||T x − y||
2
= ||x + y||
2
, and
||T y − x||
2
= ||x − y||
2
. Hence,
||T x − y||
2
+ ||T y − x||
2
− 2||T x − T y||
2
= −4xy ≥ 0.
6
(2) Similar to the above, we know that T is a T J-1 mapping, a T J-2 map-
ping, a hybrid mapping, (α, β)-generalized hybrid mapping, and T is a α-
nonexpansive mapping.
On the other hand, the following iteration process is known as Mann’s type iteration
process [10] which is defined as
x

n+1
= α
n
x
n
+ (1 − α
n
)T x
n
, n ∈ N,
where the initial guess x
0
is taken in C arbitrarily and {α
n
} is a sequence in [0, 1].
In 1974, Ishikawa [11] gave an iteration process which is defined recursively by









x
1
∈ C chosen arbitrary,
y
n

:= (1 − β
n
)x
n
+ β
n
T x
n
,
x
n+1
:= (1 − α
n
)x
n
+ α
n
T y
n
,
where {α
n
} and {β
n
} are sequences in [0, 1].
In 1995, Liu [12] introduced the following modification of the iteration method
and he called Ishikawa iteration method with errors: for a normed space E, and
T : E → E a given mapping, the Ishikawa iteration method with errors is the
following sequence










x
1
∈ E chosen arbitrary,
y
n
:= (1 − β
n
)x
n
+ β
n
T x
n
+ u
n
,
x
n+1
:= (1 − α
n
)x
n

+ α
n
T y
n
+ v
n
,
where {α
n
} and {β
n
} are sequences in [0, 1], and {u
n
} and {v
n
} are sequences in
E with

∞
n=1
||u
n
|| < ∞ and

∞
n=1
||v
n
|| < ∞.
In 1998, Xu [13] introduced an Ishikawa iteration method with errors which ap-

pears to be more satisfactory than the one introduced by Liu [12]. For a nonempty
convex subset C of E and T : C → C a given mapping, the Ishikawa iteration
method with errors is generated by









x
1
∈ C chosen arbitrary,
y
n
:= a
n
x
n
+ b
n
T x
n
+ c
n
u
n
,

x
n+1
:= a

n
x
n
+ b

n
T y
n
+ c

n
v
n
,
7
where {a
n
}, {b
n
}, {c
n
}, {a

n
}, {b


n
}, {c

n
} are sequences in [0, 1] with a
n
+b
n
+c
n
= 1
and a

n
+ b

n
+ c

n
= 1, and {u
n
} and {v
n
} are bounded sequences in C.
Motivated by the above studies, we consider an Ishikawa iteration method with
errors for mapping with condition (B).
We also consider the following iteration for mappings with condition (B). Let C
be a nonempty closed convex subset of a real Hilbert space H. Let G : C × C → R
be a function. Let T : C → C be a mapping. Let {a

n
}, {b
n
}, and {θ
n
} be sequences
in [0, 1] with a
n
+ b
n
+ θ
n
= 1. Let {ω
n
} be a bounded sequence in C. Let {r
n
} be
a sequence of positive real numbers. Let {x
n
} be defined by u
1
∈ H



x
n
∈ C such that G(x
n
, y) +

1
r
n
y − x
n
, x
n
− u
n
 ≥ 0 ∀y ∈ C;
u
n+1
:= a
n
x
n
+ b
n
T x
n
+ θ
n
ω
n
.
Furthermore, we observed that Phuengrattana [14] studied approximating fixed
points of for a nonlinear mapping T with condition (C) by the Ishikawa itera-
tion metho d on uniform convex Banach space with Opal property. Here, we also
consider the Ishikawa iteration method for a mapping T with condition (C) and
improve some conditions of Phuengrattana’s result.

In this article, a new type of mappings that satisfies condition (B) is introduced.
We study Pazy’s type fixed point theorems, demiclosed principles, and ergodic
theorem for mappings with condition (B). Next, we consider the weak convergence
theorems for equilibrium problems and the fixed points of mappings with condition
(B).
2 Preliminaries
Throughout this article, let N be the set of positive integers and let R be the set of
real numbers. Let H be a (real) Hilbert space with inner product ·, · and norm
|| · ||, respectively. We denote the strongly convergence and the weak convergence
of {x
n
} to x ∈ H by x
n
→ x and x
n
 x, respectively. From [15], for each x, y ∈ H
and λ ∈ [0, 1], we have
||λx + (1 − λ)y||
2
= λ||x||
2
+ (1 − λ)||y||
2
− λ(1 − λ)||x − y||
2
.
8
Hence, we also have
2x − y, u − v  = ||x − v||
2

+ ||y − u||
2
− ||x − u||
2
− ||y − v||
2
for all x, y, u, v ∈ H. Furthermore, we know that
||αx+βy +γz||
2
= α||x||
2
+β||y||
2
+γ||z||
2
−αβ||x−y||
2
−αγ||x−z||
2
−βγ||y −z||
2
for each x, y, z ∈ H and α, β, γ ∈ [0, 1] with α + β + γ = 1 [16].
Let 
∞
be the Banach space of bounded sequences with the supremum norm.
Let µ be an element of (
∞
)
∗
(the dual space of 

∞
). Then, we denote by µ(f)
the value of µ at f = (x
1
, x
2
, x
3
, . . .) ∈ 
∞
. Sometimes, we denote by µ
n
x
n
the
value µ(f ). A linear functional µ on 
∞
is called a mean if µ(e) = ||µ|| = 1, where
e = (1, 1, 1, . . .). For x = (x
1
, x
2
, x
3
, . . .), A Banach limit on 
∞
is an invariant
mean, that is, µ
n
x

n
= µ
n
x
n+1
for any n ∈ N. If µ is a Banach limit on 
∞
, then
for f = (x
1
, x
2
, x
3
, . . .) ∈ 
∞
,
lim inf
n→∞
x
n
≤ µ
n
x
n
≤ lim sup
n→∞
x
n
.

In particular, if f = (x
1
, x
2
, x
3
, . . .) ∈ 
∞
and x
n
→ a ∈ R, then we have µ(f) =
µ
n
x
n
= a. For details, we can refer [17].
Lemma 2.1. [17] Let C be a nonempty closed convex subset of a Hilbert space H,
{x
n
} be a bounded sequence in H, and µ be a Banach limit. Let g : C → R be
defined by g(z) := µ
n
||x
n
− z||
2
for all z ∈ C. Then there exists a unique z
0
∈ C
such that g(z

0
) = min
z∈C
g(z).
Lemma 2.2. [17] Let C be a nonempty closed convex subset of a Hilbert space H.
Let P
C
be the metric projection from H onto C. Then for each x ∈ H, we have
x − P
C
x, P
C
x − y ≥ 0 for all y ∈ C.
Lemma 2.3. [17] Let D be a nonempty closed convex subset of a real Hilbert space
H. Let P
D
be the metric projection from H onto D, and let {x
n
}
n∈N
be a sequence
in H. If x
n
 x
0
and P
D
x
n
→ y

0
, then P
D
x
0
= y
0
.
9
Lemma 2.4. [18] Let D be a nonempty closed convex subset of a real Hilbert space
H. Let P
D
be the metric projection from H onto D. Let {x
n
}
n∈N
be a sequence in
H with ||x
n+1
− u||
2
≤ (1 + λ
n
)||x
n
− u||
2
+ δ
n
for all u ∈ D and n ∈ N, where

{λ
n
} and {δ
n
} are sequences of nonnegative real numbers such that
∞

n=1
λ
n
< ∞
and
∞

n=1
δ
n
< ∞. Then {P
D
x
n
} converges strongly to an element of D.
Lemma 2.5. [19] Let {s
n
} and {t
n
} be two nonnegative sequences satisfying s
n+1
≤
s

n
+ t
n
for each n ∈ N. If
∞

n=1
t
n
< ∞, then lim
n→∞
s
n
exists.
The equilibrium problem is to find z ∈ C such that
G(z, y) ≥ 0 foreach y ∈ C.(2.1)
The solution set of equilibrium problem (2.1) is denoted by (EP ). For solving the
equilibrium problem, let us assume that the bifunction G : C × C → R satisfies
the following conditions:
(A1) G(x, x) = 0 for each x ∈ C;
(A2) G is monotone, i.e., G(x, y) + G(y, x) ≤ 0 for any x, y ∈ C;
(A3) for each x, y, z ∈ C, lim
t↓0
G(tz + (1 − t)x, y) ≤ G(x, y);
(A4) for each x ∈ C, the scalar function y → G(x, y) is convex and lower semicon-
tinuous.
Lemma 2.6. [20]Let C be a nonempty closed convex subset of a real Hilbert space
H. Let G : C × C → R be a bifunction which satisfies conditions (A1)–(A4). Let
r > 0 and x ∈ C. Then there exists z ∈ C such that
G(z, y) +

1
r
y − z, z − x ≥ 0 for all y ∈ C.
Furthermore, if
T
r
(x) := {z ∈ C : G(z, y) +
1
r
y − z, z − x ≥ 0 for all y ∈ C},
then we have:
10
(i) T
r
is single-valued;
(ii) T
r
is firmly nonexpansive, that is, ||T
r
x − T
r
y||
2
≤ T
r
x − T
r
y, x − y for each
x, y ∈ C;
(iii) (EP ) is a closed convex subset of C;

(iv) (EP ) = F (T
r
).
3 Fixed point theorems
Proposition 3.1. Let C be a nonempty closed convex subset of a real Hilbert space
H. Let T : C → C be a mapping with condition (B). Then for each x, y ∈ C, we
have:
(i) ||T x − T
2
x||
2
+ ||x − T
2
x||
2
≤ ||x − T x ||
2
;
(ii) ||T x − T
2
x|| ≤ ||x − T x|| and ||x − T
2
x|| ≤ ||x − T x||;
(iii) either
1
2
||x − T x|| ≤ ||x − y|| or
1
2
||T x − T

2
x|| ≤ ||T x − y|| holds;
(iv) either
||T x − T y||
2
+ ||x − Ty||
2
≤ ||T x − y||
2
+ ||x − y||
2
or
||T
2
x − T y||
2
+ ||T x − T y||
2
≤ ||T
2
x − y||
2
+ ||T x − y||
2
holds;
(v) lim
n→∞
||T
n
x − T

n+2
x|| = 0.
Proof Since
1
2
||x − T x|| ≤ ||x − T x||, it is easy to see (i) and (ii) are satisfied. (iii)
Suppose that
1
2
||x − T x|| > ||x − y|| and
1
2
||T x − T
2
x|| > ||T x − y||
holds. So,
||x − T x|| ≤ ||x − y|| + ||y − T x||
11
<
1
2
||x − T x|| +
1
2
||T x − T
2
x||
≤
1
2

||x − T x|| +
1
2
||x − T x|| = ||x − T x||.
This is a contradiction. Therefore, we obtain the desired result. Next, it is easy to
get (iv) by (iii).
(v): By (i), we know that
||T
n+1
x − T
n+2
x||
2
+ ||T
n
x − T
n+2
x||
2
≤ ||T
n
x − T
n+1
x||
2
.
Then {||T
n
x − T
n+1

x||} is a decreasing sequence, and lim
n→∞
||T
n
x − T
n+1
x|| exists.
Furthermore, we have:
lim
n→∞
||T
n
x − T
n+2
x||
2
≤ lim
n→∞
||T
n
x − T
n+1
x||
2
− lim
n→∞
||T
n+1
x − T
n+2

x||
2
= 0.
So, lim
n→∞
||T
n
x − T
n+2
x|| = 0. 
Proposition 3.2. Let C be a nonempty closed convex subset of a real Hilbert space
H. Let T : C → C be a mapping with condition (B). Then for each x, y ∈ C,
T x − T y, y − T y ≤ x − y, T y − y + ||T
2
x − x|| · ||T y − y||.
Proof By Proposition 3.1(iv), for each x, y ∈ C, either
||T x − T y||
2
+ ||x − Ty||
2
≤ ||T x − y||
2
+ ||x − y||
2
or
||T
2
x − T y||
2
+ ||T x − T y||

2
≤ ||T
2
x − y||
2
+ ||T x − y||
2
holds. In the first case, we have
||T x − T y||
2
+ ||x − Ty||
2
≤ ||T x − y||
2
+ ||x − y||
2
⇒ ||T x − T y||
2
+ ||x − y||
2
+ 2x − y, y − T y + ||T y − y||
2
≤ ||T x − T y||
2
+
2T x − T y, T y − y + ||T y − y ||
2
+ ||x − y||
2
⇒ x − y , y − Ty ≤ Tx − T y, T y − y

12
⇒ T x − T y, y − T y ≤ x − y, T y − y.
In the second case, we have
||T
2
x − T y||
2
+ ||T x − T y||
2
≤ ||T x − y||
2
+ ||T
2
x − y||
2
⇒ ||T
2
x − y||
2
+ 2T
2
x − y, y − T y + ||y − T y||
2
+ ||T x − T y||
2
≤ ||T x − T y||
2
+
2T x − T y, T y − y + ||y − T y ||
2

+ ||T
2
x − y||
2
⇒ T x − T y, y − T y ≤ T
2
x − y, T y − y
≤ x − y, T y − y + ||T
2
x − x|| · ||T y − y||.
Therefore, the proof is completed. 
Remark 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H.
Let T : C → C be a mapping with condition (B). Then for each x, y ∈ C, we have:
(a) ||T x − T y||
2
+ ||x − Ty||
2
≤ ||x − y||
2
+ ||T x − y||
2
+ ||T
2
x − x|| · ||T y − y||.
(b) T x − T y, y − T y ≤ x − y, T y − y + ||T x − x|| · ||T y − y||.
Proof By Proposition 3.2, it is easy to prove Remark 3.1. 
The following theorem shows that demiclosed principle is true for mappings
with condition (B).
Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space
H, and let T : C → C be a mapping with condition (B). Let {x

n
} be a sequence
in C with x
n
 x and lim
n→∞
||x
n
− T x
n
|| = 0. Then T x = x.
Proof By Remark 3.1, we get:
T x
n
− T x, x − T x ≤ x
n
− x, T x − x + ||x
n
− T x
n
|| · ||x − T x||
for each n ∈ N. By assumptions, x − T x, x − T x ≤ 0. So, T x = x. 
13
Theorem 3.2. Let C be a nonempty closed convex subset of a real Hilbert space
H, and let T : C → C be a mapping with condition (B). Then {T
n
x} is a bounded
sequence for some x ∈ C if and only if F (T ) = ∅.
Proof For each n ∈ N, let x
n

:= T
n
x. Clearly, {x
n
} is a bounded sequence. By
Lemma 2.1, there is a unique z ∈ C such that µ
n
||x
n
− z||
2
= min
y∈C
µ
n
||x
n
− y||
2
. By
Proposition 3.2, for each n ∈ N,
x
n+1
− T z, z − T z ≤ x
n
− z, T z − z + ||x
n
− x
n+2
|| · ||z − T z||

⇒
1
2
||x
n+1
− T z||
2
+
1
2
||T z − z||
2
−
1
2
||x
n+1
− z||
2
≤
1
2
||x
n
− z||
2
+
1
2
||z − T z||

2
−
1
2
||x
n
− T z||
2
+ ||x
n
− x
n+2
|| · ||z − T z||
⇒ µ
n
||x
n
− T z||
2
≤ µ
n
||x
n
− z||
2
+ µ
n
||x
n
− x

n+2
|| · ||z − T z||.
By Proposition 3.1(v), µ
n
||x
n
− T z||
2
≤ µ
n
||x
n
− z||
2
. This implies that T z = z and
F (T ) = ∅. Conversely, it is easy to see. 
Corollary 3.1. Let C be a nonempty bounded closed convex subset of a real Hilbert
space H, and let T : C → C be a mapping with condition (B). Then F (T ) = ∅.
The following theorem shows that Ballion’s type Ergodic’s theorem is also true
for the mapping with condition (B).
Theorem 3.3. Let C be a nonempty closed convex subset of a real Hilbert space H,
and let T : C → C be a mapping with condition (B). Then the following conditions
are equivalent:
(i) for each x ∈ C, S
n
x =
1
n
n−1


k=0
T
k
x converges weakly to an element of C;
(ii) F (T ) = ∅.
In fact, if F (T ) = ∅, then for each x ∈ C, we know that S
n
x  υ, where v =
lim
n→∞
P
F (T )
T
n
x and P
F (T )
is the metric projection from H onto F (T ).
Proof (i)⇒ (ii): Take any x ∈ C and let x be fixed. Then there exists υ ∈ C such
that S
n
x  υ. By Proposition 3.2, for each k ∈ N, we have:
14
T T
k
x − T υ, υ − T υ ≤ T
k
x − υ, T υ − υ + ||T
2
T
k

x − T
k
x|| · ||T υ − υ||
⇒ T
k+1
x − T υ, υ − T υ ≤ T
k
x − υ, T υ − υ + ||T
k+2
x − T
k
x|| · ||T υ − υ||
⇒
n−2

k=0
T
k+1
x − T υ, υ − T υ
≤
n−2

k=0
T
k
x − T υ, T υ − υ +
n−2

k=0
||T

k+2
x − T
k
x|| · ||T υ − υ||
⇒ nS
n
x − x − (n − 1)T υ, υ − T υ ≤ (n − 1)S
n−1
x − (n − 1)T υ, T υ − υ +
n−2

k=0
||T
k+2
x − T
k
x|| · ||T υ − υ||
⇒ 
n
n−1
S
n
x −
x
n−1
− T υ, υ − T υ
≤ S
n−1
x − T υ, T υ − υ +
1

n − 1
n−2

k=0
||T
k+2
x − T
k
x|| · ||T υ − υ||.
By Proposition 3.1(v), lim
k→∞
||T
k+2
x − T
k
x|| = 0. This implies that
lim
n→∞
1
n − 1
n−2

k=0
||T
k+2
x − T
k
x|| = 0.
Since S
n

x  υ, we have:
υ − T υ, υ − T υ ≤ υ − T υ, T υ − υ.
So, T υ = υ .
(ii)⇒ (i): Take any x ∈ C and u ∈ F (T ), and let x and u be fixed. Since T satisfies
condition (B), ||T
n
x − u|| ≤ ||T
n−1
x − u|| for each n ∈ N. Hence, lim
n→∞
||T
n
x − u||
exists and this implies that {T
n
x} is a bounded sequence. By Lemma 2.4, there
exists z ∈ F (T ) such that lim
n→∞
P
F (T )
T
n
x = z. Clearly, z ∈ F (T ). Besides, we have:
||S
n
x − u|| ≤
1
n
n−1


k=0
||T
k
x − u|| ≤ ||x − u||.
So, {S
n
x} is a bounded sequence. Then there exist a subsequence {S
n
i
x} of {S
n
x}
and υ ∈ C such that S
n
i
x  υ. By the above proof, we have:

n
n−1
S
n
x −
x
n−1
− Tυ, υ − T υ ≤ S
n−1
x − T υ, T υ − υ +
1
n − 1
n−2


k=0
||T
k+2
x −
T
k
x|| · ||T υ − υ||.
15
This implies that

n
i
n
i
−1
S
n
i
x −
x
n
i
−1
− T υ, υ − T υ
≤ S
n
i
−1
x −

x
n
i
−1
− T υ, T υ − υ +
1
n
i
− 1
n
i
−2

k=0
||T
k+2
x − T
k
x|| · ||T υ − υ||.
Since S
n
i
x  υ, {T
n
x} is a bounded sequence, and lim
n→∞
1
n − 1
n−2


k=0
||T
k+2
x−T
k
x|| =
0, it is easy to see that T υ = υ. So, υ ∈ F (T ).
By Lemma 2.2, for each k ∈ N, T
k
x − P
F (T )
T
k
x, P
F (T )
T
k
x − u ≥ 0. This
implies that
T
k
x − P
F (T )
T
k
x, u − z
≤ T
k
x − P
F (T )

T
k
x, P
F (T )
T
k
x − z
≤ ||T
k
x − P
F (T )
T
k
x|| · ||P
F (T )
T
k
x − z||
≤ ||T
k
x − z|| · ||P
F (T )
T
k
x − z||
≤ ||x − z|| · ||P
F (T )
T
k
x − z||.

Adding these inequalities from k = 0 to k = n − 1 and dividing by n, we have

S
n
x −
1
n
n−1

k=0
P
F (T )
T
k
x, u − z

≤
||x − z||
n
n−1

k=0
||P
F (T )
T
k
x − z||.
Since S
n
k

x  υ and P
F (T )
T
k
x → z, we get υ − z, u − z ≤ 0. Since u is any point
of F (T ), we know that υ = z = lim
n→∞
P
F (T )
T
n
x.
Furthermore, if {S
n
i
x} is a subsequence of {S
n
x} and S
n
i
x  q, then q = υ
by following the same argument as in the ab ove pro of. Therefore, S
n
x  υ =
lim
n→∞
P T
n
x, and the proof is completed. 
4 Weak convergence theorems with errors

Theorem 4.1. Let C be a nonempty closed convex subset of a real Hilbert space
H, and let T
1
, T
2
: C → C be two mappings with condition (B) and Ω := F (T
1
) ∩
16
F (T
2
) = ∅. Let {a
n
}, {b
n
}, {c
n
}, {d
n
}, {θ
n
}, and {λ
n
} be sequences in [0, 1] with
a
n
+ c
n
+ θ
n

= b
n
+ d
n
+ λ
n
= 1, n ∈ N.
Let {u
n
} and {v
n
} be bounded sequence in C. Let {x
n
} and {y
n
} be defined by









x
1
∈ C chosen arbitrary,
y
n

:= a
n
x
n
+ c
n
T
1
x
n
+ θ
n
u
n
,
x
n+1
:= b
n
x
n
+ d
n
T
2
y
n
+ λ
n
v

n
.
Assume that:
(i) lim inf
n→∞
a
n
c
n
> 0 and lim inf
n→∞
b
n
d
n
> 0;
(ii)
∞

n=1
θ
n
< ∞ and
∞

n=1
λ
n
< ∞.
Then x

n
 z and y
n
 z, where z = lim
n→∞
P
Ω
x
n
.
Proof Take any w ∈ Ω and let w be fixed. Then for each n ∈ N, we have:
||y
n
− w||
2
= ||a
n
x
n
+ c
n
T
1
x
n
+ θ
n
u
n
− w||

2
= a
n
||x
n
− w||
2
+ c
n
||T
1
x
n
− w||
2
+ θ
n
||u
n
− w||
2
−a
n
c
n
||x
n
− T
1
x

n
||
2
− a
n
θ
n
||x
n
− w||
2
− c
n
θ
n
||T
1
x
n
− u
n
||
2
≤ a
n
||x
n
− w||
2
+ c

n
||x
n
− w||
2
+ θ
n
||u
n
− w||
2
−a
n
c
n
||x
n
− T
1
x
n
||
2
− a
n
θ
n
||x
n
− w||

2
− c
n
θ
n
||T
1
x
n
− u
n
||
2
≤ ||x
n
− w||
2
+ θ
n
||u
n
− w||
2
,
and
||x
n+1
− w||
2
= ||b

n
x
n
+ d
n
T
2
y
n
+ λ
n
v
n
− w||
2
= b
n
||x
n
− w||
2
+ d
n
||T
2
y
n
− w||
2
+ λ

n
||v
n
− w||
2
−b
n
d
n
||x
n
− T
2
y
n
||
2
− b
n
λ
n
||x
n
− v
n
||
2
− d
n
λ

n
||T
2
y
n
− v
n
||
2
17
≤ b
n
||x
n
− w||
2
+ d
n
||y
n
− w||
2
+ λ
n
||v
n
− w||
2
−b
n

d
n
||x
n
− T
2
y
n
||
2
− b
n
λ
n
||x
n
− v
n
||
2
− d
n
λ
n
||T
2
y
n
− v
n

||
2
≤ b
n
||x
n
− w||
2
+ d
n
(||x
n
− w||
2
+ θ
n
||u
n
− w||
2
) + λ
n
||v
n
− w||
2
−b
n
d
n

||x
n
− T
2
y
n
||
2
− b
n
λ
n
||x
n
− v
n
||
2
− d
n
λ
n
||T
2
y
n
− v
n
||
2

≤ ||x
n
− w||
2
+ d
n
θ
n
||u
n
− w||
2
+ λ
n
||v
n
− w||
2
−b
n
d
n
||x
n
− T
2
y
n
||
2

− b
n
λ
n
||x
n
− v
n
||
2
− d
n
λ
n
||T
2
y
n
− v
n
||
2
≤ ||x
n
− w||
2
+ d
n
θ
n

||u
n
− w||
2
+ λ
n
||v
n
− w||
2
.
By Lemma 2.5, lim
n→∞
||x
n
− w || exists. So, {x
n
} is a bounded sequence. Now, we
set lim
n→∞
||x
n
− w|| = t. Besides,
b
n
d
n
||x
n
− T

2
y
n
||
2
+ b
n
λ
n
||x
n
− v
n
||
2
+ d
n
λ
n
||T
2
y
n
− v
n
||
2
≤ ||x
n
− w||

2
+ d
n
θ
n
||u
n
− w||
2
+ λ
n
||v
n
− w||
2
− ||x
n+1
− w||
2
.
This implies that
lim
n→∞
b
n
d
n
||x
n
− T

2
y
n
||
2
= 0.
By assumption, lim
n→∞
||x
n
− T
2
y
n
|| = 0. Furthermore, we have:
||x
n+1
− w||
2
≤ b
n
||x
n
− w||
2
+ d
n
||y
n
− w||

2
+ λ
n
||v
n
− w||
2
.
This implies that
b
n
d
n
(||x
n
− w||
2
+ θ
n
||u
n
− w||
2
− ||y
n
− w||
2
)
≤ d
n

(||x
n
− w||
2
+ θ
n
||u
n
− w||
2
− ||y
n
− w||
2
)
≤ (1 − b
n
)||x
n
− w||
2
+ d
n
θ
n
||u
n
− w||
2
− d

n
||y
n
− w||
2
≤ ||x
n
− w||
2
− ||x
n+1
− w||
2
+ λ
n
||v
n
− w||
2
+ d
n
θ
n
||u
n
− w||
2
.
Hence, lim
n→∞

b
n
d
n
(||x
n
− w||
2
+ θ
n
||u
n
− w||
2
− ||y
n
− w||
2
) = 0. By assumption,
lim
n→∞
(||x
n
− w||
2
+ θ
n
||u
n
− w||

2
− ||y
n
− w||
2
) = 0.
18
Since lim
n→∞
θ
n
||u
n
− w||
2
= 0,
lim
n→∞
(||x
n
− w||
2
− ||y
n
− w||
2
) = 0.
Hence, lim
n→∞
||y

n
− w|| = lim
n→∞
||x
n
− w|| = t. Similar to the above proof, we also get
lim
n→∞
||x
n
− T
1
x
n
|| = 0.
Besides,
||y
n
− x
n
|| = ||a
n
x
n
+ c
n
T
1
x
n

+ θ
n
u
n
− x
n
||
≤ c
n
||T
1
x
n
− x
n
|| + θ
n
||x
n
− u
n
||
≤ ||T
1
x
n
− x
n
|| + θ
n

||x
n
− u
n
||.
This implies that lim
n→∞
||y
n
− x
n
|| = 0 and lim
n→∞
||y
n
− T
2
y
n
|| = 0. Since {x
n
} is a
bounded sequence, there exists a subsequence {x
n
k
} of {x
n
} such that x
n
k

 z.
By Theorem 3.1, z = T
1
z.
If x
n
j
is a subsequence of {x
n
} and x
n
j
 q, then T
1
q = q. Suppose that q = z.
Then we have:
lim inf
k→∞
||x
n
k
− z|| < lim inf
k→∞
||x
n
k
− q|| = lim
n→∞
||x
n

− q|| = lim
j→∞
||x
n
j
− q||
< lim inf
j→∞
||x
n
j
− z|| = lim
n→∞
||x
n
− z|| = lim inf
k→∞
||x
n
k
− z||.
And this leads to a contradiction. Then every weakly convergent subsequence of
x
n
has the same limit. So, x
n
 z ∈ F (T
1
). Since x
n

 z and lim
n→∞
||x
n
− y
n
|| = 0,
y
n
 z. By Theorem 3.1, z ∈ F (T
2
). Hence, z ∈ Ω.
Next, by Lemma 2.4, P
Ω
x
n
converges. Then there exists υ ∈ Ω such that
lim
n→∞
P
Ω
x
n
= υ. By Lemma 2.3, P
Ω
z = υ. Since z ∈ Ω, z = υ = lim
n→∞
P
Ω
x

n
, and the
proof is completed. 
In Theorem 4.1, if θ
n
= λ
n
= 0 for each n ∈ N, then we have the following
result.
19
Theorem 4.2. Let C be a nonempty closed convex subset of a real Hilbert space
H, and let T
1
, T
2
: C → C be two mappings with condition (B) and Ω := F (T
1
) ∩
F (T
2
) = ∅. Let {a
n
} and {b
n
} be two sequences in [0, 1]. Let {x
n
} be defined by










x
1
∈ C chosen arbitrary,
y
n
:= a
n
x
n
+ (1 − a
n
)T
1
x
n
,
x
n+1
:= b
n
x
n
+ (1 − b
n

)T
2
y
n
.
Assume that lim inf
n→∞
a
n
(1 − a
n
) > 0 and lim inf
n→∞
b
n
(1 − b
n
) > 0. Then x
n
 z and
y
n
 z, where z = lim
n→∞
P
Ω
x
n
.
Furthermore, we also have the following corollaries from Theorem 4.2.

Corollary 4.1. Let C be a nonempty closed convex subset of a real Hilbert space
H, and let T : C → C be a mapping with condition ( B) and F (T ) = ∅. Let {a
n
}
and {b
n
} be two sequences in [0, 1]. Let {x
n
} be defined by









x
1
∈ C chosen arbitrary,
y
n
:= a
n
x
n
+ (1 − a
n
)T x

n
,
x
n+1
:= b
n
x
n
+ (1 − b
n
)T y
n
.
Assume that lim inf
n→∞
a
n
(1 − a
n
) > 0 and lim inf
n→∞
b
n
(1 − b
n
) > 0. Then x
n
 z and
y
n

 z, where z = lim
n→∞
P
F (T )
x
n
.
Corollary 4.2. Let C be a nonempty closed convex subset of a real Hilbert space
H, and let T : C → C be a mapping with condition (B) and F (T ) = ∅. Let {b
n
}
be a sequence in [0, 1]. Let {x
n
} be defined by



x
1
∈ C chosen arbitrary,
x
n+1
:= b
n
x
n
+ (1 − b
n
)T x
n

.
Assume that lim inf
n→∞
b
n
(1 − b
n
) > 0. Then x
n
 z, where z = lim
n→∞
P
F (T )
x
n
.
Theorem 4.3. Let C be a nonempty closed convex subset of a real Hilbert space
H. Let G : C × C → R be a function satisfying (A1)–(A4). Let T : C → C be a
mapping with condition (B) and Ω := F (T ) ∩ (EP ) = ∅. Let {a
n
}, {b
n
}, and {θ
n
}
20
be sequences in [0, 1] with a
n
+ b
n

+ θ
n
= 1. Let {ω
n
} be a bounded sequence in C.
Let {r
n
} ⊆ [a, ∞) for some a > 0. Let {x
n
} be defined by u
1
∈ H



x
n
∈ C such that G(x
n
, y) +
1
r
n
y − x
n
, x
n
− u
n
 ≥ 0 ∀y ∈ C;

u
n+1
:= a
n
x
n
+ b
n
T x
n
+ θ
n
ω
n
.
Assume that: lim inf
n→∞
a
n
b
n
> 0, and
∞

n=1
θ
n
< ∞. Then x
n
 z, where z =

lim
n→∞
P
(EP )∩F (T )
x
n
.
Proof Take any w ∈ Ω and let w be fixed. Putting x
n
= T
r
n
u
n
for each n ∈ N.
Then we have:
||x
n+1
− w||
2
= ||T
r
n+1
u
n+1
− w||
2
≤ ||u
n+1
− w||

2
≤ ||a
n
x
n
+ b
n
T x
n
+ θ
n
ω
n
− w||
2
≤ a
n
||x
n
− w||
2
+ b
n
||T x
n
− w||
2
+ θ
n
||ω

n
− w||
2
− a
n
b
n
||x
n
− T x
n
||
2
≤ a
n
||x
n
− w||
2
+ b
n
||x
n
− w||
2
+ θ
n
||ω
n
− w||

2
− a
n
b
n
||x
n
− T x
n
||
2
≤ ||x
n
− w||
2
+ θ
n
||ω
n
− w||
2
− a
n
b
n
||x
n
− T x
n
||

2
.
By Lemma 2.5, lim
n→∞
||x
n
− w|| exists. So, {x
n
} is bounded. Furthermore, we have:
(a) lim
n→∞
a
n
b
n
||x
n
− T x
n
||
2
= 0;
(b) lim
n→∞
||x
n
− T x
n
|| = 0;
(c) u

n+1
− x
n
 = b
n
T x
n
− b
n
x
n
+ θ
n
w
n
− θ
n
x
n
 ≤ T x
n
− x
n
 + θ
n
w
n
− x
n
;

(d) lim
n→∞
||u
n+1
− x
n
|| = 0;
(e) lim
n→∞
||u
n
− w|| = lim
n→∞
||x
n
− w||.
Following the same argument as the proof of Theorem 4.2, there exists z ∈ C such
that x
n
 z and T z = z. Besides, we also have
21
||x
n+1
− w||
2
= ||T
r
n+1
u
n+1

− w||
2
= ||T
r
n+1
u
n+1
− T
r
n+1
w||
2
≤ T
r
n+1
u
n+1
− T
r
n+1
w, u
n+1
− w
≤ x
n+1
− w, u
n+1
− w
=
1

2
||x
n+1
− w||
2
+
1
2
||u
n+1
− w||
2
−
1
2
||x
n+1
− u
n+1
||
2
.
This implies that
||x
n+1
− u
n+1
||
2
≤ ||u

n+1
− w||
2
− ||x
n+1
− w||
2
.
By (e), lim
n→∞
||x
n
− u
n
|| = 0. Next, we want to show that z ∈ (EP ). Since
x
n
= T
r
n
u
n
,
G(x
n
, y) +
1
r
n
y − x

n
, x
n
− u
n
 ≥ 0 ∀y ∈ C.
By (A2),
1
r
n
y − x
n
, x
n
− u
n
 ≥ G(y, x
n
) ∀y ∈ C.
By (A4), (i), and lim
n→∞
||x
n
− u
n
|| = 0, we get
0 ≥ lim
n→∞
G(y, x
n

) ≥ G(y, z) ∀y ∈ C.
By (A2), G(z, y) ≥ 0 for all y ∈ C. So, z ∈ (EP ) ∩ F (T ) = Ω. By Lemma 2.4,
there exists υ ∈ (EP ) ∩ F (T ) such that lim
n
→∞
P
(EP )∩F (T )
x
n
= υ. By Lemma 2.3,
z = P
(EP )∩F (T )
z = υ, and the proof is completed. 
In Theorem 4.3, if θ
n
= 0 for each n ∈ N, then we have the following result.
Theorem 4.4. Let C be a nonempty closed convex subset of a real Hilbert space
H. Let G : C × C → R be a function satisfying (A1)–(A4). Let T : C → C be a
mapping with condition (B) and Ω := F (T ) ∩ (EP ) = ∅. Let {a
n
} be a sequence
in [0, 1]. Let {x
n
} be defined by u
1
∈ H



x

n
∈ C such that G(x
n
, y) +
1
r
n
y − x
n
, x
n
− u
n
 ≥ 0 ∀y ∈ C;
u
n+1
:= a
n
x
n
+ (1 − a
n
)T x
n
.
Assume that: {r
n
} ⊆ [a, ∞) for some a > 0 and lim inf
n→∞
a

n
(1 − a
n
) > 0. Then
x
n
 z, where z = lim
n→∞
P
(EP )∩F (T )
x
n
.
22
Competing interests
The authors declare that they have no competing interests, except Prof. L J.
Lin was supported by the National Science Council of Republic of China while
he worked on the publish, and C. S. Chuang was supported as postdoctor by the
National Science Council of the Republic of China while he worked on this problem.
Authors’ contributions
L-JL is responsible for problem resign, coordinator, discussion, revise the important
part, and submit. C-SC is responsible for the important results of this article,
discuss, and draft. Z-TY is responsible for giving the examples of this types of
problems, discussion. All authors read and approved the final manuscript.
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