Báo cáo toán học: " Weak convergence theorem for the three-step iterations of non-Lipschitzian nonself mappings in Banach spaces" pptx

RESEARC H Open Access
Weak convergence theorem for the three-step
iterations of non-Lipschitzian nonself mappings in
Banach spaces
Lanping Zhu, Qianglian Huang
*
and Xiaoru Chen
* Correspondence:

College of Mathematics, Yangzhou
University, Yangzhou 225002, China
Abstract
In this article, we introduce a new three-step iterative scheme for the mappings
which are asymptotically nonexpansive in the intermediate sense in Banach spaces.
Weak convergence theorem is established for this three-step iterative scheme in a
uniformly convex Banach space that satisfies Opial’s condition or whose dual space
has the Kadec-Klee property. Furthermore, we give an example of the nonself
mapping which is asymptotically nonexpansive in the intermediate sense but not
asymptotically nonexpansive. The results obtained in this article extend and improve
many recent results in this area.
AMS classification: 47H10; 47H09; 46B20.
Keywords: asymptotically nonexpansive in the intermediate sense non-self mapping,
Kadec-Klee property, Opial?’?s condition, common fixed point
1 Introduction
Fixed- point iterations process for nonexpansive and asymptotically nonexpansive map-
pings in Banach spaces have been studied extensively by various authors [1-13]. In
1991, Schu [4] considered the following modified Mann iteration process for an
asymptotically nonexpansive map T on C and a sequence {a
n
} in [0, 1]:
x

1
∈ C, x
n+1
= α
n
x
n
+
(
1 − α
n
)
T
n
x
n
, n ≥ 1
.
(1:1)
Since then, Schu’s iteration process (1.1) has been widely used to approximate fixed
points of asymptotically nonexpansive mappings in Hilbert spaces or Banach spaces
[7,8,10-13]. Noor, in 2000, introduced a three-step iterative scheme and studied the
approximate solutions of variational inclusion in Hilbert spaces [6]. Later, Xu and
Noor [7], Cho et al. [8], Suantai [9], Plubtieng et al. [12] studied t he convergence of
the three-step iterations for asymptotically nonexpansive mappings in a uniformly con-
vex Banach space which satisfies Opial’ s condition or whose norm is Fréchet
differentiable.
In most of these articles, the operator T remains a self-mapping of a nonempty
closed convex subset C of a uniformly convex Banach space X. If, however, the domain
of T, D(T), is a proper subset of X (and this is the case in several applications), and T

maps D(T)intoX, then the iterative sequence {x
n
} may fail to be well defined. One
method that has been used to overcome this is to introduce a retraction P. A subset C
Zhu et al. Fixed Point Theory and Applications 2011, 2011:106
/>© 2011 Zhu et al; licensee Springer. This is a n Open Access article distributed under the terms of t he Creative Commons Attribution
License ( which permits unrestricted use, distribution, and reproduction in any medium,
provided the original work is properly cited .
of X is said to be retract if there exists continuous mapping P : X ® C such that Px =
x for all x Î C and P is said to be a retraction. Recent results on approximation of
fixed points of nonexp ansive or asymptotically nonexpansive nonself mappings can be
found in [14-19] and the references cited therein. For example, in 2003, Chidume et al.
[16] introduced the following modified Mann iteration process and got the conver-
gence theorems for asymptotically nonexpansive nonself-mapping:
x
1
∈ C, x
n+1
= P[α
n
x
n
+
(
1 − α
n
)
T
(
PT

)
n−1
x
n
], n ≥ 1
.
(1:2)
Recently, Thianwan [18] generalized the iteration process (1.2) as follows: x
1
Î C,
x
n+1
= P[α
n
y
n
+(1− α
n
)T
1
(PT
1
)
n−1
y
n
];
y
n
= P[β

n
x
n
+
(
1 − β
n
)
T
2
(
PT
2
)
n−1
x
n
]
.
(1:3)
Obviously, if b
n
=1foralln ≥ 1, then (1.3) reduces to (1.2). Thianwan [18] proved
the weak convergence theorem of the iteration process (1.3) in uniformly convex
Banach spaces that satisfy Opial’s condition.
The concept of asymptotical ly nonexpansive in the intermediate sense nonself map-
pings was introduced by Chidume et al. [20] as an important generalization of asymp-
totically nonexpansive in the intermediate sense self-mappings.
Definition 1.1 Let C be a nonempty subset of a Banach space X. Let P : X ® Cbea
nonexpansive retraction of X onto C. A nonself mapping T : C ® X is called asymptoti-

cally nonexpansive in the intermediate sense if T is continuous and the following
inequality holds:
lim sup
n→+∞
sup
x,
y
∈C
( T(PT)
n−1
x − T(PT)
n−1
y −x − y ) ≤ 0
.
It should be noted that in [20- 22], the asymptotically nonexpansive in the intermedi-
ate sense mapping is required to be uniformly continuous. In this article, we assume
the continuity of T instead of un iform continuity. Chidume et al. [ 20] gave the w eak
convergence theorem for u niforml y continuous nonself mapping which is asymptoti-
cally nonexpansive in the intermediate sense in uniformly convex Banach space whose
dual space has the Kadec-Klee property.
Inspired and motivated by [16,18,20], we investigate the weak convergence theo rem
of three-step iteration process for co ntinuous nonself mappings which are asymptoti-
cally nonexpansive in the intermediate sense in this article. Since the asymptotically
nonexpansive in the intermediate sense mappings are non-Lipschitzian and Bruck ’s
Lemma [23] do not extend beyond Lipschitzian mappi ngs, new tec hniques are needed
for this more general case. Utilizing the technique of the modulus of convexity and a
new demiclosed principle for nonself-maps of Kazor [24], we establish the we ak con-
vergence theorem of the three-step iterative scheme in a uniformly convex Banach
space that satisfies Opial’s condition or whose dual space has the Kadec-Klee property,
which extends an d improves the recently announced ones in [4,16,18-20]. It should be

noted that our theorems are new even in the case that the space has a Fréchet differ-
entiable norm. In the end, to illustrate our theorem, we give a nonself mapping which
is asymptotically nonexpansive in the intermediate sense but not asymptotical ly
nonexpansive.
Zhu et al. Fixed Point Theory and Applications 2011, 2011:106
/>Page 2 of 13
2 Preliminaries
Let X be a Banach space and X* be its dual, then the value of x* Î X*atx Î X will be
denoted by 〈x, x*〉 and we associate the set
J(
x
)
= {x
∗
∈ X
∗
: x, x
∗
 = ||x||
2
= ||x
∗
||
2
}
.
It follows from the Hahn-Bana ch theorem that J(x) ≠ ∅ for any x Î X.Thenthe
multi-valued operator J : X ↦ X* is called the normalized duality mapping of X.Recall
that a Banach space X is said to be uniformly convex if for each ε Î [0, 2], the modu-
lus of convexity of X defined by

δ(ε)=inf{1 −
1
2
 x + y : x ≤ 1,  y ≤ 1,  x − y ≥ ε}
,
satisfies the inequality δ(ε) >0 for all ε >0. Note that every closed convex subset of a uni-
formly convex Banach space is a retract. We say that X has the Kadec-Klee property if for
every sequence {x
n
} ⊂ X, whenever x
n
⇀ x with ||x
n
|| ® ||x||, it follows that x
n
® x.We
would li ke to remark that a reflexive Banach space X with a Fréchet differentiable norm
implies that its dual X* has Kadec-Klee property, while the converse implication fails [25].
Recall that a Banach space X is said to satisfies Opial’s condition if x
n
⇀ x and x ≠ y
implies that
lim sup
n
→+∞
 x
n
− x < lim sup
n
→+∞

 x
n
− y 
.
The following lemmas are needed to prove our main results in next section.
Lemma 2.1 [5]Let the nonnegative number sequences {c
n
} and {w
n
} satisfy
c
n
+1
≤ c
n
+ w
n
, n ∈
N
If
+∞

n=1
w
n
< +
∞
, then
lim
n

→+∞
c
n
exists.
Lemma 2.2 [4] Suppose that X is a uniformly convex Banach space and for al l posi-
tive integers n,0<p≤ t
n
≤ q<1. If {x
n
} and {y
n
} are two sequences of X such that
lim sup
n
→+∞
 y
n
≤
r
,
lim sup
n
→+∞
 y
n
≤
r
and
lim
n

→+∞
 t
n
x
n
+(1− t
n
)y
n
=
r
hold for some r ≥ 0. Then
lim
n
→+∞
 x
n
− y
n
=
0
.
Lemma 2.3 [3]Let X be a uniformly convex Banach space. If ||x|| ≤ 1, ||y|| ≤ 1 and
||x - y|| ≥ ε >0, then for all l Î [0, 1],
 λx +
(
1 − λ
)
y ≤ 1 − 2λ
(

1 − λ
)
δ
(
ε
).
Lemma 2.4 [26]Let X be a Banach space and J be the normalized duali ty mapping.
Then for given x, y Î X and j(x + y) Î J(x + y), we have
 x + y
2
≤ x
2
+2y, j
(
x + y
)

.
Lemma 2.5 (De miclosed principle for nonself-map [ 24]) Let C be a nonempty closed
convex subset of a uniformly convex Banach space X and T : C ® X be a nonself map-
ping which is continuous and asymptotically nonexpansive in the interme diate sense. If
{x
n
} is a sequence in C converging weakly to x and
Zhu et al. Fixed Point Theory and Applications 2011, 2011:106
/>Page 3 of 13
lim
k→+∞
lim sup
n

→+∞
 x
n
− T(PT)
k−1
x
n
=0
,
then x Î F(T), i.e., Tx = x.
3 Main results
Let C be a nonempty closed convex subset of a uniformly convex Banach space X and
P : X ® C beanonexpansiveretractionfromX onto C.LetT
1
, T
2
, T
3
: C ® X be
three continuous nonself mappings which are asymptotically nonexpansive in the inter-
mediate sense. Suppose that
r
n
=max{0, sup
x,
y
∈C;i=1,2,3.
 T
i
(PT

i
)
n−1
x − T
i
(PT
i
)
n−1
y −x − y }
,
then r
n
≥ 0,
lim
n
→+∞
r
n
=
0
and for all x, y Î C and n Î N,
 T
i
(
PT
i
)
n−1
x − T

i
(
PT
i
)
n−1
y −x − y ≤ r
n
, i =1,2,3
.
For a given x
1
Î C, we define the sequence {x
n
} ⊂ C by
x
n+1
= P[α
(1)
n
z
n
+(1− α
(1)
n
)T
1
(PT
1
)

n−1
z
n
];
z
n
= P[α
(2)
n
y
n
+(1− α
(2)
n
)T
2
(PT
2
)
n−1
y
n
];
y
n
= P[α
(3)
n
x
n

+
(
1 − α
(3)
n
)
T
3
(
PT
3
)
n−1
x
n
]
.
(3:1)
where
{
α
(
i
)
n
}
is in 0[1] with
0 <
p
≤ α

(i)
n
≤
q
<
1
, i =1,2,3.
We also assume that the sequence {r
n
} satisfies
+∞

n
=1
r
n
< +
∞
andthesetofcommon
fixed points of
{T
i
}
3
i
=
1
is nonempty, i.e.,
F = ∩
3

i
=1
F( T
i
)={x ∈ C : T
1
x = T
2
x = T
3
x = x} = ∅
.
Lemma 3.1
lim
n
→+∞
 x
n
− f = lim
n
→+∞
 y
n
− f = lim
n
→+∞
 z
n
− f =
r

(3:2)
exists for all f Î F.
Proof. For all f Î F,
 y
n
− f = P [α
(3)
n
x
n
+(1− α
(3)
n
)T
3
(PT
3
)
n−1
x
n
] − f

≤ [α
(3)
n
x
n
+(1− α
(3)

n
)T
3
(PT
3
)
n−1
x
n
] − f 
≤ α
(3)
n
 x
n
− f  +(1 − α
(3)
n
)  T
3
(PT
3
)
n−1
x
n
− f 
= x
n
−

f
 +r
n
.
Hence
 z
n
− f = P [α
(2)
n
y
n
+(1− α
(2)
n
)T
2
(PT
2
)
n−1
y
n
] − f

≤ [α
(2)
n
y
n

+(1− α
(2)
n
)T
2
(PT
2
)
n−1
y
n
] − f 
≤ y
n
− f  +r
n
≤ x
n
−
f
 +2r
n
.
(3:3)
Zhu et al. Fixed Point Theory and Applications 2011, 2011:106
/>Page 4 of 13
Thus
 x
n+1
− f = P [α

(1)
n
z
n
+(1− α
(1)
n
)T
1
(PT
1
)
n−1
z
n
] − f

≤ [α
(1)
n
z
n
+(1− α
(1)
n
)T
1
(PT
1
)

n−1
z
n
] − f 
≤ z
n
− f  +r
n
≤ x
n
−
f
 +3r
n
.
(3:4)
Put w
n
=3r
n
, then we can obtain
+∞

n
=1
w
n
< +
∞
and

 x
n+1
−
f
≤ x
n
−
f
 +w
n
.
By Lemma 2.1, we can conclude that
lim
n
→+∞
 x
n
− f =
r
exists. Combining it with (3.4), we have
lim
n
→+∞
 z
n
− f = r
.
Hence by (3.3), we get
lim
n

→+∞
 y
n
− f = r
.
This completes the proof.
Lemma 3.2
lim
k→+∞
lim sup
n
→+∞
 x
n
− T
i
(PT
i
)
k
−1
x
n
=0, i = 1,2,3
.
Proof. By (3.2) and (3.4), we can get
r = lim
n→+∞
 [α
(1)

n
z
n
+(1− α
(1)
n
)T
1
(PT
1
)
n−1
z
n
] − f 
= lim
n
→+
∞
 (1 − α
(1)
n
)[T
1
(PT
1
)
n−1
z
n

− f]+α
(1)
n
(z
n
− f)

Then it follows from Lemma 2.2 and
lim sup
n→+∞
 T
1
(PT
1
)
n−1
z
n
− f ≤ r
that
lim
n
→+
∞
 T
1
(PT
1
)
n−1

z
n
− z
n
=0
.
(3:5)
According to (3.3), we have
r = lim
n→+∞
 [α
(2)
n
y
n
+(1− α
(2)
n
)T
2
(PT
2
)
n−1
y
n
] − f 
= lim
n
→+

∞
 (1 − α
(2)
n
)[T
2
(PT
2
)
n−1
y
n
− f]+α
(2)
n
(y
n
− f)

Noting
lim sup
n
→+∞
 T
2
(PT
2
)
n−1
y

n
− f ≤
r
, by Lemma 2.2 again, we can get
lim
n
→+∞
 T
2
(PT
2
)
n−1
y
n
− y
n
=0
.
(3:6)
Similarly, we can obtain
lim
n
→+
∞
 T
3
(PT
3
)

n−1
x
n
− x
n
=0
.
(3:7)
Zhu et al. Fixed Point Theory and Applications 2011, 2011:106
/>Page 5 of 13
Hence, it follows from
 y
n
− x
n
 = P[α
(3)
n
x
n
+(1− α
(3)
n
)T
3
(PT
3
)
n−1
x

n
] − x
n

≤ [α
(3)
n
x
n
+(1− α
(3)
n
)T
3
(PT
3
)
n−1
x
n
] − x
n

≤ T
3
(
PT
3
)
n−1

x
n
− x
n
 .
that
lim
n
→+∞
 y
n
− x
n
=
0
. Also, we can see
 z
n
− y
n
 = P[α
(
2
)
n
y
n
+(1− α
(
2

)
n
)T
2
(PT
2
)
n−1
y
n
] − y
n

≤ [α
(2)
n
y
n
+(1− α
(2)
n
)T
2
(PT
2
)
n−1
y
n
] − y

n

≤ T
2
(
PT
2
)
n−1
y
n
− y
n

and
 x
n+1
− z
n
 = P[α
(1)
n
z
n
+(1− α
(1)
n
)T
1
(PT

1
)
n−1
z
n
] − z
n

≤ [α
(1)
n
z
n
+(1− α
(1)
n
)T
1
(PT
1
)
n−1
z
n
] − z
n

≤ T
1
(

PT
1
)
n−1
z
n
− z
n
 .
It follows from (3.5) and (3.6) that
lim
n
→+
∞
 x
n+1
− z
n
= lim
n
→+
∞
 z
n
− y
n
=0
.
Hence
lim

n
→+∞
 x
n+1
− x
n
= lim
n
→+∞
 z
n
− x
n
=
0
. Thus for any fixed k Î N,
lim
n
→+∞
 x
n+k
− x
n
=0
.
Noting (3.7) and
 x
n
− T
3

(PT
3
)
k−1
x
n

≤ x
n
− x
n+k
 +  x
n+k
− T
3
(PT
3
)
n+k−1
x
n+k
 +  T
3
(PT
3
)
n+k−1
x
n+
k

−T
3
(PT
3
)
n+k−1
x
n
 +  T
3
(PT
3
)
n+k−1
x
n
− T
3
(PT
3
)
k−1
x
n

≤ x
n
− x
n+k
 +  x

n+k
− T
3
(PT
3
)
n+k−1
x
n+k
 +  x
n+k
− x
n
 +r
n+k
+  T
3
(
PT
3
)
n−1
x
n
− x
n
 +r
k
we have
lim sup

n
→+∞
 x
n
− T
3
(PT
3
)
k−1
x
n
≤ r
k
, which implies
lim
k→+∞
lim sup
n
→+∞
 x
n
− T
3
(PT
3
)
k−1
x
n

=0
.
Combining (3.6) with
 T
2
(PT
2
)
n−1
x
n
− x
n

≤ T
2
(PT
2
)
n−1
x
n
− T
2
(PT
2
)
n−1
y
n

 +  T
2
(PT
2
)
n−1
y
n
− y
n
 +  y
n
− x
n

≤ 2  x
n
− y
n
 +  T
2
(
PT
2
)
n−1
y
n
− y
n

 +r
n
,
we can see
lim
n
→+∞
 T
2
(PT
2
)
n−
1
x
n
− x
n
=
0
. Thus
 x
n
− T
2
(PT
2
)
k
−1

x
n

≤ x
n
− x
n+k
 +  x
n+k
− T
2
(PT
2
)
n+k−1
x
n+k
 +  T
2
(PT
2
)
n+k−1
x
n+
k
−T
2
(PT
2

)
n+k−1
x
n
 +  T
2
(PT
2
)
n+k−1
x
n
− T
2
(PT
2
)
k−1
x
n

≤ x
n
− x
n+k
 +  x
n+k
− T
2
(PT

2
)
n+k−1
x
n+k
 +  x
n+k
− x
n
 +r
n+k
+  T
2
(
PT
2
)
n−1
x
n
− x
n
 +r
k
Zhu et al. Fixed Point Theory and Applications 2011, 2011:106
/>Page 6 of 13
which implies
lim
k→+∞
lim sup

n
→+∞
 x
n
− T
2
(PT
2
)
k−1
x
n
=0
.
Combining (3.5) with
 T
1
(PT
1
)
n−1
x
n
− x
n

≤ T
1
(PT
1

)
n−1
x
n
− T
1
(PT
1
)
n−1
z
n
 +  T
1
(PT
1
)
n−1
z
n
− z
n
 +  x
n
− x
n

≤ 2  x
n
− z

n
 +  T
1
(
PT
1
)
n−1
z
n
− z
n
 +r
n
,
we can see
lim
n
→+∞
 T
1
(PT
1
)
n−1
x
n
− x
n
=

0
. Thus
 x
n
− T
1
(PT
1
)
k−1
x
n

≤ x
n
− x
n+k
 +  x
n+k
− T
1
(PT
1
)
n+k−1
x
n+k
 +  T
1
(PT

1
)
n+k−1
x
n+
k
−T
1
(PT
1
)
n+k−1
x
n
 +  T
1
(PT
1
)
n+k−1
x
n
− T
1
(PT
1
)
k−1
x
n


≤ x
n
− x
n+k
 +  x
n+k
− T
1
(PT
1
)
n+k−1
x
n+k
 +  x
n+k
− x
n
 +r
n+k
+  T
1
(
PT
1
)
n−1
x
n

− x
n
 +r
k
which implies
lim
k→+∞
lim sup
n
→+∞
 x
n
− T
1
(PT
1
)
k−1
x
n
=0
.
This completes the proof.
Define the operator W
n
: C ® C by
W
n
x = P[α
(

1
)
n
x
(1)
+(1− α
(
1
)
n
)T
1
(PT
1
)
n−1
x
(1)
]
;
x
(1)
= P[α
(2)
n
x
(2)
+(1− α
(2)
n

)T
2
(PT
2
)
n−1
x
(2)
]
;
x
(2)
= P[α
(3)
n
x +
(
1 − α
(3)
n
)
T
3
(
PT
3
)
n−1
x],
where x Î C. Then by (3.1), x

n+1
= W
n
x
n
and for all x, y Î C, we have
 x
(2)
− y
(2)
≤α
(3)
n
 x − y  +(1 − α
(3)
n
)  T
3
(PT
3
)
n−1
x − T
3
(PT
3
)
n−1
y


≤ x − y  +r
n
,
 x
(1)
−
y
(1)
≤x
(2)
−
y
(2)
 +r
n
≤ x −
y
 +2r
n
and

W
n
x − W
n
y 
≤

x −
y 

+3r
n
=

x −
y 
+w
n
.
For any f Î F, we get W
n
f = f. Set
S
n
,
m
= W
n+m−1
W
n+m−2
···W
n+1
W
n
: C → C
,
then x
n+m
= S
n,m

x
n
and for all f Î F, S
n,m
f = f. Note that for any x, y Î C,
 S
n,m
x − S
n,m
y ≤ x − y  +
(
w
n
+ ···+ w
n+m−1
).
(3:8)
Lemma 3.3 Let f, g Î F and l Î [0, 1], then
h(λ) = lim
n
→+∞
 λx
n
+(1− λ)f − g

exists.
Zhu et al. Fixed Point Theory and Applications 2011, 2011:106
/>Page 7 of 13
Proof. It follows from Lemma 3.1 that
lim

n
→+∞
 x
n
− f  =
r
exists. If l =0,1orr =0,
then the c onclusion holds. Assume that r>0andl Î (0, 1), then for any ε >0, there
exists d>0(d<ε) such that
(r + d)[1 − 2λ(1 − λ)δ(
ε
r
+
d
)] < r − d
,
(3:9)
where δ(·) is the modulus of convexity of the norm. Hence there exists a positive
integer n
0
such that for all n>n
0
,
r −
d
4
≤ x
n
− f ≤ r +
d

4
(3:10)
and
+∞

i
=
n
w
i
≤ λ(1 − λ)
d
4
<
ε
4
(3:11)
Now we claim that for all n>n
0
,
 S
n,m
[λx
n
+
(
1 − λ
)
f ] − [λS
n,m

x
n
+
(
1 − λ
)
f ] ≤ ε. ∀m ∈
N
Otherwise, we can suppose that there are some n>n
0
and some m Î N such that
 S
n,m
[λx
n
+
(
1 − λ
)
f ] − [λS
n,m
x
n
+
(
1 − λ
)
f ] ≥ ε
.
Put z = lx

n
+(1-l)f, x =(1-l)(S
n,m
z - f ), and y = l(S
n,m
x
n
- S
n,m
z), then by (3.8),
(3.10), and (3.11), we have
 x  =(1− λ)  S
n,m
z − f 
≤ (1 − λ)[ z − f  +(w
n+m−1
+ ···+ w
n+1
+ w
n
)]
≤ λ(1 − λ)( x
n
− f  +
d
4
)
≤ λ(1 − λ)(r + d),
 y  = λ  S
n,m

x
n
− S
n,m
z 
≤ λ[ x
n
− z  +(w
n+m−1
+ ···+ w
n+1
+ w
n
)]
≤ λ(1 − λ)( x
n
− f  +
d
4
)
≤ λ(1 − λ)(r + d),
 x − y  = S
n,m
[λx
n
+
(
1 − λ
)
f ] − [λS

n,m
x
n
+
(
1 − λ
)
f ] ≥
ε
and
λx +
(
1 − λ
)
y = λ
(
1 − λ
)(
S
n,m
x
n
− f
).
So by Lemma 2.3, we get
λ(1 − λ)  S
n,m
x
n
− f  = λx +(1− λ)y 

≤ λ(1 − λ)(r + d)[1 − 2λ(1 − λ)δ(
ε
λ(1 − λ)(r + d)
)
]
≤ λ(1 − λ)(r + d)[1 − 2λ(1 − λ)δ(
ε
r
+
d
)]
Zhu et al. Fixed Point Theory and Applications 2011, 2011:106
/>Page 8 of 13
and then by (3.10),
r − d ≤ x
n+m
− f = S
n,m
x
n
− f 
≤ ( r + d)[1 − 2λ(1 − λ)δ(
ε
r
+
d
)]
,
which contradicts (3.9). Thus we can conclude that for all n>n
0

,
 S
n,m
[λx
n
+
(
1 − λ
)
f ] − [λS
n,m
x
n
+
(
1 − λ
)
f ] ≤ ε, ∀m ∈ N
.
Hence by (3.11), for all n>n
0
,
 λx
n+m
+(1− λ)f − g 
= λS
n,m
x
n
+(1− λ)f − g 

≤ [λS
n,m
x
n
+(1− λ)f ] − S
n,m
[λx
n
+(1− λ)f ]  +  S
n,m
[λx
n
+(1− λ)f ] − g

≤ ε+  λx
n
+(1− λ)f − g  +(w
n+m−1
+ ···+ w
n+1
+ w
n
)
≤ 2ε+  λx
n
+
(
1 − λ
)
f − g  .

For any fixed n>n
0
, we can take the limsup for m and obtain
lim sup
m
→+∞
 λx
m
+(1− λ)f − g ≤ λx
n
+(1− λ)f − g  +2ε
.
Hence
lim sup
m
→+∞
 λx
m
+(1− λ)f − g ≤ lim inf
n→+∞
 λx
n
+(1− λ)f − g  +2ε
.
Since ε >0 is arbitrary, this implies that
h(λ) = lim
n
→+∞
 λx
n

+(1− λ)f − g

exists. This completes the proof.
Remark 3.1 If the mappings are asymptotically nonexpansive, we can use Bruck’s
Lemma [23]to prove Lemma 3.3. While Bruck’s Lemma is not valid for non-Lipschitzian
mappings, we must introduce new technique to establish a similar inequality. In [20],
Chidume et al. also proved that
lim
n
→+∞
 λx
n
+(1− λ)f − g

exists (Lemma 3.12 in
[20]). As we have seen, our proof is completely different from theirs in [20].
Lemma 3.4 If f Î ω
ω
({x
n
}) and
lim
n
→+∞
 λx
n
+(1− λ)f − g

exists, then
h(λ) = lim

n
→+∞
 λx
n
+(1− λ)f − g ≤ f − g 
.
Proof. For any ε >0, there exists n
0
such that for all n ≥ n
0
,
 λx
n
+
(
1 − λ
)
f − g ≤ h
(
λ
)
+ ε
.
Then for all n ≥ n
0
,
λx
n
+
(

1 − λ
)
f − g, J
(
f − g
)
≤f − g 
(
h
(
λ
)
+ ε
).
Since f Î ω
ω
({x
n
}), there exists a subsequence
{x
n
i
}⊂{x
n
}
with
x
n
i


f
.Hence
f ∈
¯
co{x
n
i
, i ≥ n
0
}
and
{λf +
(
1 − λ
)
f − g, J
(
f − g
)
}≤f − g 
(
h
(
λ
)
+ ε
),
i.e., ||f - g||
2
≤ ||f - g||(h(l)+ε). Therefore ||f - g|| ≤ h(l). This completes the proof.

Now we can prove the weak convergence theorem of the iterative sequence (3.1).
Zhu et al. Fixed Point Theory and Applications 2011, 2011:106
/>Page 9 of 13
Theorem 3.1 Let C be a nonempty close d conve x subset of uniformly conve x Banach
space X which satisfies the Opial’s condition or whose dual X* has the Kadec-Klee prop-
erty. Let P : X ® C be a nonexpansive retraction from X onto C. Let T
1
, T
2
, T
3
: C ® X
be three asymptotically nonexpansive in the intermediate sense nonself mappings with F
≠ ∅ and the nonnegative sequence {r
n
} satisfy
+∞

n=1
r
n
< +
∞
. Let {x
n
} be defined by: x
1
Î
C and
x

n+1
= P[α
(1)
n
z
n
+(1− α
(1)
n
)T
1
(PT
1
)
n−1
z
n
]
;
z
n
= P[α
(2)
n
y
n
+(1− α
(2)
n
)T

2
(PT
2
)
n−1
y
n
];
y
n
= P[α
(3)
n
x
n
+
(
1 − α
(3)
n
)
T
3
(
PT
3
)
n−1
x
n

].
where
{
α
(i)
n
}
is in [0, 1] with
0 <
p
≤ α
(
i
)
n
≤
q
<
1
, i =1,2,3.Then {x
n
}, {y
n
}, and {z
n
}
converge weakly to a common fixed point of
{T
i
}

3
i
=
1
.
Proof. It suffices to show that { x
n
} converges weakly to a common fixed point of
{T
i
}
3
i
=
1
.Tothisaim,weonlyneedtoprovethatthesetω
ω
({x
n
}) is singleton. Since X is
refl exive and C is bounded, we obtain ω
ω
({x
n
}) ≠ ∅ . Assume that f, g Î ω
ω
({x
n
}), then
there exist two subsequences

{
x
n
i
}
and
{x
n
j
}
in {x
n
}suchthat
x
n
i

f
and
x
n
j
 g
.Inthe
following, we shall show f = g. By Lemmas 2.5 and 3.2, f, g Î F. On one hand, if X
satisfies the Opial’s condition and f ≠ g, then by the Lemma 3.1, we get
r = lim
n→+∞
 x
n

− f = lim
i→+∞
 x
n
i
− f 
< lim
i→+∞
 x
n
i
− g = lim
n→+∞
 x
n
− g = lim
j→+∞
 x
n
j
− g

< lim
j
→+∞
 x
n
j
− f = lim
n→+∞

 x
n
− f = r.
This contraction implies f = g.Ontheotherhand,ifX* has Kadec- Klee property,
then from Lemmas 2.4, 3.3, and 3.4, we have
 λx
n
+(1− λ)f − g
2
≤f − g
2
+2λx
n
− f,J
(
λx
n
+
(
1 − λ
)
f − g
)
and for all l Î [0, 1],
lim inf
n
→+∞
x
n
− f,J(λx

n
+(1− λ)f − g)≥0
.
Hence
lim inf
j
→+∞
x
n
j
− f,J(λx
n
j
+(1− λ)f − g)≥0
.
Thus for arbitrary k Î N, there exists j
k
≥ k,{j
k
} ↑, such that
x
n
j
k
− f,J(
1
k
x
n
j

k
+(1−
1
k
)f − g)≥−
1
k
.
(3:12)
Obviously
x
n
j
k

g
. Put
j
k
= J(
1
k
x
n
j
k
+(1−
1
k
)f − g)

,
then we may assume that, without loss of generality, j
k
is weakly convergent to some
Zhu et al. Fixed Point Theory and Applications 2011, 2011:106
/>Page 10 of 13
point j Î X*. Therefore
 j ≤ lim inf
k
→+∞
 j
k
= f − g 
. Noting
f − g, j
k
 =
1
k
x
n
j
k
+(1−
1
k
)f − g
2
−
1

k
x
n
j
k
− f,j
k

and passing the limit for k, we have 〈f - g, j〉 =||f - g||
2
. Hence ||j|| ≥ || f - g|| and

f
−
g
,
j
 =
f
−
g

2
=
j

2
,
which means j = J(f - g). Thus we can conclude j
k

⇀ j and ||j
k
|| ® ||f - g|| = ||j ||.
Since X* has Kadec-Kl ee property, we have j
k
® j. Taking the limit in (3.12), we get 〈g
- f, j〉 ≥ 0, i.e., ||f - g||
2
≤ 0, which implies f = g. This completes the proof.
Remark 3.2 Theorem 3.1 extends the main results in [4,16,18,20]to the case of
asymptotically nonexpansive in the intermediate sense mappings and it seems to be new
even in the case that the space has a Fréchet differentiable norm.
In the fo llowing, we shall give a n onself mapping which is asympto tically nonexpan-
sive in the intermediate sense but not asymptotically nonexpansive.
Example 3.1 Let Δ be the Cantor ternary set. Define the Cantor ternary function
τ (x)=
⎧
⎨
⎩
+∞

n=1
b
n
2
n
x =
+∞

n=1

2b
n
3
n
∈ ,(b
n
=0,1
)
sup{τ (y), y ≤ x, y ∈ } x ∈ [0, 1]\
then τ :[0,1]® [0, 1] is a continuous and inc reasing but not absolutely continuous
function with τ(0) = 0,
τ (
1
2
)=
1
2
(see [27]). Since a Lipschitzian function is absolutely
continuous, τ is non-Lipschitzian. Define : R ® R by
ϕ(x)=
⎧
⎨
⎩
0 x < 0orx >
1
x
2
0 ≤ x ≤
1
2

1
2
τ (1 − x)
1
2
< x ≤ 1
It is easy to see that  is continuous and for all x, y Î R,
|
ϕ(x) − ϕ(y)|≤
1
2
.Italso
can be verified that the n-fold composition mapping 
n
is defined by
ϕ
n
(x)=
⎧
⎨
⎩
0 x < 0orx >
1
x
2
n
0 ≤ x ≤
1
2
1

2
n
τ (1 − x)
1
2
< x ≤ 1
Since τ is non-Lipschitzian, so is 
n
and for all x, y Î R,
|
ϕ
n
(x) − ϕ
n
(y)|≤
1
2
n
.
Taking X = R
2
with the norm
 (x, y)  =

x
2
+ y
2
,(x, y) Î X and C = R ×{0},we
define the nonself mapping T : C ® X by

T
(
x,0
)
=
(
ϕ
(
x
)
, x
)
,
(
x,0
)
∈ C
,
then T is continuous and (0, 0) is a fixed point of T. Define P : X ® C by
P
(
x, y
)
=
(
x,0
)
,
(
x, y

)
∈ X
,
then P is a nonexpansive retraction from X onto C. Hence for all (x, 0), (y,0)Î C,
T
(
PT
)
n−1
(
x,0
)
=
(
ϕ
n
(
x
)
, ϕ
n−1
(
x
)),
Zhu et al. Fixed Point Theory and Applications 2011, 2011:106
/>Page 11 of 13
which means T(PT)
n-1
is is non-Lipschitzian and
 T(PT)

n−
1
(x,0)− T(PT)
n−
1
(y,0) 
= (ϕ
n
(x), ϕ
n−1
(x)) − (ϕ
n
(y), ϕ
n−1
(y)) 
=

(ϕ
n
(x) − ϕ
n
(y))
2
+(ϕ
n−1
(x) − ϕ
n−1
(y))
2
≤|ϕ

n
(x) − ϕ
n
(y)| + |ϕ
n−1
(x) − ϕ
n−1
(y)|
≤ (x,0)− (y,0)  +
3
2
n
.
Therefore, we can conclude that T is asymptotically nonexpansive in the intermedi-
ate sense but not an asymptotically nonexpansive.
If T
1
, T
2
, and T
3
are nonexpansive, we can prove the following theorem.
Theorem 3.2 Let C be a nonempty close d conve x subset of uniformly conve x Banach
space X which satisfies the Opial’s condition or whose dual has the Kadec-Klee prop-
erty. Let P : X ® C be a nonexpansive retraction from X o nto C. Let T
1
, T
2
, T
3

: C ®
X be three nonexpansive nonself mappings and {x
n
} be defined by: x
1
Î C and
x
n+1
= P[α
(1)
n
z
n
+(1− α
(1)
n
)T
1
z
n
]
;
z
n
= P[α
(2)
n
y
n
+(1− α

(2)
n
)T
2
y
n
];
y
n
= P[α
(3)
n
x
n
+
(
1 − α
(3)
n
)
T
3
x
n
].
where
{
α
(
i

)
n
}
is in 0[1]with
0 <
p
≤ α
(i)
n
≤
q
<
1
, i =1,2,3.Then {x
n
}, {y
n
} and {z
n
}
converge weakly to a common fixed point of
{T
i
}
3
i
=
1
.
Remark 3.3 We would like to remark that if the so-called error terms are added in

our recursion formula and are assumed to be bounded, then the results of this article
still hold. Thus we can get the main results in [19].
Acknowledgements
This research is supported by the National Natural Science Foundation of China (10971182), the Natural Science
Foundation of Jiangsu Province (BK2009179 and BK2010309), the Tianyuan Youth Foundation (11026115), the Jiangsu
Government Scholarship for Overseas Studies, the Natural Science Foundation of Jiangsu Education Committee
(09KJB110010 and 10KJB110012) and the Natural Science Foundation of Yangzhou University.
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Received: 20 August 2011 Accepted: 30 December 2011 Published: 30 December 2011
References
1. Mann, WR: Mean value methods in iteration. Proc Amer Math Soc. 4(3), 506–510 (1953). doi:10.1090/S0002-9939-1953-
0054846-3
2. Ishikawa, S: Fixed points and iteration of a nonexpansive mapping in a Banach space. Proc Amer Math Soc. 59(1),
65–71 (1967)
3. Groetsch, CW: A note on segmenting Mann iterates. J Math Anal Appl. 40, 369–372 (1972). doi:10.1016/0022-247X(72)
90056-X
4. Schu, J: Weak and strong convergence to fixed points of asymptotically non-expansive mappings. Bull Austral Math
Soc. 43(1), 153–159 (1991). doi:10.1017/S0004972700028884
5. Tan, KK, Xu, HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J Math Anal
Appl. 178, 301–308 (1993). doi:10.1006/jmaa.1993.1309
6. Noor, MA: New approximation schemes for general variational inequalities. J Math Anal Appl. 251, 217–229 (2000).
doi:10.1006/jmaa.2000.7042
7. Xu, B, Noor, MA: Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces. J Math Anal Appl.
267(2), 444–453 (2002). doi:10.1006/jmaa.2001.7649
Zhu et al. Fixed Point Theory and Applications 2011, 2011:106
/>Page 12 of 13
8. Cho, Y, Zhou, H, Guo, G: Weak and strong convergence theorems for three-step iterations with errors for asymptotically

nonexpansive mappings. Comput Math Appl. 47(4-5), 707–717 (2004). doi:10.1016/S0898-1221(04)90058-2
9. Suantai, S: Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings. J Math
Anal Appl. 311, 506–517 (2005). doi:10.1016/j.jmaa.2005.03.002
10. Shahzad, N, Udomene, A: Fixed point solutions of variational inequalities for asymptotically nonexpansive mappings in
Banach spaces. Nonlinear Anal: Theory Methods Appl. 64(3), 558–567 (2006). doi:10.1016/j.na.2005.03.114
11. Shahzad, N, Udomene, A: Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in
Banach spaces. Fixed Point Theory and Applications 2006, 10 (2006). Article ID 18909
12. Plubtieng, S, Wangkeeree, R, Punpaeng, R: On the convergence of modified Noor iterations with errors for
asymptotically nonexpansive mappings. J Math Anal Appl. 322, 1018–1029 (2006). doi:10.1016/j.jmaa.2005.09.078
13. Cai, G, Hu, C: On strong convergence by the hybrid method for equilibrium and fixed point problems for an inifnite
family of asymptotically nonexpansive mappings. Fixed Point Theory and Applications 2009, 20 (2009). Article ID
798319
14. Chidume, CE, Zegeye, H, Shahzad, N: Convergence theorems for a common fixed point of finite family of nonself
nonexpansive mappings. Fixed Point Theory Appl. 2005,1–9 (2005)
15. Shahzad, N: Approximating fixed points of non-self nonexpansive mappings in Banach spaces. Nonlinear Anal: Theory,
Methods Appl. 61, 1031–1039 (2005). doi:10.1016/j.na.2005.01.092
16. Chidume, CE, Ofoedu, EU, Zegeye, H: Strong and weak convergence theorems for asymptotically nonexpansive
mappings. J Math Anal Appl. 280, 364–374 (2003). doi:10.1016/S0022-247X(03)00061-1
17. Wang, C, Zhu, J: Convergence theorems for common fixed points of nonself asymptotically quasi-non-expansive
mappings. Fixed Point Theory and Applications 2008, 11 (2008). Article ID 428241
18. Thianwan, S: Common fixed points of new iterations for two asymptotically nonexpansive nonself-mappings in a
Banach space. J Comput Appl Math. 224, 688–695 (2009). doi:10.1016/j.cam.2008.05.051
19. Huang, QL, Hu, Y, Gao, SY: Convergence theorems for three-step iterations with errors of asymptotically nonexpansive
nonself mappings in Banach spaces. Math Appl. 24, 540–547 (2011)
20. Chidume, CE, Shahzad, N, Zegeye, H: Convergence theorems for mappings which are asymptotically nonexpansive in
the intermediate sense. Numer Funct Anal Optimiz. 25, 239–257 (2004)
21. Kim, GE, Kim, TH: Mann and Ishikawa iterations with errors for non-Lipschitzian mappings in Banach spaces. Comput
Math Appl. 42, 1565–1570 (2001). doi:10.1016/S0898-1221(01)00262-0
22. Plubtieng, S, Wangkeeree, R: Strong convergence theorems for three-step iterations with errors for non-Lipschitzian
nonself-mappings in Banach Spaces. Comput Math Appl. 51, 1093–1102 (2006). doi:10.1016/j.camwa.2005.08.035

23. Bruck, RE: A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces. Israel J Math. 32,
107–116 (1979). doi:10.1007/BF02764907
24. Kaczor, W: A nonstandard proof of a generalized demiclosedness principle. pp. 43–50. Annales Universitatis Mariae
Curie-Sklodowska Lublin-Polonia LIX (2005)
25. Kaczor, W: Weak convergence of almost orbits of asymptotically nonexpansive commutative semigroups. J Math Anal
Appl. 272, 565–574 (2002). doi:10.1016/S0022-247X(02)00175-0
26. Chang, SS: Some problems and results in the study of nonlinear analysis. Nonlinear Anal: Theory Methods Appl. 30,
4197–4208 (1997). doi:10.1016/S0362-546X(97)00388-X
27. Royden, HL: Real Analysis, 3rd edn.Pearson Education (2004)
doi:10.1186/1687-1812-2011-106
Cite this article as: Zhu et al.: Weak convergence theorem for the three-step iterations of non-Lipschitzian
nonself mappings in Banach spaces. Fixed Point Theory and Applications 2011 2011:106.
Submit your manuscript to a
journal and benefi t from:
7 Convenient online submission
7 Rigorous peer review
7 Immediate publication on acceptance
7 Open access: articles freely available online
7 High visibility within the fi eld
7 Retaining the copyright to your article
Submit your next manuscript at 7 springeropen.com
Zhu et al. Fixed Point Theory and Applications 2011, 2011:106
/>Page 13 of 13