Some spaces of sequences of interval numbers defined by a modulus function

Research Paper

Some spaces of sequences of interval numbers defined by a modulus function

Ayhan Esi

^∗

, Sibel Yasemin

Adiyaman University, Science and Art Faculty, Department of Mathematics, 02040, Adiyaman, Turkey

*Corresponding author E-mail: [email protected]

Copyright c°2014 Ayhan Esi, Sibel Yasemin. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The main purpose of the present paper is to introduce c

o

(f, p, s) , c (f, p, s) l

∞

(f, p, s) and l

p

(f, p, s) of sequences of interval numbers defined by a modulus function. Furthermore, some inclusion theorems related to these spaces are given.

Keywords: Complete space, interval number, modulus function.

1. Introduction

Interval arithmetic was first suggested by Dwyer [8] in 1951. Development of interval arithmetic as a formal system and evidence of its value as a computational device was provided by Moore [11] in 1959 and Moore and Yang [13] 1962. Furthermore, Moore and others [9], [10], [11] and [14] have developed applications to differential equations.

Chiao in [6] introduced sequence of interval numbers and defined usual convergence of sequences of interval number. S¸eng¨on¨ ul and Eryilmaz in [7] introduced and studied bounded and convergent sequence spaces of interval numbers and showed that these spaces are complete metric space. Recently, Esi in [1], [2], [3], [4] and [5] defined and studied different properties of interval numbers.

We denote the set of all real valued closed intervals by IR. Any elements of IR is called interval number and denoted by A = [x

l

, x

r

] . Let x

l

and x

r

be first and last points of x interval number, respectively. For A

1

, A

2

∈IR, we have A

1

= A

2

⇔ x

1l

=x

2l

,x

1r

=x

2r

. A

1

+ A

2

= {x ∈ R : x

1l

+ x

2l

≤ x ≤ x

1r

+ x

2r

} ,and if α ≥ 0, then αA = {x ∈ R : αx

1l

≤ x ≤ αx

1r

} and if α < 0, then αA = {x ∈ R : αx

1r

≤ x ≤ αx

1l

} ,

A

1

.A

2

= {x ∈ R : min {x

1l

.x

2l

, x

1l

.x

2r

, x

1r

.x

2l

, x

1r

.x

2r

} ≤ x ≤ max {x

1l

.x

2l

, x

1l

.x

2r

, x

1r

.x

2l

, x

1r

.x

2r

}} . The set of all interval numbers IR is a complete metric space defined by

d ¡ A

1

, A

2

¢ = max {|x

1l

− x

2l

| , |x

1r

− x

2r

|} [12].

In the special case A

1

= [a, a] and A

2

= [b, b] , we obtain usual metric of R. Let us define transformation f : N → R by k → f (k) = A, A = ¡

A

_k

¢

. Then A = ¡ A

_k

¢

is called sequence of interval numbers. The A

_k

is called k

^th

term of sequence A = ¡

A

k

¢ . w denotes the set of all interval numbers with real terms and the algebraic properties of w can be found in [14].

Now we give the definition of convergence of interval numbers:

Definition 1.1 ([6]) A sequence A = ¡ A

k

¢ of interval numbers is said to be convergent to the interval number x

o

if for each ε > 0 there exists a positive integer k

o

such that d ¡ A

k

, A

o

¢ < ε for all k ≥ k

o

and we denote it by lim

k

A

k

= A

o

.

Thus, lim

k

A

k

= A

o

⇔ lim

k

A

kl

= A

ol

and lim

k

A

kr

= A

or

.

We recall that modulus function is a function f : [0, ∞) → [0, ∞) such that (a) f (x) = 0 if and only if x = 0,

(b) f (x + y) ≤ f (x) + f (y) for all x, y ≥ 0, (c) f is increasing,

(d) f is continuous from the right at zero.

It follows from (a) and (d) that f must be continuous everywhere on [0,∞).

Let p = (p

k

) be a bounded sequence of strictly positive real numbers. If H = sup

k

p

k

, then for any complex numbers a

k

and b

k

|a

k

+ b

k

|

^pk

≤ C (|a

k

|

^pk

+ |b

k

|

^pk

) (1)

where C = max(1, 2

^H−1

).

Definition 1.2 A set of X sequences of interval numbers is said to be solid (or normal) if ¡ B

k

¢ ∈ X whenever d ¡

B

k

, 0 ¢

≤ d ¡ A

k

, 0 ¢

for all k ∈ N, for some ¡ A

k

¢ ∈ X.

In this paper, we essentially deal with the metric spaces c

o

(f, p, s) , c (f, p, s) , l

∞

(f, p, s) and l

p

(f, p, s) of sequences of interval numbers defined by a modulus function which are generalization of the metric spaces c

o

, c, l

∞

and l

p

of sequences of interval numbers. We state and prove some topological and inclusion theorems related to those sets.

2. Main results

Let f be a modulus function and s ≥ 0 be a real number and p = (p

k

) be a sequence of strictly positive real numbers. We introduce the sets of sequences of interval numbers as follows:

c

o

(f, p, s) =

½ A = ¡

A

k

¢ : lim

k

k

^−s

£ f ¡

d ¡

A

k

, 0 ¢¢¤

p_k

= 0

¾ ,

c (f, p, s) =

½ A = ¡

A

k

¢ : lim

k

k

^−s

£ f ¡

d ¡ A

k

, A

o

¢¢¤

pk

= 0

¾ ,

l

∞

(f, p, s) =

½ A = ¡

A

k

¢ : sup

k

k

^−s

£ f ¡

d ¡

A

k

, 0 ¢¢¤

p_k

< ∞

¾

and

l

p

(f, p, s) = (

A = ¡ A

k

¢ : X

k

k

^−s

£ f ¡

d ¡

A

k

, 0 ¢¢¤

p_k

< ∞ )

.

Now, we may begin with the following theorem.

Theorem 2.1 The sets c

o

(f, p, s) , c (f, p, s) , l

∞

(f, p, s) and l

p

(f, p, s) of sequences of interval numbers are closed under the coordinatewise addition and scalar multiplication.

Proof. It is easy, so we omit the detail.

Theorem 2.2 The sets c

_o

(f, p, s) , c (f, p, s) , l

_∞

(f, p, s) and l

_p

(f, p, s) of sequences of interval numbers are com- plete metric spaces with respect to the metrics

d

∞

(A, B) = sup

k

k

^−s

£ f ¡

d ¡ A

k

, A

o

¢¢¤

^pk

M

and

d

p

(A, B) =

( X

k

k

^−s

£ f ¡

d ¡ A

k

, B

k

¢¢¤

^pk

)

¹

M

respectively, where A = ¡ A

k

¢ and B = ¡ B

k

¢ are elements of the sets c

o

(f, p, s) , c

o

(f, p, s) , l

∞

(f, p, s) and l

p

(f, p, s) and M = max (1, sup

_k

p

k

= H)

Proof. We consider only the space c

o

(f, p, s) , since the proof is similar for the spaces c (f, p, s) , l

∞

(f, p, s) and l

_p

(f, p, s) . One can easily establish that d

_∞

defines a metric on c

_o

(f, p, s) which is a routine verification, so we omit it. It remains to prove the completeness of the space c

o

(f, p, s) . Let ³

A

ⁱ

´

be any Cauchy sequence in the space c

o

(f, p, s) , where A

ⁱ

=

³

A

⁽ⁱ⁾_o

, A

⁽ⁱ⁾₁

, A

⁽ⁱ⁾₂

, ...

´

. Then, for a given ε > 0 there exists a positive integer n

o

(ε) such that

d

∞

³ A

ⁱ

, A

^j

´

= sup

k

k

^−s

h

f

³ d

³

A

⁽ⁱ⁾_k

, A

^(j)_k

´´i

^pk

M

< ε (2)

for all i, j > n

o

(ε) . We obtain for each fixed k ∈ N from (2) that

k

^−s

h f ³

d ³

A

⁽ⁱ⁾_k

, A

^(j)_k

´´i

^pk

M

< ε (3)

for all i, j > n

o

(ε) . (3) means that

i,j→∞

lim k

^−s

h f ³

d ³

A

⁽ⁱ⁾_k

, A

^(j)_k

´´i

^pk

M

= 0. (4)

Since k

^−s

6= 0 for all k ∈ N and f is continuous, we have from (4) that f

·

i,j→∞

lim

³ d

³

A

⁽ⁱ⁾_k

, A

^(j)_k

´´¸

= 0. (5)

Therefore, since f is a modulus function one can derive by (5) that

i,j→∞

lim d

³

A

⁽ⁱ⁾_k

, A

^(j)_k

´

= 0 (6)

which means that

³ A

⁽ⁱ⁾_k

´

is a Cauchy sequence in IR for every fixed k ∈ N. Since IR is complete, it converges, say A

⁽ⁱ⁾_k

→ A

k

as i → ∞. Using these infinitely many limits, we defined the interval sequence ¡

A

k

¢ = ¡

A

o

, A

1

, A

2

, ... ¢ . Let us pass to limit firstly as j → ∞ and nextly taking supremum over k ∈ N in (3) we obtain d

∞

³ A

ⁱ

, A

k

´

≤ ε.

Since

³ A

⁽ⁱ⁾_k

´

∈ c

o

(f, p, s) for each i ∈ N, there exists k

o

∈ N such that

k

^−s

h

f

³ d

³ A

⁽ⁱ⁾_k

, 0

´´i

_pk

< ε

for every k ≥ k

o

(ε) and for each fixed i ∈ N. Therefore, since k

^−s

£

f ¡ d ¡

A

k

, 0 ¢¢¤

pk

≤ Ck

^−s

h

f

³ d

³ A

⁽ⁱ⁾_k

, A

k

´´i

_pk

+ Ck

^−s

h

f

³ d

³ A

⁽ⁱ⁾_k

, 0

´´i

_pk

hold by triangle inequality for all i, k ∈ N, where C = max(1, 2

^H−1

). Now for all k ≥ k

_o

(ε) , we have k

^−s

£

f ¡ d ¡

A

k

, 0 ¢¢¤

pk

≤ 2ε.

This shows that ¡ A

k

¢ ∈ c

o

(f, p, s) . Since

³ A

⁽ⁱ⁾_k

´

was an arbitrary Cauchy sequence, the the space c

o

(f, p, s) is complete.

Theorem 2.3 The spaces c

o

(f, p, s) , l

∞

(f, p, s) and l

p

(f, p, s) are solid.

Proof. Let X (f, p, s) denotes the anyone of the spaces c

o

(f, p, s) , l

∞

(f, p, s) and l

p

(f, p, s) . Suppose that d ¡

B

k

, 0 ¢

≤ d ¡ A

k

, 0 ¢

(7) holds for some ¡

A

k

¢ ∈ X (f, p, s) . Since the modulus function is increasing, one can easily see by (7) that

lim

k

k

^−s

£ f ¡

d ¡

B

k

, 0 ¢¢¤

pk

≤ lim

k

k

^−s

£ f ¡

d ¡

A

k

, 0 ¢¢¤

pk

,

sup

k

k

^−s

£ f ¡

d ¡

B

k

, 0 ¢¢¤

pk

≤ sup

k

k

^−s

£ f ¡

d ¡

A

k

, 0 ¢¢¤

pk

and X

k

k

^−s

£ f ¡

d ¡

B

k

, 0 ¢¢¤

pk

≤ X

k

k

^−s

£ f ¡

d ¡

A

k

, 0 ¢¢¤

pk

.

The above inequalities yield the desired that ¡ B

k

¢ ∈ X (f, p, s) .

Theorem 2.4 Let inf

k

p

k

= h > 0. Then a) ¡

A

k

¢ ∈ c implies ¡ A

k

¢ ∈ c (f, p, s) , b) ¡

A

k

¢ ∈ c (p, s) implies ¡ A

k

¢ ∈ c (f, p, s) , c) β = lim

tf (t)

t

> 0 implies c (p, s) = c (f, p, s) . Proof. a) Suppose that ¡

A

k

¢ ∈ c. Then lim

k

d ¡ A

k

, A

o

¢ = 0. As f is modulus function, then

lim

k

f ¡ d ¡

A

k

, A

o

¢¢ = f

· lim

k

¡ d ¡ A

k

, A

o

¢¢ ¸

= f (0) = 0.

As inf

k

p

k

= h > 0, then lim

k

£ f ¡ d ¡

A

k

, A

o

¢¢¤

_h

= 0. So, for 0 < ε < 1, ∃k

o

such that for all k > k

o

£ f ¡ d ¡

A

k

, A

o

¢¢¤

_h

<

ε < 1, an as p

k

≥ h for all k,

£ f ¡ d ¡

A

k

, A

o

¢¢¤

pk

≤ £ f ¡

d ¡ A

k

, A

o

¢¢¤

h

< ε < 1, then we obtain

lim

k

£ f ¡ d ¡

A

k

, A

o

¢¢¤

_pk

= 0.

As (k

^−s

) is bounded, we can write lim

k

k

^−s

£

f ¡ d ¡

A

k

, A

o

¢¢¤

pk

= 0.

Therefore ¡ A

k

¢ ∈ c (f, p, s) . b) Let ¡

A

k

¢ ∈ c (p, s) , then lim

k

k

^−s

¡ d ¡

A

k

, A

o

¢¢

pk

= 0. Let ε > 0 and choose δ with 0 < δ < 1, such that f (t) < ε for 0 ≤ t ≤ δ. Now we write

I

ı

= ©

k ∈ N : d ¡ A

k

, A

o

¢ ≤ δ ª

and I

2

= ©

k ∈ N : d ¡ A

k

, A

o

¢ > δ ª . For d ¡

A

k

, A

o

¢ > δ

d ¡ A

_k

, A

_o

¢

< d ¡ A

_k

, A

_o

¢

δ

⁻¹

< 1 + £¯

¯d ¡

A

_k

, A

_o

¢¯

¯ ¤

where k ∈ I

2

and [|t|] denotes the integer of t. By using properties of modulus function, for d ¡ A

k

, A

o

¢ > δ, we have

f £ d ¡

A

k

, A

o

¢¤ < 1 + £¯

¯d ¡ A

k

, A

o

¢¯ ¯ ¤

f (1) ≤ 2f (1) d ¡ A

k

, A

o

¢ δ

⁻¹

.

For d ¡ A

k

, A

o

¢ ≤ δ, f £ d ¡

A

k

, A

o

¢¤ < ε, where k ∈ I

1

. Hence

k

^−s

£ f ¡

d ¡ A

k

, A

o

¢¢¤

pk

= k

^−s

£ f ¡

d ¡ A

k

, A

o

¢¢¤

pk

c

k∈I1

+ k

^−s

£ f ¡

d ¡ A

k

, A

o

¢¢¤

pk

c

k∈I2

≤ k

^−s

ε

^H

+ £

2f (1) δ

⁻¹

¤

H

k

^−s

£ f ¡

d ¡ A

k

, A

o

¢¢¤

pk

→ 0 as k → ∞.

Then we obtain ¡ A

k

¢ ∈ c (f, p, s) .

c) In (b), it was shown that c (p, s) ⊂ c (f, p, s) . We must show that c (f, p, s) ⊂ c (p, s) . For any modulus function, the existence of positive limit given by β in Maddox[16, Proposition 1]. Now, β > 0 and let ¡

A

k

¢ ∈ c (f, p, s) . As β > 0 for every t > 0, we write f (t) ≥ βt. From this inequality, it is easy seen that ¡

A

k

¢ ∈ c (p, s) .

Theorem 2.5 Let f and g be two modulus functions and s, s

1

, s

2

≥ 0. Then a) c (f, p, s) ∩ c (g, p, s) ⊂ c (f + g, p, s) ,

b) s

1

≤ s

2

implies c (f, p, s

1

) ⊂ c (f, p, s

2

) . Proof. a) Let ¡

A

k

¢ ∈ c (f, p, s) ∩ c (g, p, s) . From (1), we have

£ (f + g) ¡ d ¡

A

k

, A

o

¢¢¤

pk

= £ f ¡

d ¡ A

k

, A

o

¢¢ + g ¡ d ¡

A

k

, A

o

¢¢¤

pk

≤ C £ f ¡

d ¡ A

k

, A

o

¢¢¤

p_k

+ C £ g ¡

d ¡ A

k

, A

o

¢¢¤

p_k

. As (k

^−s

) is bounded, we can write

k

^−s

£

(f + g) ¡ d ¡

A

k

, A

o

¢¢¤

pk

≤ Ck

^−s

£ f ¡

d ¡ A

k

, A

o

¢¢¤

pk

+ Ck

^−s

£ g ¡

d ¡ A

k

, A

o

¢¢¤

pk

. Hence we obtain ¡

A

k

¢ ∈ c (f + g, p, s) .

b) Let s

1

≤ s

2

. Then k

^−s2

≤ k

^−s1

for all k ∈ N. Hence k

^−s2

£

f ¡ d ¡

A

k

, A

o

¢¢¤

_pk

≤ k

^−s1

£ f ¡

d ¡ A

k

, A

o

¢¢¤

_pk

. This inequality implies that c (f, p, s

1

) ⊂ c (f, p, s

2

) . Theorem 2.6 Let f be a modulus function, then

a) l

∞

⊂ l

∞

(f, p, s) ,

b) If f is bounded then l

∞

(f, p, s) = w.

Proof. a) Let ¡ A

_k

¢

∈ l

_∞

. Then there exists a positive integer M such that d ¡ A

_k

, 0 ¢

≤ M. Since f is bounded then f £

d ¡ A

k

, 0 ¢¤

is also bounded. Hence k

^−s

£

f ¡ d ¡

A

k

, 0 ¢¢¤

pk

≤ k

^−s

[M f (1)]

^pk

≤ k

^−s

[M f (1)]

^H

< ∞.

Therefore ¡ A

k

¢ ∈ l

∞

(f, p, s) .

b) If f is bounded, then for any ¡ A

_k

¢

∈ w, k

^−s

£

f ¡ d ¡

A

k

, 0 ¢¢¤

pk

≤ k

^−s

L

^pk

≤ k

^−s

L

^H

< ∞.

Hence l

∞

(f, p, s) = w.

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[3]

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[4]

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[9]

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