I- convergent sequence spaces of fuzzy real numbers defined by sequence of modulus functions

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_____________ *

Corresponding author Received March 15, 2014

1

I- CONVERGENT SEQUENCE SPACES OF FUZZY REAL NUMBERS DEFINED

BY SEQUENCE OF MODULUS FUNCTIONS

MANMOHAN DAS1,* AND BIPUL SARMA2

1

Deptt. of Mathematics, Bajali College (Gauhati University), Assam, India

2_{Deptt. of Mathematics, Gauhati University, Assam, India}

permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract: In this article our aim to introduce some new I- convergent sequence spaces of fuzzy real numbers defined by sequence of modulus functions and studies some topological and algebraic properties. Also we establish some inclusion relations.

Keywords: Fuzzy real number, I- convergence, modulus function, sequence of fuzzy real numbers.

2010 AMS Subject Classification: 40A05, 40D25, 46A45, 46E30.

1. INTRODUCTION

The notion of fuzzy sets was introduced by Zadeh [16]. After that many authors have

studied and generalized this notion in many ways, due to the potential of the introduced

notion. Also it has wide range of applications in almost all the branches of studied in

particular science, where mathematics is used. It attracted many workers to introduce

different types of fuzzy sequence spaces.

Bounded and convergent sequences of fuzzy numbers were studied by Matloka [8].

Later on sequences of fuzzy numbers have been studied by Kaleva and Seikkala [2], Tripathy

and Sarma ([13], [14]) and many others.

Eng. Math. Lett. 2014, 2014:8

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I-convergence of real valued sequence was studied at the initial stage by Kostyrko, Šalát and Wilczyński [4] which generalizes and unifies different notions of convergence of

sequences. The notion was further studied by Šalát, Tripathy and Ziman [9].

Let X be a non-empty set, then a non-void class I⊆2X(power set of X ) is called an ideal

if I is additive (i.e. A, B∈I⇒A∪B∈I) and hereditary (i.e. A∈I and B⊆A⇒B∈I). An ideal I

⊆2X is said to be non-trivial if I ≠ 2X. A non-trivial ideal I is said to be admissible if I

contains every finite subset of N. A non-trivial ideal I is said to be maximal if there does not exist any non-trivial ideal J ≠I containing I as a subset.

Let X be a non-empty set, then a non-void class F ⊆2X is said to be a filter in X if φ∉

F ; A, B∈F⇒A ∩B∈F and A∈F, A ⊆ B ⇒B∈F . For any ideal I,there is a filter Ψ(I) corresponding to I, given by

Ψ (I) = {K⊆ N : N \ K∈I }.

A modulus function f is a function from [0,∞) to [0,∞) such that : (i) 𝑓(𝑥) = 0 𝑖𝑓𝑓 𝑥= 0

(ii)𝑓(𝑥+𝑦)≤ 𝑓(𝑥) +𝑓(𝑦) for all 𝑥,𝑦 ≥0.

(iii) 𝑓 is increasing.

(iv) 𝑓 is continous from the right at 0 .

It follows that 𝑓must be continous everywhere on [0,∞) and a modulus function

may be bounded or not bounded .

Let X be a linear metric space. A function p X: →Ris called paranorm if (1) p x( )≥0 for all x∈X

(2) p(− =x) p x( ) for all x∈X

(3) p x( +y)≤ p x( )+ p y( ) for all x y, ∈X

(4) If

( )

λ_n be a sequence of scalars such that λ →_n 0as n→ ∞ and

( )

xn be a sequence

of vectors with p x( n −x)→0 as n→ ∞, then p(λnxn −λx)→0 as n→ ∞.

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2. Definitions and Background

Let D denote the set of all closed and bounded intervals X = [a₁,b₁] on the real line R.

For X = [a₁,b₁]∈D and Y = [a₂,b₂]∈D, define d( X, Y ) by

d( X, Y ) = max ( | a₁- b₁|, |a₂-b₂| ).

It is known that (D, d ) is a complete metric space.

A fuzzy real number X is a fuzzy set on R i.e. a mapping X : R → L(= [0,1] ) associating each real number t with its grade of membership X(t).

The α- level set [ ]X α set of a fuzzy real number X for 0 < α ≤ 1, defined as α

X = { t ∈R : X(t) ≥α}.

A fuzzy real number X is called convex, if X(t) ≥X(s) ∧X(r) = min ( X(s), X(r) ), where

s < t < r.

If there exists t0 ∈R such that X(t0) = 1, then the fuzzy real number X is called normal.

A fuzzy real number X is said to be upper semi- continuous if for each ε > 0, −1

X ([0, a

+ ε )), for all a∈L is open in the usual topology of R.

The set of all upper semi-continuous, normal, convex fuzzy number is denoted by L (R). The absolute value |X| of X ∈L(R) is defined as (see for instance Kaleva and Seikkala [2] )

|X| (t) = max { X(t), X(-t) } , if t ≥0

= 0 , if t < 0 . Let d : L(R) ×L(R) →R be defined by

d ( X, Y ) =

1 0

sup ≤ ≤α d (

α

X ,Yα).

Then d defines a metric on L(R).

A sequence X = (Xk) of fuzzy numbers is a function X from the set N of all positive integers into L(R). The fuzzy number Xk denotes the value of the function at k∈N and is called the k-th term or general term of the sequence. The set of all sequences of fuzzy numbers is denoted by F

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A sequence (Xk) of fuzzy real numbers is said to be convergent to the fuzzy real

numberX₀, if for every ε >0, there exists k₀∈N such that d (X_k,X₀) < ε for all k ≥ k₀.

A sequence space EF is said to be symmetric if (X_{π( )}k )∈

F

E , whenever (Xk)∈

F

E ,

π is a permutation on N.

A sequence X = (Xk) of fuzzy numbers is said to be I- convergent if there exists a fuzzy

numberX₀ such that for all ε >0, the set {k∈N : d (Xk,X0) ≥ε }∈I. We write I-limXk= X0.

A sequence (Xk) of fuzzy numbers is said to be I- bounded if there exists a real number

µ such that the set {k∈N : d (Xk, 0) > µ}∈I.

If I = I_f , then I_f convergence coincides with the usual convergence of fuzzy

sequences. If I =I_d (I_δ), then I_d(I_δ ) convergence coincides with statistical convergence

(logarithmic convergence) of fuzzy sequences. If I =I_u , I_u convergence is said to be

uniform convergence of fuzzy sequences.

Throughout I(F)

c , ( ) 0

I F

c and I F( )

∞

 denote the spaces of fuzzy real-valued I- convergent,

I-null and I- bounded sequences respectively.

It is clear from the definitions that c₀I F( )⊂ cI(F)⊂ I F_∞( )and the inclusions are proper.

It can be easily shown that I F( )

∞

 is complete with respect to the metric ρ defined by f

( X, Y ) = n

sup d (Xk, Yk) , where X = (Xk) , Y = (Yk) ∈

( )

I F ∞

 .

Lemma 2.1:

(a) The condition sup _k( ) , 0 k

f t < ∞ >t hold iff there exists t₀ >0such that sup _k( )₀ . k

f t < ∞

(b) The condition inf _k( ) , 0

k f t > ∞ >t hold iff there exists t0 >0such that infk f tk( )0 > ∞. Lemma 2.2: Let (𝛼_𝑘) and (𝛽_𝑘) be sequences of real or complex numbers and (𝑝_𝑘) be a

bounded sequence of positive real numbers , then

|𝛼𝑘+𝛽𝑘|𝑝𝑘 ≤ 𝐷(|𝛼𝑘|𝑝𝑘+ |𝛽𝑘|𝑝𝑘)

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where D = 𝑚𝑎𝑥(1, |𝜆|𝐻−1),𝐻 =𝑠𝑢𝑝𝑝_𝑘 , 𝜆 is any real or complex number.

Let F =(f_k)be a sequence of modulus functions, p=(p_k) be a bounded sequence of strictly positive real numbers and v=(v_k)a sequence of positive real numbers. We define the

following new sequence spaces as:

( )

( , , )

{

( ) : lim ([ ( , 0)] ) 0, 0 ( )

}

k

F

I F p

k k k k

p v X X w I f d v X X forX L R I

c

F = = ∈ − = ∈ ∈

( )

0 ( , , )

{

( ) : lim ([ ( , 0)] ) 0

}

k F

I _F _p

k k k k

p v X X w I f d v X I

c

F = = ∈ − = ∈

( )

( , , )

{

( ) : sup ([ ( , 0)] )

}

k

F

I F p

k k k k

k

p v X X w I f d v X I

l

∞ F = = ∈ − < ∞ ∈

Some special cases:

a. If F = f_k( )x = x for all k, then we have

( )

( , )

{

( ) : lim[ ( , 0)] 0, 0 ( )

}

k

F

I F p

k k k

p v X X w I d v X X forX L R I

c

= = ∈ − = ∈ ∈

( )

0 ( , )

{

( ) : lim[ ( , 0)] 0

}

k F

I F p

k k k

p v X X w I d v X I

c

= = ∈ − = ∈

( )

( , )

{

( ) : sup[ ( , 0)]

}

k F

I F p

k k k

k

p v X X w I d v X I

l

∞ = = ∈ − < ∞ ∈

b. If (p_k)=1 and ( )v_k =1 for all k∈N , we have

( )

{

( ) : lim ([ ( , 0)]) 0, 0 ( )

}

F

I _F

k k k

X X w I f d X X forX L R I

c

F = = ∈ − = ∈ ∈

( )

0 ( )

{

( ) : lim ([ ( , 0)]) 0

}

F

I F

k k k

X X w I f d X I

c

F = = ∈ − = ∈

( )

{

( ) : sup ([ ( , 0)])

}

F

I _F

k k k

k

X X w I f d X I

l

∞ F = = ∈ − < ∞ ∈

c. If F = f_k( )x =x and (p_k)=1 for all k∈N, then

( )

{

( ) : lim[ ( , 0)] 0, 0 ( )

}

F

I _F

k k k

v X X w I d v X X forX L R I

c

= = ∈ − = ∈ ∈

( )

0 ( )

{

( ) : lim[ ( , 0)] 0

}

F

I F

k k k

v X X w I d v X I

c

= = ∈ − = ∈

( )

{

( ) : sup[ ( , 0)]

}

F

I _F

k k k

k

v X X w I d v X I

l

∞ = = ∈ − < ∞ ∈

d. If F = f_k( )x =x , (p_k)=1 and ( )v_k =1 for all k∈N, then

( )

{

( ) : lim[ ( , 0)] 0, 0 ( )

}

F

I _F

k k

X X w I d X X forX L R I

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( )

0

{

( ) : lim[ ( , 0)] 0

}

F

I _F

k k

X X w I d X I

c

= = ∈ − = ∈

( )

{

( ) : sup[ ( , 0)]

}

F

I F

k k

k

X X w I d X I

l

∞ = = ∈ − < ∞ ∈

e. If ( )v_k =1 for all k∈N_{, then-}

( )

( , )

{

( ) : lim ([ ( , 0)] ) 0, 0 ( )

}

k

F

I F p

k k k

p X X w I f d X X forX L R I

c

F = = ∈ − = ∈ ∈

( )

₀ ( , )

{

( ) : lim ([ ( , 0)] )k 0

}

F

I _F _p

k k k

p X X w I f d X I

c

F = = ∈ − = ∈

( )

( , )

{

( ) : sup ([ ( , 0)] )k

}

F

I F p

k k k

k

₀ ( , , )

F I k

Y = Y ∈

_c

F p v . For scalarsα β, ∈C, there exist integers a_α

and b_β such that α ≤a_α and β ≤b_β. Since F =(f_k)be a sequence of modulus functions, we have –

([ ( ( ), 0)] )pk ( )H ([ ( , 0)] )pk ( )H ([ ( , 0)] )pk

k k k k k k k k k k

f d v αX +βY ≤D a_α f d v X +D b_β f d v Y →0 as k→ ∞.

Therefore,

( )

0 ( , , )

F I

k k

X Y

_c

p v

α +β ∈ F . This completes the proof.

( )

( , )

( )

( , , ).

F F

I I

p v p v

l

∞ ⊂

l

∞ F Proof: Let ( )

( )

( , )

F I k

X = X ∈

_l

_∞ p v , then we have sup ([ ( , 0)] )pk .

k k k

k

I− f d v X < ∞ Let

0

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, ( ,0) , ( ,0)

sup ([ ( , 0)] )k sup ([ ( , 0)] )k sup ([ ( , 0)] )k

k k

p p p

k k k k k k k k k

k k d X k d X

I f d v X I f d v X I f d v X

δ δ

≤ >

− = − + −

sup

(

( k k, 0)

)

pk

k

M

d v X ε

δ  

≤ + _ _

 by properties of modulus function.

< ∞

Hence ( )

( )

( , , ).

F I k

X = X ∈

_l

_∞ F p v This completes the proof.

Theorem 3.3: Let F =(f_k)be a sequence of modulus functions and lim k( ) 0 t

f t t

α

→∞

t

α

≤ .

Thus,

sup([ ( , 0)] )pk 1sup ([ ( , 0)] )pk

k k k k k

k k

I d v X I f d v X

α

− ≤ −

< ∞

This follows that ( )

( )

( , ).

F I k

X = X ∈

_l

_∞ p v

( )

0 ( , , )

F F

I I

v p v

l

∞ ⊂

c

F if lim k( ) 0

t→∞ f t = for t>0. Proof: It is easy to prove, so omitted.

Theorem 3.5: Let F =(f_k)be a sequence of modulus functions and if lim k( )

t→∞ f t = ∞ for

0

t > then

( )

0

( , , ) ( ).

F F

I I

p v v

l

∞ F ⊂

c

Proof: Let lim k( )

t→∞ f t = ∞ for t >0. If ( )

( )

( , , ). F

I k

X = X ∈

_l

_∞ F p v

Then,

([ ( , 0)] )pk

k k k

f d v X ≤M < ∞ for all k .

If possible let ( )

( )

₀ ( ).

F I k

X = X ∉

_c

v Then for some ε >0there exists a positive integer k₀

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( ) ([ ( , 0)] )pk

k k k k

f ε ≥ f d v X ≤M for k ≥k₀.

This contradicts to our assumption that lim _k( )

t→∞ f t = ∞ for t >0 and hence

( )

0

( ) ( ).

F I k

X = X ∈

_c

v

Theorem 3.6: Let F =(f_k)be a sequence of modulus functions then

( )

₀ ( , , )

F I

p v

c

F and

( )

( , , )

F I

p v

l

∞ F are paranormed spaces with the paranorm

{

}

1

( ) sup [ ( , 0)]pk M

k k k

k

h X = f d v X

Where max 1, sup

{ }

_k

k

M = p

Proof: Obviously h X( )= −h( X) for all

( )

₀ ( , , )

F I

X∈

_c

F p v

It is trivial that v X_k _k =0 for X =0.

Since pk 1

M ≤ , since d is translation invariant and by using Minkowski’s inequality , we

have

{

} {

}

1 1 1

[ ( ( ), 0)]pk M [ ( , 0)]pk M [ ( , 0)]pk M

k k k k k k k k k k

f d v X +Y ≤ f d v X + f d v Y

Hence,

h X( +Y)≤h X( )+h Y( )

Finally to check the continuity of scalar multiplication, let λbe any scalar, by definition we have

{

}

1

( ) sup [ ( , 0)] k ( )

H

p M M

k k k

k

h λX = f d v λX ≤K_λ h X

where sup k k

H = p < ∞.

Where K_λis positive integer such that λ ≤K_λ. Let λ→0 for any fixed X with h X( )=0.

By definition for λ ≤1, we have

sup

{

[ ( , 0)]pk

}

k k k

k

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Also for 1≤ ≤n N_{by taking}λ small enough, since f_k is continuous, we get

sup

{

[ ( , 0)]pk

}

.

k k k

k

f d v λX ≤ε

Implies that h(λX)→0 as λ→0. This completes the proof.

∞ F are complete metric spaces under the metric –

{

}

1

( , ) sup [ ( , )]pk M

k k k

k

h X Y = f d X Y

Where max 1, sup

{ }

_k

k

M = p

Proof: It is easy to see that his a metric on

( )

( , )

F I

p

c

F . To show completeness. Let

( )

Xi be a Cauchy sequence in

( )

( , )

F I

p

c

F where

( ) ( )

Xi = X_ki . Therefore for each ε >0 there exists i₀∈Nsuch that

h X( i,X j)<ε for all i j, ≥i0.

i.e

{

}

1

sup [ ( i, j)]pk M

k k k

k

f d X X <ε for all i j, ≥i₀.

This means

sup( [ ( i, j)] )pk

k k k

k

f d X X <ε for all i j, ≥i₀.

Since f is modulus function, so choosing suitable ε >₁ 0and we obtain

( i, j) ₁

k k

d X X <ε for all i j, ≥i₀ and for each k.

i.e

( )

X_ki is a Cauchy sequence in L R

( )

for each k.

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sup( [ ( i, )] )pk

k k k

k

f d X X <ε for all i≥i₀.

That means,

(

i,

)

h X X <ε for all i≥i₀.

Next to show

( )

( , )

F I

X ∈

_c

F p , for which the proof as follows: Since

( )

( , )

F I i

k

X ∈

_c

F p for i∈N , so for ,i j , there exist L Li, j∈L R( ) and k ki, j∈N Such that

sup( [ ( i, i)] )pk

k k

k

f d X L <ε for all k ≥k_i.

And

sup( [ ( j, j)] )pk

k k

k

f d X L <ε for all k ≥k_j.

Now let k₀ =max

(

k k_i, _j

)

and i j, ≥i₀, we have

sup( [ ( ,i j)] )pk sup( [ ( ,i i)] )pk

k k k

k k

f d L L ≤C f d L X

sup( [ ( i, j)] )pk

k k k

k

C f d X X

+

sup( [ ( j, j)] )pk

k k

k

C f d X L

+

<3Cε for all i j, ≥i₀ and k ≥k₀.

Hence

( )

Li is a Cauchy sequence in L R

( )

. So there exists L∈L R( ) such that

i

L →L as i→ ∞

Now keeping i fixed and letting j→ ∞, once can find

sup( [ ( , )] )i pk 3

k k

f d L L < Cε for all i≥i0 and k ≥k0.

Therefore,

_sup( _{[ (} _{, )] )}pk _sup( _{[ (} _, i0_{)] )}pk

k k k k k

k k

f d X L ≤C f d X X

+sup( [ ( i, i)] )pk

k k

k

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+_sup( _{[ (} i0_{, )] )}pk k

k

f d L L

2 1

2Cε 3C ε ε

< + ≅ for all k ≥k₀.

This implies that

( )

( , )

F I k

X = X ∈

_c

F p . This completes the proof.

Conflict of Interests

The author declares that there is no conflict of interests.

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[15] B.C. Tripathy and P.C Das; Some classes of difference sequences of fuzzy real numbers , FASCICULI MATHEMATICI , 40(2008) : 105-117.