Volume 2010, Article ID 189751, 7 pagesdoi:10.1155/2010/189751 Research Article On the Weak Relatively Nonexpansive Mappings in Banach Spaces 1 Department of Mathematics, Hebei North Uni
Trang 1Volume 2010, Article ID 189751, 7 pages
doi:10.1155/2010/189751
Research Article
On the Weak Relatively Nonexpansive Mappings in Banach Spaces
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,suyongfu@tjpu.edu.cn
Received 23 March 2010; Accepted 20 May 2010
Academic Editor: Billy Rhoades
Copyrightq 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 l2and L p 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
φ
x, y
x2− 2x, Jy
y2
Following Alber1, the generalized projection ΠC from E onto C is defined by
ΠC x arg min
y∈C φ
y, x
The generalized projectionΠC from E onto C is well defined, single value and satisfies
x −y2≤ φx, y
≤x y2
If E is a Hilbert space, then φy, x y − x2, andΠC is the metric projection of E onto C.
Trang 2Let 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 {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 FT FT.
The hybrid algorithms for fixed point of relatively nonexpansive mappings and applications have been studied by many authors, for example2 7
In recent years, the definition of weak relatively nonexpansive mapping has been presented and studied by many authors5 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{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 FT FT.
Remark 1.1 In6, 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 ≤ 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
monotone mapping with A−10 / ∅, then it is proved in 2 that J r : J rA−1J, for r > 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.
Trang 32 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 x0/ 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,
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, 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 x0is an asymptotic fixed point of T In fact since {x n } converges weakly to x0,
Tx n − x n
n 1 n x n − x n
n 11 x n −→ 0 2.2
as n → ∞, so, x0 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 → z0andz n − Tz n → 0 as n → ∞, from the conditions
of Theorem 2.1, there exists sufficiently large nature number N such that zn / x m, for any
n, m > N Then Tz n −z n for n > N, it follows from z n − Tz n → 0 that 2z n → 0,and hence
z n → z0 0 Observe that
φ 0, Tx Tx2≤ 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
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 − Tλ m x n
n 1 n x n λ m x n
n 1 n λ m x n ≥
n
n 1 x n > 0, 2.4 then T is not continuous.
Trang 43 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}∞
i1for all n ≥ 1.
Therefore we have
n − Tδ n x n , Jx 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 n , J 1 − δ n x n
n
n 1 δ n
1
1− δ n n x n , J 1 − δ n x n
n
n 1 δ n 1
1− δ n 1 − δ n x n2
n
n 1 δ n
1
1− δ n x n − δ n x n2.
2.5
Sincen/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
n − Tδ n x n , Jx 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 n , J 1 − δ n x n
n
n 1 δ n
1
1− δ n n x n , J 1 − δ n x n
n
n 1 δ n
1
1− δ n 1 − δ n x n2
n
n 1 δ n
1
1− δ n x n − δ n x n2
.
2.6
Sincen/n 1 δ n 1/1 − δ n → −∞ as n → ∞, we know that T is not monotone
accretive
In this section, we will give an example which is a weak relatively nonexpansive mapping but not a relatively nonexpansive mapping
Trang 5Example 3.1 Let E l2, where
l2
ξ ξ1, ξ2, ξ3, , ξ n , :∞
n1
|x n|2< ∞
,
ξ
∞
n1
|ξ n|2
1/2
, ∀ξ ∈ l2,
ξ, η
∞
n1
ξ n η n , ∀ξ ξ1, ξ2, ξ3, , ξ n , , η η1, η2, η3, , η n ,
∈ l2.
3.1
It is well known that l2is a Hilbert space, so thatl2∗ l2 Let{x n } ⊂ 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, ,
x n ξ n,1 , ξ n,2 , ξ n,3 , , ξ n,k , ,
3.2
where
ξ n,k
⎧
⎨
⎩
1 if k 1, n 1,
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,
Conclusion 1 {x n } converges weakly to x0
Proof For any f ζ1, ζ2, ζ3, , ζ k , ∈ l2 l2∗, we have
f x n − x0 n − x0 ∞
k2
as n → ∞ That is, {x n } converges weakly to x0
The following conclusion is obvious
Trang 6Conclusion 2 x n − x m √2 for any n / m.
It follows fromTheorem 2.1and 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
monotoneaccretive
4 An Example in Banach Space Lp0, 1 1 < p < ∞
Let E L p 0, 1 1 < p < ∞, and
x n 1 − 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
x n1 − x n if x n1 x n
2 ≤ x < x n1 ,
4.2
for all n ≥ 1 Firstly, we can see, for any x ∈ 0, 1, that
x
0
f n tdt −→ 0
x
0
where f0x ≡ 0 It is wellknown that the above relation 4.3 is equivalent to {f n x} which converges weakly to f0x in uniformly smooth Banach space L p 0, 1 1 < p < ∞ On the other hand, for any n / m, we have
f n − f m 1
0
f n x − f m xp
dx
1/p
x n1
x n
f n x − f m xp
dx
x m1
x m
f n x − f m xp
dx
1/p
x n1
x n
f n xp
dx
x m1
x m
f m xp
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
2p
x n1 − x np−1 2p
x m1 − x mp−1
1/p
≥ 2p 2p1/p
> 0.
4.4
Trang 7It is obvious that u n converges weakly to u0x ≡ 1 and
u n − u m f n − f m ≥ 2p 2p1/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,
Since4.6 holds, by usingTheorem 2.1, we know that T : 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 monotoneaccretive
Acknowledgments
This project is supported by the Zhangjiakou City Technology Research and Development Projects Foundation0811024B-5, Hebei Education Department Research Projects Founda-tion2009103, and Hebei North University Research Projects Foundation 2009008
References
1 Y I Alber, “Metric and generalized projection operators in Banach spaces: properties and
applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol 178
of Lecture Notes in Pure and Applied Mathematics, pp 15–50, Marcel Dekker, New York, NY, USA, 1996.
2 S Matsushita and W Takahashi, “A strong convergence theorem for relatively nonexpansive mappings
in a Banach space,” Journal of Approximation Theory, vol 134, no 2, pp 257–266, 2005.
3 W Takahashi and K Zembayashi, “Strong and weak convergence theorems for equilibrium problems
and relatively nonexpansive mappings in Banach spaces,” Nonlinear Analysis Theory, Methods &
Applications, vol 70, no 1, pp 45–57, 2009.
4 X Qin and Y Su, “Strong convergence theorems for relatively nonexpansive mappings in a Banach
space,” Nonlinear Analysis Theory, Methods & Applications, vol 67, no 6, pp 1958–1965, 2007.
5 Y Su, J Gao, and H Zhou, “Monotone CQ algorithm of fixed points for weak relatively nonexpansive
mappings and applications,” Journal of Mathematical Research and Exposition, vol 28, no 4, pp 957–967,
2008
6 H Zegeye and N Shahzad, “Strong convergence theorems for monotone mappings and relatively
weak nonexpansive mappings,” Nonlinear Analysis Theory, Methods & Applications, vol 70, no 7, pp.
2707–2716, 2009
7 Y Su, D Wang, and M Shang, “Strong convergence of monotone hybrid algorithm for hemi-relatively
nonexpansive mappings,” Fixed Point Theory and Applications, Article ID 284613, 8 pages, 2008.
8 Y Su, Z Wang, and H Xu, “Strong convergence theorems for a common fixed point of two
hemi-relatively nonexpansive mappings,” Nonlinear Analysis Theory, Methods & Applications, vol 71, no 11,
pp 5616–5628, 2009
... which is a weak relatively nonexpansive mapping but not a relatively nonexpansive mapping Trang 5Example... Zegeye and N Shahzad, “Strong convergence theorems for monotone mappings and relatively
weak nonexpansive mappings, ” Nonlinear Analysis Theory, Methods & Applications, vol 70, no 7, pp.... “Strong and weak convergence theorems for equilibrium problems
and relatively nonexpansive mappings in Banach spaces,” Nonlinear Analysis Theory, Methods &
Applications,