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 nl.

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 }i1 be a countable collection of subsets of N N such that {Pi }i1 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

zX .

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

zX ,

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 . m1 m1 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

jN

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

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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

[7] [8] [9]

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.

[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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68 | Page

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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