New Types of Generalized Difference Double A-sequence Spaces Defined by Orlicz function

Vol. 8, No. 2, 2015, 201-213

ISSN 1307-5543 – www.ejpam.com

New Types of Generalized Difference Double

A

-sequence Spaces

Defined by Orlicz Function

Bipan Hazarika

_{, Ayhan Esi}

1_{Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh-791 112, Arunachal}

Pradesh, India

2_{Adiyaman University, Science and Art Faculty, Department of Mathematics, 02040, Adiyaman,}

Turkey

Abstract. In this paper we introduce some new generalized difference double sequence spaces defined by Orlicz function and study different topological properties of these spaces and also establish some inclusion results among them.

2010 Mathematics Subject Classifications: 40A05,40B05, 46A45.

Key Words and Phrases: Orlicz function, difference space, double sequence,P-convergence.

1. Introduction

In 1971 Lindenstrauss and Tzafriri [6] used the idea of Orlicz function to construct the sequence space for single sequences as follows:

l_M =

(

x = x_k:

∞

X

k=1

M ‚

x_k

ρ

Œ

<∞, for someρ >0

)

,

which is a Banach space normed by

x_k

=inf

(

ρ >0 : ∞

X

k=1

M ‚

x_k

ρ

Œ

≤1

)

.

Definition 1. An Orlicz function is a function M :[0,∞)→[0,∞)which is continuous,

non-decreasing and convex with M(0) =0, M(x)>0for x>0and M(x)_{→ ∞}as x _{→ ∞}.

∗_{Corresponding author.}

Email addresses:[email protected](B. Hazarika),[email protected](A. Esi)

An Orlicz function M is said to satisfy the∆₂-conditionfor all values ofu, if there exists a constantK>0, such thatM(2u)≤K M(u),u≥0. Note that, if 0< λ <1, then

M(λx)_≤λM(x), for all x≥0.

In the later stage different Orlicz sequence spaces were introduced and studied by Parashar and Choudhary[8], Et and Colak[2]and many others.

Kizmaz[5]introduced the notion of difference sequence spaces as follows:

X(∆) =x= x_k: ∆x_k∈X

forX =l_∞,candc_o. Later on, the notion was generalized by Et and Colak[2]as follows:

X(∆m) =x= x_k: ∆mx_k∈X

forX =l_∞,c andco, where∆mx = ∆mxk

= ∆m−1_x

k−∆m−1xk+1

,∆0_{x =x and also this} generalized difference notion has the following binomial representation:

∆mxk= m

X

i=0

(₋1)i

 m

i ‹

xk+i for allk∈N.

Definition 2 ([9]). A double sequence x = x_k,l

has a Pringsheim limit L (denoted by P−

limx =L) provided that given anǫ >0there exists an N ∈N_{such that}x_k,l−L

< ǫwhenever

k,l>N . We shall describe such an x= x_k,lmore briefly as "P-convergent".

The four dimensional matrixAis said to beRH-regularif it maps every boundedP-convergent

sequence into aP-convergentsequence with the sameP-limit. The assumption of boundedness was made because a double sequence which is P-convergent is not necessarily bounded. Using this definition Robison and Hamilton, independently, both presented the following Silverman-Toeplitz type characterization ofRH-regularity.

Lemma 1([4, 10]). The four dimensional matrix A is RH-regularif and only if

RH1: P−limm,nam,n,k,l =0for each k and l;

RH2: P−limm,n

P∞,∞

k,l=1,1am,n,k,l =1;

RH3: P−limm,n

P∞,∞

k,l=1,1

a_m,n,k,l

=0for each l;

RH4: P−limm,n

P∞,∞

k,l=1,1

a_m,n,k,l

=0, for each k;

RH₅: P∞_k,l=,∞_1,1a_m,n,k,l

is P-convergent;

RH₆: There exist finite positive integers E and F such thatP_k,l>Fa_m,n,k,l

2. New Generalized Difference Double Sequence Spaces

Let M be an Orlicz function,p= pk,l

be a factorable double sequence of strictly positive real numbers andA= a_m,n,k,lbe a nonnegativeRH-regularsummability matrix method. We now define the following new difference double sequence spaces (for someρ >0 andL):

w2_oA,M,p(∆r) =

(

x = x_k,l

:P−lim m,n

∞_X,∞

k,l=0,0

a_m,n,k,l

–

M ‚

∆rx_k,l

ρ

Ϊpk,l

=0,

)

,

w2A,M,p(∆r) =

(

x = x_k,l:P−lim

m,n ∞,∞

X

k,l=0,0

a_m,n,k,l –

M ‚

∆rx_k,l−L

ρ

Ϊpk,l

=0,

)

,

w2_∞A,M,p(∆r) =

(

x = x_k,l: sup m,n

∞,∞

X

k,l=0,0

a_m,n,k,l –

M ‚

∆rx_k,l

ρ

Ϊpk,l

<∞,

)

.

WhenM(x) =x, we have the following difference sequence spaces:

w2_oA,p(∆r) =

(

x = x_k,l:P−lim

m,n ∞_X,∞

k,l=0,0

a_m,n,k,l∆rx_k,l

pk,l =0

)

,

w2A,p(∆r) =

(

x = x_k,l:P−lim

m,n ∞,∞

X

k,l=0,0

a_m,n,k,l∆rx_k,l−L

pk,l =0, for someL )

,

w2_∞A,p(∆r) =

(

x = x_k,l: sup m,n

∞,∞

X

k,l=0,0

a_m,n,k,l∆rx_k,l

pk,l <∞

)

.

Some spaces are defined by specializingA,M,r andp= pk,l

. For example, if

A= (C, 1, 1)the difference sequence spaces defined above becomew2_oM,p(∆r),w2M,p(∆r)

andw2_∞M,p(∆r)which are as follows (for someρ >0 andL):

w2_oM,p(∆r) =

(

x= x_k,l∈w2:P−lim

m,n 1

mn

m−1,n−1

X

k,l=0,0

–

M ‚

∆rx_k,l

ρ

Ϊpk,l

=0,

)

,

w2M,p(∆r) =

(

x= x_k,l:P−lim

m,n 1

mn

m−1,n−1

X

k,l=0,0

–

M ‚

∆rx_k,l−L

ρ

Ϊpk,l

=0,

)

,

w2_∞M,p(∆r) =

(

x= x_k,l: sup m,n

1

mn

m−1,n−1

X

k,l=0,0

–

M ‚

∆rx_k,l

ρ

Ϊpk,l

<_∞,

)

.

LetA= (C, 1, 1),p_k,l=1, for allk,l∈NandM(x) =x, we obtain the following difference sequence spaces:

w2_o(∆r) =

(

x= xk,l

:P−lim m,n

1

mn

m−1,n−1

X

k,l=0,0

∆rx_k,l

=0

)

w2(∆r) =

(

x= x_k,l

:P−lim m,n

1

mn

m−1,n−1

X

k,l=0,0

∆rx_k,l−L

=0, for someL )

,

w2_∞(∆r) =

(

x= x_k,l: sup m,n

1

mn

m−1,n−1

X

k,l=0,0

∆rx_k,l

<∞

)

.

If r=1 the we obtain the following difference sequence spaces (for someρ >0 andL):

w2_oA,M,p(∆) =

(

x = x_k,l:P−lim

m,n ∞,∞

X

k,l=0,0

a_m,n,k,l –

M ‚

∆x_k,l

ρ

Ϊpk,l

=0,

)

,

w2A,M,p(∆) =

(

x = x_k,l:P−lim

m,n ∞,∞

X

k,l=0,0

a_m,n,k,l –

M ‚

∆x_k,l−L

ρ

Ϊpk,l

=0,

)

,

w2_∞A,M,p(∆) =

(

x = x_k,l: sup m,n

∞,∞

X

k,l=0,0

a_m,n,k,l –

M ‚

∆x_k,l

ρ

Ϊpk,l

<_∞,

)

which were defined and studied by Esi[1].

3. Main Results

In this section we shall establish some basic properties for the difference sequence spaces defined above.

Theorem 1. Let p= p_k,l

be bounded. The classes of sequences w2_oA,M,p(∆r_{), w}2

A,M,p(∆r₎

and w2_∞A,M,p(∆r)are linear spaces.

Proof. The proof of the theorem is easy, so omitted.

Theorem 2. If 0 < h= infpk,l ≤ suppk,l = H < ∞, then for any Orlicz function M and a

nonnegative RH-regularsummability matrix method A, then w2A,p(∆r)_⊂w2A,M,p(∆r).

Proof. Let 0< h= infp_k,l ≤supp_k,l = H < ∞and x = x_k,l∈ w2A,p(∆r)and let

0< ǫ <1 andδwith 0< δ <1 such that M(t)< ǫfor 0_≤ t< δ. We can write for eachm

andn

∞_X,∞

k,l=0,0

a_m,n,k,l

–

M ‚

∆rx_k,l−L

ρ

Ϊpk,l

=

∞_X,∞

k,l=0,0 &|∆r_xk,l−L|_≤δ

a_m,n,k,l

–

M ‚

∆rx_k,l−L

ρ

Ϊpk,l

+

∞_X,∞

k,l=0,0 &|∆r_x k,l−L|>δ

a_m,n,k,l –

M ‚

∆rx_k,l−L

ρ

Ϊpk,l

.

Then

∞_X,∞

k,l=0,0 &|∆r_xk,l−L|_≤δ

a_m,n,k,l –

M ‚

∆rx_k,l−L

ρ

Ϊpk,l

≤ǫh ∞_X,∞

k,l=0,0

On the other hand, we use the fact that

∆rx_k,l−L

<1+





∆rx_k,l−L

ρ





where[|t|]denotes the integer part of t. SinceM is Orlicz function we have

M ‚

∆rx_k,l−L

ρ

Œ

≥M(1).

Now, let us consider the second part where the sum is taken over∆rx_k,l−L

> δ. Thus

X

k,l=0,0 &|∆r_x k,l−L|>δ

∞,∞_a m,n,k,l

–

M ‚

∆rx_k,l−L

ρ

Ϊpk,l

≤

∞,∞

X

k,l=0,0 &|∆r_x k,l−L|>δ

a_m,n,k,l 

M



1+





∆rx_k,l−L

ρ



 

 



pk,l

≤ 2M(1)δ−1H ∞_X,∞

k,l=0,0

a_m,n,k,l

‚

∆rx_k,l−L

ρ

Œpk,l

This inequality and from (1) andRH-regularity of A, we are granted that

x = x_k,l∈w2A,M,p(∆r)and this completes the proof.

Theorem 3. w2_oA,M,p(∆r), w2A,M,p(∆r) and w2_∞A,M,p(∆r) are complete linear topological spaces with the paranorm

g x_k,l=

r

X

k=1

|x_k,1|+

r

X

l=1 |x_1,l|

+inf







ρpk,lT >_{0 : sup} m,n

∞_X,∞

k,l=0,0

am,n,k,l

M

|∆rx_k,l|

ρ

pk,l!

1 T

≤1







.

where T =max(1,H), H=sup_k,lpk,l.

Proof. Clearly g(0) =0, g(₋x) =g(x). Letx = xk,l

, y = yk,l

∈w2_∞A,M,p(∆r). Then there exist someρ₁andρ₂ such that

sup m,n

∞_X,∞

k,l=0,0

am,n,k,l

–

M ‚

∆rx_k,l

ρ₁

Ϊpk,l!

1 T

and

sup m,n

∞,∞

X

k,l=0,0

a_m,n,k,l –

M ‚

∆ry_k,l

ρ₂

Ϊpk,l!1_T

≤1.

Letρ=ρ₁+ρ₂. Then we have

sup m,n

∞_X,∞

k,l=0,0

am,n,k,l

–

M ‚

∆r x_k,l+y_k,l

ρ

Ϊpk,l!

1 T

sup m,n

∞,∞

X

k,l=0,0

a_m,n,k,l –

M ‚

∆rx_k,l+ ∆ry_k,l

ρ₁+ρ₂

Ϊpk,l!

1 T

≤sup m,n

∞_X,∞

k,l=0,0

a_m,n,k,l –

ρ₁ ρ₁+ρ₂M

‚ ∆rx_k,l

ρ₁

Œ

+ ρ2

ρ₁+ρ₂M

‚ ∆ry_k,l

ρ₂

Ϊpk,l!_T1

By Minkowsky’s inequality

≤

ρ₁ ρ₁+ρ₂

sup m,n

∞,∞

X

k,l=0,0

a_m,n,k,l –

M ‚

∆rx_k,l

ρ₁

Ϊpk,l!

1 T

+

ρ₂ ρ₁+ρ₂

sup m,n

∞_X,∞

k,l=0,0

a_m,n,k,l

–

M ‚

∆ry_k,l

ρ₂

Ϊpk,l!

1 T

≤1.

Now

g x_k,l+ y_k,l=

r

X

k=1

|x_k,1|+|y_k,1|+

r

X

l=1

|x_1,l|+|y_1,l|

+inf







ρpk,lT _{>0 : sup} m,n

∞_X,∞

k,l=0,0

a_m,n,k,l –

M ‚

∆r x_k,l+y_k,l

ρ

Ϊpk,l!

1 T ≤1    ≤ r X

k=1

|x_k,1|+

r

X

k=1

|y_k,1|+

r

X

l=1 |x_1,l|+

r

X

l=1 |y_1,l|

+inf    ρ pk,l T

1 >0 : sup m,n

∞_X,∞

k,l=0,0

am,n,k,l

–

M ‚

∆rx_k,l

ρ₁

Ϊpk,l!

1 T    +inf    ρ pk,l T

2 >0 : sup m,n

∞_X,∞

k,l=0,0

am,n,k,l

–

M ‚

∆ry_k,l

ρ₂

Ϊpk,l!

1 T







=g xk,l

+g yk,l

Letλ∈C_{, then the continuity of the product follows from the following equality:}

g λ x_k,l

=

r

X

k=1

|λx_k,1|+

r

X

l=1 |λx1,l|

+inf







ρpk,lT >_{0 : sup} m,n

∞_X,∞

k,l=0,0

am,n,k,l

–

M ‚

λ∆rx_k,l

ρ

Ϊpk,l!

1 T

≤1,ρ >0







=_|λ_|

r

X

k=1

|xk,1|+|λ| r

X

l=1 |x1,l|

+inf







(|λ|r)pk,lT _{>0 : sup} m,n

∞,∞

X

k,l=0,0

a_m,n,k,l –

M ‚

λ∆rx_k,l

ρ

Ϊpk,l!

1 T

≤1,r>0







=|λ|g x_k,l

where 1_r = |λ_ρ|.

Now let€xs_k,lŠbe a Cauchy sequence inw_∞2 A,M,p(∆r). Then

g€€x_ks_,l−x_kt_,lŠŠ→0 ass,t → ∞.

For given ǫ > 0, choose b > 0 and x_o > 0 be such that _{b x}ǫ

o > 0 and M

€_{b x}

o 2

Š

≥ 1. Now

g€€xs_k,l−x_kt_,lŠŠ→0 ass,t→ ∞implies that there existsn_o∈N_{such that}

g€€xs_k,l−x_kt_,lŠŠ< ǫ

b x_o for alls,t≥no.

⇒ r

X

k=1

x

s k,1−x

t k,1 + r X

l=1

x

s 1,l−x

t 1,l +inf       

ρpk,lT >_{0 : sup} m,n



 

∞_X,∞

k,l=0,0

am,n,k,l

  M    ∆

r_xs k,l−∆

r_xt k,l

ρ       pk,l   1 T

l eq1

       < ǫ

b x_o

(2) This implies that

r

X

k=1

x

s k,1−x

t k,1 + r X

l=1

x

s 1,l−x

t 1,l

< ǫ, f or al l s,t≥n0.

This shows that €xs_k,1Š, €x_1,t_lŠare Cauchy sequences of real numbers. As the set of real numbers is complete so there exists real numbers x_k,1,x_1,l such that

lim s→∞x

s

Now from 0(2) we have,

M 

 

∆

r_xs k,l−∆

r_xt k,l

ρ





≤1≤M

_{b x}

o 2

‹

⇒

∆

r_xs k,l−∆

r_xt k,l

g€€x_ks_,l−x_kt_,lŠŠ ≤ b xo

2

⇒

∆

r_xs k,l−∆

r_xt k,l

<

b xo 2 .

ǫ

b xo

= ǫ

2.

This implies that€∆rx_ks_,lŠis a Cauchy sequence of real numbers. Let lim_s→∞∆rxs_k,l=z_k,l for allk,l∈N.

Letk,l=1, we have lim_s→∞∆r_xs

1,1=lims→∞ r

P

i=0 r

P

j=0

(₋1)i+j r i

_r

j

x1+i,1+j=z1,1. Similarly we have lim_s→∞∆r_xs

k,l=lims→∞xks,l=zk,l fork,l=1, 2, . . . ,r.

Thus we have lim_s→∞xs_1+r,1+r exists. Let lim_s→∞x₁s_+r,1+r = x_1+r,1+r. Proceeding in this way inductively we conclude that lim_s→∞xs_k,l =xk,l exists for eachk,l∈N. Using continuity ofM, we have

lim t→∞M



 

∆

r_xs k,l−∆

r_xt k,l

ρ





≤1

⇒M



 

∆

r_xs

k,l−∆xk,l

ρ





≤1.

Lets≥no, then taking the infimum of suchρ′swe haveg

€€

x_ks_,l−xk,l

ŠŠ

< ǫ. Thus

(xs_k,l−xk,l)∈w2_∞

A,M,p(∆r_{). By linearity of the spacew}2 ∞

A,M,p(∆r_{)we have}

x_k,l∈w2_∞A,M,p(∆r). Hencew2_∞A,M,p(∆r)is complete space.

Proposition 1.

(a) w2A,M,p(∆r_)⊂w2

∞

A,M,p(∆r_),

(b) w2_oA,M,p(∆r)_⊂w2_∞A,M,p(∆r).

Proof. The proof is easy.

Proof. The proof is clear in view of Theorem 3 and Proposition 1.

Theorem 5.

(a) If0<h=infpk,l<pk,l ≤1, then

w2A,M,p(∆r)⊂w2[A,M] (∆r).

(b) If1_≤p_k,l≤supp_k,l <_∞, then

w2[A,M] (∆r)_⊂w2A,M,p(∆r).

Proof. (a)Let x = x_k,l∈w2A,M,p(∆r), since 0<h=infp_k,l < p_{k,l ≤}1, we obtain

the following: ∞,∞

X

k,l=0,0

a_m,n,k,lM ‚

∆rx_k,l−L

ρ

Œ

≤ ∞,∞

X

k,l=0,0

a_m,n,k,l –

M ‚

∆rx_k,l−L

ρ

Ϊpk,l

thus x= xk,l

∈w2[A,M] (∆r_).

(b)Let p_k,l ≥1 for eachk,l and supp_k,l<∞and letx = x_k,l∈w2[A,M] (∆r). Then for each 0< ǫ <1 there exists a positive integerK such that

∞,∞

X

k,l=0,0

a_m,n,k,lM ‚

∆rx_k,l−L

ρ

Œ

≤ǫ <1

for alln,m≥K. This implies that ∞_X,∞

k,l=0,0

a_m,n,k,l

–

M ‚

∆rx_k,l−L

ρ

Ϊpk,l

≤ ∞_X,∞

k,l=0,0

a_m,n,k,lM

‚

∆rx_k,l−L

ρ

Œ

.

Thus x = x_k,l∈w2A,M,p(∆r). This completes the proof.

Corollary 1. Let A= (C, 1, 1). Then (a) If0<h=infpk,l<pk,l ≤1, then

w2M,p(∆r)⊂w2[M] (∆r).

(b) If1_≤pk,l≤suppk,l <∞, then

Theorem 6. Ifsuppk,l_p

i,j <∞for all k≥i , l≥ j, then w

2

A,M,p⊂w2A,M,p(∆r_{)and the}

inclusion is strict, where

w2A,M,p=

(

x = x_k,l∈w2:P−lim

m,n ∞_X,∞

k,l=0,0

a_m,n,k,l –

M ‚

x_k,l−L

ρ

Ϊpk,l

=0,

for someρ >0and L .

Proof. Let x = x_k,l∈w2A,M,p. Then

P−lim

m,n ∞,∞

X

k,l=0,0

a_m,n,k,l –

M ‚

x_k,l−L

ρ

Ϊpk,l

=0, for someρ >0 andL. (3)

Since suppk,l

pi,j <∞, so there existsC >0 such thatpk,l <C pi,j for allk≥i,l≥ j. Thus from (3) we have,

P−lim

m,n ∞,∞

X

k,l=0,0

a_m,n,k,l –

M ‚

x_k,l−L

ρ

Ϊpk,l+1

=0,

P−lim

m,n ∞_X,∞

k,l=0,0

am,n,k,l

–

M ‚

x_k,l−L

ρ

Ϊpk+1,l

=0,

P−lim

m,n ∞,∞

X

k,l=0,0

a_m,n,k,l –

M ‚

x_k,l−L

ρ

Ϊpk+1,l+1

=0.

Now for ∆rx_k,l

=

∆r−1x_k,l−∆r−1x_k,l+1−∆r−1x_k+1,l+ ∆r−1x_k+1,l+1+L−L+L−L

we

have,

P−lim

m,n ∞,∞

X

k,l=0,0

a_m,n,k,l –

M ‚

∆rx_k,l−L

ρ

Ϊpk,l

≤P−lim m,n

∞_X,∞

k,l=0,0

a_m,n,k,l –

M ‚

∆r−1x_k,l−L

ρ +

∆r−1x_k+1,l−L

ρ +

∆r−1x_k,l+1−L

ρ

+

∆r−1x_k+1,l+1−L

ρ

Ϊpk,l

≤D2P−lim m,n

∞,∞

X

k,l=0,0

a_m,n,k,l 

 ‚

M ‚

∆r−1x_k,l−L

ρ

ŒŒpk,l

+

‚

M ‚

∆r−1x_k+1,l−L

ρ

ŒŒpk,l

+

‚

M ‚

∆r−1x_k,l+1−L

ρ

ŒŒpk,l

+

‚

M ‚

∆r−1x_k+1,l+1−L

ρ

ŒŒpk,l

≤D2P−lim m,n

∞_X,∞

k,l=0,0

a_m,n,k,l



 ‚

M ‚

∆r−1x_k,l−L

ρ

ŒŒpk,l

+

‚

M ‚

∆r−1x_k+1,l−L

ρ

ŒŒpk+1,l

+

‚

M ‚

∆r−1x_k,l+1−L

ρ

ŒŒpk,l+1

+

‚

M ‚

∆r−1x_k+1,l+1−L

ρ

ŒŒpk+1,l+1 

=0

where D=max 1, 2H−1. Thusx = x_k,l∈w2A,M,p(∆r). This completes the proof. The inclusion is strict follows from the following example.

Example 1. Let A= (C, 1, 1), M(x) =xp, p_k,l =1for all k odd and for all l ∈N_{and pk,l =2}

otherwise. Consider the sequence x = x_k,ldefined by x_k,l = (k+l)r for all k,l ∈N_{. We have}

∆rxk,l =0for all k,l∈N. Hence x= xk,l

∈w2A,M,p(∆r)but x= xk,l

/

∈w2A,M,p.

Let Ebe a sequence space. ThenEis called

(a) solid (or normal) if(α_kxk)∈Ewhenever(xk)∈E for all sequences(αk)of scalars with |α_{k| ≤}1 for allk∈N;

(b) monotone providedEcontains the canonical preimages of all its step spaces. It is a well known result that ifEis normal then it is monotone.

Theorem 7. The spaces w2_oA,M,p(∆r_{)and w}2 ∞

A,M,p(∆r_{)are normal as well as}

mono-tone.

Proof. Let(α_k,l)be a double sequences of scalars such that_|α_{k,l| ≤}1 for allk,l_∈N. Since

M is monotone, we get for someρ >0 ∞_X,∞

k,l=0,0

am,n,k,l

M

|∆r(α_k,lx_k,l)|

ρ

pk,l

≤ ∞_X,∞

k,l=0,0

am,n,k,l

M

sup|α_k,l||∆

r_x k,l|

ρ

pk,l

≤ ∞,∞

X

k,l=0,0

a_m,n,k,l

M

|∆rx_k,l|

ρ

pk,l

which leads us to the desired results.

4. Double

∆

−

Statistical Convergence

The concept of statistical convergence for single sequences was introduced by Fast[3]in 1951. Later, Mursaleen and Edely [7] defined the statistical analogue for double sequence

x = xk,l

as follows: A real double sequencex = xk,l

is said to be P-statistically convergent to Lprovided that for eachǫ >0

P−lim

m,n 1

mn

the number of (k,l):k<m,l <n;x_k,l−L

In this case, we writest₂−lim_k,lx_k,l= Land we denote the set of allP-statistically convergent

double sequences byst₂.

Definition 3. A real double sequence x= xk,l

is said to be P-statistically∆r_{-convergent to L}

provided that for eachǫ >0

P−lim

m,n 1

mn

the number of (k,l):k<m,l<n;∆rx_k,l−L

≥ǫ =0.

In this case, we write st2(∆r)−limk,l xk,l = L and we denote the set of all P-statistically∆r

-convergentdouble sequences by st2(∆r).

Theorem 8. If M be an Orlicz function, then w2[M] (∆r)⊂st₂(∆r).

Proof. Suppose that x = x_k,l∈w2[M] (∆r)andǫ >0, then we obtain the following for

everynandm

1

mn

m−1,n−1

X

k,l=0,0

M ‚

∆rx_k,l−L

ρ

Œ

≥ 1

mn

m−1,n−1

X

k,l=0,0 &|∆r_x k,l−L|≥ǫ

M ‚

∆rx_k,l−L

ρ

Œ

≥M(ǫ)

mn

the number of (k,l):k<m,l<n;∆rx_k,l−L

≥ǫ .

Hence x= xk,l

∈st2(∆r).

Theorem 9. st2(∆r) =w2o[M] (∆

r_{)if and only if the Orlicz function M is bounded.}

Proof. Suppose that M is bounded and x = xk,l

∈st2(∆r). Since M is bounded then there exists an integerK such thatM(x)_≤K, for all x ≥0. Then for eachmandn, we have

1

mn

m−1,n−1

X

k,l=0,0

M ‚

∆rx_k,l

ρ

Œ

= 1

mn

m−1,n−1

X

k,l=0,0 &|∆r_x k,l−L|≥ǫ

M ‚

∆rx_k,l

ρ

Œ

+ 1

mn

m−1,n−1

X

k,l=0,0 &|∆r_x k,l−L|<ǫ

M ‚

∆rx_k,l

ρ

Œ

≤ K

mn

the number of (k,l):k<m,l<n;∆rx_k,l

≥ǫ +M(ǫ)

and thus the Pringsheim’s limit onmandngrant us the result.

Conversely, suppose thatM is unbounded so that there is a positive double sequence z_mn

withM z_mn= (mn)2form,n=1, 2, . . .. Now the sequence x= x_k,l

defined by

∆rx_k,l =z_mnifk,l= (mn)2form,n=1, 2, . . . and∆rx_k,l=0, otherwise. Then we have 1

mn

the number of (k,l): k<m,l<n;∆rx_k,l

≥ǫ ≤

p

mn

mn →0, asm,n→ ∞.

References

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