Vol 2, No 1 (2011)

International Journal of Mathematical Archive-2(1), Jan. - 2011, Page: 154-158

Available online through

www.ijma.info

A NEW TYPE OF DIFFERENCE SEQUENCE SPACES OF FUZZY REAL NUMBERS

T. BILGIN and V.A. Khan

Department of Mathematics Faculty of Sciences and Arts, Yüzüncü Yil University, 65080-Van –TURKEY E-mail: [email protected]

*Department of Mathematics, A.M.U. Aligarh (INDIA) E-mail: [email protected]

(Received on; 26-11-10; Accepted on: 06-12-10)

--- ABSTRACT

T

he Idea of difference sequence sets introduced by Kizmaz [1]. In this paper, we introduce certain new difference sequence spaces of fuzzy real numbers and give some topological properties and inclusion relations.

Keyword and phrases: Difference sequence, Fuzzy real number, Solid space, Symmetric space.

AMS Mathematical Subject Classification: 40A05, 40A25, 40A30, 40C05.

---1. INTRODUCTION:

The concept of fuzzy was introduced by Zadeh [2]. Latter on sequences of fuzzy number have been discussed by Matloka [3]. Tripathy and Nanda [4], Nuray and Savas [5], Bilgin [6], Altin, Et, and colak [7], Kwon [8] and many others.

Let D denote the set of all closed bounded intervals A= [*A, A*] on the real line R, where *A and A* denote the end point of A. For A, B

∈

D define A B and iff *A * B and A* B*, d (A,B) = max (|*A -*B|, A* -B*|). It is well known that (D,d) is a complete metric space and d(A,B) is called the distance between the intervals A and B. Also it is easy to see that defined above is a partial order relation in D (see Matloka [3]).

A fuzzy number is a fuzzy subset of the real line R

which is bounded, convex and normal. Let R (I) denote the set of all fuzzy numbers which are upper semi continuous and have compact support. For X ∈R(I), theα -level set Xα for 0 <

α 1 is defined by, Xα = {t ∈ R: X(t) α }. The 0-level i.e. X0

is the closure of strong 0-cut, i.e. X0= cl{t ∈ R: X (t) > 0}. The absolute value of X∈ R(I) i.e. |X| is defined as (see Kaleva and Seikhala [9]).

|X| (t)=

{

( ) ( )

}

<

≥

−

t

t

for

t

X

t

X

0

,

0

0

,

,

max

For r ∈

R

,

r

∈

R

(

I

)

is defined by

( )

=

=

otherwise

r

t

for

t

r

,

0

1

For r∈ R and X ∈R(I) we define rX(t) =

( )

=

≠

−

0

,

0

0

1

r

for

r

for

t

r

X

---

*Corresponding author: V.A. Khan E-mail: [email protected]

Define d: R(I) x R(I) → R By

(

X

,

Y

)

₀sup ₁

d

(

X

,

Y

)

,

for

X

,

Y

R

(

I

)

d

=

_<α≤ α α

∈

The it is well known that (R(I),

d

) is a complete metric space. A sequence X=(Xk) of fuzzy numbers is said to be converge to a fuzzy number

X

0 if for if for every ε > 0 there is a positive integer No such that d(Xk, X0) < ε for k > N0. and X=(Xk) of fuzzy numbers is said to be Cauchy sequence if for every

ε > 0 there is a positive integer No such that

X = (Xk, Xl)< ε for k,l > N0.

A sequence space E is said to be solid if (Yn) ∈ E, whenever

(Xn) ∈ E and |Yn|<|Xn|, for all n∈ N.A sequence space E is

said to be monotone if E contains the canonical pre-images of all its step space, Let X = (Xn) be a sequence, the S(X) denotes the set of all permutations of the elements of (Xn) i.e. S(X) = {(Xπ(n)): π is a permutation of N}. A sequence spaces E is said to be symmetric if S(X) ⊂E for all X ∈E.A sequence space E is said to be convergence-free if (Yn)∈E wherever (Xn) E and

Xn =

0

implies Yn =

0

.

REMARK: A sequence space E is solid implies that E is monotone. Let 0 be the set of all complex sequences and l∞,c

and c0 be the sets of all bounded, convergent and null

sequences

x

=

(

x

_k

)

with complex terms, respectively.

The idea of difference sequence spaces was introduced by Kizmaz [1]. In 1981, Kizmaz [1] define the sequence spaces.

l∞(∆) = {x = {xk}∈

0

: (∆xk) ∈l∞},

c(∆) = {x = {xk}∈

0

: (∆xk) ∈c}, and

c0(∆) = {x = {xk}∈

0

T. BILGIN and V.A. Khan*/ A new type of difference sequence spaces of fuzzy real numbers/IJMA- 2(1), Jan.-2011, Page: 154-158

where ∆x = (xk – xk+1). There are Banach spaces with the norm

||x||∆ = |x1| + ||∆x||∞.

Then Et and Colak [10] generalized the above sequence spaces, to the sequence spaces

X(∆r) = {x ={xk} ∈

0

: ∆rxk∈ X},

for X= l∞, c and c0, where r ∈|x|,

∆0x = (xk), ∆x = (xk – xk+1), ∆

r

x = (∆rxk - ∆

r

xk+1)

and so that

∆rxk = _{k i}

r

i

i

_x

i

r

+ =

−

0

)

1

(

Difference sequence spaces have been studied by Colak and Et [11], Tripathy and Esi [12], Et and Esi [13], Et, Altin, and altinok [14], Khan [15, 16] and many others.

Let c denote the space whose elements are finite sets of distinct positive integers. Given any element σ of C, we denote by c(σ) the sequence {cn (σ)} which is such that cn (σ) = 1 if n ∈σ, cn (σ) = 0 otherwise. Further

:

( ) (

[

17

]

)

1

see

s

c

C

C

n n s

=

∈

∞

=

σ

σ

,

the set of those σ whose support has cardinality at most s, and

{ }

∈ > ∆ ≥ ∆ ≤

(

=

)

= =

Φ : 1 0, 0 0 1,2...

0

k k and k k

k

φ φ

φ φ

φ

where ∆φk = φk - φk – 1; and

0

is the set of all real sequences.

For φ ∈

Φ

, we define the following sequence space, introduce in [18],

m(φ) =

=

{ }

∈

<

∞

∈ ∈

≥

σ

σ

_φ

k k s C s

k

x

x

x

s

|

|

1

1 sup sup : 0

The space m(φ) was extended to m(φ,p) by Tripathy and Sen [19] as follows:

m(φ,p) :=

=

{ }

∈

<

∞

∈ ∈

≥

p

k k s C s

k

x

x

x

s

|

|

1

1 sup sup : 0

σ

σ

_φ

Let u be a fixed positive integer and u = (uk) be any fixed scalars sequence of non zero complex numbers (see [20,21,13]. Khan [16] generalized this sequence space and

introduced the sequence space m

(

∆

u_u

,

φ

,

p

)

:

defined as follows:

m

(

∆

)

=

{ }

∈

(

∆

)

<

∞

<

∞

∈ ∈

≥

x

p

x

x

p

k

p k u u s C s k u

u

s

0

,

|

|

1

:

,

,

1 sup sup : 0

σ

σ

_φ

φ

where

(

_{k k}

)

,

k u

u

x

=

u

x

∆

(

−

_{+1 +1}

)

,

=

∆

_u

x

_k

u

_k

x

_k

u

_k

x

_k

(

1

)

1 1

+ − −

∆

−

∆

=

∆

u_u

x

_k u_u

x

_k u_u

x

_k

and so that

( )

k i k i u

i i k

u

u

u

x

i

u

x

_{+ +}

=

−

=

∆

0

1

We introduce the sequence space m

(

∆

u_u

,

φ

,

p

)

Fof fuzzy real numbers as follows;

(

∆

)

=

=

( )

(

∆

)

>

∞

<

<

∞

∈ ∈ ≥

p

X

d

X

X

p

m

p

k

k u

u C

s k F

u u

s

0

,

0

,

1

,

,

1 sup sup :

σ

σ

φ

φ

2. MAIN RESULTS:

In this section, we prove some results involving the sequence

space

m

(

∆

u_u

,

φ

,

p

)

F with twos values of p such that 0 < p <

∞

Theorem: 2.1. (a) The sequence space

m

(

∆

u_u

,

φ

,

p

)

F for1 <

p < ∞ is a complete metric space by the metric,

(

)

(

)

(

(

)

)

p

k

p k u

u k u

u s

C s u

i

i

i

Y

d

X

Y

X

d

Y

X

s

/ 1

1 sup sup : 1

,

1

,

,

∈ ∈ ≥ =

∆

∆

+

=

σ

σ

φ

ρ

(2.1.1)

for X,Y ∈

m

(

∆

u_u

,

φ

,

p

)

F.

(b) The sequence space

m

(

∆

u_u

,

φ

,

p

)

F for 0 < p < 1 is a complete metric space by the metric

(

)

(

)

(

(

)

)

p

k

k u

u k u

u s

C s u

i

i

i

Y

d

X

Y

X

d

Y

X

s ∈

∈ ≥ =

∆

∆

+

=

σ

σ

φ

η

,

,

1

,

1 sup sup : 1

,

(2.1.2)

for X,Y ∈

m

(

∆

u_u

,

φ

,

p

)

F.

Proof: It is clear that

m

(

∆

u_u

,

φ

,

p

)

F is a metric space by (2.1.1) for 1 < p < ∞ and (2.1.2) for 0 < p < 1. We need to show that

m

(

∆

u_u

,

φ

,

p

)

F is complete.

T. BILGIN and V.A. Khan*/ A new type of difference sequence spaces of fuzzy real numbers/IJMA- 2(1), Jan.-2011, Page: 154-158

Let (X(l)) be a Cauchy sequence in

m

(

∆

u_u

,

φ

,

p

)

F where Xl =

( ) (

,

1

,....

)

2

X

X

X

l

k k l

k

=

∈

(

)

F u

u

p

m

∆

,

φ

,

for each l ∈ |x|. Then for given ε > 0 there exists n0∈ |x| such that

( ) ( )

(

,

)

ε

,

η

X

l

X

t

<

for all l, t > n0.

Hence

( ) ( )

(

_,

)

1

₍

₍

( )

_,

( )

₎

₎

_,

1 sup sup : 1

ε

φ

σ

σ

<

∆

∆

+

∈ ∈ ≥ =

p

k

t k u

u l k u

u s

C s u

i

t i l

i

X

d

X

X

X

d

s

(2.1.3) for all l,t > n0.

Now we obtain

( ) ( )

(

,

)

ε

,

σ

<

∈

k

t i l i

X

X

d

for all l,t > n0.

which implies that

( ) ( )

(

,

t

)

<

ε

,

i l i

X

X

d

for all l,t > n0 for i=1,2,3…u.

Hence,

(

X

_i( )l

)

is a Cauchy sequence in R(I), so it is convergent in R(I), by the completeness property of R(I), for i = 1,2,3,….u

Also,

( ) ( )

(

)

(

)

ε

φ

_σ

σ

∆

∆

<

∈ ∈ ≥

k

p t k u

u l k u

u s

C

s s

d

X

,

X

1

sup sup

1 , for all l, t > n0.

On taking s = 1, we have,

( ) ( )

(

u_u

X

_kl

,

u_u

X

_kt

)

(

₁

)

1/p

,

d

∆

∆

<

εφ

for all l,t > n0, k∈|x|

Which implies that for each fixed k (1 < k <∞), the sequence ( )

(

l

)

k u

u

X

∆

is a Cauchy sequence in R(I), hence converges in

R(I).

Hence, we get, lim

∆

u_u

X

_k( )l

=

∆

u_u

X

_k for k ∈ |x|

Taking limit as t →∞ in (2.1.3), we get,

( )

(

)

(

(

( )

)

)

_ε

φ

σ σ

<

∆

∆

+

∈ ∈ ≥

= k

p k u

u l k u

u s

C s u

i

i l

i

X

d

X

X

X

d

s

,

1

,

1 sup sup 1

,

for all l > n0. (2.1.4)

( )

(

)

ε

η

X

l

,

X

<

, for all l > n0.

Since (X(l)) ∈

m

(

∆

u_u

,

φ

,

p

)

F and by (2.1.4), for all l > n0.

we have,

η(X(l), θ) < η(X(l), θ) + η(X(l),θ) < ∞.

Hence, X ∈

m

(

∆

u_u

,

φ

,

p

)

F. Hence,

m

(

∆

u_u

,

φ

,

p

)

F is a complete metric space. This completes the proof of the theorem.

Theorem 2.2:

m

(

∆

u_u

,

φ

)

F⊂

m

(

∆

u_u

,

φ

,

p

)

F, for all 1 < p < ∞. Proof: Let X∈

m

(

∆

u_u

,

φ

)

F, then we have

(

)

(

)

∈ ∈ ≥

∞

<

=

∆

σ

σ

φ

k

k u

u s

C s

K

X

d

s

0

,

1

1 sup sup

Hence, for each fixed s, we have

(

)

∈

=

∆

σ

φ

k

s k

u

u

X

K

d

,

0

σ∈φs for each fixed s. Hence

(

)

(

,

0

)

,

/ 1

s p

k

p k u

u

X

K

d

φ

σ

<

∆

∈

σ∈φs for each p o and 1 < p < ∞.

Thus X ∈

m

(

∆

u_u

,

φ

,

p

)

F.

Theorem 2.3: For any two sequence

( )

φ

_s and

( )

Ψ

_s of real numbers

(

∆

u_u

p

)

F

⊂

m

,

φ

,

(

u

)

F

u

p

m

∆

,

Ψ

,

if and only if

∞

<

Ψ

≥

s s s

φ

sup 1

Proof: Let X∈

m

(

∆

u_u

,

φ

,

p

)

. Then

(

)

(

∆

)

<

∞

∈ ∈ ≥

σ σ

φ

k

p k u

u s

C

s s

d

X

,

0

1

sup sup

1

Suppose that

∞

<

Ψ

≥

s s s

φ

sup

1 .

Then φs < Kψs and so that s

ψ

1

<

s

K

φ

for some positive

number K and for all s. Therefore we have

(

)

(

)

∈

∆

σ

ψ

k

p k u

v s

X

d

,

0

1

<

(

(

)

)

∈

∆

σ

φ

k

p k u

v s

X

d

,

0

1

for each s.

Now

(

)

(

)

∈ ∈

≥

∆

σ σ

ψ

k

p k u

u s

C

s s

d

X

,

0

1

sup sup

1

(

(

)

)

∈ ∈

≥

∆

σ σ

φ

k

p k u u s

C

s

d

X

K

s

,

0

1

sup sup

T. BILGIN and V.A. Khan*/ A new type of difference sequence spaces of fuzzy real numbers/IJMA- 2(1), Jan.-2011, Page: 154-158 Hance

(

)

(

∆

)

<

∞

Ψ

∈

∈ ≥

σ σ

k

p k u

u s

C

s s

d

X

,

0

1

sup sup

1

Therefore X

∈

∈

m

(

∆

uu

Ψ

,

p

)

F.

Conversely, let

m

(

∆

u_u

φ

,

p

)

F⊆

m

(

∆

u_u

Ψ

,

p

)

F and suppose that

∞

=

Ψ

≥

s s s

φ

sup

1 .

Then there exists a increasing sequence (si) of naturals number

such that lim

i i

s s

ψ

φ

= ∞. Now for every B ∈ R+, the set of

positive real numbers, there exists i0∈|x| such that

>

i i

s s

ψ

φ

B

for all si >i0. Hence

i i s

s

B

ψ

ψ

>

1

and so that

i

s

ψ

1

(

)

(

)

∈

∆

σ k

p k u

u

X

d

,

0

i

s

B

ψ

∈

(

(

)

)

∆

σ k

p k u

u

X

d

,

0

for all si > i0. Now taking supremum over si > i0and σ∈Cs we get

(

)

(

d

X

)

B

k

p k u

u s

C s

i

s

∆

>

∈ ∈ ≥

σ σ

ψ

,

0

1

sup sup

1

(

(

,

0

)

)

.

1

sup sup

1

∈ ∈

≥

∆

σ

σ

_φ

k

p k u

u s

C

s

d

X

i s

(2.2.1)

Since (2.2.1) holds for all B∈R+ (we may take B sufficiently large) we have

(

)

(

∆

)

=

∞

∈ ∈

≥

σ σ

ψ

k

p k u

u s

C

s

d

X

i

s

,

0

1

sup sup

1

When X

∈

∈

m

(

∆

u_u

φ

,

p

)

F with

(

)

(

∆

)

<

∞

<

∈ ∈ ≥

σ σ

φ

k

p k u

u s

C

s

d

X

i

s

,

0

1

0

sup₁sup

Therefore X∉

m

(

∆

u_u

ψ

,

p

)

F. This contradict to

m

(

∆

u_u

φ

,

p

)

F⊆

m

(

∆

u_u

ψ

,

p

)

F.

Hence sup_≥1

<

∞

.

s s s

ψ

φ

Form Theorem 2.3, we get the following result.

Corollary 2.4:

m

(

∆

u_u

φ

,

p

)

F=

m

(

∆

u_u

ψ

,

p

)

F if and only if

.

0

<

inf_≥1

≤

sup_≥1

<

∞

s s s s s s

ψ

φ

ψ

φ

Theorem 2.5:

l

_p

( )

∆

u_u F

⊆

m

(

∆

u_u

,

φ

,

p

)

F

⊆

l

_∞

( )

∆

u_u F

Proof: Since

m

(

∆

u_u

,

φ

,

p

)

F

=

1

_p

( )

∆

u_u F for φn= 1, for all |x|, then

( )

(

_u

)

F

u F

u u

p

m

p

l

∆

⊆

∆

,

φ

,

Now suppose that X∈

m

(

∆

u_u

,

φ

,

p

)

F. Then we have

For s = 1,

(

,

X

,

0

)

K

φ

₁

,

d

∆

u_{u k}

<

for all k ∈ N and for some positive integer K.

Thus X ∈

l

_∞

( )

∆

u_u F. This completes the proof of Theorem.

The proof of the following result is obvious.

Corollary 2.6: If 0 < p < q, then

m

(

∆

u_u

,

φ

,

q

)

F⊆

m

(

∆

u_u

,

φ

,

p

)

F.

Theorem 2.7: The sequence space

m

(

∆

u_u

,

φ

,

p

)

F is not solid for 0 < p < ∞.

Proof: The proof follows from the following example.

Take u = 3, u = 1, p = 2 and φs = 1, for all s ∈ |x|. Let Xk =

I

for all k ∈N.Then we have,

(

3

,

0

)

k

X

d

∆

= 0 for all k ∈ N. Hence

(

)

(

,

0

)

0

1

sup sup

1

∆

=

∈ ∈ ≥

σ σ

φ

k

p k u

u s

C

s

d

X

i

s .

This implies that, (Xk) ∈

(

)

F

m

3

,

2

φ

∆

. Consider the sequence (αk) of scalars defined by

α

_k

=

{

1_0,,_otherwiseforkiseven_. ,

So

(

_∆

3

,

0

)

₌

1

k k

X

d

α

(

)

[

∆

]

=

[ ]

=

≥ ∈

=

∞

∈ ∈ ≥ ∈

∈

≥

d

X

s s

s

i

s s C

k p C

s k

p k u u s

C s

sup sup

1 sup

sup 1 sup

sup

1

1

1

1

0

,

1

σ σ

σ σ

σ

φ

.

Implies that

(

α

_k

x

_k

)

∉

m

(

∆

3

,

φ

,

2

)

F. Hence

m

(

∆

u_u

,

φ

,

p

)

Fis not solid.

T. BILGIN and V.A. Khan*/ A new type of difference sequence spaces of fuzzy real numbers/IJMA- 2(1), Jan.-2011, Page: 154-158 Proof: The proof follows from the following example.

Take u = 1 φs = 1, for all s ∈ |x|. Let Xk =

I

for all k ∈ N Then we have,

d

(

∆

X

_k

,

0

)

=1 for all k ∈ N. Let

( )

Y

_k be rearrangement of

(

X

_k

)

such that

(Yk) = (X1, X2, X4, X3, X9, X5, X16, X6, X25….)

This implies that

(

)

[

∆

]

=

∞

∈ ∈ ≥

σ σ

φ

k

p k u

u C

s s

d

Y

,

0

1

sup sup

1 .

Hence

( )

Y

_k

∉

m

(

∆

u_u

,

φ

,

p

)

F. Hence

m

(

∆

u_u

,

φ

,

p

)

Fis not symmetric.

REFERENCES:

[1] Kizmaz, H. On certain sequence spaces, Canad. Math. Bull.24 (2) (1981): 169-176.

[2] Zadeh, L.A. Fuzzy sets, information and control, vol 8(3) (1965): 338-353

[3] Matloka, M. sequence of fuzzy numbers, BUSEFAL, vol 28(1986): 28-37.

[4] Tripathy, B.C. and Nanda, S. Absolute value of fuzzy real number and fuzzy sequence spaces, Jour. Fuzzy Math., 8(4) (2000), 883-892.

[5] Nuray, F. and Savas, E. statistical convergence of sequence of fuzzy real numbers, Math. Slovaca, 45(3) (1995), 269-273.

[6] Bilgin, T. Lacunary strongly ∆- convergent sequnces of fuzzy numbers, Information Science. 160 (1-4) (2004), 201-206.

[7] Altin, Y., Et, M. and Colak, R. Lacunary statistical and Lacunary strongly convergence of generalized difference sequence of fuzzy numbers, Comput. Math. Appl., 52(6-7) (2006), 1011-1020.

[8] Know., J.S. On statistical and p- Cesaro convergence of fuzzy numbers, Korean J. Compute. & Appl. Math., 7(1) (2000), 195-203.

[9] Kaleva, O. and Seikkala, S. On fuzzy metric spaces, Fuzzy Sets and Ststems, 12 (1984), 215-229.

[10] Et, M. and Colak, R. On some generalized difference sequence spaces, Soochow J. of Math.21 (1995), 337-386. [11] Colak, R. Et, M. on some difference sequence sets and their topological properties, Bull. Malaysian Math. Sci. Soc. 28(2) (2005): 125-130.

[12] Tripathy, B.C. and Esi, A. A new type of difference sequence spaces, International Journal of Science and Technology, 1(1) (2006), 11-14.

[13] Et, M. and Esi, A. On Köthe Toeplitz duals of generalized difference sequence spaces, Bull. Malaysian Math. Sci. Soc., 24 (11) (2000): 25-32.

[14] Et, M .Altin, Y., and Altinok, H. On almost statistical convergence of generalized difference sequences of Fuzzy numbers, Math. Model. Anal., 10(4) (2005), 345-352.

[15] Khan, V.A. New Lacunary Strongly Sammable

Difference Sequences and

∆

m_u

−

Lacunary Almost Statistical Convergence, Vietnam Journal of Mathematics 36 (4) (2008): 405-413.

[16] Khan, V.A. A new type of difference sequence spaces. Appl. Sci. 12 (2010), 102-108.

[17] Mursaleen. Some geometrical properties of a new sequence space related to lp , Bull. Aust. Math. Soc., 67

(2003), 343-347.

[18] Sargent, W.L.C. Some sequence spaces related to the lp spaces, J. London Math. Soc., 35 (1960), 161-171.

[19] Tripathy, B.C. and Sen, M. On a new class of sequences related to the space lp ,Tamkang J. Math. 33(2), (2002), 167-171.