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Abstract and Applied Analysis
Volume 2011, Article ID 574756, 8 pages
doi:10.1155/2011/574756
Research Article
Modular Locally Constant Mappings in Vector
Ultrametric Spaces
Kamal Fallahi1 and Kourosh Nourouzi1, 2
1
Department of Mathematics, K. N. Toosi University of Technology, P.O. Box 16315-1618,
Tehran 1541849611, Iran
2
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746,
Tehran, Iran
Correspondence should be addressed to Kourosh Nourouzi, nourouzi@kntu.ac.ir
Received 16 November 2010; Revised 15 February 2011; Accepted 23 February 2011
Academic Editor: Norimichi Hirano
Copyright q 2011 K. Fallahi and K. Nourouzi. 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.
We give some sufficient conditions for mappings defined on vector ultrametric spaces to be
modular locally constant.
1. Introduction and Preliminaries
A metric space X, d in which the triangle inequality is replaced by
d x, y ≤ max dx, z, d z, y ,
x, y, z ∈ X ,
1.1
is called an ultrametric space. Generalized ultrametric spaces were given in 1, 2 via
partially ordered sets and some applications of them appeared in logic programming 3,
computational logic 4, and quantitative domain theory 5.
In 6, the notion of a metric locally constant function on an ultrametric space was
given in order to investigate certain groups of isometries and describe various Galois
groups over local fields. Locally constant functions also appear in contexts such as higher
ramification groups of finite extensions of Qp , and the Fontaine ring BdR
. Also, metric locally
constant functions were studied in 7, 8. On the other hand, vector ultrametric spaces are
given in 9 as vectorial generalizations of ultrametrics. Hence, locally constant functions,
in modular sense, can play the same role in vector ultrametric spaces as they do in usual
ultrametric spaces.
2
Abstract and Applied Analysis
In this paper, we introduce modular locally constant mappings in vector ultrametric
spaces. Some sufficient conditions are given for mappings defined on vector ultrametric
spaces to be modular locally constant.
We first present some basic notions.
Recall that a modular on a real linear space A is a real valued functional ρ on A
satisfying the conditions:
1 ρx 0 if and only if x 0,
2 ρx ρ−x,
3 ραx βy ≤ ρx ρy, for all x, y ∈ A and α, β ≥ 0, α β 1.
Then, the linear subspace
Aρ x ∈ A : ραx −→ 0 as α −→ 0
1.2
of A is called a modular space.
A sequence xn ∞
n1 in Aρ is called ρ-convergent briefly, convergent to x ∈ Aρ if ρxn −
x → 0 as n → ∞, and is called Cauchy sequence if ρxm − xn → 0 as m, n → ∞. By
a ρ-closed briefly, closed set in Aρ we mean a set which contains the limit of each of its
convergent sequences. Then, Aρ is a complete modular space if every Cauchy sequence in Aρ
is convergent to a point of Aρ . We refer to 10, 11 for more details.
A cone P in a complete modular space Aρ is a nonempty set such that
i P is ρ-closed, and P /
{0};
ii a, b ∈ Ê, a, b ≥ 0, x, y ∈ P ⇒ ax by ∈ P;
iii P ∩ −P {0}, where −P {−x : x ∈ P}.
y whenever
Let be the partial order on Aρ induced by the cone P, that is, x
y − x ∈ P. The cone P is called normal if
0
x
y ⇒ ρx ≤ ρ y ,
x, y ∈ Aρ .
1.3
The cone P is said to be unital if there exists a vector e ∈ P with modular 1 such that
x
ρxe,
x ∈ P.
1.4
Example 1.1. Consider the real vector space C0, 1 consisting of all real-valued continuous
functions on 0, 1 equipped with the modular ρ defined by
ρx max |xt|2 ,
t∈0,1
x ∈ C0, 1.
1.5
It is not difficult to see that C0, 1 is a complete modular space and
P {x ∈ C0, 1 : xt ≥ 0, ∀t ∈ 0, 1}
is a normal cone in C0, 1.
1.6
Abstract and Applied Analysis
3
Example 1.2. The vector space C1 0, 1 consisting of all continuously differentiable real-valued
functions on 0, 1 equipped with the modular ρ defined by
ρx max |xt| max x′ t,
t∈0,1
t∈0,1
x ∈ C1 0, 1
1.7
constitutes a complete modular space. The subset
P x ∈ C1 0, 1 : xt ≥ 0, ∀t ∈ 0, 1
1.8
is a unital cone in C1 0, 1 with unit 1. The cone P is not normal since, for example, xt
tn 1, for n ≥ 1 does not imply that ρx ≤ ρ1.
Throughout this note, we suppose that P is a cone in complete modular space Aρ , and
is the partial order induced by P.
Definition 1.3. A vector ultrametric on a nonempty set X is a mapping d : X × X → Aρ
satisfying the conditions:
CUM1 dx, y
0 for all x, y ∈ X and dx, y 0 if and only if x y;
CUM2 dx, y dy, x for all x, y ∈ X;
CUM3 If dx, z
p and dy, z
p, then dx, y
p, for any x, y, z ∈ X, and p ∈ P.
Then the triple X, d, P is called a vector ultrametric space. If P is unital and normal, then
X, d, P is called a unital-normal vector ultrametric space.
For unital-normal vector ultrametric space X, d, P, since
d x, y
ρ d x, y e,
d y, z
ρ d y, z e,
1.9
from CUM3 we get
dx, z
max ρ d x, y , ρ d y, z e,
1.10
and therefore
ρdx, z ≤ max ρ d x, y , ρ d y, z .
1.11
Let X, d, P be a unital-normal vector ultrametric space. If x ∈ X and p ∈ P \ {0}, the ball
Bx, p centered at x with radius p is defined as
B x, p : y ∈ X : ρ d x, y ≤ ρ p .
1.12
The unital-normal vector ultrametric space X, d, P is called spherically complete if every chain
of balls with respect to inclusion has a nonempty intersection.
4
Abstract and Applied Analysis
The following lemma may be easily obtained.
Lemma 1.4. Let X, d, P be a unital-normal vector ultrametric space.
1 If a, b ∈ X, 0
p and b ∈ Ba, p, then Ba, p Bb, p.
2 If a, b ∈ X, 0 ≺ p
q, then either Ba, p
Bb, q ∅ or Ba, p ⊆ Bb, q.
Definition 1.5. Let X, d, P be a unital-normal vector ultrametric space. A mapping f : X →
P\{0} is said to be modular locally constant provided that for any x ∈ X and any y ∈ Bx, fx
one has ρfx ρfy.
2. Main Theorem
Theorem 2.1. Let X, d, P be a spherically complete unital-normal vector ultrametric space and
T : X → X be a mapping such that for every x, y ∈ X, x
/ y, either
ρ d Tx, Ty < max ρdx, Tx, ρ d y, Ty
2.1
ρ d Tx, Ty ≤ ρ d x, y .
2.2
or
Then there exists a subset B of X such that T : B → B and the mapping
fx dx, Tx,
x ∈ B
2.3
is modular locally constant.
Proof. Let E {Ba }a∈X where Ba Ba, da, Ta. Consider the partial order ⊑ on E defined
by
Ba ⊑ Bb
iff Bb ⊆ Ba ,
2.4
where a, b ∈ X. If E1 is any chain in E, then the spherical completeness of X, d, P implies
that the intersection Ω of elements of E1 is nonempty.
Abstract and Applied Analysis
5
Suppose that 2.1 holds. Let b ∈ Ω and Ba ∈ E1 . Obviously b ∈ Ba , so ρda, b ≤ ρda, Ta.
For any x ∈ Bb , we have
ρdx, b ≤ ρdb, Tb
≤ max ρdb, a, ρda, Ta, ρdTa, Tb
< max ρdb, a, ρda, Ta, max ρda, Ta, ρdb, Tb
by 2.1
2.5
≤ max ρdb, a, ρda, Ta, ρdb, Tb
≤ max ρda, Ta, ρdb, Tb
ρda, Ta,
ρdx, a ≤ max ρdx, b, ρdb, a ≤ ρda, Ta.
So for every Ba ∈ E1 , Bb ⊆ Ba ; that is, Bb is an upper bound in E for the family E1 . By Zorn’s
lemma, there exists a maximal element in E1 , say Bz . If b ∈ Bz , ρdb, z ≤ ρdz, Tz, and
we get
ρdb, Tb ≤ max ρdb, z, ρdz, Tz, ρdTz, Tb
by 2.1
< max ρdb, z, ρdz, Tz, max ρdz, Tz, ρdb, Tb
≤ max ρdb, z, ρdz, Tz, ρdb, Tb
2.6
≤ max ρdz, Tz, ρdb, Tb
ρdz, Tz.
Then
ρdb, Tb ≤ ρdz, Tz.
2.7
Since b ∈ Bb ∩ Bz , we have Bb ⊆ Bz by Lemma 1.4. But Tb ∈ Bb , so T : Bz → Bz . Now we show
that ρfb ρfz for every b ∈ Bz . It is clear that ρdb, Tb ≤ ρdz, Tz, for all b ∈ Bz .
Suppose ρdb, Tb < ρdz, Tz for some b ∈ Bz . We have ρdb, z ≤ ρdz, Tz, and
ρdz, Tz ≤ max ρdz, b, ρdb, Tb, ρdTb, Tz
by 2.1
< max ρdb, z, ρdb, Tb, max ρdb, Tb, ρdz, Tz
≤ max ρdb, z, ρdb, Tb, ρdz, Tz
2.8
≤ max ρdb, Tb, ρdz, Tz
ρdz, Tz.
which is a contradiction. Thus f is modular locally constant on Bz .
6
Abstract and Applied Analysis
Suppose that 2.2 holds. As above, let b ∈ Ω and Ba ∈ E1 . Obviously b ∈ Ba , so ρda, b ≤
ρda, Ta. For any x ∈ Bb , we have
ρdx, b ≤ ρdb, Tb
≤ max ρdb, a, ρda, Ta, ρdTa, Tb
by 2.2
≤ max ρdb, a, ρda, Ta
2.9
ρda, Ta.
Thus
ρdx, a ≤ max ρdx, b, ρdb, a ≤ ρda, Ta,
ρdx, a ≤ max ρdx, b, ρdb, a ≤ ρda, Ta.
2.10
So, for every Ba ∈ E1 , Bb ⊆ Ba ; that is, Bb is an upper bound for the family E1 . Again, by Zorn’s
lemma there exists a maximal element in E1 , say Bz . For any b ∈ Bz , we have
ρdb, Tb ≤ max ρdb, z, ρdz, Tz, ρdTz, Tb
≤ max ρdb, z, ρdz, Tz, ρdz, b
by 2.2
2.11
ρdz, Tz.
This implies that b ∈ Bb ∩ Bz , and Lemma 1.4 gives Bb ⊆ Bz . Since Tb ∈ Bb , so T : Bz → Bz .
If z Tz, then fx 0 on Bz and this yields the result. If z /
Tz, we show that
ρfb ρfz for every b ∈ Bz . Since ρdb, Tb ≤ ρdz, Tz for any b ∈ Bz , let us
suppose that for some b ∈ Bz , ρdb, Tb < ρdz, Tz. So ρdb, z ≤ ρdz, Tz and
ρdz, Tz ≤ max ρdz, b, ρdb, Tb, ρdTb, Tz
≤ max ρdb, z, ρdb, Tb, ρdz, b
by 2.2
2.12
ρdb, z,
thus ρdb, z ρdz, Tz. But ρdb, z ρdz, Tz > ρdb, Tb implies that z ∈ Bz ,
but z ∈
/ Bb and hence Bb Bz which contradicts the maximality of Bz . This completes the
proof.
In the following, we assume that X, d, P is a spherically complete unital-normal
vector ultrametric space.
Corollary 2.2. Let T : X → X be a mapping such that for all x, y ∈ X, x /
y,
ρ d Tx, Ty < max ρ d y, Tx , ρ d x, Ty
.
2.13
Then there exists a subset B of X such that T : B → B and the mapping f defined in 2.3 is modular
locally constant.
Abstract and Applied Analysis
7
Proof. Since
ρ d y, Tx ≤ max ρ d y, x , ρdx, Tx ,
ρ d x, Ty ≤ max ρ d x, y , ρ d y, Ty
,
2.14
for all x, y ∈ X, x /
y, we get
ρ d x, y ≤ max ρdx, Tx, ρ d Tx, Ty , ρ d Ty, y
2.15
for all x, y ∈ X, x /
y. Now, if
max ρdx, Tx, ρ d y, Ty
< ρ d Tx, Ty ,
2.16
then
ρ d Tx, Ty < max ρ d y, Tx , ρ d x, Ty
by 2.13
≤ max ρ d x, y , ρdx, Tx, ρ d y, Ty
by 2.14
≤ max ρdx, Tx, ρ d Tx, Ty , ρ d Ty, y
ρ d Tx, Ty ,
by 2.16
2.17
which is a contradiction. Thus ρdTx, Ty ≤ max{ρdx, Tx, ρdy, Ty}, and so
ρ d x, y ≤ max ρdx, Tx, ρ d y, Ty
.
2.18
Therefore
ρ d Tx, Ty < max ρ d y, Tx , ρ d x, Ty
,
≤ max ρ d x, y , ρdx, Tx, ρ d y, Ty
≤ max ρdx, Tx, ρ d y, Ty
.
by 2.13
by 2.14
by 2.18
2.19
Now, Theorem 2.1 completes the proof.
Corollary 2.3. Let T : X → X be a mapping such that for all x, y ∈ X, x /
y,
ρ d Tx, Ty < ρ d x, y .
2.20
Then there exists a subset B of X such that T : B → B and the mapping f defined in 2.3 is modular
locally constant.
8
Abstract and Applied Analysis
Proof. We have
ρ d x, y ≤ max ρdx, Tx, ρ d Tx, Ty , ρ d Ty, y
< max ρ d x, y , ρdx, Tx, ρ d Ty, y
by 2.20
≤ max ρdx, Tx, ρ d y, Ty
,
2.21
for all x, y ∈ X, x /
y. Again, Theorem 2.1, completes the proof.
Acknowledgments
The authors would like to thank the referee for his/her valuable comments on this paper. The
second author’s research was in part supported by a grant from IPM No. 89470128.
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