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

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.

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)={kn:| x k |ε} has asymptotic density zero, i.e.,

lim n 1 n | { k n : | x k | ε } | =0.

In this case we write Slimx= 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 cdab if ca and db 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 αR,

  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:

(P2 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 tR,

(P2 N 2 ) F(x,y;t) H 0 (t) if x and y are linearly independent,

(P2 N 3 ) F(x,y;t)=F(y,x;t), for all x,yX,

(P2 N 4 ) F(αx,y;t)=F(x,y; t | α | ), for every t>0, α0 and x,yX,

(P2 N 5 ) F(x+y,z;t)τ(F(x,z;t),F(y,z;t)), whenever x,y,zX.

If (P2 N 5 ) is replaced by

(P2 N 6 ) F(x+y,z; t 1 + t 2 )F(x,z; t 1 )F(y,z; t 2 ), for all x,y,zX 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 (tx,y) for every x,yX, t>0 and ab=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 ab=ab, a,b[0,1]. For all xX, t>0 and nonzero zX, 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)>1η, whenever k,l n 0 and for nonzero zX. 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)>1η, whenever k,pN and l,qM and for nonzero zX.

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 zX 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 limx=, 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 KN×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 limx=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 zX 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 limx= 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 =mn 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. Ifx=( x k , l )is a sequence in X, then for everyε>0, η(0,1)and for nonzerozX, 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)1η})=0;

  3. (iii)

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

Then, for any t>0 and for nonzero zX, 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 ) >(1a)(1a).

It follows that

F( 1 2 ,z;t)>(1ε).

Since ε>0 was arbitrary, we get F( 1 2 ,z;t)=0 for all t>0 and nonzero zX. 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, andx=( x k , l )andy=( 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 limc 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. Ifx=( x k , l )is a sequence in X such thatFlim x k , l =, then S λ ¯ 2 R 2 N lim x k , l =.

Proof Let Flim x k , l =. Then for every ε>0, t>0 and nonzero zX, there is a positive integer n 0 and m 0 such that

F( x k ,z;t)>1ε

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 ab=ab for all a,b[0,1]. Let F(x,y;t)= t t + x , y , for all x,zX, 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 Flim x k , l 0.

Theorem 4 Let(X,F,)be a random 2-normed space. Ifx=( 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 subsetK={( k n , l n ): k 1 < k 2 ,; l 1 < l 2 ,} N × N such that δ λ ¯ (K)=1andF 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 zX, 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),Flim 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 Flim 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 zX, we can find a positive integer n 0 such that

F( x k , l ,z;t)>1ε

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 ) ) 11=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 zX, 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

(1a)(1a)>1ε.
(3.2)

For t>0 and for nonzero zX, 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 ) >1a.
(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 zX, we have

F( x k , l x p , q ,z;t)1εandF ( x k , l , z ; t 2 ) >1a.
(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 zX, 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

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

Since

F( x k , l x N M ,z;t)F ( x k , l , z ; t 2 ) F ( x N , M , z ; t 2 ) >(1a)(1a)>1ε,

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. □

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Savas, E. On generalized double statistical convergence in a random 2-normed space. J Inequal Appl 2012, 209 (2012). https://doi.org/10.1186/1029-242X-2012-209

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Keywords

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