Some I-convergent lambda-summable difference sequence spaces of fuzzy real numbers defined by a sequence of Orlicz functions

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Mathematical and Computer Modelling

journal homepage:www.elsevier.com/locate/mcm

Some I-convergent lambda-summable difference sequence spaces of fuzzy real numbers defined by a sequence of Orlicz functions

Bipan Hazarika

^a,∗

, Ekrem Savas

^b

aDepartment of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh-791112, Arunachal Pradesh, India

bDepartment of Mathematics, Istanbul Ticaret University, Uskudar-Istanbul, Turkey

a r t i c l e i n f o

Article history:

Received 13 June 2011 Accepted 24 July 2011

Keywords:

Ideal I-convergent Orlicz function Fuzzy number Difference space

a b s t r a c t

In this paper we introduce certain new sequence spaces of fuzzy numbers defined by I- convergence using sequences of Orlicz functions and a difference operator of order m. We study some basic topological and algebraic properties of these spaces. Also we investigate the relations related to these spaces.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The concept of fuzzy sets was initially introduced by Zadeh [1]. It has a wide range of applications in almost all branches of study, in particular in science, where mathematics is used. Now the notion of fuzziness is used by many researchers in cybernetics, artificial intelligence, expert systems and fuzzy control, pattern recognition, operations research, decision making, image analysis, projectiles, probability theory, agriculture, weather forecasting, etc. The fuzziness of all the subjects of mathematical sciences has been investigated. It has attracted many workers on sequence spaces and summability theory to introduce different types of fuzzy sequence spaces and to study their different properties. Our studies are based on the linear spaces of sequences of fuzzy numbers which are very important for higher level studies in quantum mechanics, particle physics and statistical mechanics, etc. Different classes of sequences of fuzzy real numbers have been discussed by Nanda [2], Nuray and Savas [3], Matloka [4], Altinok et al. [5], Colak et al. [6] and many others.

The notion of I-convergence was initially introduced by Kostyrko et al. [7]. Later on, it was further investigated from the sequence space point of view and linked with the summability theory by ˘Salàt et al. [8,9], Tripathy and Hazarika [10,11], Kumar and Kumar [12], Savas [13,14] and many other authors.

Let X be a non-empty set, then a family of sets I

⊂

2^X(the class of all subsets of X ) is called an ideal if and only if for each A

,

^B

∈

I, we have A

∪

B

∈

I and for each A

∈

I and each B

⊂

A

,

^{we have B}

∈

I. A non-empty family of sets F

⊂

2^X is a filter on X if and only ifΦ

̸∈

F , for each A

,

^B

∈

F , we have A

∩

B

∈

F and for each A

∈

F and each A

⊂

B, we have B

∈

F . An ideal I is called a non-trivial ideal if I

̸=

_Φand X

̸∈

I. Clearly I

⊂

2^Xis a non-trivial ideal if and only if F

=

F

(

^I

) = {

^X

−

A

:

A

∈

I

}

is a filter on X . A non-trivial ideal I

⊂

2^X is called admissible if and only if

{{

x

} :

x

∈

X

} ⊂

I. A non-trivial ideal I is maximal if there cannot exist any non-trivial ideal J

̸=

I containing I as a subset. Further details on ideals of 2^Xcan be found in [7].

Let

w

F be the set of all sequences of fuzzy numbers. The operator∆ⁿ

: w

F

→ w

Fis defined by

(

∆^0X

)

k

=

X_k

; (

∆^1X

)

k

=

∆^Xk

=

X_k

−

X_k+1

; (

∆^nX

)

k

=

_∆ⁿX_k

=

_∆ⁿ⁻¹X_k

−

_∆ⁿ⁻¹X_k+1

, (

ⁿ

≥

2

)

^{for all n}

∈

N. The generalized difference has the following

∗Corresponding author. Tel.: +91 360 2278512; fax: +91 360 2277881.

E-mail addresses:bh_rgu@yahoo.co.in(B. Hazarika),ekremsavas@yahoo.com(E. Savas).

0895-7177/$ – see front matter©2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.mcm.2011.07.026

B. Hazarika, E. Savas / Mathematical and Computer Modelling 54 (2011) 2986–2998 2987

binomial expression for n

≥

0,

∆^nxk

=

n

−

ν=0



_n

ν  (−

¹

)

^νxk+ν

.

⁽¹⁾

Recall in [15] that an Orlicz function M is a continuous, convex, nondecreasing function defined for x

>

0 such that M

(

⁰

) =

^{0 and M}

(

^x

) >

0. If the convexity of the Orlicz function is replaced by M

(

^x

+

y

) ≤

^M

(

^x

) +

^M

(

^y

)

then this function is called the modulus function and was characterized by Ruckle [16]. An Orlicz function M is said to satisfy∆2-condition for all values of u, if there exists K

>

0 such that M

(

^2u

) ≤

^KM

(

^u

),

^u

≥

0.

Lemma 1.1. Let M be an Orlicz function which satisfies the∆2- condition and let 0

< δ <

1. Then for each t

≥ δ

^{, we have} M

(

^t

) <

^K

δ

^−1M

(

²

)

for some constant K

>

^0.

Two Orlicz functions M₁and M₂are said to be equivalent if there exist positive constants

α, β

^{and x}0such that M₁

(α) ≤

^M2

(

^x

) ≤

^M1

(β)

for all x with 0

≤

x

<

^x0.

Lindenstrauss and Tzafriri [17] studied some Orlicz type sequence spaces defined as follows:

ℓ

M

=



(

^xk

) ∈ w :

∞

−

k=1

M

 |

_xk

| ρ



< ∞,

^{for some}

ρ >

⁰

 .

The space

ℓ

Mwith the norm

‖

x

‖ =

inf

 ρ >

⁰

:

∞

−

k=1

M

 |

x_k

| ρ



≤

1



becomes a Banach space which is called an Orlicz sequence space. The space

ℓ

Mis closely related to the space

ℓ

pwhich is an Orlicz sequence space with M

(

^t

) = |

^t

|

^p, for 1

≤

p

< ∞

^.

The following well-known inequality will be used throughout the article. Let p

= (

^pk

)

be any sequence of positive real numbers with 0

≤

p_k

≤

sup_kp_k

=

G

,

^H

=

max

{

1

,

^2G−1

}

then

|

a_k

+

b_k

|

^pk

≤

H

(|

^ak

|

^pk

+ |

b_k

|

^pk

)

for all k

∈

N and a_k

,

^bk

∈

C. Also

|

a

|

^pk

≤

max

{

1

, |

^a

|

^G

}

for all a

∈

C.

In a later stage different classes of Orlicz sequence spaces were introduced and studied by Parashar and Choudhary [18], Tripathy and Sarma [19], Mursaleen and Basarir [20] and many others.

Throughout the article

w

^F denotes the class of all fuzzy real-valued sequence spaces. Also N and R denote the set of positive integers and set of real numbers respectively. The zero sequence is denoted by

θ

^.

In this paper we study some new sequence spaces of fuzzy numbers defined by I-convergence using the sequence of Orlicz functions and the difference operator. We establish an inclusion relation between the sequence spaces

w

_λ^I(F)

[

M

,

∆^m

,

^p

] , w

^I_λ^(F)

[

M

,

∆^m

,

^p

]

₀

, w

_λ^F

[

M

,

∆^m

,

^p

]

_∞and

w

_λ^I(F)

[

M

,

∆^m

,

^p

]

_∞where p

= (

^pk

)

denote the sequence of positive real numbers for all n

∈

N, a non-decreasing sequence

λ = (λ

n

)

of positive real numbers such that

λ

n+1

≤ λ

n

+

1

, λ

1

=

1

, λ

n

→ ∞ (

ⁿ

→ ∞ )

^{and M}

= (

^Mk

)

are sequences of Orlicz functions. In addition, we study some algebraic and topological properties of these spaces.

2. Definitions and notations

We now give a brief introduction about the sequences of fuzzy real numbers. Let D denote the set of all closed and bounded intervals X

= [

x₁

,

^x2

]

on the real line R. For X

,

^Y

∈

D, we define X

≤

Y if and only if x₁

≤

y₁and x₂

≤

y₂,

d

(

^X

,

^Y

) =

^max

{|

x₁

−

y₁

| , |

^x2

−

y₂

|} ,

^{where X}

= [

x₁

,

^x2

]

and Y

= [

y₁

,

^y2

] .

Then it can be easily seen that d defines a metric on D and

(

^D

,

^d

)

is a complete metric space (see [4]). Also the relation

‘‘

≤

’’ is a partial order on D. A fuzzy number X is a fuzzy subset of the real line R i.e. a mapping X

:

R

→

J

(=[

⁰

,

¹

] )

associating each real number t with its grade of membership X

(

^t

)

. A fuzzy number X is 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 number X is called normal. A fuzzy number X is said to be upper semicontinuous if for each

ϵ >

⁰

,

^X−1

([

⁰

,

^a

+ ϵ))

^{for all a}

∈ [

0

,

¹

]

is open in the usual topology in R. Let R

(

^J

)

denote the set of all fuzzy numbers which are upper semicontinuous and have compact support, i.e. if X

∈

R

(

^J

)

^{the for} any

α ∈ [

⁰

,

¹

] , [

^X

]

^αis compact, where

[

X

]

^α

= {

t

∈

R

:

X

(

^t

) ≥ α,

^if

α ∈ [

⁰

,

¹

]} ,

[

X

]

⁰

=

closure of

({

^t

∈

R

:

X

(

^t

) > α,

^if

α =

⁰

} ).

The set R of real numbers can be embedded in R

(

^J

)

if we define r

∈

R

(

^J

)

^by r

(

^t

) =



1

,

^{if t}

=

r

;

0

,

^{if t}

̸=

r

.

The additive identity and multiplicative identity of R

(

^J

)

are defined by 0 and 1 respectively.

The arithmetic operations on R

(

^J

)

are defined as follows:

(

^X

⊕

Y

)(

^t

) =

^sup

{

X

(

^s

) ∧

^Y

(

^t

−

s

)},

^t

∈

R

, (

^X

⊖

Y

)(

^t

) =

^sup

{

X

(

^s

) ∧

^Y

(

^s

−

t

)},

^t

∈

R

, (

^X

⊗

Y

)(

^t

) =

^sup



X

(

^s

) ∧

^Y



t s



,

^t

∈

R

,



X Y



(

^t

) =

^sup

{

X

(

^st

) ∧

^Y

(

^s

)},

^t

∈

R

.

Let X

,

^Y

∈

R

(

^J

)

^{and the}

α

-level sets be

[

_X

]

^α

= [

_x^α₁

,

^xα2

] , [

^Y

]

^α

= [

_y^α₁

,

^yα2

] , α ∈ [

⁰

,

¹

]

. Then the above operations can be defined in terms of

α

-level sets as follows:

[

X

⊕

Y

]

^α

= [

x^α₁

+

y^α₁

,

^xα2

+

y^α₂

] , [

X

⊖

Y

]

^α

= [

x^α₁

−

y^α₁

,

^xα2

−

y^α₂

] , [

X

⊗

Y

]

^α

=

[

min

i∈{1,2}x^α_iy^α_i

,

^max

i∈{1,2}x^α_iy^α_i

]

,

[

X⁻¹

]

^α

= [ (

^xα2

)

⁻¹

, (

^xα1

)

⁻¹

] ,

^xαi

>

⁰

,

^{for each 0}

< α ≤

¹

.

For r

∈

R and X

∈

R

(

^J

)

, the product rX is defined as follows:

rX

(

^t

) =



X

(

^r−1t

),

^{if r}

̸=

0

;

0

,

^{if r}

=

0

.

The absolute value,

|

X

|

of X

∈

R

(

^J

)

is defined by (for details see [21])

|

X

| (

^t

) =



max

{

X

(

^t

),

^X

(−

^t

)},

^{if t}

≥

0

;

0

,

^{if t}

<

⁰

.

Define a mapping_d

¯ :

R

(

^J

) ×

^R

(

^J

) →

^R+

∪ {

0

}

by d

¯ (

^X

,

^Y

) =

^sup

0≤α≤1

d

([

^X

]

^α

, [

^Y

]

^α

).

It is known that

(

^R

(

^J

), ¯

^d

)

is a complete metric space (for details see [21]).

A metric on R

(

^J

)

us said to be translation invariant if d

¯ (

^X

+

Z

,

^Y

+

Z

) = ¯

^d

(

^X

,

^Y

),

^{for X}

,

^Y

,

^Z

∈

R

(

^J

).

A sequence X

= (

^Xk

)

of fuzzy numbers is said to converge to a fuzzy number X₀if for every

ϵ >

0, there exists a positive integer n₀such that_d

¯ (

^Xk

,

^X0

) < ϵ

^{for all n}

≥

n₀.

A sequence X

= (

^Xk

)

of fuzzy numbers is said to be bounded if the set

{

X_k

:

k

∈

N

}

of fuzzy numbers is bounded.

A sequence X

= (

^Xk

)

of fuzzy numbers is said to be I-convergent to a fuzzy number X₀if for each

ϵ >

0 such that A

= {

k

∈

N

: ¯

d

(

^Xk

,

^X0

) ≥ ϵ} ∈

^I

.

The fuzzy number X₀is called the I-limit of the sequence

(

^Xk

)

of fuzzy numbers and we write I

−

lim X_k

=

X₀. A sequence X

= (

^Xk

)

of fuzzy numbers is said to be I-bounded if there exists M

>

0 such that

{

k

∈

N

: ¯

d

(

^Xk

, ¯

⁰

) >

^M

} ∈

I

.

Let E_F denote the sequence space of fuzzy numbers.

A sequence space E_Fis said to be solid (or normal) if

(

^Yk

) ∈

^EFwhenever

(

^Xk

) ∈

^EFand

|

Y_k

| ≤ |

X_k

|

for all k

∈

N.

A sequence space E_Fis said to be symmetric if

(

^Xk

) ∈

^EFimplies

(

^Xπ(^k)

) ∈

^EFwhere

π

is a permutation of N.

A sequence space E_Fis said to be monotone if E_Fcontains the canonical pre-images of all its step spaces.

Example 2.1. If we take I

=

I_f

= {

A

⊆

N

:

A is a finite subset

}

. Then I_f is a non-trivial admissible ideal of N and the corresponding convergence coincides with the usual convergence.

Example 2.2. If we take I

=

I_δ

= {

A

⊆

N

: δ(

^A

) =

⁰

}

where

δ(

^A

)

denotes the asymptotic density of the set A

,

^{then I}δis a non-trivial admissible ideal of N and the corresponding convergence coincides with the statistical convergence.

Lemma 2.1. A sequence space E_Fis normal implies that E_F is monotone. (For the crisp set case, one may refer to Kamthan and Gupta [22, page 53]).

Lemma 2.2 (Kostyrko et al. [7, Lemma 5

.

^{1]). If I}

⊂

2^Nis a maximal ideal, then for each A

⊂

N we have either A

∈

I or N

−

A

∈

I.

B. Hazarika, E. Savas / Mathematical and Computer Modelling 54 (2011) 2986–2998 2989

3. Some new sequence spaces of fuzzy numbers

In this section using the sequence of Orlicz functions and difference operator∆^mdefined by I-convergence; we introduce the following new sequence spaces and examine some properties of the resulting sequence spaces. Let I be an admissible ideal of N and let p

= (

^pk

)

be a sequence of positive real numbers for all k

∈

N, and a nondecreasing sequence

λ = (λ

n

)

^of positive real numbers such that

λ

n+1

≤ λ

n

+

1

, λ

1

=

1

, λ

n

→ ∞ (

ⁿ

→ ∞ )

^{. Let M}

= (

^Mk

)

be a sequence Orlicz functions and X

= (

^Xk

)

be a sequence of fuzzy numbers, we define the following sequence spaces as:

w

_λ^I(F)

[

M

,

∆^m

,

^p

] =



(

^Xk

) ∈ w

^F

: ∀ ε >

⁰

,



n

∈

N

:

¹

λ

n

−

k∈In

[

M_k



_d

¯ (

∆^mXk

,

^X0

) ρ

]

pk

≥ ε



∈

I

,

for some

ρ >

^{0 and X}0

∈

R

(

^J

)

 ,

w

_λ^I(F)

[

M

,

∆^m

,

^p

]

₀

=



(

^Xk

) ∈ w

^F

: ∀ ε >

⁰

,



n

∈

N

:

¹

λ

n

−

k∈In



M_k

 ¯

_d



∆^mXk

, ¯

⁰

 ρ



pk

≥ ε



∈

I

,

for some

ρ >

⁰

 ,

w

_λ^F

[

M

,

∆^m

,

^p

]

_∞

=



(

^Xk

) ∈ w

^F

:

_sup

n

1

λ

n

−

k∈In



M_k

 ¯

_d



∆^mXk

, ¯

⁰

 ρ



pk

< ∞,

^{for some}

ρ >

⁰

 ,

and

w

_λ^I(F)

[

M

,

∆^m

,

^p

]

_∞

=



(

^Xk

) ∈ w

^F

: ∃

K

>

^{0 s.t.}



n

∈

N

:

¹

λ

n

−

k∈In



M_k

 ¯

_d



∆^mXk

, ¯

⁰

 ρ



pk

≥

K



∈

I

,

for some

ρ >

⁰

 ,

where I_n

= [

n

− λ

n

+

1

,

ⁿ

]

.

From the definitions it is clear that

w

_λ^I(F)

[

M

,

∆^m

,

^p

]

₀

⊂ w

_λ^I(F)

[

M

,

∆^m

,

^p

] ⊂ w

^F_λ

[

M

,

∆^m

,

^p

]

_∞

.

Also that the inclusions are strict follows from the following example.

Example 3.1. Let M

(

^x

) =

^{x2for all x}

∈ [

0

, ∞),

^pk

=

1 for all k

∈

N. Consider the sequence

(

^Xk

)

of fuzzy numbers to be defined as follows:

For k

=

2ⁱ

,

ⁱ

=

1

,

²

,

³

, . . .

^.

X_k

(

^t

) =



 



 



4

k

+

1

,

^if

−

^k

4

≤

t

≤

0

;

−

⁴

k

+

1

,

^{if 0}

<

^t

≤

^k 4

;

0

,

^otherwise

and X_k

(

^t

) = ¯

0, otherwise.

It is easy to prove that the sequences

(

^Xk

)

^and

(

∆^mXk

)

are bounded but these are not I-convergent.

Some classes are obtained by specializing m

,

^M

, (λ

n

),

^{I and}

(

^pk

)

. Here are some examples:

(i) If m

=

0, then we obtain

w

_λ^I(F)

[

M

,

^p

] =



(

^Xk

) ∈ w

^F

: ∀ ε >

⁰

,



n

∈

N

:

¹

λ

n

−

k∈In

[

M_k



_d

¯ (

^Xk

,

^X0

) ρ

]

pk

≥ ε



∈

I

,

for some

ρ >

^{0 and X}0

∈

R

(

^J

)

 ,

w

_λ^I(F)

[

M

,

^p

]

₀

=



(

^Xk

) ∈ w

^F

: ∀ ε >

⁰

,



n

∈

N

:

¹

λ

n

−

k∈In



M_k

 ¯

_d



X_k

, ¯

⁰



ρ



pk

≥ ε



∈

I

,

^{for some}

ρ >

⁰

 ,

w

_λ^F

[

M

,

^p

]

_∞

=



(

^Xk

) ∈ w

^F

:

sup

n

1

λ

n

−

k∈In



M_k

 ¯

_d



X_k

, ¯

⁰



ρ



pk

< ∞,

^{for some}

ρ >

⁰



and

w

^I_λ^(F)

[

M

,

^p

]

_∞

=



(

^Xk

) ∈ w

^F

: ∃

K

>

^{0 s.t.}



n

∈

N

:

¹

λ

n

−

k∈In



M_k

 ¯

_d



X_k

, ¯

⁰



ρ



pk

≥

K



∈

I

,

^{for some}

ρ >

⁰

 .

(ii) If M_k

(

^x

) =

x for all k

∈

N, then we obtain

w

^I_λ^(F)

[

_∆^m

,

^p

] =



(

^Xk

) ∈ w

^F

: ∀ ε >

⁰

,



n

∈

N

:

¹

λ

n

−

k∈In



_d

¯ 

∆^mXk

,

^X0



pk

≥ ε



∈

I

,

^{for some X}0

∈

R

(

^J

)

 ,

w

^I_λ^(F)

[

_∆^m

,

^p

]

₀

=



(

^Xk

) ∈ w

^F

: ∀ ε >

⁰

,



n

∈

N

:

¹

λ

n

−

k∈In



_d

¯ 

∆^mXk

,

^X0



pk

≥ ε



∈

I

 ,

w

^F_λ

[

_∆^m

,

^p

]

_∞

=



(

^Xk

) ∈ w

^F

:

sup

n

1

λ

n

−

k∈In



_d

¯ 

∆^mXk

,

^X0



pk

< ∞



and

w

^I_λ^(F)

[

_∆^m

,

^p

]

_∞

=



(

^Xk

) ∈ w

^F

: ∃

K

>

^{0 s.t.}



n

∈

N

:

¹

λ

n

−

k∈In



_d

¯ 

∆^mXk

,

^X0



pk

≥

K



∈

I

 .

(iii) If

(λ

n

) = (

¹

,

²

,

³

, . . .)

, then we obtain

w

^I(F)

[

M

,

∆^m

,

^p

] =



(

^Xk

) ∈ w

^F

: ∀ ε >

⁰

,



n

∈

N

:

¹ n

n

−

k=1

[

M_k



_d

¯ (

∆^mXk

,

^X0

) ρ

]

^pk

≥ ε



∈

I

,

for some

ρ >

^{0 and X}0

∈

R

(

^J

)

 ,

w

^I(F)

[

M

,

∆^m

,

^p

]

₀

=



(

^Xk

) ∈ w

^F

: ∀ ε >

⁰

,



n

∈

N

:

¹ n

n

−

k=1



M_k

 ¯

_d



∆^mXk

, ¯

⁰

 ρ



pk

≥ ε



∈

I

,

for some

ρ >

⁰

 ,

w

^F

[

M

,

∆^m

,

^p

]

_∞

=



(

^Xk

) ∈ w

^F

:

sup

n

1 n

n

−

k=1



M_k

 ¯

_d



∆^mXk

, ¯

⁰

 ρ



pk

< ∞,

^{for some}

ρ >

⁰



and

w

^I(F)

[

M

,

∆^m

,

^p

]

_∞

=



(

^Xk

) ∈ w

^F

: ∃

K

>

^{0 s.t.}



n

∈

N

:

¹ n

n

−

k=1



M_k

 ¯

_d



∆^mXk

, ¯

⁰

 ρ



pk

≥

K



∈

I

,

for some

ρ >

⁰

 .

(iv) If p

= (

^pk

) = (

¹

,

¹

,

¹

, . . .)

, then we obtain

w

^I_λ^(F)

[

M

,

∆^m

] =



(

^Xk

) ∈ w

^F

: ∀ ε >

⁰

,



n

∈

N

:

¹

λ

n

−

k∈In

[

M_k



_d

¯ (

∆^mXk

,

^X0

) ρ

]

≥ ε



∈

I

,

for some

ρ >

^{0 and X}0

∈

R

(

^J

)

 ,

w

^I_λ^(F)

[

M

,

∆^m

]

₀

=



(

^Xk

) ∈ w

^F

: ∀ ε >

⁰

,



n

∈

N

:

¹

λ

n

−

k∈In



M_k

 ¯

_d



∆^mXk

, ¯

⁰

 ρ



≥ ε



∈

I

,

for some

ρ >

⁰



,

B. Hazarika, E. Savas / Mathematical and Computer Modelling 54 (2011) 2986–2998 2991

w

_λ^F

[

M

,

∆^m

]

_∞

=



(

^Xk

) ∈ w

^F

:

sup

n

1

λ

n

−

k∈In



M_k

 ¯

_d



∆^mXk

, ¯

⁰

 ρ



< ∞,

^{for some}

ρ >

⁰



and

w

_λ^I(F)

[

M

,

∆^m

]

_∞

=



(

^Xk

) ∈ w

^F

: ∃

K

>

^{0 s.t.}



n

∈

N

:

¹

λ

n

−

k∈In



M_k

 ¯

_d



∆^mXk

, ¯

⁰

 ρ



≥

K



∈

I

,

for some

ρ >

⁰

 .

(v) If I

=

I_f, then we obtain

w

_λ^F

[

M

,

∆^m

,

^p

] =



(

^Xk

) ∈ w

^F

:

lim

n→∞

1

λ

n

−

k∈In

[

M_k



_d

¯ (

∆^mXk

,

^X0

) ρ

]

pk

=

0

,

^{for some}

ρ >

^{0 and X}0

∈

R

(

^J

)

 ,

w

_λ^F

[

M

,

∆^m

,

^p

]

₀

=



(

^Xk

) ∈ w

^F

:

lim

n→∞

1

λ

n

−

k∈In



M_k

 ¯

_d



∆^mXk

, ¯

⁰

 ρ



pk

=

0

,

^{for some}

ρ >

⁰

 ,

w

_λ^F

[

M

,

∆^m

,

^p

]

_∞

=



(

^Xk

) ∈ w

^F

:

sup

n

1

λ

n

−

k∈In



M_k

 ¯

_d



∆^mXk

, ¯

⁰

 ρ



pk

< ∞,

^{for some}

ρ >

⁰

 .

If X

= (

^Xk

) ∈ w

^F_λ

[

M

,

∆^m

,

^p

]

then we say that X

= (

^Xk

)

is strongly∆^m_λ(p)-Cesàro convergent with respect to the sequence of Orlicz functions M.

4. Main results

In this section we examine the basic topological and algebraic properties of the new sequence spaces and obtain the inclusion relation related to these spaces.

Theorem 4.1. Let

(

^pk

)

be a bounded sequence. Then the sequence spaces

w

^I_λ^(F)

[

M

,

∆^m

,

^p

] , w

_λ^I(F)

[

M

,

∆^m

,

^p

]

₀, and

w

_λ^I(F)

[

M

,

∆^m

,

^p

]

_∞are linear spaces.

Proof. We will prove the result for the space

w

_λ^I(F)

[

M

,

∆^m

,

^p

]

₀only and the others can be proved in a similar way.

Let X

= (

^Xk

)

^{and Y}

= (

^Yk

)

be two elements in

w

_λ^I(F)

[

M

,

∆^m

,

^p

]

₀. Then there exist

ρ

1

>

^{0 and}

ρ

2

>

0 such that Aε

2

=



n

∈

N

:

¹

λ

n

−

k∈In

[

M_k



_d

¯ (

∆^mXk

, ¯

⁰

) ρ

1

]

pk

≥ ε

2



∈

I and

Bε

2

=



n

∈

N

:

¹

λ

n

−

k∈In

[

M_k



_d

¯ (

∆^mYk

, ¯

⁰

) ρ

2

]

pk

≥ ε

2



∈

I

.

Let

α, β

be two scalars. By the continuity of the function M

= (

^Mk

)

the following inequality holds:

1

λ

n

−

k∈In

[

M_k



_d

¯ (

∆^m

(α

^Xk

+ β

^Yk

, ¯

⁰

))

| α|ρ

1

+ | β|ρ

2

]

pk

≤

D1

λ

n

−

k∈In

[ | α|

| α|ρ

1

+ | β|ρ

2

M_k



_d

¯ (

∆^mXk

, ¯

⁰

) ρ

1

]

pk

+

D1

λ

n

−

k∈In

[ | β|

| α|ρ

1

+ | β|ρ

2

M_k



_d

¯ (

∆^mYk

, ¯

⁰

) ρ

2

]

^pk

≤

DK 1

λ

n

−

k∈In

[

M_k



_d

¯ (

∆^mXk

, ¯

⁰

) ρ

1

]

pk

+

DK 1

λ

n

−

k∈In

[

M_k



_d

¯ (

∆^mYk

, ¯

⁰

) ρ

2

]

pk

,

where K

=

max



1

, 

|α|

|α|ρ1+|β|ρ2



G

, 

|α|

|β|ρ1+|β|ρ2



G



.

From the above relation we obtain the following:



n

∈

N

:

¹

λ

n

−

k∈In

[

M_k



_d

¯ (

∆^m

(α

^Xk

+ β

^Yk

, ¯

⁰

))

| α|ρ

1

+ | β|ρ

2

]

pk

≥ ε



⊆



n

∈

N

:

DK 1

λ

n

−

k∈In

[

M_k



_d

¯ (

∆^mXk

, ¯

⁰

) ρ

1

]

pk

≥ ε

2



∪



n

∈

N

:

_DK ¹

λ

^m

−

k∈In

[

M_k



_d

¯ (

∆^mYk

, ¯

⁰

) ρ

2

]

pk

≥ ε

2



∈

_I

.

This completes the proof. 

Remark 4.1. It is easy to verify that the space

w

^F_λ

[

M

,

∆^m

,

^p

]

_∞is a linear space.

Theorem 4.2. The spaces

w

_λ^I(F)

[

M

,

∆^m

,

^p

] , w

^I_λ^(F)

[

M

,

∆^m

,

^p

]

₀, and

w

^I_λ^(F)

[

M

,

∆^m

,

^p

]

_∞

w

_λ^F

[

M

,

∆^m

,

^p

]

_∞ are paranormed spaces (not totally paranormed) with the paranorm g_∆defined by

g_∆

(

^X

) = −

k∈In

d

¯ (

^Xk

, ¯

⁰

) +

^inf



ρ

^pnH

:

sup

k

[

M_k



_d

¯ (

∆^mXk

, ¯

⁰

) ρ

]

^pk

≤

1

,

^{for some}

ρ >

⁰

,

ⁿ

=

1

,

²

,

³

, . . .

 ,

where H

=

max

{

1

,

^supkp_k

}

.

Proof. Clearly g_∆

(−

^X

) =

^g∆

(

^X

)

^{and g}∆

(θ) =

^{0. Let X}

= (

^Xk

)

^{and Y}

= (

^Yk

)

be two elements in

w

^I_λ^(F)

[

M

,

∆^m

,

^p

]

₀. Then for

ρ >

^{0 we put}

A₁

=



ρ >

⁰

:

sup

k

M_k



_d

¯ (

∆^mXk

, ¯

⁰

) ρ



≤

1



and A₂

=



ρ >

⁰

:

sup

k

M_k



_d

¯ (

∆^mYk

, ¯

⁰

) ρ



≤

1

 .

Let

ρ

1

∈

A₁and

ρ

2

∈

A₂. If

ρ = ρ

1

+ ρ

2then we obtain the following M_k



_d

¯ (

∆^m

(

^Xk

+

_Yk

), ¯

⁰

) ρ



≤ ρ

1

ρ

1

+ ρ

2

M_k



_d

¯ (

∆^mXk

, ¯

⁰

) ρ

1

 + ρ

2

ρ

1

+ ρ

2

M_k



_d

¯ (

∆^mYk

, ¯

⁰

) ρ

2

 .

Thus we have

sup

k

[

M_k



_d

¯ (

∆^m

(

^Xk

+

Y_k

), ¯

⁰

) ρ

]

^pk

≤

1 and

g_∆

(

^X

+

Y

) = −

k∈In

d

¯ (

^Xk

+

Y_k

, ¯

⁰

) +

^inf

(ρ

1

+ ρ

2

)

^pnH

: ρ

1

∈

A₁

, ρ

2

∈

A₂



≤ −

k∈In

d

¯ (

^Xk

, ¯

⁰

) +

^inf



ρ

1^pnH

: ρ

1

∈

A₁



+ −

k∈In

d

¯ (

^Yk

, ¯

⁰

) +

^inf



ρ

2^pnH

: ρ

2

∈

A₂



=

g_∆

(

^X

) +

^g∆

(

^Y

).

Let t^m

→

t where t^m

,

^t

∈

C and let g_∆

(

^Xm

−

X

) →

^{0 as m}

→ ∞

. To prove that g_∆

(

^tmXm

−

tX

) →

^{0 as m}

→ ∞

, we put

A₃

=



ρ

m

>

⁰

:

sup

k

[

M_k



_d

¯ (

∆^mX_k^m

, ¯

⁰

) ρ

m

]

^pk

≤

1



and

A₄

=



ρ

m^/´

>

⁰

:

sup

k



M_k

 ¯

_d

(

∆^m

(

^Xk^m

−

X_k

), ¯

⁰

) ρ

^/m^´



pk

≤

1



.