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ISSN 2229 – 5046
ENTIRE RATE SEQUENCE SPACE OF INTERVAL NUMBERS
Dr. D. CHRISTIAJEBA KUMARI
Assistant professor, Department of Mathematics,
Pope’s College, Sawyerpuram-628251, Thoothukudi District,
Affiliated to Manonmaniam Sundaranar University,
Tirunelveli-627012, Tamilnadu, India.
(Received On: 17-04-18; Revised & Accepted On: 13-08-18)
ABSTRACT
I
n this paper we introduced the new concept of interval valued sequence spaceΓ
π(
IR
)
where( )
π
k is a sequence ofpositive numbers. We present the different properties like completeness, solidness, AB property etc.
2000 Mathematics subject classification: 46A 45.
Keywords and Phrases: Banach space, AB space, AK property, interval numbers.
1. INTRODUCTION
The power of interval arithmetic lies in its implementation on computers. In particular, outwardly rounded interval arithmetic allows rigorous enclosures for the ranges of operations and functions. This makes a qualitative difference in scientific computations, since the results are now intervals in which the exact result must lie. It also enables the use of computations for proving automated theorem.
Interval arithmetic is a tool in numerical computing where the rules for the arithmetic of intervals are explicitly stated. It was first suggested by P.S.Dwyer [10] in 1951. Development of interval arithmetic as a formal system and evidence of its value as a computational device was provided by R.E.Moore [34], [35] in 1959 and 1962.
A set consisting of a closed interval of real numbers
x
such thata
≤
x
≤
b
is called an interval number. A real interval can also be considered as a set. We denote the set of all real valued closed intervals byI
ℜ
. Any element ofℜ
I
may be called closed interval and denoted byx
ˆ
. That is,x
ˆ
=
[
x
l,
x
r]
=
{
x
∈
ℜ
:
x
l≤
x
≤
x
r}.
An intervalnumber
x
ˆ
is a closed subset of real numbers. Letx
land
x
r be respectively referred to as the infimum (lower bound) and supremum (upper bound) of the interval numberx
ˆ
. Ifx
ˆ
=
[
0
,
0
]
, thenx
ˆ
is said to be a zero interval. It is denotedby
0ˆ
.For
x
1,
x
2∈
I
ℜ
, we definex
1=
x
2 if and only ifx
1l=
x
2land
x
1r=
x
2r)}
:
{
1 2 1 22
1
x
x
x
lx
lx
x
rx
rx
+
=
∈
ℜ
+
≤
≤
+
)}
,
,
,
max(
)
,
,
,
min(
:
{
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 22
1
x
x
x
lx
lx
lx
rx
rx
lx
rx
rx
x
lx
lx
lx
rx
rx
lx
rx
rx
×
=
∈
ℜ
≤
≤
The set of all interval numbers
I
ℜ
is a complete metric space defined by}
,
max{
)
,
(
x
1x
2x
1lx
2lx
1rx
2rd
=
−
−
In the special case
x
1=
[
a
,
a
]
andx
2=
[
b
,
b
]
, we obtain usual metric ofℜ
.Corresponding Author: Dr. D. ChristiaJebakumari
Assistant professor, Department of Mathematics,
Pope’s College, Sawyerpuram-628251, Thoothukudi District,
Let us define transformation f: N
→ ℜ
, k→
f(k)=x
k,
thenx
=
(
x
k)
is called sequence of interval numbers.x
k iscalled kth term of sequence
x
=
(
x
k)
,i
ω
denotes the set of all interval numbers with real terms and the algebraic properties ofω
i are in[7].A sequence
x
=
(
x
k)
of interval numbers is said to be convergent to the interval numberx
0if for eachε
>0 thereexists a positive integer k0 such that
d
(
x
k,
x
0)
<
ε
for all k≥
k0 and we denote it bylim
x
kx
0k
=
. Equivalently0
lim
x
kx
k
=
ifflim
kx
kl=
x
0land
lim
kx
kr=
x
0r.2. MAIN RESULTS
The entire sequence space of symmetric interval numbers is denoted by
Γ
π(
IR
)
where( )
π
k is a sequence of positive numbers and the set of functional of interval numbers is denoted byΛ
π(
IR
)
.
=
∈
= =
Γ ,~0 ~0
~ lim : ) ( ~ ~ ) (
k k k
k
k x
D IR
w x x IR
π
π
π
∞ <
∈
= =
Λ ,~0
~ sup : ) ( ~ ~ ) (
k k k
k
k x
D IR
w x x IR
π
π
π
where
−
−
=
kk k k k
k k k
k k
k
k
y
x
y
x
y
x
D
1 1
max
~
,
~
π
π
π
π
The metricd~is defined by
=
−
−
=
k k
k k
k k
k k k k
k k k
k k k
k
k
y
x
y
x
y
D
x
y
x
d
π
π
π
π
π
π
~
,
~
sup
max
sup
~
,
~
~
1 1
(2.1)
which satisfies the metric space axioms .
Theorem 2.1: The sequence space of interval numbers
Γ
π(
IR
)
is a complete metric space with respect to the metric defined by (2.1).Proof: Let
( ) ( )
n n
x
π
~
be ainterval number fundamental sequence in
Γ
π(
IR
)
.Then for a givenε
>
0
there exists a positive integern
0 such that( ) ( )
( ) ( )
( ) ( )
( ) ( )
ε
π
π
π
π
<
=
m k
m k n k
n k k
m k
m k n k
n
k
x
D
x
x
x
d
~
,
~
sup
~
,
~
~
for all
n
,
m
≥
n
0For each
k
, we have( )
( ) ( )
( )
ε
π
π
<
m k
m k n k
n
k
x
x
D
~
,
~
for all
n
,
m
≥
n
0( ) ( ) ( ) ( )
ε
π
π
<
−
−
kk m k n k k
k m n
k
x
x
x
x
1 1,
max
for alln
,
m
≥
n
0( ) ( ) ( ) ( )
k k
k m k n k k
k
k m k n
k
x
x
and
x
x
ε
π
ε
π
<
−
<
−
1 1for all
n
,
m
≥
n
0This leads to the fact
( ) ( )n k
n k
x
π
~
is a interval number fundamental sequence in
IR
.SinceIR
is a complete metric space,( ) ( )n k
n k
x
π
~
is convergent
( ) ( )
k k n k
n k n
x
x
π
π
~
~
Hence
( )
( )
π
ε
π
<
k k n k
n k k
x
x
D
~
,
~
sup
So
( )
( )
(
)
~
~
IR
in
n
as
x
x
k k n k
n k
π
π
π
→
→
∞
Γ
We have to show that
(
)
~
~
x
x
IR
k k
k
π
∈
Γ
π
=
Since
~
x
k∈
Γ
π(
IR
)
, we have( )
( )
ε
π
<
0
~
,
~
~
n k
n k
x
d
Consider
( ) ( )
( ) ( )
ε
ε
ε
π
π
π
π
π
2
0
~
,
~
sup
~
,
~
sup
0
~
,
~
sup
0
~
,
~
~
=
+
<
+
≤
=
n k
n k k
k k n k
n k k
k k k
k k
x
D
x
x
D
x
D
x
d
Hence
( )
~
x
k∈
Γ
π(
IR
)
.This completes the proof.Theorem 2.2: A necessary and sufficient condition that
∑
,
~
0
~
~
k k
k
k
y
x
D
π
π
should be convergent for every
k k
x
π
~
in
which
,
~
0
0
~
lim
=
k k k
x
D
π
is that
0
~
,
~
k k
y
D
π
should be bounded.Proof: Suppose
0 ~ , ~
k k
y D
π
is bounded. Then there is an M so that0
1
~
,
~
≥
≤
k
for
M
y
D
k k
π
.Since
,
~
0
0
~
~
→
k k
x
D
π
ask
→
∞
,
there exists a positive integerk
0so that0
,
2
1
0
~
,
~
k
k
M
x
D
k
k
≤
≥
π
k k k k k k k
k k k
k
y
D
x
D
y
x
D
,
0
~
]
~
[
]
0
~
,
~
[
]
0
~
,
~
~
[
≤
π
π
π
π
k k k
M
M
2
1
2
1
=
<
Then
∑
kk k k k
y
x
D
,
~
0
~
~
π
π
convergesConversely,
Suppose
0
~
,
~
k k
y
D
π
is not bounded .Then there is an increasing sequence{
k
p}
of integers such that,..
2
,
1
,
0
~
,
~
=
≥
p
p
y
D
p p
k k
That is,
,
~
0
,
1
,
2
,..
~
=
≥
p
p
y
D
pp
p
p k
k
k k
π
Take
≠
=
=
p p k
k k
k
k
if
k
k
if
p
x
]
0
,
0
[
1
,
0
~
π
Then
,
0
~
0
~
lim
=
k k k
x
D
π
But
~
,
0
~
~
,
~
0
1
( )
11
.
1
1
=
=
=
pp p p p p p p p
p
k k k k k k k
k k
k k k
k
k
p
p
p
p
y
D
x
D
π
π
fork
=
k
pSo that
∑
kk k
k
k
y
x
D
,
0
~
~
~
π
π
does not converge.Hence
0
~
,
~
k k
y
D
π
is bounded.Theorem 2.3: The sequence spaces of interval numbers
Γ
π(
IR
)
andΛ
π(
IR
)
are solid.Proof: Consider,
Γ
π(
IR
)
. Now let
≤
0
~
,
~
~
0
~
,
~
~
k k
k
k
d
x
y
d
π
π
for allk
∈
N
and for some~
x
∈
Γ
π(
IR
)
.Then,
{
k k}
{
k k k k}
k
k
y
x
x
y
1 1max
1 1max
≤
, k k kk
x
and
y
x
y
≤
≤
≤
0
~
,
~
0
~
,
~
k k k
k
x
D
y
D
π
π
0
0
~
,
~
lim
0
~
,
~
lim
=
≤
∞ → ∞
→
k k k
k k k
x
D
y
D
π
π
It is clear that
~
y
k∈
Γ
π(
IR
)
Therefore
Γ
π(
IR
)
is solid.Similarly, it can be proved that
Λ
π(
IR
)
is solid.Theorem 2.4: The sequence of interval numbers
(
e
~
1,
e
~
2....
~
e
k,..)
is schauder base forΓ
π(
IR
)
, where~
e
k=
{
~
0
,
~
0
,...[
1
,
1
],
~
0
,...}
.
Proof: Let
(
)
~
~
x
IR
x
k k k
π
∈
Γ
π
=
. Therefore for everyε
>
0
there exists a positive integern
∈
N
such thatk
≥
n
,ε
π
π
<
=
0
~
,
~
sup
0
~
,
~
~
k k k
k
k
x
D
x
d
It is enough to show that
,
~
0
0
~
~
~
~
lim
=
−
∑
∞ →
k k k k
k k
x
e
x
d
Now,
]
0
~
]),
,
],...[
,
[
],
,
([
],...)
,
],...[
,
[
],
,
[([
~
0
~
,
~
~
~
~
2 2 1 1 2 2 1 1 1 n n k k n k k k k kk
e
x
d
x
x
x
x
x
x
x
x
x
x
x
x
x
d
=
−
−
∑
=π
π
0
]
~
]),
,
[
],
,
,...[
0
~
,
0
~
[([
~
2 2 11 + + +
+
=
d
x
nx
nx
nx
n
{
}
∞
→
→
=
+≥
x
x
as
n
k k k k n k
0
~
max
sup
1 11
We have ,
∑
∞ =
=
1~
~
~
k k k k kk
e
x
x
π
π
(2.2)Let us show the uniqueness of the representation given by (5.2) for
(
)
~
~
x
IR
x
k k
π
π
∈
Γ
=
Suppose that there exists a representation
∑
∞ =
=
1~
~
~
~
k k k k kk
e
y
x
x
π
π
, then,∑
∑
= =
−
=
−
n k k k k k k k n k k k kk
y
e
d
x
y
e
x
d
1 10
~
,
~
~
~
~
0
~
,
~
~
~
~
π
π
π
π
∞
→
→
−
−
=
+≥
as
n
x
y
x
y
k k k k k k k k n k0
max
sup
1 11
π
π
∞
→
→
−
→
−
n
as
x
y
and
x
y
k k k k k k k k0
0
1 1π
π
Therefore k k k
k
x
and
y
x
y
=
=
.That is.
~
~
π
π
x
y
=
Theorem 2.5: The
β
dual of the sequence space of interval numbersΓ
π(
IR
)
isΛ
π(
IR
)
.Proof: Let us suppose that
(
)
~
~
y
y
IR
k k
π
π
∈
Λ
=
for every(
)
~
~
x
IR
x
k k
π
π
∈
Γ
=
, then[
]
=
=
∑
∑
∑
= = = n k k k k k k k k k k k k k k k k k n n k k k k k n n k k k k k nx
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
D
x
x
y
y
D
y
x
D
1 1 10
~
,
)
,
,
,
max(
,
)
,
,
,
min(
lim
)
0
~
],
,
[
],
,
[
(
lim
0
~
,
~
~
lim
π
π
1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
lim max
,
,
,
lim max
,
,
,
k k k
n n n n
k k k k k
k k k
n
k k k k
n k n k n k n
k
k k k k k
k k k
n
k k k k
y x
y x
y x
y x
y x
y x
y x
y x
= = = = = = = =
=
≤
∑
∑
∑
∑
∑
∑
∑
∑
=
∑
∑
= = n k n k k k k kn
M
x
x
1 1 1 1
,
max
lim
Where
M
=
max{
M
1,
M
2}
k kk k k k
y
M
y
1 1
1 1
lim
, 0
lim
, 0
, 0
, 0
n n
k k k
n n
k k k k k
k k
k k k k
x
y
x
D
M
D
x
x
MD
M
D
π π
π
π
π
= =
∞ ∞
= =
≤
=
=
< ∞
∑
∑
∑
∑
Therefore, we get
~
x
k~
y
k∈
cs
(
IR
)
.Hence
( )
~
x
k∈
[
Γ
π(
IR
)]
β β π π(
IR
)
⊂
[
Γ
(
IR
)]
Λ
(2.3)Let π β
π
[
(
)
]
~
~
y
y
IR
k
k
∈
Γ
=
, then∑
0
~
,
~
~
k k k
k
y
x
D
π
π
converges for every(
)
~
~
x
x
IR
k k
π
π
∈
Γ
=
By theorem 2.2,
0
~
,
~
k k
y
D
π
is bounded.
0
~
,
~
sup
k k k
y
D
π
exists, then ( ) ~~y y IR
k k
π
π
∈Λ =
)
(
)]
(
[
Γ
πIR
β⊂
Λ
πIR
(2.4)
From equations (2.3) and (2.4), [Γπ(IR)]β =Λπ (IR)
REFERENCES
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4. EsiA., Strongly almost λ-convergence and statistically almost λ-convergence of interval numbers, Scientia Magna,7(2)(2011), 117-122.
5. EsiA., A new class of interval numbers, Journal of Qafqaz University, Mathematics and Computer science,31(2011),98-102.
6. Kuo-Ping, Chiao, Fundamental properties of interval vector max-norm, Tamsui Oxford Journal of Mathematics, 18(2) (2002), 219-233.
7. Markov S., Quasilinear spaces and their relation to vector spaces, Electronic Journal on Mathematics of Computation, 2(1)(2005).
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Source of support: Nil, Conflict of interest: None Declared.
