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Some generalized difference statistically convergent sequence spaces in 2-normed space

Abstract

In this paper, we define a new generalized difference matrix B ( m ) n and introduce some B ( m ) n -difference statistically convergent sequence spaces in a real linear 2-normed space. We also investigate some topological properties of these spaces.

MSC:40A05, 46A45, 46E30.

1 Introduction

We shall write w for the set of all real sequences x=( x k )= ( x k ) k = 0 . Let c, c 0 , c ¯ , c ¯ 0 , l , m and m 0 denote the sets of all convergent, null, statistically convergent, statistically null, bounded, bounded statistically convergent and bounded statistically null sequences, respectively. The difference sequence spaces l (Δ), c(Δ) and c 0 (Δ) were first defined by Kızmaz in [1]. The idea of difference sequences is generalized by Et and Çolak [2] as

Z ( Δ n ) = { x = ( x k ) w : ( Δ n x k ) Z } (nN)

for Z= l ,c, c 0 , where Δ n x k = Δ n 1 x k Δ n 1 x k + 1 and Δ 0 x k = x k for all kN, the difference operator is equivalent to the following binomial representation:

Δ n x k = v = 0 n ( 1 ) v ( n v ) x k + v .

Et and Başarır [3] generalized these spaces to E( Δ n ), where E= l (p),c(p), c 0 (p) are Maddox’s sequence spaces. Tripathy and Esi [4], who studied the spaces l ( Δ m ), c( Δ m ) and c 0 ( Δ m ), gave a new type of generalization of the difference sequence spaces, where Δ m x=( Δ m x k )=( x k x k + m ). Tripathy et al. [5] generalized this notion as follows:

Z ( Δ m n ) = { x = ( x k ) w : ( Δ m n x k ) Z } (n,mN),

where Δ m n x=( Δ m n x k )=( Δ m n 1 x k Δ m n 1 x k + m ) and Δ m 0 x k = x k for all kN, which is equivalent to the following binomial representation:

Δ m n x k = v = 0 n ( 1 ) v ( n v ) x k + m v .

The difference sequence spaces have been studied by several authors, [3, 625].

The concept of 2-normed spaces has been initially introduced by Gähler in the 1960s [26], as an interesting non-linear generalization of a normed linear space, which has been subsequently studied by many authors [2729]. Since then, a lot of activities have been started to study summability, sequence spaces and related topics on 2-normed spaces [3033]. Recently, some difference sequence spaces have been introduced in 2-normed spaces by several authors [30, 31, 34].

Dutta [34] introduced the sequence spaces c ¯ (,, Δ ( m ) n ,p), c ¯ 0 (,, Δ ( m ) n ,p), l (,, Δ ( m ) n ,p), m(,, Δ ( m ) n ,p) and m 0 (,, Δ ( m ) n ,p), where m,nN and Δ ( m ) n x=( Δ ( m ) n x k )=( Δ ( m ) n 1 x k Δ ( m ) n 1 x k m ), and Δ ( m ) 0 x k = x k for all kN, which is equivalent to the following binomial representation:

Δ ( m ) n x k = v = 0 n ( 1 ) v ( n v ) x k m v .
(1.1)

In [35], Başar and Altay introduced the generalized difference matrix B(r,s)=( b n k (r,s)) which is a generalization of Δ ( 1 ) 1 -difference operator as follows:

b n k (r,s)={ r ( k = n ) , s ( k = n 1 ) , 0 ( 0 k < n 1 )  or  ( k > n )

for all k,nN, r,sR{0}. Recently, Başarır and Kayıkçı [36] have defined the generalized difference matrix B n of order n, which reduced the difference operator Δ ( 1 ) n in case r=1, s=1 and the binomial representation of this operator is

B n x k = v = 0 n ( n v ) r n v s v x k v ,
(1.2)

where r,sR{0} and nN.

Thus, for any sequence space Z, the space Z( B n ) is more general and more comprehensive than the corresponding consequences of the space Z( Δ ( 1 ) n ). For details, one may refer to [6, 15, 3540].

The idea of statistical convergence was given by Zygmund [41] in 1935. The concept of statistical convergence was introduced by Fast [42] and Schoenberg [43], independently for the real sequences. Later on, it was further investigated from sequence point of view and linked with the summability theory by Fridy [44] and generalized to the concept of 2-normed space by Gürdal and Pehlivan [45]. The idea is based on the notion of natural density of subsets of , the set of positive integers, which is defined as follows: the natural density of a subset E of is denoted by

δ(E)= lim n 1 n |{kE:kn}|,

where the vertical bar denotes the cardinality of the enclosed set.

2 Definitions and preliminaries

A sequence space E is said to be solid (or normal) if ( x k )E implies ( α k x k )E for all sequences of scalars ( α k ) with | α k |1 for all kN.

A linear topological space X over the real field R is said to be a paranormed space if there is a sub-additive function g:XR such that g(θ)=0, g(x)=g(x), g(x+y)g(x)+g(y) and scalar multiplication is continuous, i.e. | λ n λ|0 and g( x n x)0 imply that g( λ n x n λx)0 for all λ’s in and all x’s in X, where θ is the zero vector in the linear space X.

The following inequality will be used throughout the paper:

Let p=( p k ) be a positive sequence of real numbers with inf k p k =h, sup k p k =H and D=max{1, 2 H 1 }. Then for all a k , b k C for all kN, we have

| a k + b k | p k D { | a k | p k + | b k | p k }

and |λ | p k max{|λ | h ,|λ | H } for λC.

A 2-norm on a vector space X of d dimension, where d2, is a function ,:X×XR, which satisfies 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 = x 2 , x 1 ,

  3. (3)

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

  4. (4)

    x+ x , x 1 x, x 1 + x , x 1 .

The pair (X,,) is then called a 2-normed space. For example, standard and Euclidean 2-norms on R 2 are respectively given by

x 1 , x 2 S = | x 1 , x 1 x 1 , x 2 x 2 , x 1 x 2 , x 2 | 1 2

and

x 1 , x 2 E =abs ( | x 11 x 12 x 21 x 22 | ) , x i =( x i 1 , x i 2 ) R 2 (i=1,2),
(2.1)

where , stands for the inner product on X [27].

Now we will give the following known example for 2-normed spaces.

Example 2.1 Consider the space Z for l , c and c 0 . Let us define:

x,y= sup i N sup j N | x i y j x j y i |,

where x=( x 1 , x 2 ,) and y=( y 1 , y 2 ,)Z. Then , is a 2-norm on Z.

A sequence ( x k ) in a 2-normed space (X,,) is said to be convergent to some LX in the 2-norm if

lim k x k L,z=0for every zX [45].

A sequence ( x k ) in a 2-normed space (X,,) is said to be Cauchy sequence with respect to the 2-norm if

lim k , l x k x l ,z=0for every zX [45].

If every Cauchy sequence in X converges to some LX, then X is said to be complete with respect to the 2-norm. Any complete 2-normed space is said to be a 2-Banach space [29].

Let recall that a sequence ( x k ) is said to be statistically convergent to L if for every ε>0 the set {kN: x k L,zε} has natural density zero for each nonzero z in X, in other words ( x k ) statistically converges to L in 2-normed space (X,,) if

lim k 1 k | { k N : x k L , z ε } |=0,

for each nonzero z in X. For L=0, we say this is statistically null [45].

Firstly, we give the following lemma, which we need to establish our main results.

Lemma 2.2 [34]

Every closed linear subspace F of an arbitrary linear normed space E, different from E, is a nowhere dense set in E.

Throughout the paper w(X), c(X), c 0 (X), c ¯ (X), c ¯ 0 (X), l (X), m(X) and m 0 (X) denote the spaces of all, convergent, null, statistically convergent, statistically null, bounded, bounded statistically convergent and bounded statistically null X valued sequence spaces, where (X,,) is a real 2-normed space. By θ=(θ,θ,θ,), we mean the zero element of X.

3 Main results

In this section, we define the generalized difference matrix B ( m ) n and introduce difference sequence spaces c ¯ ( B ( m ) n ,p,,), c ¯ 0 ( B ( m ) n ,p,,), m( B ( m ) n ,p,,), m 0 ( B ( m ) n ,p,,), c( B ( m ) n ,p,,), c 0 ( B ( m ) n ,p,,), l ( B ( m ) n ,p,,), W( B ( m ) n ,p,,), which are defined on a real linear 2-normed space. We investigate some topological properties of the spaces c ¯ 0 ( B ( m ) n ,p,,), c ¯ ( B ( m ) n ,p,,), m( B ( m ) n ,p,,) and m 0 ( B ( m ) n ,p,,) including linearity, existence of paranorm and solidity. Further, we show that the sequence spaces m( B ( m ) n ,p,,) and m 0 ( B ( m ) n ,p,,) are complete paranormed spaces when the base space is a 2-Banach space. Moreover, we give some inclusion relations.

By the notation x k stat 0, we will mean that x k is statistically convergent to zero, throughout the paper. Let m, n be non-negative integers and p=( p k ) be a sequence of strictly positive real numbers. Then we define new sequence spaces as follows:

c ¯ ( B ( m ) n , p , , ) = { x = ( x k ) w ( X ) : B ( m ) n x k L , z p k stat 0 , c ¯ ( B ( m ) n , p , , ) = for every nonzero  z X  and some  L X } , c ¯ 0 ( B ( m ) n , p , , ) = { x = ( x k ) w ( X ) : B ( m ) n x k , z p k stat 0 ,  for every nonzero  z X } , l ( B ( m ) n , p , , ) = { x = ( x k ) w ( X ) : sup k 1 ( B ( m ) n x k , z p k ) < , l ( B ( m ) n , p , , ) = for every nonzero  z X } , c ( B ( m ) n , p , , ) = { x = ( x k ) : lim k B ( m ) n x k L , z p k = 0 , c ( B ( m ) n , p , , ) = for every nonzero  z X  and some  L X } , c 0 ( B ( m ) n , p , , ) = { x = ( x k ) : lim k B ( m ) n x k , z p k = 0 ,  for every nonzero  z X } , W ( B ( m ) n , p , , ) = { x = ( x k ) w ( X ) : lim j 1 j k = 1 j B ( m ) n x k L , z p k = 0 , W ( B ( m ) n , p , , ) = for every nonzero  z X  and some  L X } , m ( B ( m ) n , p , , ) = c ¯ ( B ( m ) n , p , , ) l ( B ( m ) n , p , , )

and

m 0 ( B ( m ) n , p , , ) = c ¯ 0 ( B ( m ) n , p , , ) l ( B ( m ) n , p , , ) ,

where B ( m ) n x= B ( m ) n x k =r B ( m ) n 1 x k +s B ( m ) n 1 x k m and B ( m ) 0 x k = x k for all kN, which is equivalent to the binomial representation as follows:

B ( m ) n x k = v = 0 n ( n v ) r n v s v x k m v .

In this representation, we obtain the matrix B ( 1 ) n defined in [36] for n>1 and in [35] for n=1.

  1. (1)

    If we take n=0 then the above sequence spaces are reduced to c ¯ (p,,), c ¯ 0 (p,,), l (p,,), c(p,,), c 0 (p,,), W(p,,), m(p,,) and m 0 (p,,), respectively.

  2. (2)

    If we take r=1, s=1, then the sequence spaces c ¯ ( B ( m ) n ,p,,), c ¯ 0 ( B ( m ) n ,p,,), l ( B ( m ) n ,p,,), W( B ( m ) n ,p,,), m( B ( m ) n ,p,,), m 0 ( B ( m ) n ,p,,) are reduced to c ¯ ( Δ ( m ) n ,p,,), c ¯ 0 ( Δ ( m ) n ,p,,), l ( Δ ( m ) n ,p,,), W( Δ ( m ) n ,p,,), m( Δ ( m ) n ,p,,) and m 0 ( Δ ( m ) n ,p,,), respectively, which are studied in [34].

  3. (3)

    By taking p k =1 for all kN, then these sequence spaces are denoted by c ¯ ( B ( m ) n ,,), c ¯ 0 ( B ( m ) n ,,), l ( B ( m ) n ,,), c( B ( m ) n ,,), c 0 ( B ( m ) n ,,), W( B ( m ) n ,,), m( B ( m ) n ,,) and m 0 ( B ( m ) n ,,), respectively.

  4. (4)

    If we replace the base space X, which is a real linear 2-normed space by , complete normed linear space, and take m=1 and take r=1, s=1, then the above sequence spaces are denoted by c ¯ ( Δ ( 1 ) n ,p), c ¯ 0 ( Δ ( 1 ) n ,p), l ( Δ ( 1 ) n ,p), c( Δ ( 1 ) n ,p), c 0 ( Δ ( 1 ) n ,p), W( Δ ( 1 ) n ,p), m( Δ ( 1 ) n ,p) and m 0 ( Δ ( 1 ) n ,p), respectively.

  5. (5)

    If we take r=1, s=1, p k =1 for all kN, then these sequence spaces are denoted by c ¯ ( Δ ( m ) n ,,), c ¯ 0 ( Δ ( m ) n ,,), l ( Δ ( m ) n ,,), c( Δ ( m ) n ,,), c 0 ( Δ ( m ) n ,,), W( Δ ( m ) n ,,), m( Δ ( m ) n ,,) and m 0 ( Δ ( m ) n ,p,,), respectively.

  6. (6)

    If we replace the base space X, which is a real linear 2-normed space by , we obtain the spaces c ¯ ( B ( m ) n ,p), c ¯ 0 ( B ( m ) n ,p), l ( B ( m ) n ,p), c( B ( m ) n ,p), c 0 ( B ( m ) n ,p), W( B ( m ) n ,p), m( B ( m ) n ,p) and m 0 ( B ( m ) n ,p), respectively.

  7. (7)

    Moreover, if we take X=C, n=0 and p k =1 for all kN, we get the spaces c ¯ , c ¯ 0 , l , c, c 0 , W, m and m 0 , respectively.

Theorem 3.1 Let p=( p k ) be a sequence of strictly positive real numbers. Then the sequence spaces Z( B ( m ) n ,p,,) are linear spaces where Z= c ¯ , c ¯ 0 , l ,W,m, m 0 .

Proof The proof of the theorem can be obtained by similar techniques in [34]. □

Theorem 3.2 For any two sequences p=( p k ) and t=( t k ) of positive real numbers and for any two 2-norms , 1 and , 2 on X we have Z( B ( m ) n ,p, , 1 )Z( B ( m ) n ,p, , 2 ), where Z= c ¯ , c ¯ 0 ,m, m 0 .

Proof The proof follows from the fact that the zero element belongs to each of the sequence spaces involved in the intersection. □

Theorem 3.3 Let (X,,) be a 2-Banach space. Then the spaces m( B ( m ) n ,p,,), m 0 ( B ( m ) n ,p,,) are complete paranormed sequence spaces, paranormed by

g(x)= sup k N , z X ( B ( m ) n x k , z p k M ) ,
(3.1)

where M=max{1,H} and H= sup k p k , h= inf k p k .

Proof We will prove the theorem for the sequence space m 0 ( B ( m ) n ,p,,). It can be proved for the space m( B ( m ) n ,p,,) similarly.

Clearly g(x)=g(x) and g(θ)=0. From the following inequality, we have

g ( x + y ) = sup k N , z X ( B ( m ) n ( x k + y k ) , z p k M ) sup k N , z X ( B ( m ) n x k , z p k M ) + sup k N , z X ( B ( m ) n y k , z p k M ) .

This implies that g(x+y)g(x)+g(y).

To prove the continuity of scalar multiplication, assume that ( x n ) be any sequence of the points in m 0 ( B ( m ) n ,p,,) such that g( x n x)0 and ( λ n ) be any sequence of scalars such that λ n λ. Since the inequality

g ( x n ) g(x)+g ( x n x )

holds by subadditivity of g, (g( x n )) is bounded. Thus, we have

g ( λ n x n λ x ) = sup k N , z X ( B ( m ) n λ n x k n λ x k , z p k M ) ( max { | λ n λ | h , | λ n λ | H } ) 1 M sup k N , z X ( B ( m ) n x k , z p k M ) + ( max { | λ | h , | λ | H } ) 1 M sup k N , z X ( B ( m ) n ( x k n x ) , z p k M ) = ( max { | λ n λ | h , | λ n λ | H } ) 1 M g ( x n ) + ( max { | λ | h , | λ | H } ) 1 M g ( x n x )

which tends to zero as n. Hence, g is a paranorm on the sequence space m 0 ( B ( m ) n ,p,,).

To prove that m 0 ( B ( m ) n ,p,,) is complete, assume that ( x i ) is a Cauchy sequence in m 0 ( B ( m ) n ,p,,). Then for a given ε (0<ε<1), there exists a positive integer N 0 such that g( x i x j )<ε, for all i,j N 0 . This implies that

sup k N , z X ( B ( m ) n x k i B ( m ) n x k j , z p k M ) <ε,

for all i,j N 0 . It follows that for every nonzero zX,

B ( m ) n x k i B ( m ) n x k j , z <ε,

for each k1 and for all i,j N 0 . Hence ( B ( m ) n x k i ) is a Cauchy sequence in X for all kN. Since X is a 2-Banach space, ( B ( m ) n x k i ) is convergent in X for all kN, so we write ( B ( m ) n x k i )( B ( m ) n x k ) as i. Now we have for all i,j N 0 ,

sup k N , z X ( B ( m ) n ( x k i x k j ) , z p k M ) < ε lim j { sup k N , z X ( B ( m ) n ( x k i x k j ) , z p k M ) } < ε sup k N , z X ( B ( m ) n ( x k i x k ) , z p k M ) < ε ,

for all i N 0 . It follows that ( x i x) m 0 ( B ( m ) n ,p,,). Since ( x i ) m 0 ( B ( m ) n ,p,,) and m 0 ( B ( m ) n ,p,,) is a linear space, so we have x= x i ( x i x) m 0 ( B ( m ) n ,p,,). This completes the proof. □

Theorem 3.4

  1. (1)

    If Z 1 Z 2 , then Z 1 ( B ( m ) n ,p,,) Z 2 ( B ( m ) n ,p,,) and the inclusion is strict, where Z 1 , Z 2 =c, c 0 , l .

  2. (2)

    If n 1 < n 2 , then Z( B ( m ) n 1 ,p,,)Z( B ( m ) n 2 ,p,,) and the inclusion is strict, where Z=c, c 0 , l .

Proof The parts of proof Z 1 ( B ( m ) n ,p,,) Z 2 ( B ( m ) n ,p,,) and Z 1 ( B ( m ) n 1 ,p,,) Z 2 ( B ( m ) n 2 ,p,,) are easy. To show the inclusions are strict, choose Z 1 = c 0 , Z 2 =c, x=( x k )=( k 2 , k 2 ) and consider the 2-norm as defined in (2.1), let p k =1 for all kN, m=1, n=2, r=1, s=1, then xc( B ( 1 ) 2 ,,) but x c 0 ( B ( 1 ) 2 ,,). If we choose Z=c, x=( x k )=( k 2 , k 2 ) and p k =1 for all kN, m=1, n=2, r=1, s=1, then xc( B ( 1 ) 2 ,,) but xc( B ( 1 ) 1 ,,). These complete the proofs of parts (1) and (2) of the theorem, respectively. □

Theorem 3.5

  1. (1)

    c( B ( m ) n ,,) c ¯ ( B ( m ) n ,,) and the inclusion is strict.

  2. (2)

    c ¯ (,) c ¯ ( B ( m ) n ,,) and the inclusion is strict.

  3. (3)

    c ¯ ( B ( m ) n ,,) and l ( B ( m ) n ,,) overlap but neither one contains the other.

Proof

  1. (1)

    It is clear that c( B ( m ) n ,,) c ¯ ( B ( m ) n ,,). To show that the inclusion is strict, choose the sequence x=( x k ) such that,

    B ( m ) n x k ={ ( 0 , k ) , k = n 2 , ( 0 , 0 ) , k n 2 ,
    (3.2)

where nN{0}, and consider the 2-norm as defined in (2.1). Then we obtain B ( m ) n x k c ¯ (,), but B ( m ) n x k c(,). That is, x k c ¯ ( B ( m ) n ,p,,), but x k c( B ( m ) n ,p,,).

  1. (2)

    It is easy to see that c ¯ (,) c ¯ ( B ( m ) n ,,). To show that the inclusion is strict, let us take x=( x k )=(k,k) and consider the 2-norm as defined in (2.1), m=1, n=1, r=1, s=1, then x c ¯ ( B ( 1 ) 1 ,,) but x c ¯ (,).

  2. (3)

    Since the sequence x=θ belongs to each of the sequence spaces, the overlapping part of the proof is obvious. For the other part of the proof, consider the sequence defined by (3.2) and the 2-norm as defined in (2.1). Then x c ¯ ( B ( m ) n ,,), but x l ( B ( m ) n ,,). Conversely if we choose ( B ( m ) n x k )=( 1 ¯ , 0 ¯ , 1 ¯ , 0 ¯ ,) where k ¯ =(k,k) for all k=0,1, then B ( m ) n x k l (,) but B ( m ) n x k c ¯ (,). That is, x l ( B ( m ) n ,,) but x c ¯ ( B ( m ) n ,,).

 □

Theorem 3.6 The space Z( B ( m ) n ,p,,) is not solid in general, where Z= c ¯ , c ¯ 0 ,m, m 0 .

Proof To show that the space is not solid in general, consider the following examples. □

Example 3.7 Let m=3, n=1, r=1, s=1 and consider the 2-normed space as defined in Example 2.1. Let p k =5 for all kN. Consider the sequence ( x k ), where x k =( x k i ) is defined by ( x k i )=(k,k,k,) for each fixed kN. Then x k Z( B ( 3 ) 1 ,p,,) for Z= c ¯ ,m. Let α k = ( 1 ) k , then ( α k x k )Z( B ( 3 ) 1 ,p,,) for Z= c ¯ ,m. Thus Z( B ( 3 ) 1 ,p,,) for Z= c ¯ ,m is not solid in general.

Example 3.8 Let m=3, n=1, r=1, s=1 and consider the 2-normed space as defined in Example 2.1. Let p k =1 for all odd k and p k =2 for all even k. Consider the sequence ( x k ), where x k =( x k i ) is defined by ( x k i )=(3,3,3,) for each fixed kN. Then x k Z( B ( 3 ) 1 ,p,,) for Z= c ¯ 0 , m ¯ 0 . Let α k = ( 1 ) k , then ( α k x k )Z( B ( 3 ) 1 ,p,,) for Z= c ¯ 0 , m ¯ 0 . Thus Z( B ( 3 ) 1 ,p,,) for Z= c ¯ 0 , m ¯ 0 is not solid in general.

Theorem 3.9 The spaces m 0 ( B ( m ) n ,p,,) and m( B ( m ) n ,p,,) are nowhere dense subsets of l ( B ( m ) n ,p,,).

Proof From Theorem 3.3, it follows that m 0 ( B ( m ) n ,p,,) and m( B ( m ) n ,p,,) are closed subspaces of l ( B ( m ) n ,p,,). Since the inclusion relations

m 0 ( B ( m ) n , p , , ) l ( B ( m ) n , p , , ) ,m ( B ( m ) n , p , , ) l ( B ( m ) n , p , , )

are strict, the spaces m 0 ( B ( m ) n ,p,,) and m( B ( m ) n ,p,,) are nowhere dense subsets of l ( B ( m ) n ,p,,) by Lemma 2.2. □

Theorem 3.10 Let p=( p k ) be a sequence of non-negative bounded real numbers such that inf k p k >0. Then

W ( B ( m ) n , p , , ) l ( B ( m ) n , p , , ) m ( B ( m ) n , p , , ) .

Proof Let ( x k )W( B ( m ) n ,p,,) l ( B ( m ) n ,p,,). Then for a given ε>0, we have

1 j k = 1 j B ( m ) n x k L , z p k 1 j k = 1 j B ( m ) n x k L , z p k ε B ( m ) n x k L , z p k ε 1 j | { k j : B ( m ) n x k L , z p k ε } | .

If we take the limit for j, it follows that ( x k )c( B ( m ) n ,p,,) from the inequality above. Since ( x k ) l ( B ( m ) n ,p,,), we have the result. □

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Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors would like to thank the anonymous reviewers for their comments and suggestions to improve the quality of the paper.

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Correspondence to Şükran Konca.

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Başarır, M., Konca, Ş. & Kara, E.E. Some generalized difference statistically convergent sequence spaces in 2-normed space. J Inequal Appl 2013, 177 (2013). https://doi.org/10.1186/1029-242X-2013-177

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Keywords

  • statistical convergence
  • generalized difference sequence space
  • 2-norm
  • paranorm
  • completeness
  • solidity