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Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2010, Article ID 189751, 7 pages
doi:10.1155/2010/189751
Research Article
On the Weak Relatively Nonexpansive Mappings in
Banach Spaces
Yongchun Xu
1
and Yongfu Su
2
1
Department of Mathematics, Hebei North University, Zhangjiakou 075000, China
2
Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
Correspondence should be addressed to Yongfu Su,
Received 23 March 2010; Accepted 20 May 2010
Academic Editor: Billy Rhoades
Copyright q 2010 Y. Xu and Y. Su. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
In recent years, the definition of weak relatively nonexpansive mapping has been presented and
studied by many authors. In this paper, we give some results about weak relatively nonexpansive
mappings and give two examples which are weak relatively nonexpansive mappings but not
relatively nonexpansive mappings in Banach space l
2
and L
p
0, 11 <p<∞.
1. Introduction
Let E be a smooth Banach space, and let C be a nonempty closed convex subset of E.We
denote by φ the function defined by
φ
x, y
x
2
− 2
x, Jy
y
2
for x, y ∈ E.
1.1
Following Alber 1, the generalized projection Π
C
from E onto C is defined by
Π
C
x
arg min
y∈C
φ
y, x
, ∀x ∈ E.
1.2
The generalized projection Π
C
from E onto C is well defined, single value and satisfies
x
−
y
2
≤ φ
x, y
≤
x
y
2
for x, y ∈ E.
1.3
If E is a Hilbert space, t hen φy, xy − x
2
,andΠ
C
is the metric projection of E onto C.
2 Fixed Point Theory and Applications
Let C be a closed convex subset of E,andletT be a mapping from C into itself. We
denote by FT the set of fixed points of T.Apointp in C is said to be an asymptotic fixed point
of T 2–4 if C contains a sequence {x
n
} which converges weakly to p such that lim
n →∞
Tx
n
−
x
n
0. The set of asymptotic fixed point of T will be denoted by b
FT.
Following Matsushita and Takahashi 2, a mapping T of C into itself is said to be
relatively nonexpansive if the following conditions are satisfied:
1 FT is nonempty;
2 φu, Tx ≤ φu, x, for all u ∈ FT,x∈ C;
3
FTFT.
The hybrid algorithms for fixed point of relatively nonexpansive mappings and applications
have been studied by many authors, for example 2–7
In recent years, the definition of weak relatively nonexpansive mapping has been
presented and studied by many authors 5–8, but they have not given the example which
is weak relatively nonexpansive mapping but not relatively nonexpansive mapping. In this
paper, we give an example which is weak relatively nonexpansive mapping but not relatively
nonexpansive mapping in Banach space l
2
.
Apointp in C is said to be a strong asymptotic fixed point of T 5, 6 if C contains a
sequence {x
n
} which converges strongly to p such that lim
n →∞
Tx
n
− x
n
0. The set of
strong asymptotic fixed points of T will be denoted by
FT. A mapping T from C into itself
is called weak relatively nonexpansive if
1 FT is nonempty;
2 φu, Tx ≤ φu, x, for all u ∈ FT,x∈ C;
3
FTFT.
Remark 1.1. In 6, the weak relatively nonexpansive mapping is also said to be relatively
weak nonexpansive mapping.
Remark 1.2. In 7, the authors have given the definition of hemirelatively nonexpansive
mapping as follows. A mapping T from C into itself is called hemirelatively nonexpansive if
1 FT is nonempty;
2 φu, Tx ≤ φ
u, x, for all u ∈ FT,x∈ C.
The following conclusion is obvious.
Conclusion 1. A mapping is closed hemi-relatively nonexpansive if and only if it is weak
relatively nonexpansive.
If E is strictly convex and reflexive Banach space, and A ⊂ E × E
∗
is a continuous
monotone mapping with A
−1
0
/
∅, then it is proved in 2 that J
r
:J rA
−1
J,forr>0
is relatively nonexpansive. Moreover, if T : E → E is relatively nonexpansive, then using
the definition of φ, one can show that FT is closed and convex. It is obvious that relatively
nonexpansive mapping is weak relatively nonexpansive mapping. In fact, for any mapping
T : C → C, we have FT ⊂
FT ⊂
FT. Therefore, if T is relatively nonexpansive mapping,
then FT
FT
FT.
Fixed Point Theory and Applications 3
2. Results for Weak Relatively Nonexpansive Mappings in
Banach Space
Theorem 2.1. Let E be a smooth Banach space and C a nonempty closed convex and balanced subset
of E.Let{x
n
} be a sequence in C such that {x
n
} converges weakly to x
0
/
0 and x
n
− x
m
≥r>0
for all n
/
m. Define a mapping T : C → C as follows:
T
x
⎧
⎨
⎩
n
n 1
x
n
if x x
n
∃n ≥ 1
,
−x if x
/
x
n
∀n ≥ 1
.
2.1
Then the following conclusions hold:
1 T is a weak relatively nonexpansive mapping but not relatively nonexpansive mapping;
2 T is not continuous;
3 T is not pseudo-contractive;
4 if {x
n
}⊂intC,thenT is also not monotone (accretive), where intC is the interior of C.
Proof. 1 It is obvious that T has a unique fixed point 0, that is, FT{0}. Firstly, we show
that x
0
is an asymptotic fixed point of T. In fact since {x
n
} converges weakly to x
0
,
Tx
n
− x
n
n
n 1
x
n
− x
n
1
n 1
x
n
−→ 0 2.2
as n →∞,so,x
0
is an asymptotic fixed point of T. Secondly, we show that T has a unique
strong asymptotic fixed point 0, so that, FT
FT. In fact, for any strong convergent
sequence, {z
n
}⊂C such that z
n
→ z
0
and z
n
− Tz
n
→0asn →∞, from the conditions
of Theorem 2.1, there exists sufficiently large nature number N such that z
n
/
x
m
, for any
n, m > N. Then Tz
n
−z
n
for n>N, it follows from z
n
−Tz
n
→0that2z
n
→ 0,and hence
z
n
→ z
0
0. Observe that
φ
0,Tx
Tx
2
≤
x
2
φ
0,x
, ∀x ∈ C.
2.3
Then T is a weak relatively nonexpansive mapping. On the other hand, since x
0
is an
asymptotic fixed point of T but not fixed point, hence T is not a relatively nonexpansive
mapping.
2 For any x
n
/
0, we can take 0 ≤ λ
m
→ 0 such that λ
m
x
n
∈{x
n
}
∞
n1
, then we have
x
n
− λ
m
x
n
−→ 0,m−→ ∞,
Tx
n
− T
λ
m
x
n
n
n 1
x
n
λ
m
x
n
n
n 1
λ
m
x
n
≥
n
n 1
x
n
> 0,
2.4
then T is not continuous.
4 Fixed Point Theory and Applications
3 Since x
n
− x
m
≥r>0 for all n
/
m, without loss of generality, we assume that
x
n
/
0 for all n ≥ 1. In this case, we can take 1 ≥ δ
n
→ 1 such that δ
n
x
n
∈{x
i
}
∞
i1
for all n ≥ 1.
Therefore we have
Tx
n
− T
δ
n
x
n
,J
x
n
− δ
n
x
n
n
n 1
x
n
δ
n
x
n
,J
x
n
− δ
n
x
n
n
n 1
δ
n
x
n
,J
1 − δ
n
x
n
n
n 1
δ
n
1
1 − δ
n
1 − δ
n
x
n
,J
1 − δ
n
x
n
n
n 1
δ
n
1
1 − δ
n
1 − δ
n
x
n
2
n
n 1
δ
n
1
1 − δ
n
x
n
− δ
n
x
n
2
.
2.5
Since n/n1δ
n
1/1−δ
n
→ ∞as n →∞, we know that T is not pseudo-contractive.
4 In the same as 2, we can take 1 ≤ δ
n
→ 1 such that δ
n
x
n
∈{x
i
}
∞
i1
for all n ≥ 1.
Therefore we have
Tx
n
− T
δ
n
x
n
,J
x
n
− δ
n
x
n
n
n 1
x
n
δ
n
x
n
,J
x
n
− δ
n
x
n
n
n 1
δ
n
x
n
,J
1 − δ
n
x
n
n
n 1
δ
n
1
1 − δ
n
1 − δ
n
x
n
,J
1 − δ
n
x
n
n
n 1
δ
n
1
1 − δ
n
1 − δ
n
x
n
2
n
n 1
δ
n
1
1 − δ
n
x
n
− δ
n
x
n
2
.
2.6
Since n/n 1δ
n
1/1 − δ
n
→−∞as n →∞, we know that T is not monotone
accretive.
3. An Example in Banach Space l
2
In this section, we will give an example which is a weak relatively nonexpansive mapping
but not a relatively nonexpansive mapping.
Fixed Point Theory and Applications 5
Example 3.1. Let E l
2
, where
l
2
ξ
ξ
1
,ξ
2
,ξ
3
, ,ξ
n
,
:
∞
n1
|
x
n
|
2
< ∞
,
ξ
∞
n1
|
ξ
n
|
2
1/2
, ∀ξ ∈ l
2
,
ξ, η
∞
n1
ξ
n
η
n
, ∀ξ
ξ
1
,ξ
2
,ξ
3
, ,ξ
n
,
,η
η
1
,η
2
,η
3
, ,η
n
,
∈ l
2
.
3.1
It is well known that l
2
is a Hilbert space, so that l
2
∗
l
2
.Let{x
n
}⊂E be a sequence defined
by
x
0
1, 0, 0, 0,
,
x
1
1, 1, 0, 0,
,
x
2
1, 0, 1, 0, 0,
,
x
3
1, 0, 0, 1, 0, 0,
,
.
.
.
x
n
ξ
n,1
,ξ
n,2
,ξ
n,3
, ,ξ
n,k
,
,
3.2
where
ξ
n,k
⎧
⎨
⎩
1ifk 1,n 1,
0ifk
/
1,k
/
n 1,
3.3
for all n ≥ 1. Define a mapping T : E → E as follows:
T
x
⎧
⎨
⎩
n
n 1
x
n
if x x
n
∃n ≥ 1
,
−x if x
/
x
n
∀n ≥ 1
.
3.4
Conclusion 1. {x
n
} converges weakly to x
0
.
Proof . For any f ζ
1
,ζ
2
,ζ
3
, ,ζ
k
, ∈ l
2
l
2
∗
, we have
f
x
n
− x
0
f, x
n
− x
0
∞
k2
ζ
k
ξ
n,k
ζ
n1
−→ 0,
3.5
as n →∞.Thatis,{x
n
} converges weakly to x
0
.
The following conclusion is obvious.
6 Fixed Point Theory and Applications
Conclusion 2. x
n
− x
m
√
2 for any n
/
m.
It follows from Theorem 2.1 and the above two conclusions that T is a weak relatively
nonexpansive mapping but not relatively nonexpansive mapping. We have also the following
conclusions: 1 T is not continuous; 2 T is not pseudo-contractive; 3 T is also not
monotone accretive.
4. An Example in Banach Space L
p
0, 11 <p<∞
Let E L
p
0, 11 <p<∞,and
x
n
1 −
1
2
n
,n 1, 2, 3,
. 4.1
Define a sequence of functions in L
p
0, 1 by the following expression:
f
n
x
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
2
x
n1
− x
n
if x
n
≤ x<
x
n1
x
n
2
,
−2
x
n1
− x
n
if
x
n1
x
n
2
≤ x<x
n1
,
0 otherwise
4.2
for all n ≥ 1. Firstly, we can see, for any x ∈ 0, 1,that
x
0
f
n
t
dt −→ 0
x
0
f
0
t
dt,
4.3
where f
0
x ≡ 0. It is wellknown that the above relation 4.3 is equivalent to {f
n
x} which
converges weakly to f
0
x in uniformly smooth Banach space L
p
0, 11 <p<∞.Onthe
other hand, for any n
/
m, we have
f
n
− f
m
1
0
f
n
x
− f
m
x
p
dx
1/p
x
n1
x
n
f
n
x
− f
m
x
p
dx
x
m1
x
m
f
n
x
− f
m
x
p
dx
1/p
x
n1
x
n
f
n
x
p
dx
x
m1
x
m
f
m
x
p
dx
1/p
2
x
n1
− x
n
p
x
n1
− x
n
2
x
m1
− x
m
p
x
m1
− x
m
1/p
2
p
x
n1
− x
n
p−1
2
p
x
m1
− x
m
p−1
1/p
≥
2
p
2
p
1/p
> 0.
4.4
Fixed Point Theory and Applications 7
Let
u
n
x
f
n
x
1, ∀n ≥ 1. 4.5
It is obvious that u
n
converges weakly to u
0
x ≡ 1and
u
n
− u
m
f
n
− f
m
≥
2
p
2
p
1/p
> 0, ∀n ≥ 1.
4.6
Define a mapping T : E → E as follows:
T
x
⎧
⎨
⎩
n
n 1
u
n
if x u
n
∃n ≥ 1
,
−x if x
/
u
n
∀n ≥ 1
.
4.7
Since 4.6 holds, by using Theorem 2.1,weknowthatT : L
p
0, 1 → L
p
0, 1 is a weak
relatively nonexpansive mapping but not relatively nonexpansive mapping. We have also
the following conclusions: 1 T is not continuous; 2 T is not pseudo-contractive; 3 T is
also not monotone accretive.
Acknowledgments
This project is supported by the Zhangjiakou City Technology Research and Development
Projects Foundation 0811024B-5, Hebei Education Department Research Projects Founda-
tion 2009103, and Hebei North University Research Projects Foundation 2009008.
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