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:
[email protected]
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

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
http://www.fixedpointtheoryandapplications.com/content/2011/1/106
© 2011 Zhu et al; licensee Springer. This is a n Open Access article distributed under the terms of t he Creative Commons Attribution
License (http://creativecommons.org/licenses/by/2.0), 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
(

= 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

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
http://www.fixedpointtheoryandapplications.com/content/2011/1/106
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

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
→+∞
 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
)

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.

)
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

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

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
+(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 

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
http://www.fixedpointtheoryandapplications.com/content/2011/1/106
Page 4 of 13

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

.
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

)
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

] − 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

3
(PT
3
)
n−1
x
n
− x
n
=0
.
(3:7)
Zhu et al. Fixed Point Theory and Applications 2011, 2011:106
http://www.fixedpointtheoryandapplications.com/content/2011/1/106
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
 .
that
lim
n
→+∞
 y
n
− x
n
=
0
. Also, we can see
 z
n
− y
n
 = P[α
(
2
)
n

)
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

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

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

≤ 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

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

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

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

http://www.fixedpointtheoryandapplications.com/content/2011/1/106
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

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

: 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

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

+
(
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

[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

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
,

)
≤ 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

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

}⊂{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

< +
∞
. 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

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

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


< 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
)

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

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
−


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
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

(
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

(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

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

}
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

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