on -statistical convergence of order ˛ in random 2-normed space

Miskolc Mathematical Notes Vol. 16 (2015), No. 2, pp. 1003–1015

HU e-ISSN 1787-2413 DOI: 10.18514/MMN.2015.821

ON -STATISTICAL CONVERGENCE OF ORDER ˛ IN RANDOM 2-NORMED SPACE CHAWLA MEENAKSHI, M. S. SAROA, AND V. KUMAR Received 24 September, 2013 Abstract. Let  D .n / be a non-decreasing sequence of positive real numbers tending to 1 with nC1  n C1; 1 D 1: In the present paper, we introduce the notion of -statistical convergence of order ˛, -statistical Cauchy sequences of order ˛ in random 2-normed spaces and obtain some results. We display examples which show that our method of convergence is more general in random 2-normed space. 2010 Mathematics Subject Classification: 40A05; 46A70; 46A99; 60B99 Keywords: statistical convergence, -statistical convergence, t -norm, 2-norm, random 2-normed space

1. I NTRODUCTION AND BACKGROUND The idea of the statistical convergence was given by Zygmund [25] in the first edition of his monograph published in Warsaw in 1935. The concept of statistical convergence was introduced by Fast [5] and Steinhaus [23] and then reintroduced by Schoenberg [20] independently. Over the years, statistical convergence has been developed in [3, 6, 7, 12, 16, 17, 24] and turned out very useful to resolve many convergence problems arising in Analysis. Definition 1 ([5]). A sequence x D .xk / of numbers is said to be statistically convergent to a number L if for every  > 0, 1 lim jfk  n W jxk Lj > gj D 0; n!1 n where vertical bars denotes the cardinality of enclosed set. In this case, we write S limk!1 xk D L. In literature, several interesting generalizations of statistical convergence have been appeared. One among these is -statistical convergence given by Mursaleen [14] with the help of a non-decreasing sequence  D .n /. Let  D .n / be a nondecreasing sequence of positive real numbers tending to 1 with nC1  n C 1; 1 D 1: c 2015 Miskolc University Press

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CHAWLA MEENAKSHI, M. S. SAROA, AND V. KUMAR

The idea of -statistical convergence can be connected to the generalized de la Vall´eePoussin mean. It is defined by 1 X tn .x/ D xk ; where In D Œn n C 1; n for n D 1; 2; : : : n k2In

Definition 2 ([14]). A sequence x D .xk / of numbers is said to be -statistically convergent to a number L provided that for every  > 0, 1 jfk 2 In W jxk Lj  gj D 0: lim n!1 n In this case, the number L is called -statistical limit of the sequence x D .xk / and we write S limk!1 xk D L. Recently, for ˛ 2 .0; 1 C ¸ olak and Bektas¸[2] generalized Definition 2 in terms of -statistical convergence of order ˛ and obtained some analogous results. Definition 3 ([2]). Let  D .n / be a non-decreasing sequence of positive real numbers as defined above and 0 < ˛  1 be given. A sequence x D .xk / of numbers is said to be -statistically convergent of order ˛ if there is a number L such that 1 lim ˛ jfk 2 In W jxk Lj  gj D 0: n!1  n In this case, we write S˛

limk!1 xk D L.

We next quote the following definition due to Mursaleen and Noman [19] on convergent series. Definition 4 ([19]). Let  D .k / be a sequence of positive real numbers tending to infinity such that 0 < 0 < 1 < 2 : : : and k ! 1 as k ! 1: Then a sequence x D .xk / of numbers is said to be -convergent to a number l if xk ! l as k ! 1, where k 1 X xk D .i k

i

1 /xi

i D0

Esi and Braha [4] used Definition 4 to introduce a new notion called -statistical convergence in random 2-normed spaces and studied some of its properties. Before we proceed further it would be better to recall the ideas of probabilistic and random 2-normed spaces which are of much interest in the study of random operator equations. The concept of probabilistic normed spaces was initially introduced by A. N. Sherstnev [22] in 1962. Menger [13] introduced the notion of probabilistic metric spaces in 1942. The idea of Menger [13] was to use distribution function instead of non-negative real numbers as values of the metric. In last few years these spaces

ON -STATISTICAL CONVERGENCE OF ORDER ˛ IN R2N SPACE

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are grown up rapidly and many detereministic results of linear normed spaces are obtained for probabilistic normed spaces. For a detailed study on probabilistic functional analysis, we refer [1, 10, 11, 18, 21]. In 2005, Golet¸ [9] used the concept of 2-norm of G¨ahler [8] and presented generalized probabilistic normed space which he called random 2-normed space. C Let R denotes the set of reals and RC 0 D Œ0; 1/: A function f W R ! R0 is called a distribution function if it is non-decreasing and left-continuous with inf t 2R f .t / D 0 and sup t 2R f .t / D 1. We will denote the set of all distribution functions by D. Also, a distance distribution function is a non decreasing function F defined on RC D Œ0; 1 that satisfies F .0/ D 0 and F .1/ D 1; and is left continuous on .0; 1/. Let DC denotes the set of all distance distribution functions. A triangular norm, briefly t-norm, is a binary operation  on Œ0; 1 which is continuous, commutative, associative, non-decreasing and has 1 as neutral element, i.e., it is the continuous mapping  W Œ0; 1  Œ0; 1 ! Œ0; 1 such that for all a; b; c 2 Œ0; 1: (1) a  1 D a, (2) a  b D b  a, (3) c  d  a  b if c  a and d  b, (4) .a  b/  c D a  .b  c/. The  operations a  b D max fa C b 1; 0g, a  b D ab, and a  b D mi nfa; bg on Œ0; 1 are t-norms. In following, we quote some needful definitions. Definition 5 ([8]). Let X be a real vector space of dimension d > 1 (d may be infinite). A real valued function jj
;
jj W X 2 ! R satisfying the following conditions: (1) jjx1 ; x2 jj D 0, if and only if x1 ; x2 are linearly dependent. (2) jjx1 ; x2 jj D jjx2 ; x1 jj for all x1 ; x2 2 X , (3) jj˛x1 ; x2 jj D j˛jjjx1 ; x2 jj, for any ˛ 2 R and (4) jjx1 C x2 ; x3 jj  jjx1 ; x3 jj C jjx2 ; x3 jj is called a 2-norm and the pair .X; jj
;
jj/ is called a 2-normed space. For example, if we take X D R2 with 2-norm jjx1 ; x2 jj D area of parallelogram spanned by the vectors x1 ; x2 which may be given explicitly by the formula

jjx1 ; x2 jj D jdet .xij /j D abs: det xi ; xj where xi D .xi1 ; xi 2 / 2 R2 for each i D 1; 2. Then .X; jj
;
jj/ is a 2-normed space. Definition 6 ([9]). Let X be a real linear space of dimension d > 1 (d may be infinite),  be a triangle function(a binary operation on DC which is associative, commutative, nondecreasing and 0 as a unit) and F W X  X ! DC (for x; y 2 X , F .x; yI t / is the value of F .x; y/ at t 2 R). Then F is called a probabilistic norm and .X; F ;  / a probabilistic 2-normed space if the following conditions are satisfied:

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(1) F .x; yI t / D H0 .t / if x; y are linearly dependent, where H0 .t / D 0 if t  0 and H0 .t / D 1 if t > 0. (2) F .x; yI t / ¤ H0 .t / if x; y are linearly independent, (3) F .x; yI t / D F .y; xI t / for all x; y 2 X, t (4) F .˛x; yI t / D F .x; yI j˛j / for every t > 0, ˛ ¤ 0 and x; y 2 X, (5) F .x C y; ´I t /   .F .x; ´I t /; F .y; ´I t //, where x; y; ´ 2 X . If (5) is replaced by F .x C y; ´I t1 C t2 /  F .x; ´I t1 /  F .y; ´I t2 / for all x; y; ´ 2 X and t1 ; t2 2 RC 0 then .X; F ; / is called a random 2-normed space. Example 1. Every 2-normed space .X; jj
;
jj/ can be made a random 2-normed space by setting F .x; yI t / D H0 .t jjx; yjj/ where  0; if t  a; Ha .t / D 1; if t > a for all x; y 2 X, t > 0 and a  b D ab; a; b 2 Œ0; 1. Example 2. Let .X; jj
;
jj/ be a 2-normed space with jjx; ´jj D jx1 ´2 x2 ´1 j; x D .x1 ; x2 /, ´ D .´1 ; ´2 / and a  b D ab for all a; b 2 Œ0; 1. For every x; y 2 X and t t > 0 we define F .x; yI t / D t Cjjx;yjj , then .X; F ; / is a random 2-normed space. Definition 7 ([15]). Let .X; F ; / be a random 2-normed space. Then a sequence x D .xk / is said to be convergent to x0 2 X with respect to the norm F if for every  > 0, t 2 .0; 1/ and  ¤ ´ 2 X, there exists a positive integer k0 such that F .xk x0 ; ´I / > 1 t whenever k  k0 . It is denoted by F lim xk D x0 . Definition 8. p[15]] Let .X; F ; / be a random 2-normed space. Then a sequence x D .xk / is said to be statistically convergent or S R2N convergent to x0 2 X with respect to the norm F if for every  > 0, t 2 .0; 1/ and  ¤ ´ 2 X, ı .fk 2 N W F .xk In this case, we write S R2N

x0 ; ´I /  1

tg/ D 0:

lim xk D x0 .

Definition 9 ([4]). Let .X; F ; / be a random 2-normed space. Then a sequence x D .xk / is said to be -statistically convergent with respect to the norm F provided that, for every  > 0, t 2 .0; 1/ and  ¤ ´ 2 X, ı .fk 2 In W F .xk

x0 ; ´I /  1

tg/ D 0;

i:e: lim

n!1

1 jfk 2 In W F .xk n

R2N In this case, we write S

lim xk D x0 :

x0 ; ´I /  1

t gj D 0:

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Definition 10 ([4]). Let .X; F ; / be a random 2-normed space. Then a sequence x D .xk / is said to be -statistically cauchy with respect to the norm F provided that, for every  > 0, t 2 .0; 1/ and  ¤ ´ 2 X, there exists a positive integer k0 ./ such that for all k; l  k0 ı .fk 2 In W F .xk

xl ; ´I /  1

tg/ D 0:

In present paper, we study quite natural and new notion of -statistical convergence of order ˛ in random 2-normed spaces. 2. M AIN RESULTS In this section, we begin with the following definitions of statistical and -statistical convergence of order ˛ in random 2-normed spaces. Definition 11. A sequence x D .xk / in a random 2-normed space .X; F ; / is said to be statistically convergent of order ˛ .0 < ˛  1/ to x0 2 X provided that, for every  > 0, t 2 .0; 1/ and  ¤ ´ 2 X; 1 lim jfk 2 N W F .xk x0 ; ´I /  1 t gj D 0; n!1 n˛ or equivalently 1 lim ˛ jfk 2 N W F .xk x0 ; ´I / > 1 t gj D 1: n!1 n In this case, we write S ˛ limk!1 xk D x0 . Let S ˛ .X / denotes the set of all statistically convergent sequences of order ˛ in a random 2-normed space .X; F ; /. Definition 12. Let  D .n / be a non-decreasing sequence of positive real numbers tending to 1 with nC1  n C 1; 1 D 1. A sequence x D .xk / in a random 2normed space .X; F ; / is said to be -statistically convergent of order ˛ (0 < ˛  1) to x0 2 X provided that, for every  > 0, t 2 .0; 1/ and  ¤ ´ 2 X; 1 lim ˛ jk 2 In W F .xk x0 ; ´I /  1 tgj D 0; n!1  n or equivalently lim

n!1

1 jk 2 In W F .xk ˛n

x0 ; ´I / > 1

tgj D 1;

where ˛n denote the ˛th power of n , i:e:, .˛n / D .˛1 ; ˛2 ; ˛3 ;





/. In this case, ˛ we write S limk!1 xk D x0 . ˛ Let S .X / denotes the set of all -statistically convergent sequences of order ˛ in a random 2-normed space .X; F ; /. For the particular choice ˛ D 1, Definition 12 coincides with the notion of -statistical

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CHAWLA MEENAKSHI, M. S. SAROA, AND V. KUMAR

convergence of [4]; For n D n, Definition 12 coincides with the notion of statistical convergence of order ˛ in random 2-normed space; For n D n and ˛ D 1, Definition 12 coincides with the notion of statistical convergence in random 2-normed space[15]. We next give an example that shows Definition 12 is well defined for .0 < ˛  1/ but not for ˛ > 1. In view of this, we need the following theorem with lemma. Lemma 1. Let  D .n / be a non-decreasing sequence as defined above and .X; F ; / be a random 2-normed space. Let 0 < ˛  1 and x D .xk / be a sequence in X. Then, for  > 0, t 2 .0; 1/ and  ¤ ´ 2 X, the following statements are equivalent: ˛ limk!1 xk D x0 , (1) S (2) limn!1 1˛ jk 2 In W F .xk x0 ; ´I /  1 t gj D 0; n

(3) limn!1 1˛ jk 2 In W F .xk x0 ; ´I / > 1 n ˛ (4) S limk!1 F .xk x0 ; ´I / D 1.

t gj D 1;

Theorem 1. Let .X; F ; / be a random 2-normed space and 0 < ˛  1 be given. ˛ If S limk!1 xk D x0 , then x0 must be unique. ˛ Proof. Suppose S limk!1 xk D y0 where y0 ¤ x0 . Given  > 0 and t > 0, choose  > 0 such that .1 /  .1 / > 1 . For  ¤ ´ 2 X, define     t 1  I K1 .; t / D k 2 In W F xk x0 ; ´I 2     t 1  : K2 .; t / D k 2 In W F xk y0 ; ´I 2 ˛ Since S

˛ limk!1 xk D x0 and S

lim

n!1

limk!1 xk D y0 , it follows for every t > 0,

1 jK1 .; t /j D 0 and ˛n

lim

n!1

1 jK2 .; t /j D 0 ˛n

1 jK.; t /j D 0 which im˛ n c Let k 2 K .; t / D K1c .; t /\K2c .; t /.

Let K.; t / D K1 .; t / [ K2 .; t /, then clearly limn!1

mediately implies limn!1 1˛ jK c .; t /j D 1. n Now one can write,       t t t F x0 y0 ; ´I  F xk x0 ; ´I  F xk y0 ; ´I 2 2 2 > .1 /  .1 / > 1 :
Since  is arbitrary, it follows that F x0 y0 ; ´I 2t D 1, for t > 0 and  ¤ ´ 2 X . This shows that x0 D y0 .  Example 3. Let X D R2 with the 2-norm jjx; ´jj D jjx1 ´2 x2 ´1 jj where x D t .x1 ; x2 /, ´ D .´1 ; ´2 / and a  b D ab for all a; b 2 Œ0; 1. Let F .x; ´I t / D t Cjjx;´jj

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where x 2 X; t 2 .0; 1/ and  ¤ ´ 2 X . Then .R2 ; F ; / is a random 2-normed space. We define a sequence x D .xk / as follows:  .1; 0/; if k is even, xk D .0; 0/; if k is odd: For  > 0, t 2 .0; 1/, if we define K.; t / D fk 2 In W F .xk ; ´I t/  1 g ;  D .0; 0/   t D k 2 In W 1  t C jjxk ; ´jj   t D k 2 In W jjxk ; ´jj  >0 1  D fk 2 In W xk D .1; 0/g D fk 2 In W k is eveng I then,

p 1 Œ n  C 1 1 jK.; t /j D lim ˛ jfk 2 In W k is evengj  lim D0 lim n!1  n!1 n!1 ˛ 2˛n n n

for ˛ > 1. Similarly, for  > 0 and t 2 .0; 1/ if we define B.; t / D fk 2 In W F .xk then

x0 ; ´I t/  1

g ; x0 D .1; 0/

p 1 1 Œ n  C 1 lim jB.; t /j D lim ˛ jfk 2 In W k is odd gj  lim D0 n!1 ˛ n!1  n!1 2˛n n n

for ˛ > 1. ˛ This shows that S orem 1.

limk xk is not unique and we obtain a contradiction to The-

Theorem 2. Let .X; F ; / be a random 2-normed space and 0 < ˛  1 be given. ˛ For a sequence x D .xk / in X if F limk xk D x0 , then S limk xk D x0 . However, the converse need not be true in general. Proof. Since F limk xk D x0 , so for  > 0, t 2 .0; 1/ and  ¤ ´ 2 X there exists a positive integer n0 such that F .xk x0 ; ´I t/ > 1  for all k  n0 : Hence the set K.; t / D fk 2 In W F .xk

x0 ; ´I t/  1

g  f1; 2; 3; : : : ; n0

for which we have, lim

n!1

1 j fk 2 In W F .xk ˛n

x0 ; ´I t /  1

g j D 0:

1g;

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CHAWLA MEENAKSHI, M. S. SAROA, AND V. KUMAR

˛ limk xk D x0 . We next give the following example which shows This shows that S that the converse need not be true.

Example 4. Consider the random 2-normed space as in Example 3. Define a sequence x D .xk / as follows:  p .k; 0/; if n Œ n  C 1  k  n, xk D .0; 0/; otherwise: For  > 0 and t > 0 if we define K.;˚t / D fk 2 In p W F .xk ; ´I t /  1 g, then one can write as in Example 3 K.; t / D k 2 In W n Œ n  C 1  k  n Thus, oˇ p 1 1 ˇˇn ˇ lim ˛ jK.; t /j D lim ˛ ˇ k 2 In W n Œ n  C 1  k  n ˇ n!1  n!1  n n p Œ n   lim D0 n!1 ˛ n for

1 2

˛ < ˛  1. This shows that S

limk xk D 0: But F

t D F .xk ; ´I t/ D t C jjxk ; ´jj which implies  lim F .xk ; ´I t/ D

k!1



t ; t Ck´2

1;

limk xk ¤ 0, since

p if n Œ n  C 1  k  n; otherwise:

p 0; if n Œ n  C 1  k  n; 1; otherwise: 

Theorem 3. Let .X; F ; / be a random 2-normed space and 0 < ˛  1 be given. Let x D .xk / and y D .yk / be two sequences in X. ˛ ˛ (i) If S limk!1 xk D x0 and 0 ¤ c 2 R, then S limk!1 cxk D cx0 : ˛ ˛ ˛ (ii) If S limk!1 xk D x0 and If S limk!1 yk D y0 , then S lim.xk C yk / D x0 C y0 : Proof. The proof of the Theorem is not so hard so is omitted here.



Theorem 4. Let .X; F ; / be a random 2-normed space and 0 < ˛  ˇ  1 be ˇ ˛ given. Then S .X /  S .X / and the inclusion is strict for some ˛ and ˇ such that ˛ < ˇ. Proof. If 0 < ˛  ˇ  1, then for every  > 0, t > 0 and  ¤ ´ 2 X , we have 1 jfk 2 In W F .xk l; ´I t/  1 gj ˇ n 1  ˛ jfk 2 In W F .xk l; ´I t/  1 gj I n

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ˇ

˛ .X /  S .X /. We next give an example which immediately implies the inclusion S ˇ

˛ .X /  S .X / is strict for some ˛ and ˇ with ˛ < ˇ. that shows the inclusion in S

Example 5. Let .R2 ; F ; / be a random 2-normed space as defined above. We define a sequence x D .xk / as follows:  p .1; 0/; if n Œ n  C 1  k  n, xk D .0; 0/; otherwise: ˇ

Then one can easily see S ˛ .X / S

for 0 < ˛ 

1 2.

ˇ

limk xk D 0, i:e:, x 2 S .X / for

This shows that the inclusion in

˛ .X / S



1 2

< ˇ  1 but x …

ˇ S .X /

is strict. 

Theorem 5. Let .X; F ; / be a random 2-normed space and 0 < ˛  1 be given. ˛ If x D .xk / be a sequence in X, then S limk xk D x0 if and only if there exists a subset K D fkm W k1 < k2 < : : :g of N such that limn!1 1˛ jKj D 1 and F n limk xk D x0 . ˛ Proof. First suppose that S limk xk D x0 . For t > 0,  ¤ ´ 2 X and p 2 N, if we define   1 K.p; t / D k 2 In W F .xk x0 ; ´I t/  1 p   1 M.p; t / D k 2 In W F .xk x0 ; ´I t / > 1 I p

then, lim

n!1

1 jK.p; t /j D 0: ˛n

Also, for p D 1; 2; 3; : : : M.1; t /  M.2; t /  : : : M.i; t /  M.i C 1; t /  : : : and

(2.1)

1 jM.p; t /j D 1: (2.2) ˛n Now, to prove the result in one way, it is sufficient to prove that F limk xk D x0 over M.p; t /. Suppose xk is not convergent to x0 over M.p; t / with respect to the norm F . Then, there exists some  > 0 such that lim

n!1

fk 2 N W F .xk

x0 ; ´I t /  1

g

for infinitely many terms xk . Let M.; t / D fk 2 In W F .xk

x0 ; ´I t / > 1

g

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CHAWLA MEENAKSHI, M. S. SAROA, AND V. KUMAR 1 p

for p D 1; 2; 3


. This implies that limn!1 1˛ jM.; t /j D 0. Also from n

(2.1), we have K1 .p; t /  M.; t / which gives that limn!1 1˛ jM.p; t /j D 0, this n contradicts (2.2). Hence F limk xk D x0 . Conversely, suppose that there exists a subset K D fkm W k1 < k2 < : : :g of N such that limn!1 1˛ jKj D 1 and F limk xk D x0 . Then for every t > 0,  > 0 and n  ¤ ´ 2 X there exists a positive integer k0 such that fk 2 In W F .xk

x0 ; ´I t/ > 1

for all k  k0 . Since the set fk 2 In W F .xk N fk0 C 1; k0 C 2; k0 C 3;


g therefore, 1 jfk 2 In W F .xk n!1 ˛ n

x0 ; ´I t /  1

x0 ; ´I t /  1

lim

˛ Hence, S

g g is contained in

gj D 0:

limk xk D x0 .



Definition 13. Let .X; F ; / be a random 2-normed space. A sequence x D .xk / is said to be -statistically Cauchy of order ˛ (0 < ˛  1) if for every  > 0, t 2 .0; 1/ and  ¤ ´ 2 X there exists a positive integer k0 such that for all k; l  k0 lim

1 jk 2 In W F .xk ˛n

xl ; ´I /  1

tgj D 0;

lim

1 jk 2 In W F .xk ˛n

xl ; ´I / > 1

tgj D 1:

n!1

or equivalently n!1

Theorem 6. Let .X; F ; / be a random 2-normed space and 0 < ˛  1 be given. Then a sequence x D .xk / is said to be -statistically convergent of order ˛ iff it is -statistically Cauchy of order ˛. Proof. Let x D .xk / be a -statistically convergent sequence of order ˛. Suppose ˛ that S limk xk D x0 . Let  > 0. Choose r > 0 such that .1 r/  .1 r/ > 1 . If we define     t 1 r ; K.r; t / D k 2 In W F xk x0 ; ´I 2 then   K c .r; t / D k 2 In W F xk ˛ which gives by virtue of S

t 2

 >1

 r I

limk xk D x0 ,

1 jK.r; t /j D 0 and n!1 ˛ n lim

x0 ; ´I

1 jK c .r; t /j D 1: n!1 ˛ n lim

ON -STATISTICAL CONVERGENCE OF ORDER ˛ IN R2N SPACE

Let m 2 K c .r; t /, then F xm


x0 ; ´I 2t > 1

B.; t / D fk 2 In W F .xk

1013

r. If we take xm ; ´I t /  1

g ;

then to prove the first part it is sufficient to prove that B.; t /  K.r; t /: Let k 2 B.; t /, which gives
F .xk xm ; ´I t /  1 : Suppose k … K.r; t /, then F xk x0 ; ´I 2t > 1 r. Now, we can observe that     t t  F xm x0 ; ´I 1   F .xk xm ; ´I t /  F xk x0 ; ´I 2 2  .1 r/  .1 r/ > 1 : This contradiction shows that B.; t /  K.r; t / and therefore, one way of the theorem is proved. Conversely, suppose that x D .xk / is -statistically Cauchy sequence of order ˛ but not -statistically convergent of order ˛ with respect to F . Then for every t > 0,  > 0 and  ¤ ´ 2 X there exists a positive integer m such that 1 jK.; t /j D 0 where K.; t / D fk 2 In W F .xk n!1 ˛ n lim

xm ; ´I t /  1

g :

This implies that limn!1 1˛ jK c .; t /j D 1. Choose r > 0 such that .1 r/  .1 n r/ > 1 . Let     t >1 r : B.r; t / D k 2 In W F xk x0 ; ´I 2
Let m 2 B.r; t /, then F xm x0 ; ´I 2t > 1 r: Since     t t F .xk xm ; ´I t/  F xk x0 ; ´I  F xm x0 ; ´I 2 2 > .1 r/  .1 r/ > 1 I therefore, 1 jfk 2 In W F .xk n!1 ˛ n lim

xm ; ´I t / > 1

gj D 0:

i:e: limn!1 1˛ jK c .; t /j D 0 which leads to a contradiction. Hence x D .xk / is n -statistically convergent of order ˛.  ACKNOWLEDGEMENT We would like to thank the anonymous referees of the paper for their valuable suggestions and constructive comments which definitely improve the readability of the paper.

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CHAWLA MEENAKSHI, M. S. SAROA, AND V. KUMAR

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ON -STATISTICAL CONVERGENCE OF ORDER ˛ IN R2N SPACE

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Authors’ addresses Chawla Meenakshi Maharishi Markandeshwar University, Department of Mathematics, Mullana Ambala, Haryana, India E-mail address: [email protected] M. S. Saroa Maharishi Markandeshwar University, Department of Mathematics, Mullana Ambala, Haryana, India E-mail address: [email protected] V. Kumar Haryana College of Technology and Management, Department of Mathematics, Kaithal, Haryana, India E-mail address: vjy [email protected]