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 Xu1 and Yongfu Su2
1
2
Department of Mathematics, Hebei North University, Zhangjiakou 075000, China
Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
Correspondence should be addressed to Yongfu Su, suyongfu@tjpu.edu.cn
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 l2 and Lp 0, 1 1 < 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
2
φ x, y x2 − 2 x, Jy y
for x, y ∈ E.
1.1
Following Alber 1, the generalized projection ΠC from E onto C is defined by
ΠC x arg min φ y, x ,
y∈C
∀x ∈ E.
1.2
The generalized projection ΠC from E onto C is well defined, single value and satisfies
2
2
x − y ≤ φ x, y ≤ x y
for x, y ∈ E.
1.3
If E is a Hilbert space, then φy, x y − x2 , and ΠC is the metric projection of E onto C.
2
Fixed Point Theory and Applications
Let C be a closed convex subset of E, and let T be a mapping from C into itself. We
denote by FT the set of fixed points of T . A point p in C is said to be an asymptotic fixed point
of T 2–4 if C contains a sequence {xn } which converges weakly to p such that limn → ∞ T xn −
.
xn 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, T x ≤ φu, x, for all u ∈ FT , x ∈ C;
FT .
3 FT
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 l2 .
A point p in C is said to be a strong asymptotic fixed point of T 5, 6 if C contains a
sequence {xn } which converges strongly to p such that limn → ∞ T xn − xn 0. The set of
. A mapping T from C into itself
strong asymptotic fixed points of T will be denoted by FT
is called weak relatively nonexpansive if
1 FT is nonempty;
2 φu, T x ≤ φu, x, for all u ∈ FT , x ∈ C;
FT .
3 FT
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, T x ≤ φu, 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
∅, then it is proved in 2 that Jr : J rA−1 J, for r > 0
monotone mapping with A−1 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
⊂ FT
. Therefore, if T is relatively nonexpansive mapping,
T : C → C, we have FT ⊂ FT
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
0 and xn − xm ≥ r > 0
of E. Let {xn } be a sequence in C such that {xn } converges weakly to x0 /
for all n /
m. Define a mapping T : C → C as follows:
⎧ n
⎨
xn
T x n 1
⎩−x
if x xn ∃n ≥ 1,
if x /
xn ∀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 {xn } ⊂ intC, then T 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 x0 is an asymptotic fixed point of T . In fact since {xn } converges weakly to x0 ,
n
1
x
−
x
T xn − xn xn −→ 0
n n 1 n
n1
2.2
as n → ∞, so, x0 is an asymptotic fixed point of T . Secondly, we show that T has a unique
. In fact, for any strong convergent
strong asymptotic fixed point 0, so that, FT FT
sequence, {zn } ⊂ C such that zn → z0 and zn − T zn → 0 as n → ∞, from the conditions
of Theorem 2.1, there exists sufficiently large nature number N such that zn / xm , for any
n, m > N. Then T zn −zn for n > N, it follows from zn − T zn → 0 that 2zn → 0,and hence
zn → z0 0. Observe that
φ0, T x T x2 ≤ x2 φ0, x,
∀x ∈ C.
2.3
Then T is a weak relatively nonexpansive mapping. On the other hand, since x0 is an
asymptotic fixed point of T but not fixed point, hence T is not a relatively nonexpansive
mapping.
0, we can take 0 ≤ λm → 0 such that λm xn ∈{xn }∞
2 For any xn /
n1 , then we have
xn − λm xn −→ 0, m −→ ∞,
n
n
n
xn λm xn λm xn ≥
T xn − T λm xn xn > 0,
n1
n1
n1
then T is not continuous.
2.4
4
Fixed Point Theory and Applications
m, without loss of generality, we assume that
3 Since xn − xm ≥ r > 0 for all n /
0 for all n ≥ 1. In this case, we can take 1 ≥ δn → 1 such that δn xn ∈{xi }∞
xn /
i1 for all n ≥ 1.
Therefore we have
n
xn δn xn , Jxn − δn xn n1
n
δn xn , J1 − δn xn n1
1
n
δn
1 − δn xn , J1 − δn xn n1
1 − δn
1
n
δn
1 − δn xn 2
n1
1 − δn
n
1
δn
xn − δn xn 2 .
n1
1 − δn
T xn − T δn xn , Jxn − δn xn 2.5
Since n/n1δn 1/1−δn → ∞ as n → ∞, we know that T is not pseudo-contractive.
4 In the same as 2, we can take 1 ≤ δn → 1 such that δn xn ∈{xi }∞
i1 for all n ≥ 1.
Therefore we have
n
xn δn xn , Jxn − δn xn n1
n
δn xn , J1 − δn xn n1
1
n
δn
1 − δn xn , J1 − δn xn n1
1 − δn
1
n
δn
1 − δn xn 2
n1
1 − δn
n
1
δn
xn − δn xn 2 .
n1
1 − δn
T xn − T δn xn , Jxn − δn xn 2.6
Since n/n 1 δn 1/1 − δn → −∞ as n → ∞, we know that T is not monotone
accretive.
3. An Example in Banach Space l2
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 l2 , where
l 2
∞
2
ξ ξ1 , ξ2 , ξ3 , . . . , ξn , . . . :
|xn | < ∞ ,
n1
1/2
∞
2
,
|ξn |
ξ ∀ξ ∈ l2 ,
3.1
n1
∞
ξn ηn ,
ξ, η ∀ξ ξ1 , ξ2 , ξ3 , . . . , ξn , . . ., η η1 , η2 , η3 , . . . , ηn , . . . . ∈ l2 .
n1
It is well known that l2 is a Hilbert space, so that l2 ∗ l2 . Let {xn } ⊂ E be a sequence defined
by
x0 1, 0, 0, 0, . . .,
x1 1, 1, 0, 0, . . .,
x2 1, 0, 1, 0, 0, . . .,
x3 1, 0, 0, 1, 0, 0, . . .,
3.2
..
.
xn ξn,1 , ξn,2 , ξn,3 , . . . , ξn,k , . . .,
where
ξn,k ⎧
⎨ 1 if k 1, n 1,
⎩ 0 if k 1, k n 1,
/
/
3.3
for all n ≥ 1. Define a mapping T : E → E as follows:
⎧ n
⎨
xn
T x n 1
⎩−x
if x xn ∃n ≥ 1,
if x / xn ∀n ≥ 1.
3.4
Conclusion 1. {xn } converges weakly to x0 .
Proof . For any f ζ1 , ζ2 , ζ3 , . . . , ζk , . . . ∈ l2 l2 ∗ , we have
fxn − x0 f, xn − x0 ∞
k2
as n → ∞. That is, {xn } converges weakly to x0 .
The following conclusion is obvious.
ζk ξn,k ζn1 −→ 0,
3.5
6
Fixed Point Theory and Applications
√
2 for any n /
m.
Conclusion 2. xn − xm 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 Lp 0, 1 1 < p < ∞
Let E Lp 0, 1 1 < p < ∞, and
xn 1 −
1
,
2n
4.1
n 1, 2, 3, . . . ..
Define a sequence of functions in Lp 0, 1 by the following expression:
⎧
2
⎪
⎪
⎪
⎪
x
− xn
⎪
n1
⎨
−2
fn x ⎪
⎪
⎪
xn1 − xn
⎪
⎪
⎩0
if xn ≤ x <
xn1 xn
,
2
xn1 xn
≤ x < xn1 ,
2
otherwise
4.2
if
for all n ≥ 1. Firstly, we can see, for any x ∈ 0, 1, that
x
fn tdt −→ 0 x
0
4.3
f0 tdt,
0
where f0 x ≡ 0. It is wellknown that the above relation 4.3 is equivalent to {fn x} which
converges weakly to f0 x in uniformly smooth Banach space Lp 0, 1 1 < p < ∞. On the
other hand, for any n /
m, we have
fn − fm 1
fn x − fm xp dx
1/p
0
x
n1
fn x − fm xp dx xn
x
n1
fn x − fm xp dx
1/p
xm
fn xp dx xn
xm1
2
xn1 − xn
p
xn1 − xn p−1
≥ 2p 2p 1/p > 0.
fm xp dx
xm
2p
xn1 − xn xm1
2p
1/p
2
xm1 − xm
1/p
xm1 − xm p−1
4.4
p
1/p
xm1 − xm Fixed Point Theory and Applications
7
Let
un x fn x 1,
∀n ≥ 1.
4.5
It is obvious that un converges weakly to u0 x ≡ 1 and
un − um fn − fm ≥ 2p 2p 1/p > 0,
∀n ≥ 1.
4.6
Define a mapping T : E → E as follows:
⎧ n
⎨
un
T x n 1
⎩−x
if x un ∃n ≥ 1,
if x / un ∀n ≥ 1.
4.7
Since 4.6 holds, by using Theorem 2.1, we know that T : Lp 0, 1 → Lp 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 Foundation 2009103, and Hebei North University Research Projects Foundation 2009008.
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