Open Access

On generalized double statistical convergence in a random 2-normed space

Journal of Inequalities and Applications20122012:209

https://doi.org/10.1186/1029-242X-2012-209

Received: 12 March 2012

Accepted: 22 August 2012

Published: 25 September 2012

Abstract

Recently, the concept of statistical convergence has been studied in 2-normed and random 2-normed spaces by various authors. In this paper, we shall introduce the concept of λ-double statistical convergence and λ-double statistical Cauchy in a random 2-normed space. We also shall prove some new results.

MSC:40A05, 40B50, 46A19, 46A45.

Keywords

statistical convergence λ-double statistical convergence t-norm 2-norm random 2-normed space

1 Introduction

The probabilistic metric space was introduced by Menger [1] which is an interesting and an important generalization of the notion of a metric space. The theory of probabilistic normed (or metric) space was initiated and developed in [26]; further it was extended to random/probabilistic 2-normed spaces by Goleţ [7] using the concept of 2-norm which is defined by Gähler (see [8, 9]); and Gürdal and Pehlivan [10] studied statistical convergence in 2-normed spaces. Also statistical convergence in 2-Banach spaces was studied by Gürdal and Pehlivan in [11]. Moreover, recently some new sequence spaces have been studied by Savas [1214] by using 2-normed spaces.

In order to extend the notion of convergence of sequences, statistical convergence of sequences was introduced by Fast [15] and Schoenberg [16] independently. A lot of developments have been made in this areas after the works of S̆alát [17] and Fridy [18]. Over the years and under different names, statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory and number theory. Recently, Mursaleen [19] studied λ-statistical convergence as a generalization of the statistical convergence, and in [20] he considered the concept of statistical convergence of sequences in random 2-normed spaces. Quite recently, Bipan and Savas [21] defined lacunary statistical convergence in a random 2-normed space, and also Savas [22] studied λ-statistical convergence in a random 2-normed space.

The notion of statistical convergence depends on the density of subsets of N, the set of natural numbers. Let K be a subset of N. Then the asymptotic density of K denoted by δ ( K ) is defined as
δ ( K ) = lim n 1 n | { k n : k K } | ,

where the vertical bars denote the cardinality of the enclosed set.

A single sequence x = ( x k ) is said to be statistically convergent to if for every ε > 0 , the set K ( ε ) = { k n : | x k | ε } has asymptotic density zero, i.e.,
lim n 1 n | { k n : | x k | ε } | = 0 .

In this case we write S lim x = or x k ( S ) (see [15, 18]).

2 Definitions and preliminaries

We begin by recalling some notations and definitions which will be used in this paper.

Definition 1 A function f : R R 0 + is called a distribution function if it is a non-decreasing and left continuous with inf t R f ( t ) = 0 and sup t R f ( t ) = 1 . By D + , we denote the set of all distribution functions such that f ( 0 ) = 0 . If a R 0 + , then H a D + , where
H a ( t ) = { 1 , if  t > a ; 0 , if  t a .

It is obvious that H 0 f for all f D + .

A t-norm is a continuous mapping : [ 0 , 1 ] × [ 0 , 1 ] [ 0 , 1 ] such that ( [ 0 , 1 ] , ) is an Abelian monoid with unit one and c d a b if c a and d b for all a , b , c , d [ 0 , 1 ] . A triangle function τ is a binary operation on D + , which is commutative, associative and τ ( f , H 0 ) = f for every f D + .

In [8], Gähler introduced the following concept of a 2-normed space.

Definition 2 Let X be a real vector space of dimension d > 1 (d may be infinite). A real-valued function , from X 2 into R satisfying the following conditions:
  1. (1)

    x 1 , x 2 = 0 if and only if x 1 , x 2 are linearly dependent,

     
  2. (2)

    x 1 , x 2 is invariant under permutation,

     
  3. (3)

    α x 1 , x 2 = | α | x 1 , x 2 , for any α R ,

     
  4. (4)

    x + x ¯ , x 2 x , x 2 + x ¯ , x 2

     

is called a 2-norm on X and the pair ( X , , ) is called a 2-normed space.

A trivial example of a 2-normed space is X = R 2 , equipped with the Euclidean 2-norm x 1 , x 2 E = the area of the parallelogram spanned by the vectors x 1 , x 2 which may be given explicitly by the formula
x 1 , x 2 E = | det ( x i j ) | = abs ( det ( x i , x j ) ) ,

where x i = ( x i 1 , x i 2 ) R 2 for each i = 1 , 2 .

Recently, Goleţ [7] used the idea of a 2-normed space to define a random 2-normed space.

Definition 3 Let X be a linear space of dimension d > 1 (d may be infinite), τ a triangle, and F : X × X D + . Then F is called a probabilistic 2-norm and ( X , F , τ ) a probabilistic 2-normed space if the following conditions are satisfied:

( P 2 N 1 ) F ( x , y ; t ) = H 0 ( t ) if x and y are linearly dependent, where F ( x , y ; t ) denotes the value of F ( x , y ) at t R ,

( P 2 N 2 ) F ( x , y ; t ) H 0 ( t ) if x and y are linearly independent,

( P 2 N 3 ) F ( x , y ; t ) = F ( y , x ; t ) , for all x , y X ,

( P 2 N 4 ) F ( α x , y ; t ) = F ( x , y ; t | α | ) , for every t > 0 , α 0 and x , y X ,

( P 2 N 5 ) F ( x + y , z ; t ) τ ( F ( x , z ; t ) , F ( y , z ; t ) ) , whenever x , y , z X .

If ( P 2 N 5 ) is replaced by

( P 2 N 6 ) F ( x + y , z ; t 1 + t 2 ) F ( x , z ; t 1 ) F ( y , z ; t 2 ) , for all x , y , z X and t 1 , t 2 R 0 + ;

then ( X , F , ) is called a random 2-normed space (for short, R2NS).

Remark 1 Every 2-normed space ( X , , ) can be made a random 2-normed space in a natural way by setting F ( x , y ; t ) = H 0 ( t x , y ) for every x , y X , t > 0 and a b = min { a , b } , a , b [ 0 , 1 ] .

Example 1 Let ( X , , ) be a 2-normed space with x , z = x 1 z 2 x 2 z 1 , x = ( x 1 , x 2 ) , z = ( z 1 , z 2 ) and a b = a b , a , b [ 0 , 1 ] . For all x X , t > 0 and nonzero z X , consider
F ( x , z ; t ) = { t t + x , z , if  t > 0 ; 0 , if  t 0 .

Then ( X , F , ) is a random 2-normed space.

Definition 4 A sequence x = ( x k , l ) in a random 2-normed space ( X , F , ) is said to be double convergent (or F -convergent) to X with respect to F if for each ε > 0 , η ( 0 , 1 ) , there exists a positive integer n 0 such that F ( x k , l , z ; ε ) > 1 η , whenever k , l n 0 and for nonzero z X . In this case we write F lim k , l x k , l = , and is called the F -limit of x = ( x k , l ) .

Definition 5 A sequence x = ( x k , l ) in a random 2-normed space ( X , F , ) is said to be double Cauchy with respect to F if for each ε > 0 , η ( 0 , 1 ) there exist N = N ( ε ) and M = M ( ε ) such that F ( x k , l x p , q , z ; ε ) > 1 η , whenever k , p N and l , q M and for nonzero z X .

Definition 6 A sequence x = ( x k , l ) in a random 2-normed space ( X , F , ) is said to be double statistically convergent or S 2 R 2 N -convergent to some X with respect to F if for each ε > 0 , η ( 0 , 1 ) and for nonzero z X such that
δ ( { ( k , l ) N × N : F ( x k , l , z ; ε ) 1 η } ) = 0 .
In other words, we can write the sequence ( x k , l ) double statistically converges to in random 2-normed space ( X , F , ) if
lim m , n 1 m n | { k m , l n : F ( x k , l , z ; ε ) 1 η } | = 0
or equivalently,
δ ( { k , l N : F ( x k , l , z ; ε ) > 1 η } ) = 1 ,
i.e.,
S 2 lim k , l F ( x k , l , z ; ε ) = 1 .

In this case we write S 2 R 2 N lim x = , and is called the S 2 R 2 N -limit of x. Let S 2 R 2 N ( X ) denote the set of all double statistically convergent sequences in a random 2-normed space ( X , F , ) .

In this article, we study λ-double statistical convergence in a random 2-normed space which is a new and interesting idea. We show that some properties of λ-double statistical convergence of real numbers also hold for sequences in random 2-normed spaces. We establish some relations related to double statistically convergent and λ-double statistically convergent sequences in random 2-normed spaces.

3 λ-double statistical convergence in a random 2-normed space

Recently, the concept of λ-double statistical convergence has been introduced and studied in [23] and [24]. In this section, we define λ-double statistically convergent sequence in a random 2-normed space ( X , F , ) . Also we get some basic properties of this notion in a random 2-normed space.

Definition 7 Let λ = ( λ n ) and μ = ( μ n ) be two non-decreasing sequences of positive real numbers such that each is tending to ∞ and
λ n + 1 λ n + 1 , λ 1 = 1
and
μ n + 1 μ n + 1 , μ 1 = 1 .
Let K N × N . The number
δ λ ¯ ( K ) = lim m n 1 λ ¯ m n | { k I n , l J m : ( k , l ) K } | ,

where I n = [ n λ n + 1 , n ] , J m = [ m μ m + 1 , m ] and λ ¯ n m = λ n μ m , is said to be the λ-double density of K, provided the limit exists.

Definition 8 A sequence x = ( x k , l ) is said to be λ-double statistically convergent or S λ ¯ 2 -convergent to the number if for every ε > 0 , the set N ( ε ) has λ-double density zero, where
N ( ε ) = { k I n , l J m : | x k , l | ε } .

In this case, we write S λ ¯ 2 lim x = L .

Now we define λ-double statistical convergence in a random 2-normed space (see [25]).

Definition 9 A sequence x = ( x k , l ) in a random 2-normed space ( X , F , ) is said to be λ-double statistically convergent or S λ ¯ 2 -convergent to X with respect to F if for every ε > 0 , η ( 0 , 1 ) and for nonzero z X such that
δ λ ¯ ( { k I n , l J m : F ( x k , l , z ; ε ) 1 η } ) = 0
or equivalently,
δ λ ¯ ( { k I n , l J m : F ( x k , l , z ; ε ) > 1 η } ) = 1 ,
i.e.,
S λ ¯ 2 lim k , l F ( x k , l , z ; ε ) = 1 .
In this case we write S λ ¯ 2 R 2 N lim x = or x k , l ( S λ ¯ 2 R 2 N ) and
S λ ¯ 2 R 2 N ( X ) = { x = ( x k , l ) : R , S λ ¯ 2 R 2 N lim x = } .

Let S λ ¯ 2 R 2 N ( X ) denote the set of all λ-double statistically convergent sequences in a random 2-normed space ( X , F , ) .

If λ ¯ m n = m n for every n, m then λ-double statistically convergent sequences in a random 2-normed space ( X , F , ) reduce to double statistically convergent sequences in a random 2-normed space ( X , F , ) .

Definition 9 immediately implies the following lemma.

Lemma 1 Let ( X , F , ) be a random 2-normed space. If x = ( x k , l ) is a sequence in X, then for every ε > 0 , η ( 0 , 1 ) and for nonzero z X , the following statements are equivalent:
  1. (i)

    S λ ¯ R 2 N lim k , l x k , l = ;

     
  2. (ii)

    δ λ ¯ ( { k I n , l J m : F ( x k , l , z ; ε ) 1 η } ) = 0 ;

     
  3. (iii)

    δ λ ¯ ( { k I n , l J m : F ( x k , l , z ; ε ) > 1 η } ) = 1 ;

     
  4. (iv)

    S λ ¯ lim k , l F ( x k , l , z ; ε ) = 1 .

     

Theorem 1 Let ( X , F , ) be a random 2-normed space. If x = ( x k , l ) is a sequence in X such that S λ ¯ 2 R 2 N lim x k , l = exists, then it is unique.

Proof Suppose that S λ ¯ 2 R 2 N lim k , l x k , l = 1 ; S λ ¯ 2 R 2 N lim k , l x k , l = 2 , where ( 1 2 ) .

Let ε > 0 be given. Choose a > 0 such that ( 1 a ) ( 1 a ) > 1 ε .

Then, for any t > 0 and for nonzero z X , we define

Since S λ ¯ 2 R 2 N lim k , l x k , l = 1 and S λ ¯ 2 R 2 N lim k , l x k , l = 2 , we have Lemma 1 δ λ ¯ ( K 1 ( a , t ) ) = 0 and δ λ ¯ ( K 2 ( a , t ) ) = 0 for all t > 0 .

Now, let K ( a , t ) = K 1 ( a , t ) K 2 ( a , t ) , then it is easy to observe that δ λ ¯ ( K ( a , t ) ) = 0 . But we have δ λ ¯ ( K c ( r , t ) ) = 1 .

Now, if ( k , l ) K c ( a , t ) , then we have
F ( 1 2 , z ; t ) F ( x k , l 1 , z ; t 2 ) F ( x k , l 2 , z ; t 2 ) > ( 1 a ) ( 1 a ) .
It follows that
F ( 1 2 , z ; t ) > ( 1 ε ) .

Since ε > 0 was arbitrary, we get F ( 1 2 , z ; t ) = 0 for all t > 0 and nonzero z X . Hence 1 = 2 .

This completes the proof. □

Next theorem gives the algebraic characterization of λ-statistical convergence on random 2-normed spaces. We give it without proof.

Theorem 2 Let ( X , F , ) be a random 2-normed space, and x = ( x k , l ) and y = ( y k , l ) be two sequences in X.
  1. (a)

    If S λ ¯ 2 R 2 N lim x k , l = and c ( 0 ) R , then S λ ¯ 2 R 2 N lim c x k , l = c .

     
  2. (b)

    If S λ ¯ 2 R 2 N lim x k , l = 1 and S λ ¯ R 2 N lim y k , l = 2 , then S λ ¯ 2 R 2 N lim ( x k , l + y k , l ) = 1 + 2 .

     

Theorem 3 Let ( X , F , ) be a random 2-normed space. If x = ( x k , l ) is a sequence in X such that F lim x k , l = , then S λ ¯ 2 R 2 N lim x k , l = .

Proof Let F lim x k , l = . Then for every ε > 0 , t > 0 and nonzero z X , there is a positive integer n 0 and m 0 such that
F ( x k , z ; t ) > 1 ε
for all k n 0 . Since the set
K ( ε , t ) = { k I n , l J m : F ( x k , l , z ; t ) 1 ε }

has at most finitely many terms. Since every finite subset of N × N has δ λ ¯ -density zero, finally we have δ λ ¯ ( K ( ε , t ) ) = 0 . This shows that S λ ¯ 2 R 2 N lim x k , l = . □

Remark 2 The converse of the above theorem is not true in general. It follows from the following example.

Example 2 Let X = R 2 , with the 2-norm x , z = | x 1 z 2 x 2 z 1 | , x = ( x 1 , x 2 ) , z = ( z 1 , z 2 ) and a b = a b for all a , b [ 0 , 1 ] . Let F ( x , y ; t ) = t t + x , y , for all x , z X , z 2 0 , and t > 0 . We define a sequence x = ( x k ) by
x k , l = { ( k l , 0 ) , if  n [ λ n ] + 1 k n  and  m [ μ m ] + 1 k m ; ( 0 , 0 ) , otherwise .
Now for every 0 < ε < 1 and t > 0 , we write
K n ( ε , t ) = { k I n , l J m : F ( x k , l , z ; t ) 1 ε } .
Therefore, we get
δ λ ¯ ( K ( ε , t ) ) = lim n m [ λ ¯ n m ] λ ¯ n m = 0 .

This shows that S λ ¯ 2 R 2 N lim x k , l = 0 , while it is obvious that F lim x k , l 0 .

Theorem 4 Let ( X , F , ) be a random 2-normed space. If x = ( x k , l ) is a sequence in X, then S λ ¯ 2 R 2 N lim x k , l = if and only if there exists a subset K = { ( k n , l n ) : k 1 < k 2 , ; l 1 < l 2 , } N × N such that δ λ ¯ ( K ) = 1 and F lim n x k n , l n = .

Proof Suppose first that S λ 2 R 2 N lim x k , l = . Then for any t > 0 , a = 1 , 2 , 3 , and nonzero z X , let
A ( a , t ) = { k I n ; l J m : F ( x k , l , z ; t ) > 1 1 a }
and
K ( a , t ) = { k I n ; l J m : F ( x k , l , z ; t ) 1 1 a } .
Since S λ ¯ 2 R 2 N lim x k , l = , it follows that
δ λ ¯ ( K ( a , t ) ) = 0 .
Now, for t > 0 and a = 1 , 2 , 3 ,  , we observe that
A ( a , t ) A ( a + 1 , t )
and
δ λ ¯ ( A ( a , t ) ) = 1 .
(3.1)
Now we have to show that for ( k , l ) A ( a , t ) , F lim x k , l = . Suppose that for some ( k , l ) A ( a , t ) , ( x k , l ) is not convergent to with respect to F . Then there exist some s > 0 and a positive integer k 0 , l 0 such that
{ k I n ; l J m : F ( x k , l , z ; t ) 1 s }
for all k k 0 and l l 0 . Let
A ( s , t ) = { k I n ; l J m : F ( x k , l , z ; t ) > 1 s }
for k < k 0 and l < l 0 and
s > 1 a , a = 1 , 2 , 3 , .
Then we have
δ λ ¯ ( A ( s , t ) ) = 0 .

Furthermore, A ( a , t ) A ( s , t ) implies that δ λ ¯ ( A ( a , t ) ) = 0 , which contradicts (3.1) as δ λ ¯ ( A ( a , t ) ) = 1 . Hence F lim x k , l = .

Conversely, suppose that there exists a subset K = { ( k n , l n ) : k 1 < k 2 , ; l 1 < l 2 , } N × N such that δ λ ¯ ( K ) = 1 and F lim n , m x k n , l n = . Then for every ε > 0 , t > 0 and nonzero z X , we can find a positive integer n 0 such that
F ( x k , l , z ; t ) > 1 ε
for all k , l n 0 . If we take
K ( ε , t ) = { k I n ; l J m : F ( x k , l , z ; t ) 1 ε } ,
then it is easy to see that
K ( ε , t ) N × N { ( k n 0 + 1 , l n 0 + 1 ) , ( k n 0 + 2 , l n 0 + 2 ) , } ,
and finally,
δ λ ¯ ( K ( ε , t ) ) 1 1 = 0 .

Thus S λ ¯ R 2 N lim x k , l = . This completes the proof. □

We now have

Definition 10 A sequence x = ( x k , l ) in a random 2-normed space ( X , F , ) is said to be λ-double statistically Cauchy with respect to F if for each ε > 0 , η ( 0 , 1 ) and for nonzero z X , there exist N = N ( ε ) and M = M ( ε ) such that for all k , m > N and l , n > M ,
δ λ ¯ ( { k I n ; l J m : F ( x k , l x M N , z ; ε ) 1 η } ) = 0 ,
or equivalently,
δ λ ¯ ( { k I n ; l J m : F ( x k , l x M N , z ; ε ) > 1 η } ) = 1 .

Theorem 5 Let ( X , F , ) be a random 2-normed space. Then a sequence ( x k , l ) in X is λ-double statistically convergent if and only if it is λ-double statistically Cauchy in random 2-normed space X.

Proof Let ( x k , l ) be a λ-double statistically convergent to with respect to random 2-normed space, i.e., S λ ¯ 2 R 2 N lim x k = . Let ε > 0 be given. Choose a > 0 such that
( 1 a ) ( 1 a ) > 1 ε .
(3.2)
For t > 0 and for nonzero z X , define
A ( a , t ) = { k I n ; l J m : F ( x k , l , z ; t 2 ) 1 a } .
Then
A c ( a , t ) = { k I n ; l J m : F ( x k , l , z ; t 2 ) > 1 a } .

Since S λ ¯ 2 R 2 N lim x k , l = , it follows that δ λ ¯ ( A ( a , t ) ) = 0 , and finally, δ λ ¯ ( A c ( a , t ) ) = 1 .

Let p , q A c ( a , t ) . Then
F ( x p , q , z ; t 2 ) > 1 a .
(3.3)
If we take
B ( ε , t ) = { k I n ; l J m : F ( x k , l x p , q , z ; t ) 1 ε } ,

then to prove the result it is sufficient to prove that B ( ε , t ) A ( a , t ) .

Let ( k , l ) B ( ε , t ) A c ( a , t ) , then for nonzero z X , we have
F ( x k , l x p , q , z ; t ) 1 ε and F ( x k , l , z ; t 2 ) > 1 a .
(3.4)
Now, from (3.1), (3.3) and (3.4), we get
1 ε F ( x k , l x p , q , z ; t ) F ( x k , l , z ; t 2 ) F ( x p , z ; t 2 ) > ( 1 a ) ( 1 a ) > ( 1 ε ) ,

which is not possible. Thus B ( ε , t ) A ( a , t ) . Since δ λ ¯ ( A ( a , t ) ) = 0 , it follows that δ λ ¯ ( B ( ε , t ) ) = 0 . This shows that ( x k , l ) is λ-double statistically Cauchy.

Conversely, suppose ( x k , l ) is λ-double statistically Cauchy but not λ-double statistically convergent with respect to F . Then for each ε > 0 , t > 0 and for nonzero z X , there exist a positive integer N = N ( ε ) and M = M ( ε ) such that
A ( ε , t ) = { k I n ; l J m : F ( x k , l x N M , z ; t ) 1 ε } .
Then
δ λ ¯ ( A ( ε , t ) ) = 0
and
δ λ ¯ ( A c ( ε , t ) ) = 1 .
(3.5)
For t > 0 , choose a > 0 such that
( 1 a ) ( 1 a ) > 1 ε
(3.6)
is satisfied, and we take
B ( a , t ) = { k I n ; l J m : F ( x k , l , z ; t 2 ) > 1 a } .

If N , M B ( a , t ) , then F ( x N , M , z ; t 2 ) > 1 a .

Since
F ( x k , l x N M , z ; t ) F ( x k , l , z ; t 2 ) F ( x N , M , z ; t 2 ) > ( 1 a ) ( 1 a ) > 1 ε ,
then we have
δ λ ¯ ( { x k , l : F ( x k , l x N M , z ; t ) > 1 ε } ) = 0 ,

i.e., δ λ ¯ ( A c ( ε , t ) ) = 0 , which contradicts (3.5) as δ λ ¯ ( A c ( ε , t ) ) = 1 . Hence ( x k , l ) is λ-double statistically convergent.

This completes the proof. □

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Istanbul Commerce University

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