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IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-3008, p-ISSN:2319-7676. Volume 10, Issue 2 Ver. III (Mar-Apr. 2014), PP 59-69 www.iosrjournals.org

On I 2 -Cauchy Double Sequences in p-Adic Linear 2-Normed Spaces B. Surender Reddy1, D. Shankaraiah 2 1

Department of Mathematics, University College of Science, Saifabad, Osmania University, Hyderabad –500 004, AP, INDIA 2 Department of Mathematics, CVR College of Engineering, Ibrahimpatnam, Ranga Reddy District–501510, AP, INDIA.

Abstract: In this paper, we introduce the concept of I 2  convergence which is closely related to

I 2  convergence and the concepts I 2 and I 2  Cauchy double sequences in p -adic linear 2-normed space. Also we investigate the relation between these concepts in p -adic linear 2-normed spaces. Keywords: 2-normed space, p-adic linear 2-normed space, I 2  convergence, I 2  Cauchy double sequence, I 2  Cauchy double sequence. I.

Introduction

The idea of I  convergence is based on the notion of the ideal I of subsets of N , the set of natural numbers. The notion of ideal convergence for single sequences was introduced first by P.Kostyrko et al [11, 12] as an interesting generalization of statistical convergence. F.Nuray and Ruckle [14] independently introduced the same concept as the name generalized statistical convergence. The concept of a double sequence was initially introduced by Pringsheim [16] in the 1900s and this concept has been studied by many others. A double sequence of real (or complex) numbers is a function from N  N to F (where F= R or C) and we denote a double sequence as ( xij ) where the two subscripts run through the sequence of Natural numbers independent of each other. P.Das et al [4] introduced the concept of I  convergence of double sequences in a metric space and studied some properties. Also, Pratulananda Das and Prasanta Malik [15] defined the concept of I  limit points, I  cluster points, I  limit superior and I  limit inferior of double sequences. Balakrishna Tripathy, Binod Chandra Tripathy [3] introduced the notion of I  convergence and I  Cauchy double sequences and many others discussed the properties of

I  convergence, I   convergence for double sequences and I  Cauchy double sequences (see ([5], [6], [25], for more details). The concept of linear 2-normed spaces has been investigated by Gähler in 1965 [7] and has been developed extensively in different subjects by others [9, 10, 17]. A.Sahiner et al [20] introduced I  cluster points of convergent sequences in 2-normed linear spaces and Gurdal [8] investigated the relation between I  cluster points and ordinary limit points of sequences in 2-normed spaces. The concept of I  convergence for the double sequences in 2-normed spaces introduced by Saeed Sarabadan and Sorayya Talebi [19]. Saeed Sarabadan, Fatemeh Amoee Arani and Siamak Khalehoghli [18] obtained Condition for the equivalence of I and I*-convergence for double sequences in 2-normed spaces. Mehmet Acikgoz [13] introduced a very understandable and readable connection between the concepts in p-adic numbers, p-adic analysis and linear 2-normed spaces. B.Surender Reddy [21] introduced some properties of p-adic linear 2-normed spaces and obtained necessary and sufficient conditions for p-adic 2-norms to be equivalent on p-adic linear 2-normed spaces. Recently B.Surender Reddy and D.Shankaraiah [22, 23, 24] 

introduced I  convergence, I  convergence of sequences, I  Cauchy, I  Cauchy sequences and their properties in p-adic linear 2-normed spaces and also introduced ideal convergence of double sequences in p-adic linear 2-normed spaces.

I 2  convergence of double sequences which is closely related to I 2  convergence of double sequences and the concepts I 2  Cauchy double  sequence and I 2  Cauchy double sequence in p -adic linear 2-normed space ( X , N (,) p ) . Also we investigate the relation between these concepts in p -adic linear 2-normed spaces. The main aim of this paper is we introduce the concept of

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On I 2 -Cauchy double sequences in p-adic linear 2-normed spaces II.

Preliminaries

In this paper, we will use the notations; p for a prime number, Z - the ring of rational integers,

  - the

Q - the field of rational numbers, R - the field of real numbers, R  - the positive real numbers, Z p - the ring of p-adic rational integers, Q p - the field of p-adic rational numbers, C - the field of positive integers,

complex numbers and

C p - the p-adic completion of the algebraic closure of Q p .

Definition 2.1: A double sequence if for each

x  ( xij ) is said to be convergent to a number  in the Pringsheim’s sense

  0 there exists a positive integer m such that | xij   | 

is called the Pringsheim limit of the sequence x and we write as

P  lim xij   . i , j 

n0 such that | xij  xmn |  for every i  m  n0 and j  n  n0 .

Definition 2.3: A double sequence that

i, j  m .Then the number

x  ( xij ) is said to be Cauchy sequence if for each   0 there exists a

Definition 2.2: A double sequence positive integer

whenever

x  ( xij ) is said to be bounded if there exists a real number M  0 such

| xij | M for each i and j .

Definition 2.4: Let K  N  N and

K (m, n)  {(i, j ) : (i, j )  K ; i  m, j  n} . If the sequence

 K (m, n)    has a limit in Pringsheim’s sense then we say that K has a double natural density and it is denoted  mn  K (m, n) as lim   2 (K ) . m , n  mn Definition 2.5: A double sequence

  0,

the set

x  ( xij ) is said to be statistically convergent to a number  if for each

A( )  {(i, j )  N  N :| xij   |  } has double natural density zero. If x  ( xij ) is

statistically convergent to

Definition 2.6: Let

then we write

St  lim xij   . i , j 

I 2 be an ideal in N  N . A double sequence x  ( xij ) is said to be I 2  convergent to L

in Pringsheim’s sense if for each

  0,

the set

{(i, j )  N  N :| xi j  L |  }  I 2 and L is called

I 2  limit of x  ( xij ) and we write I 2  lim xij  L . i , j 

Definition 2.7: A double sequence

x  ( xij ) is said to be I 2*  convergent to  if there exists a set

M  {(i, j ) : i, j  1,2,3,.......}  F ( I 2 ) (i.e, ( N  N )  M  I 2 ) such that lim xij   and  is i , j 

called

I  limit of x  ( xij ) and we write I  lim xij   . * 2

Definition 2.8: A double sequence

* 2

x  ( xij ) is I 2  convergent to zero in Pringsheim’s sense is called

I 2  null double sequence in Pringsheim’s sense. Definition 2.9: Let X be a linear space of dimension greater than 1 over K, where K is the real or complex numbers field. Suppose N (,) be a non-negative real valued function on X  X satisfying the following conditions: (2  N1 ) : N ( x, y)  0 and N ( x, y)  0 if and only if x and y are linearly dependent vectors, www.iosrjournals.org

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On I 2 -Cauchy double sequences in p-adic linear 2-normed spaces

(2  N 2 ) : N ( x, y)  N ( y, x) for all x, y  X , (2  N 3 ) : N (x, y)   N (x, y ) for all   K and x, y  X , (2  N 4 ) : N ( x  y, z)  N ( x, z)  N ( y, z) for all x, y, z  X . Then N (,) is called a 2-norm on X and the pair ( X , N (,)) is called a linear 2-normed space. Definition 2.10: Suppose a mapping conditions, for all

d p : X  X  X  R on a non-empty set X satisfying the following

x, y, z  X

D1 ) For any two different elements x and y in X there is an element z in X such that d p ( x, y, z )  0 .

D2 ) d p ( x, y, z )  0 when two of three elements are equal. D3 ) d p ( x, y, z )  d p ( x, z, y)  d p ( y, z, x) .

D4 ) d p ( x, y, z)  d p ( x, y, w)  d p ( x, w, z)  d p (w, y, z) for any w in X . Then d p is called p-adic 2-metric on condition

X and the pair (X, d p ) is called p-adic 2-metric space. If p-adic 2-metric also satisfies the

d p ( x, y, z )  max{ d p ( x, y, w), d p ( x, w, z), d p ( y, w, z)} for x, y, z, w  X , then d p is called

a p-adic ultra 2-metric and the pair

( X , d p ) is called a p-adic ultra 2-metric space.

Definition 2.11: Let X be a linear space of dimension greater than 1 over K, where K is the real or complex numbers field. Suppose N (, ) p be a non-negative real valued function on X  X satisfying the following conditions:

(2  pN1 ) : N ( x, z ) p  0 if and only if x and z are linearly dependent vectors. (2  pN 2 ) : N ( xy, z ) p  N ( x, z ) p .N ( y, z ) p for all x, y, z  X , (2  pN 3 ) : N ( x  y, z ) p  N ( x, z ) p  N ( y, z) p for all x, y, z  X ,

(2  pN 4 ) : N (x, z ) p   N ( x, z ) p for all   K and x, z  X . Then

N (, ) p is called a p-adic 2-norm on X and the pair ( X , N (, ) p ) is called p-adic linear 2-normed

space. For every p-adic linear 2-normed space

( X , N (,) p ) the function defined on X  X  X by

d p ( x, y, z )  N ( x  z, y  z ) p is a p-adic 2-metric. Thus every p-adic linear 2-normed space

( X , N (,) p ) will be considered to be a p-adic 2-metric space with this 2-metric. A double sequence ( xij ) of p-adic 2-metric space

( X , d p ) converges to x  X in p-adic 2-metric if for every   0 , there is an

l  1 such that d p ( xij , x, z)  N ( xij  z, x  z ) p   for every i, j  l . For the given two double sequences of p-adic 2-metric space ( X , d p ) which are

( xij ) and ( yij ) converges to x, y  X in the p-adic

2-metric space respectively, then the double sequence of sums

xij  yij and the product xij y ij converges to

x  y and to the product xy of the limits of initial double sequences. A double sequence ( xij ) of p-adic 2-metric space ( X , d p ) is a Cauchy double sequence with respect to the p-

the sum

adic 2-metric if for each for every i  m  l ,

  0,

there is an l  1 such that

d p ( xij , xmn , z )  N ( xij  z, xmn  z) p   ,

j nl.

Definition 2.12: A double sequence

x  ( xij ) in a p  adic linear 2-normed space ( X , N (,) p ) is said to

be convergent to l  X if for each

  0 there

exists m  N such that

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N ( xij  l , z ) p   for each

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On I 2 -Cauchy double sequences in p-adic linear 2-normed spaces

i, j  m and for each z  X . If x  ( xij ) is convergent to l then we write lim xij  l or i , j 

[ N ( ,  ) p ] X

xij  l . Definition 2.13: A double sequence be bounded if for each non zero

x  ( xij ) in a p  adic linear 2-normed space ( X , N (,) p ) is said to

z  X and for all i, j  N there exists M  0 such that N ( xij , z ) p  M .

Note that a convergent double sequence need not be bounded. Definition 2.14: A double sequence

x  ( xij ) in a p  adic linear 2-normed space ( X , N (,) p ) is said to

be Cauchy double sequence if for each

 0

there exists a positive integer

n0 such that

N ( xij  xmn , z ) p   for every i  m  n0 and j  n  n0 . A p-adic linear 2-normed space

( X , N (, ) p ) is called complete if every Cauchy sequence is

convergent in p-adic linear 2-normed space. A p-adic linear 2-normed space

( X , N (, ) p ) is called p-adic 2-

Banach space if p-adic linear 2-normed space is complete. Proposition 2.15: If a double sequence to x  X , then

{xij } in a p-adic linear 2-normed space ( X , N (,) p ) is convergent

lim N ( xij , z ) p  N ( x, z ) p for each z  X.

i , j 

Proposition 2.16: If

lim N ( xij , z ) p exists then we say that ( xij ) is a Cauchy sequence with respect to

i , j 

N (, ) p . Proof: Let us suppose that

 N ( x  xij , z ) p  by

using

the

 2

lim N ( xij , z ) p  x . Then we can obtain a constant M 1 such that i, j  M 1

i , j 

. If

triangle

i, j, m, n  M 1 then N ( x  xij , z ) p  inequality,

we

have

and N ( x  x mn , z ) p

, hence

2 2 N ( xij  xmn , z) p  N ( xij  x  x  xmn , z ) p

 . 2  ( xij ) is a Cauchy sequence with respect to N (, ) p .  N ( xij  x, z ) p  N ( x  xmn , z ) p 

2

III.

Main Results

I 2  convergence of double sequences which is closely related to I 2  convergence of double sequences in p-adic linear 2-normed space ( X , N (,) p ) and we In this section, we introduce the concept of

I 2  Cauchy double sequence and I 2  Cauchy double sequence in p -adic linear 2normed space ( X , N (,) p ). Also we investigate the relation between these concepts in p -adic linear 2introduce the concepts normed spaces.

I  2Y (power sets of Y ) is said to be an ideal if   I , I is additive i.e., A, B  I  A  B  I and hereditary i.e., A I , B  A  B  I . A family of sets

F  2Y is a filter on Y if and only if   F , A  B  F for each A, B  F , and any subset of an element of F is in F . An ideal I is called non-trivial if I   and Y  I . Clearly I is a non-trivial ideal if and only if F  F (I )  {Y  A : A  I } is a filter in A non empty family of sets

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On I 2 -Cauchy double sequences in p-adic linear 2-normed spaces

Y , called the filter associated with the ideal I . A non-trivial ideal I is called admissible if and only if {n} : n  Y   I . I  2Y is said to have the property (AP) if for any sequence A1 , A2 , A3 ,... of mutually disjoint sets of I there is a sequence B1 , B2 , B3 ,... of sets such that each symmetric An admissible ideal

difference Ai Bi ,

i  1,2,3...., is finite and B   Bi  I . i 1

N and N  N we shall denote the ideals of N by I and ideals of N  N by I 2 . In general, there is no connection between I and I 2 . In order to distinguish between the ideals of

A non trivial ideal

I 2 in N  N is called strongly admissible if {i}  N and N  {i} belong to I 2

for each i  N . It is clear that a strongly admissible ideal is admissible also. Let I 0

 {A  N  N : (m( A)  N )(i, j  m( A)  (i, j )  ( N  N )  A} . Then I 0 is a non

I 2 is strongly admissible if and only if I 0  I 2 . I 2  2 N  N is a non trivial ideal if and only if the class F  F ( I )  {( N  N )  A : A  I 2 } is a filter in N  N . trivial strongly admissible ideal and

I 2  2 N  N satisfies the condition ( AP2 ) if for each countable family of disjoint sets A1 , A2 , A3 ,... belongs to I 2 , there exists a countable family of sets B1 , B2 , B3 ,... such that A j B j An admissible ideal

is included in the finite union of rows and columns in N  N for each

j  N and B   B j  I 2 ( hence j 1

B j  I 2 for each j  N ). Definition 3.1: A double sequence

x  ( xij ) in a p  adic linear 2-normed space ( X , N (,) p ) is said to be

if for each   0 and I 2  convergent to l  X A( )  {(i, j )  N  N : N ( xij  l , z) p   }  I 2 and l is sequence x

non called

zero the

z  X , the I 2  limit of

set the

 ( xij ) .

x  ( xij ) is I 2  convergent to l , then we write I 2  lim xij  l or I 2  lim N ( xij  l , z ) p  0 or

If

i , j 

i , j 

I 2  lim N ( xij , z ) p  N (l , z ) p for each non zero z  X . i , j 

Now introducing the definition of related to

I 2  convergence for double sequence x  ( xij ) which is closely

I 2  convergence of double sequence x  ( xij ) in a p  adic linear 2-normed space as follows.

Definition 3.2: A double sequence

x  ( xij ) in a p  adic linear 2-normed space ( X , N (,) p ) is said to be

 2

I  convergent to l  X if there exists a set M  F ( I 2 ) ( i.e., ( N  N )  M  I 2 ) such that lim xij  l for (i, j )  M and l is called the I 2  limit of the sequence x  ( xij ) .

i , j 

x  ( xij ) is I 2  convergent to l , then we write I 2  lim xij  l or I 2  lim N ( xij  l , z ) p  0 or

If

i , j 

 2

i , j 

I  lim N ( xij , z ) p  N (l , z ) p for each non zero z  X . i , j 

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On I 2 -Cauchy double sequences in p-adic linear 2-normed spaces

{Pi }i1 be a countable collection of subsets of N  N such that {Pi }i1  F ( I 2 ) for

Lemma 3.3: Let

F ( I 2 ) is a filter associated with a strongly admissible ideal I 2 with property (AP 2 ). Then there is a set P  N  N such that P  F ( I 2 ) and the set P  Pi is finite for all i . each i , where

I 2  2 N  N be a strongly admissible ideal and ( X , N (,) p ) be a p  adic linear 2-

Lemma 3.4: Let normed space. If

I 2  lim xij  l , then I 2  lim xij  l for each non zero z  X .

Proof: Suppose I

i , j 

i , j 

 2

 lim xij  l . Then there exists H  I 2 such that for M  ( N  N )  H  F ( I 2 ) i , j 

we have

lim xij  l , (i, j )  M , for each non zero z  X

(3.5)

i , j 

Let   0 . By virtue of equation (3.5) there exists a positive integer

n1 such that N ( xij  l , z ) p   for

(i, j )  M with i, j  n1 . Let A  {1,2,3,......, n1  1} , B  {(i, j )  M : N ( xij  l , z ) p   } .Then every

it

is

clear

that

B  ( A  N )  ( N  A) and therefore B  I 2 . Obviously the set {(i, j )  N  N : N ( xij  l , z ) p   }  B  H and therefore the set

{(i, j )  N  N : N ( xij  l , z) p   }  I 2 for each non zero z  X . This implies that I 2  lim xij  l , i , j 

for each non zero

zX .

Theorem 3.6: Let

I 2  2 N  N be a strongly admissible ideal with property ( AP2 ) and ( X , N (,) p ) be a

p  adic linear 2-normed space. Then for an arbitrary double sequence x  ( xij ) of elements of X , if

I 2  lim xij  l then I 2  lim xij  l . i , j 

Proof:

i , j 

Suppose

that I 2

 lim xij  l .

Then

i , j 

for

 0

any

and

non

zero

zX ,

A( )  {(i, j )  N  N : N ( xij  l , z) p   }  I 2 . Now put

A1  {(i, j )  N  N : N ( xij  l , z ) p  1} and

1 1   Ak  (i, j )  N  N :  N ( xij  l , z ) p   for k  2 and for each non zero z  X . It is clear k k  1  that Am  An   for m  n and Am  I 2 for each m  N . By virtue of ( AP2 ) there exists a countable family of sets

B1 , B2 , B3 ,... such that

Am Bm is a included in finite union of rows and columns in N  N

for each

m  N and B   Bm  I 2 . Put M  ( N  N )  B and to prove the theorem it is sufficient to

prove that Let

m 1

lim xij  l for (i, j )  M .

i , j 

  0 . Choose k  N

such that

1   . Then we have k k

{(i, j )  N  N : N ( xij  l , z ) p   }   Am

(3.7)

m 1

Since Am Bm ,

m  1,2,3,....., k

is

a

finite

set,

there

 k   k    Bm   (i, j ) : i, j  n0 =   Am   (i, j ) : i, j  n0 .  m1   m1  www.iosrjournals.org

If

exists

i , j  n0

n0  N and

such

that

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On I 2 -Cauchy double sequences in p-adic linear 2-normed spaces k

k

m 1

m 1

then (i,

j )  B . This implies that (i, j )   Bm and therefore (i, j )   Am . Hence for every i, j  n0

and (i,

j )  M , we have by equation (3.7), N ( xij  l , z ) p   , for each z  X .

 lim xij  l , for (i, j )  M . i , j 

Thus

I 2  lim xij  l . i , j 

From the Lemma (3.4) and Theorem (3.6) we obtain the following Lemma which gives the equivalence between

I 2 -convergence and I 2  convergence in p  adic linear 2-normed spaces.

Lemma 3.8: Let

I 2  2 N  N be a strongly admissible ideal with the property ( AP2 ) and ( X , N (,) p ) be

p  adic linear 2-normed space. Then a double sequence in X is I 2 -convergent to l in X if and only if it  is I 2  convergent to l in X . a

Definition 3.9: Let

( X , N (,) p ) be a p  adic linear 2-normed space and x  X . Then x is called

accumulation point of

X if there exists a sequence ( x k ) of distinct elements of X such that xk  x for any

N ( ,  ) p

 x . k and xn  Theorem 3.10: Let normed space. If and for each

I 2  2 N  N be a strongly admissible ideal and ( X , N (,) p ) be a p  adic linear 2-

X has at least one accumulation point for any arbitrary double sequence x  ( xij ) in X

a  X , I 2  lim xij  a implies I 2  lim xij  a , then I 2 has the property ( AP2 ) . i , j 

i , j 

Proof: Suppose that a  X is accumulation point of of Put

X . Then there is a sequence (bk ) of distinct elements

N ( ,  ) p

X such that bk  a for any k , and bk  a .

 k( z )  N (bk  a, z ) p

for k  N . Let

A  j

jN

be a disjoint family of non empty sets from

I 2 . Define a

( xmn ) for all z  X such that (i) . xmn  b j if (m, n)  A j and

sequence

(ii) . Let

xmn  a if (m, n)  A j for any j .

  0 be given and z 0  X . Choose k  N

such that

 k( z )   . Then we have 0

A z0 ( )  {(m, n) : N ( xmn  a, z 0 ) p   }  A1  A2  .......  Ak . Hence A z0 ( )  I 2 and

so

I 2  lim xmn  a . m, n

By virtue of our assumption, we have I 2

 lim xmn  a . Hence there exists a set B  I 2 such that m , n

M  ( N  N )  B  F ( I 2 ) and lim xmn  a, (m, n)  M

(3.11)

m, n

Let

B j  A j  B for j  N . Then B j  I 2 for each j  N and

B

j

    A j   B and so B  B   j   j 1  j 1  

 I 2 , fix j  N .

j 1

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On I 2 -Cauchy double sequences in p-adic linear 2-normed spaces If

A j  M is not included in the finite union of rows and columns in N  N , then M must contain an

infinite sequence of elements equation (3.11). Hence

(mk , nk ) and

lim xmk nk  b j  a for all k  N , which contradicts

mk , nk 

A j  M must be contained in the finite union of rows and columns in N  N . Thus

A j B j  ( A j  B j )  ( B j  A j ) = Aj

 B j since B j  A j  

= Aj

 ( A j  B)

= Aj

 B = A j  M is also included in the finite union of rows and columns. Thus the ideal I 2 has

the property ( AP2 ) . Definition 3.12: Let

I 2  2 N  N be a strongly admissible ideal and ( X , N (,) p ) be a p  adic linear 2-

x  ( xij ) in X is said to be I 2  Cauchy double sequence in X , if for each   0 and non zero z  X , there exists m  m( , z ) , n  n( , z )  N such that {(i, j )  N  N : N ( xij  xmn , z ) p   }  I 2 . normed space. A double sequence

Definition 3.13: Let

I 2  2 N  N be a strongly admissible ideal and ( X , N (,) p ) be a p  adic linear 2-

normed space. A double sequence each

 0

and non zero

the double sequence non zero

x  ( xij ) in X is said to be I 2  Cauchy double sequence in X , if for

z  X , there exists a set M  F ( I 2 ) (i.e., H  ( N  N )  M  I 2 ) such that

( xmn ) ( m,n )M is a Cauchy sequence in X . i.e.,

lim N ( xij  xmn , z ) p  0 for each

i , j , m, n

z  X and (i, j ), (m, n)  M .

Theorem 3.14: Let

I 2  2 N  N be a strongly admissible ideal and ( X , N (,) p ) be a p  adic linear 2-

normed space. If a double sequence

X. Proof: Suppose x  ( xij ) is

x  ( xij ) is I 2  Cauchy double sequence in X , then it is I 2  Cauchy

double sequence in

I 2  Cauchy double sequence a in

space ( X , N (,) p ) . Then there exists a set

p  adic linear 2-normed

M  F ( I 2 ) (i.e., H  ( N  N )  M  I 2 ) such that

N ( xij  xmn , z ) p   for every   0 and for all (i, j),(m, n)  M ;i, j, m, n  l and l  l ( )  N . Now

A( )  {(i, j )  N  N : N ( xij  xmn , z) p   }

(3.15)  H  M  ({1,2,3,....., l  1}  N )  ( N  {1,2,3,......., l  1}) Since I 2 is a strongly admissible ideal, therefore H  M  ({1,2,3,....., l  1}  N )  ( N  {1,2,3,......., l  1}) I 2 . From equation (3.15) and by the definition of ideal, A( )  I2 .This shows that the double sequence x  ( xij ) is I 2

 Cauchy double sequence in X . Now we will prove in the following theorem that

p  adic linear 2-normed space. Theorem 3.16: Let

I 2  2 N  N be a strongly admissible ideal and ( X , N (,) p ) be a p  adic linear 2-

normed space. If a double sequence double sequence in

I 2  convergence implies I 2  Cauchy condition in

( xij ) in X is I 2  convergent to x  X , then ( xij ) is I 2  Cauchy

X. www.iosrjournals.org

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On I 2 -Cauchy double sequences in p-adic linear 2-normed spaces

( xij ) in a p  adic linear 2-normed space ( X , N (,) p ) is

Proof: Suppose a double sequence

I 2  convergent to x  X . Then there exists a set M  F ( I 2 ) (i.e., H  ( N  N )  M  I 2 ) such that lim xmn  x, (m, n)  M . It shows that there exists m, n

k 0  k 0 ( ) such that

, for every   0 , non zero z  X and m, n  k 0 2 For (i, j ), (m, n)  M and i, j, m, n  k 0 , N ( xij  xmn , z ) p  N ( xij  x  x  xmn , z) p

N ( xmn  x, z ) p 

= N (( xij

(3.17)

 x)  ( xmn  x), z ) p

 N ( xij  x, z ) p  N ( xmn  x, z ) p

Therefore hence

 2

 2

,

by equation (3.17)

  , for each   0 , non zero z  X and i, j, m, n  k 0 . lim N ( xij  xmn , z ) p  0 which implies ( xij ) (i , j )M is a Cauchy double sequence in X and

i , j , m, n

( xmn ) ( m,n )N  N is a I 2  Cauchy double sequence in X . By Theorem (3.14), ( xmn ) ( m,n )N  N is a

I 2  Cauchy double sequence in X . From Theorem (3.16) and Lemma (3.8) we have the following corollary which gives the relation between I 2  convergence and I 2  Cauchy double sequence in a p  adic linear 2-normed space ( X , N (,) p ) . Corollary 3.18: Let be a

I 2  2 N  N be a strongly admissible ideal with the property ( AP2 ) and ( X , N (,) p )

p  adic linear 2-normed space. If a double sequence ( xij ) is I 2  convergent to x in X , then ( xij ) is

I 2  Cauchy double sequence in X . Finally, we will give the following Theorem which states the equivalence of sequence and

I 2  Cauchy double

 2

I  Cauchy double sequence in a p  adic linear 2-normed space ( X , N (,) p ) in the case

I 2 has the property ( AP2 ) . I 2  2 N  N be a strongly admissible ideal with the property ( AP2 ) and ( X , N (,) p )

Theorem 3.19: Let be a

p  adic linear 2-normed space. Then a double sequence x  ( xij ) is I 2  Cauchy double sequence in

X if and only if x  ( xij ) is I 2  Cauchy double sequence in X . Proof: Suppose a double sequence space

x  ( xij ) is I 2  Cauchy double sequence in a p  adic linear 2-normed

( X , N (,) p ) .Then by Theorem (3.14) , x  ( xij ) is I 2  Cauchy double sequence in X . Now it is sufficient to prove that, if a double sequence

x  ( xij ) is I 2  Cauchy double sequence

 2

I  Cauchy double sequence in X .Suppose x  ( xij ) is I 2  Cauchy double sequence in X . Then there exists m  m( ), n  n( )  N such that A( )  {(m, n) : N ( xij  xmn , z) p   }  I 2 for each   0 and non zero z  X . in X , then it is

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On I 2 -Cauchy double sequences in p-adic linear 2-normed spaces

1 1 1 Pk  {(i, j )  N  N : N ( xij  xmk nk , z ) p  }, k  N where mk  m( ) , nk  n( ) . k k k 1 Since H k  ( N  N )  Pk  {(i, j )  N  N : N ( xij  xmk nk , z ) p  }  I 2 for each k  N and non k zero z  X , therefore Pk  F ( I 2 ) , for each k  N . Let

I 2 has the property ( AP2 ) , then by Lemma (3.3) there exists a set P  N  N such that P  F ( I 2 ) and P  Pk is finite for all k  N . Now we have to show that lim N ( xij  xmn , z ) p  0 , for Since

i , j , m, n

(i, j ), (m, n)  P and for each non zero z  X . For this, let   0 and l  N be such that l 

2

. If

(i, j ), (m, n)  P , then P  Pl is finite , so there exists    (l ) such that (i, j ), (m, n)  Pl for all i, j, m, n   (l ) . Therefore 1 1 N ( xij  xml nl , z ) p  and N ( xmn  xml nl , z ) p  for all i, j, m, n   (l ) and non zero z  X . l l Now N ( xij  xmn , z ) p  N ( xij  xml nl  xml nl  xmn , z ) p = N (( xij

 xml nl )  ( xmn  xml nl ), z) p

 N ( xij  xml nl , z ) p  N ( xmn  xml nl , z ) p

1 1 2      , for all i, j, m, n   (l ) and non zero z  X . l l l Hence for each   0 there exists    ( ) such that for i, j, m, n   ( ) and (i, j ), (m, n)  P  F ( I 2 ) , we have N ( xij  xmn , z ) p   for each non zero z  X . This shows that the sequence

x  ( xij ) is I 2  Cauchy double sequence in X . Thus a double sequence x  ( xij ) is

I 2  Cauchy double sequence in X if and only if x  ( xij ) is I 2  Cauchy double sequence in X . References [1] [2] [3] [4]

G.Bachman, Introduction to p-Adic Numbers and Valuation Theory (Academic Press, 1964). G. Bachman and L. Narici, Functional Analysis (New York and London, Academic Press, 1966). Balakrishna Tripathy and Binod Chandra Tripathy, On I-convergent double sequences ,Soochow Journal of Mathematics , 31, 2005 549-560,. P.Das ,P.Kostyrko,W.Wikzynski and P.Malik,I and I* -convergence of double sequences , Math. Slovaca, 58(5),2008,605-620.

[5]

Erdinç Dündar and Bilal Altay, On Some Properties of

I 2  Convergence and I 2  Cauchy of

Double Sequences, Gen. Math.

Notes, Vol. 7, No.1, November 2011, pp.1-12.

 Cauchy Double Sequences of Fuzzy Numbers, Gen. Math. Notes, 16(2), 2013, 103-114.

[6]

Erdinç Dündar and Özer Talo, I 2

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S. Gähler, Linear 2-normerte raume, Math. Nachr, 28, 1965, 1-45. M. Gurdal, On ideal convergent sequences in 2-normed spaces, Thai Journal of Mathematics, 4(1), 2006, 85-91. M. Gurdal and Isil Acik, On I-Cauchy sequences in2-normedspaces,Mathematical Inequalities & Applications, 11( 2) ,2008,349354. M. Gurdal, Ahmet Sahiner, Extremal I-Limit points of Double Sequences, Applied Mathematical E-Notes, 8,2008, 131-137. P. Kostyrko, M. Macaj and T. Salat, I-convergence, Real Anal. Exchange, 26(2), 2000, 669-686. P. Kostyrko, M. Macaj, T. Salat and M. Sleziak, I-convergence and extremal I-limit points, Math. Slovaca, 55, 2005, 443-464. Mehmet Acikgoz, N. Aslan, N. Koskeroglu and S. Araci, p-adic approach to linear 2-normed spaces, Mathematica Moravica, 13(2) 2009, 7-22. F.Nuray and W.H.Ruckle, Generalized statistical convergence and convergence free spaces, J.Math.Anal.Appl., 245,2000, 513-527. Pratulananda Das and Prasanta Malik, On Extrimal I-limit points of double sequences, Tatra Mt.Math.Publ. 40, 2008, 91-102. A.Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Ann., 53, 1900, 289-321. W. Raymond, R. Freese and J. Cho, Geometry of Linear 2-Normed Spaces (Nova Science Publishers, 2001). Saeed Sarabadan,Fatemeh Amoee Arani and Siamak Khalehoghli, A Condition for the Equivalence of I and I*-convergence in 2normed spaces, Int.J.Contemp. Math. Sciences, 6(43), 2011, 2147-2159. Saeed Sarabadan and Sorayya Talebi, On I-convergence of Double sequences in 2- normed spaces, Int.J.Contemp.Math.Sciences,.7(14), 2012, 673-684. A.Sahiner, M. Gurdal, S. Saltan and H. Gunawan, Ideal convergence in 2-normed spaces, Taiwanese Journal of Mathematics, 11(5) 2007, 1477-1484.

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On I 2 -Cauchy double sequences in p-adic linear 2-normed spaces [21] [22] [23] [24] [25]

B.Surender Reddy, Equivalence of p-adic 2-norms in p-adic linear 2-normed spaces, International Journal of Open Problems in Computer Science and Mathematics, 3(5),(2010, 25-38. B.Surender Reddy and D.Shankaraiah, On Ideal convergent sequences in p-adic linear 2- normed spaces, General Mathematics Notes17(1), 2013, 88-104. B.Surender Reddy and D.Shankaraiah, On I-Cauchy sequences in p-adic linear 2-normed spaces, International Journal of Pure and Applied Mathematics, 89(4), 2013, 483-496. B.Surender Reddy and D.Shankaraiah, On Ideal Convergent of Double Sequences in p-adic linear 2-normed spaces, Global Journal of Pure and Applied Mathematics, 10(1), 2014, 7-20. Vijay Kumar, On I and I* -Convergence of double sequences, Mathematical Communications 12, 2007, 171-181.

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