Some I-related properties of triple sequences

Selçuk J. Appl. Math. Selçuk Journal of Vol. 9. No. 2, pp. 9 –18 , 2008 Applied Mathematics

Some I-related properties of triple sequences

Ahmet Sahiner1, Binod Chandra Tripathy2

1_{Department of Mathematics, Suleyman Demirel University, Cunur Campus, 32260,}

Isparta, Turkey

e-mail: sahiner@fef.sdu.edu.tr

2

Mathematical Sciences Division, Institute of Advanced Study in Science and Technology, Paschim Boragaon,Garcuk; Guwahati-781 035, India e-mail: tripathybc@yahoo.com

Abstract. Ideal convergence was presented by Kostyrko et al. in 2001. This concept was extended to the double sequences by Tripathy et al. in 2006. In this paper we introduce the notions of Iconvergence, Ibounded,



I inferior and Isuperior for triple sequences. We also investigate some further properties of Ilimit superior and Ilimit inferior of triple sequences.

Key words: Double sequence; triple sequence; statistical convergence; I-convergence; double natural density; triple natural density

2000 Mathematics Subject Classification. 40A05, 26A03.

1. Preliminaries

In this article we aimed to extend the notion of statistically convergent triple sequences to I convergent triple sequences. Now we recall  some definitions and notions introduced in [14].

Definition 1. A function x : NNNR

 

C is called a real (complex) triple sequence.

Definition 2. A triple sequence

 

xnkl is said to be convergent to L

in Pringsheim's sense if for every  0, there exists n0

 

 N such that x_nklL  whenever n,k,ln₀ .

Definition 3. A triple sequence

 

x_nkl is said to be Cauchy sequence if for every  0, there exists n0

 

 N such that



  _nkl

pqr x

x whenever pnn₀,qk n₀,rln₀.

Definition 4. A triple sequence

 

x_nkl is said to be bounded if there exists M 0 such that x_nkl M for all n ,,k lN .

We denote the set of all bounded triple sequences by 3.   It can easily be shown that 3   is a normed space by ,3sup_{, ,} nkl . l k n x x

The notion of statistically convergent double sequences was introduced by Tripathy [12]. Recall that a subset E of NN is said to have density 

 

E if

 

E _E

 

n k q k p n pq q plim , 1 ,  





     exists, where _E

 

n,k is the characteristic function of the set E. Thus a double sequence

 

xnk is said to be statistically convergent to L in Pringsheim's sense if

for every  0,

 







,   :  



0

 n k N N x_nk L

[13]. The notion of statistically convergent double sequences was extended to I convergent double sequences by Tripathy in [13].  The notion of statistically convergent triple sequences was introduced by Sahiner [14]. Recall that a subset A of NNN is said to have density 

 

A if

 



n k l



pqr A _A r l q k p n r q p , , 1 lim , ,  







    

exists. For example if A





n3,k3,l3



: n,k,lN



then

 

lim



, ,



lim 0 3 3 3 , , , ,    pqr r q p pqr r q p K A r q p r q p  where



p q r

 



n k l



p n q k r l



K , ,  , , NNN :  ,  , 

and K



p,q,r



is the cardinality of K



p,q,r



. Thus a triple sequence

 

xnkl is said to be statistically convergent to L in Pringsheim's sense if

for every  0 ,











, ,    :  



0.

 n k l N N N x_nkl L

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IandBABI



. An ideal I2X is said to be non-trival if I2X.

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

In this article we aimed to introduce and examine I related  interesting properties of triple sequences.

We denote the ideals of 2 by I ; the ideals of N 2NN by I and the ₂

ideals of 2NNN by I . ₃

2. Ideal convergence of triple sequences

Definition 5. Let I be an ideal of ₃ 2NNN, then a triple sequence

 

xnkl is said to be I convergent to  L in Pringsheim's sense if for

every  0,







n,k,l NNN : x_nklL 



I₃.

If

 

x_nkl is I convergent to  L we write I - ₃ limx_nklL .

Now we give some examples of ideals and corresponding



I convergences.

(I) Let I3

 

f be the family of all finite subsets of NNN . Then

 

f

3

I is an admissible ideal in NNN and I3

 

f convergence coincides with the convergence of triple sequences in Pringsheim's sense.

(II) Let ANNN be a three dimensional set of positive integers and let A



p,q,r



be the cardinality of



n ,,k l



in A such that

p

n , kq , lr. In case of the sequence



Ap_pqrqr



r q p , , , ,

lim has a limit in Pringsheim's sense then we say that A has a triple natural density

and we denote this by lim



 , , 



 

. , , pqr A r q p A r q p  Put

 



:

 

0



. 3   ANNN  A  I Then I₃

 

 is an admissible ideal in NNN and I₃

 

 convergence coincides with the statistical convergence in Pringsheim's sense [14].

Example 1. Let II₃

 

 . Define the triple sequence

 

x_nkl by     otherwise. , 5 squares are and , if , 1 n k l x_nkl Then for every  0











, , : 5



lim 0. , ,        pqr r q p x l k n r q p nkl   N N N

This implies that I limxnkl5 in Pringsheim's sense. But, the sequence

 

xnkl is not convergent to 5 in Pringsheim's sense.

Remark 1. If I is admissible and 3

 

xnkl converges to L in

Pringsheim's sense, then

 

xnkl is I convergent to  L in Pringsheim's

sense.

3. I- limit superior and I- limit inferior for triple sequences

Definition 6. Let I be an ideal of 3 N N N 

2 . A number  is said to be an I3limit point of the triple sequence

 

xnkl provided that there

exists a set M 



n1n2...

 

 k1k2...

 

 l1l2...



NNN such that MI₃ and  

m j ikl

n

x

P lim for all i,j,m1,2,... .

Definition 7. A number  is called to be an I cluster point of the  triple sequence

 

x_nkl if for each  0 ,







n,k,l NNN: x_nkl 



I₃.

Definition 8. A real triple sequence

 

x_nkl is said to be bounded if there is a K 0 such that





n,k,l



NNN: x_nkl K



I₃.

Now let

 

x_nkl be a triple sequence and tR . Then we set the following sets to be able to give the definitions of Iliminf x and

x sup lim  I of

 

x_nkl .







n,k,l : x t



, M





n,k,l



: x t



. M_t  _nkl  t  _nkl

Definition 9. (a) If there is a tR such that MtI3, we put



:



.

sup sup

lim I3

I x tR Mt

If MtI3 holds for each tR then we put Ilimsupx.

(b )If there is a tR such that tI₃,

M we put



:



. inf inf lim I3 I   t M t x R If tI₃

M holds for each tR then we put Iliminfx.

Example 2. If we define

 

x_nkl by                nonsquare even an is , 0 nonsquare odd an is , 1 square even an is , 2 square odd an is , n n n n n xnkl or                nonsquare even an is , 0 nonsquare odd an is , 1 square even an is , 2 square odd an is , k k k k k x_nkl or                nonsquare even an is , 0 nonsquare odd an is , 1 square even an is , 2 square odd an is , l l l l l x_nkl

then, in each case,

 

xnkl is not bounded above but it is I bounded. 

Also,



tR : M_tI₃

 

 ,1



,



tR : MtI₃





 

0, and thus Ilimsupx1 , Iliminfx0 . On the other hand

 

x_nkl can not be I convergent in Pringsheim's sense and the set of  I 

cluster points in Pringsheim's sense is

 

0,1. So we have the following.

Theorem 1. (i)  Ilimsupx if and only if for each  0 ,







n,k,l NNN: xnkl  



I3

and







n,k,l NNN: xnkl 



I3

(ii)  Iliminfx if and only if for each  0 ,







n,k,l NNN: xnkl



I3

and







n,k,l NNN: xnkl 



I₃.

Proof. (i) We prove necessity first. Let  0 be given. Since ,    





 

 t : MtI₃



and







n,k,l NNN: xnkl  



I3. Similarly, since  , there exists some t such that  t and t



t : MtI3



. Thus





n,k,l



NNN: xnkl t



I₃ and







n,k,l NNN: xnkl 



I3.

Now we prove sufficiency. If  0 then





 

 t : MtI3



and

. sup

lim

3 xnkl

I On the other hand we already have 

   limsupxnkl

3

I and this means I3limsupxnkl as

desired.

(ii) can be proved analogously.

Theorem 2. For every real triple sequence

 

xnkl ,

nkl nkl x x limsup inf lim 3 3 I  I .

Proof. If

 

x_nkl is any real triple sequence we have three possibilities: (1) The case I₃limsupx_nkl  is clear.

(2) If I₃limsupx_nkl. Then we have . and 3 3 I I     t t M M t R

Thus, I₃liminf inf



: tI₃



infR

nkl t M x and nkl nkl x x limsup inf lim ₃ 3 I  I .

(3) If I3limsupx_nkl and if  I3limsupxnkl then for any tR, . and ₃ 3 I I     t t M M t 

But this means I3liminf inf



: I3



.

t

nkl t M

x

Theorem 3. For any I bounded real triple sequence 



xnkl



we

have the following inequalities.

. sup lim sup lim inf lim inf lim x_{nkl 3} x_{nkl 3} x_nkl P x_nkl P I  I   

Proof. The case Plimsupx_nkl  is straightforward. Let 

 

 x L

P limsup _nkl . Then for any t L, M_tI₃. So



: I3





 t M_t

t implies I₃limsupx_nklsup



t : M_tI₃



t and .

sup lim

3 xnklL

I This proves the last inequality. For the first one, if Pliminfx_nkl  then clearly the inequality holds. Let

  

 x T

P liminf _nkl . Then for any tT, I3.  t M So



: I3



  t M t

t implies I₃liminf x_nklsup



t : MtI₃



t and .

sup lim

3 xnklT

I

Remark 2. If I3 limxnkl exists then

 

xnkl is I bounded. 

Remark 3. Note that ideal boundedness of triple sequences implies that I₃limsup and I₃liminf are finite.

Recall that the core of a bounded double sequence x_nk, that is,

 

x_nk ,

core

P is the interval



Pliminfx_nk,Plimsupx_nk



Pcore

 

x_nk ; I core of bounded  double sequence x is the interval _nk



I₂liminf x_nk,I₂limsupx_nk



. Analogously we give the definitions of Pcore and Icore of a bounded triple sequence x_nkl.

Definition 10. We define the Pcore of bounded real triple sequence

nkl

x by

Definition 11. If

 

x_nkl is any I bounded real triple sequence then ₃ we define its Icore by



I3liminf xnkl,I3limsupxnkl



. We use I3core

 

xnkl to denote I core of 3 xnkl .

Corollary 1. If

 

x_nkl is any real triple sequence then we have

 

 

.

3core xnkl Pcore xnkl

I

Theorem 4. A real triple sequence

 

x_nkl is I₃ convergent if and only if I₃liminf x_nklI₃limsupx_nkl .

Proof. Let LI₃limx_nkl. Then







n,k,l NNN: xnklL



I3 and







n,k,l NNN: xnklL



I3. Then for any tL and ,



 

 L

t the sets M and _t Mt are in I Hence ₃.



t : M_t ₃



L

sup I and inf



t : MtI₃



L . To prove sufficiency let  0 and LI₃liminf x_nklI₃limsupx_nkl. Then since



























                   L x l k n L x l k n L x l k n nkl nkl nkl : , , : , , : , , N N N N N N N N N

we conclude that LI₃limx_nkl .

Note that if

 

x_nkl is a bounded real triple sequence then we denote the set of all I₃ cluster points of

 

x_nkl by I₃

 

_x .

Theorem 5. Suppose

 

xnkl is a bounded real triple sequence then

 

x nkl

x  

 3

3 limsup maxI

I and

 

. min inf lim 3 3 xnkl  I x I Proof. Let











3



3 limsup sup : , , : I

I  x_nkl L t n k l NNN x_nklt  . If L







n,k,l N: xnklL



I3 and this means there exists some

0   such that







n,k,l NNN: x_nklL



I₃, that is, L

 

_x .

Now, we show L is really an I₃ cluster point of

 

x_nkl . Clearly, for each  0 there exists some t



L,L



such that







n,k,l NNN: xnkl t



I3 and this implies







n,k,l NNN: xnklL 



I3.

4. Further properties

In this section we prove some further results on Ilimsup and inf

lim 

I of a triple sequence.

Theorem 6. Let I be an ideal of 3 2 . N N N 

If x

 

xnkl ,y

 

ynkl are

two I bounded triple sequences in Pringsheim's sense, then  (i) Ilimsup



xy



IlimsupxIlimsupy

(ii) Iliminf



x y



Iliminf xIliminf y.

Proof. Since the proof of

 

ii is analogous we prove only

 

i Let . ,

sup lim

1I x

 2 Ilimsupy and  0 be given. We know that both  and ₁  are finite. Let ₂











 : , , :   I3



.

 c n k l x y c

A R nkl nkl

We can also assume that A is not empty. Now since



















             _{ }        _{ } 2 1 2 1 : , , 2 : , , 2 : , ,     nkl nkl nkl nkl y x l k n y l k n x l k n we have

















. 2 : , , 2 : , , : , , 2 1 2 1       _{ }        _{ }             nkl nkl nkl nkl y l k n x l k n y x l k n







n,k,l : xnklynkl 12



I3.

If cA, then





n,k,l



: x_nkly_nkl c



I₃. We claim that .

2

1 

 

c For, otherwise we would have







n,k,l : xnklynklc

 





n,k,l



: xnklynkl12



which means





n,k,l



: xnklynklc



I3, a contradiction. Hence    ₁ ₂ c and we deduce





sup . sup lim _{1 2} 3 xnklynkl  A   I

Since  0 is arbitrary, this completes the proof.

We need the following definition for the subsequent theorem.

Definition 12. Let I be an ideal of 3 2 . N N N 

A sequence x

 

xnkl

is said to be I convergent to   in Pringsheim's sense



or 



if for every real number G0,





n,k,l



: x_nklG



I₃











or n,k,l : x_nkl G I₃



.

Theorem 7. If I is an admissible ideal and 3 Ilimsupx, then

there exists a subsequence of

 

xnkl that is I convergent to   in

Pringsheim's sense.

Proof. Since I₃ and I is admissible, we can assume that ₃

 

xnkl is a non-constant triple sequence having infinite number of

distinct elements.

Case (i) : If . Then from definition,



tR : MtI



. Hence, if K 0, then





n,k,l



: xnkl2K



I3. Since







n,k,l : xnklK

 





n,k,l



: xnkl 2K



, we have





n,k,l



: xnklK



I3 and so Ilimx.

Case (ii) : If  then



tR : MtI



R. So for any tR,







n,k,l : xnkl t



I3. Let xn₁k₁l₁ be an arbitrary term of

 

xnkl and

let





, ,



: 1



. 1 1 1 1 1 1kl  nkl  nkl  n n k l x x A Since I₃ , 1 1 1kl n A is not empty and also ₃.

1 1 1kl I

n

A We claim that there is at least one



n,k,l



A_n1k1l1 such that nn₁1,kk₁1, ll₁1. For, otherwise

  













1,1,1, 2,2,2,..., 1, 1, 1, 1 1, 1 1, 1 1



, 1 1 1  n n n n  n  n  Ankl which is a

member of I (since ₃ I is admissible) and so _{3 3},

1 1 1kl I

n

A a

contradiction. We call this



n ,,k l



as



n₂,k₂,l₂



. Thus . 1 1 1 1 2 2 2kl  nkl  n x

x Proceeding in this way we obtain a subsequence

 

xnikili of

 

xnkl with xnikili xni1ki1li1 1 for all .i Since for any

, 0  K





n k l



x K



i i ikl n i i

i, , :  is a finite set, it must belong to I 3,

because I is admissible and so _{3 3}lim .

kl

n

x

I

Case (iii) : If . By Theorem 1(i),







n,k,l : xnkl 1



I3 so that





n,k,l



: xnkl1



. We

observe that there is at least one element, say n₁k₁l₁, in this set for

which ₂1, 1 1 1kl  n x for otherwise







 





2



3 1 : , , 1 : , ,k l x_nkl   n k l x_nkl I n which is a

contradiction. Hence we have

. 1 2 1 1 1 1 1          x_nkl

Next we proceed to choose an element

2 2 2kl n x from

 

xnkl , n2 n1 , 1 2 k

k  and l2 l1 such that

2 1 2 2 2kl  n x , for otherwise







2







 

 

1 1 1





1 , , ,..., 2 , 2 , 2 , 1 , 1 , 1 : , ,k l x n n n n nkl   is a member of 3

I which contradicts

 

i of Theorem 1. Hence





( ) . 2 1 and , , : , , 1 1 1 1 1 1         _{    } say E x l l k k n n l k n nkl  nkl Now if





1 1 1 , ,k l Enkl n  always implies _₂1 nkl x then









. 4 1 : , , 2 1 : , , 1 1 1       _{ }        _{ }  _nkl  _nkl  l k n n k l x n k l x E

By

 

i of Theorem 1, the right-hand set belongs to I and so ₃ . 3 1 1 1kl I n E Since I is admissible, ₃



  

1,1,1, 2,2,2

 

,..., n1,n1,n1





I3 and thus









1,1,1

 

, 2,2,2

 

,..., , ,





. 2 1 : , , 1 1 1 1 1 1 nkl nkl n n n E x l k n         _ So







2



3 1 : , ,k l xnkl  I n , a contradiction to Theorem 1.

and l₂ l₁ such that ₂1. 2 1 2 2 2     

 xn kl Proceeding in this way we

obtain a subsequence

 

i i ikl n x of

 

x_nkl , n_i n_i1,k_i k_i1 and 1   i i l l such that i xnikili _i 1 1    

 for each .i The subsequence

 

xnikili therefore converges to  in Pringsheim's sense and is thus



I convergent to  in Pringsheim's sense by Remark 1. This proves the theorem.

The reasonings made up to now give the folowing results.

Theorem 8. If Iliminfx, then there is a subsequence of

 

xnkl

which is I convergent to   in Pringsheim's sense.

Theorem 9. Every I bounded triple sequence 

 

xnkl in

Pringsheim's sense has a subsequence which is I convergent to a  finite real number in Pringsheim's sense.

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