Open Access

On generalized difference lacunary statistical convergence in a paranormed space

Journal of Inequalities and Applications20132013:256

https://doi.org/10.1186/1029-242X-2013-256

Received: 14 December 2012

Accepted: 6 May 2013

Published: 20 May 2013

Abstract

In this article, we introduce the concept of Δ m -lacunary statistical convergence and Δ m -lacunary strong convergence in a paranormed space. Also, we establish some connections between these concepts.

Dedication

Dedicated to Professor Hari M Srivastava.

1 Introduction

In order to extend convergence of sequences, the notion of statistical convergence was introduced by Fast [1] and Steinhaus [2] and several generalizations and applications of this concept have been investigated by various authors [3, 4]. This notion was studied in normed spaces by Kolk [5], in locally convex Hausdorff topological spaces by Maddox [6], in topological Hausdorff groups by Çakallı [7] and in probabilistic normed space by Karakuş [8]. Recently, Alotaibi and Alroqi [9] extended this notion in paranormed spaces.

In this article, we study the concept of statistical convergence from difference sequence spaces which are defined over paranormed space.

2 Preliminaries and definitions

Let K be a subset of the set of natural numbers . Then the asymptotic density of K denoted by δ ( K ) = lim n 1 n | { k n : k K } | , where the vertical bars denote the cardinality of the enclosed set in [10].

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

In this case we write s t - lim x = L . This concept was studied by [11, 12].

By a lacunary θ = ( k r ) ; r = 0 , 1 , 2 ,  , where k 0 = 0 , we shall mean an increasing sequence of non-negative integers with k r k r 1 as r . The intervals determined by θ will be denoted by I r = ( k r 1 , k r ] and h r = k r k r 1 . The ratio k r k r 1 will be denoted by q r .

The notion of difference sequence space X ( Δ ) was introduced by Kızmaz [13] as follows:
X ( Δ ) = { x = ( x k ) : ( Δ x k ) X }

for X = l , c, c 0 , where Δ x k = x k x k + 1 for all k N .

The notion of difference sequence spaces was further generalized by Et and Çolak [14] as follows:
X ( Δ m ) = { x = ( x k ) w : ( Δ m x k ) X }

for X = l , c and c 0 , where m N , Δ m x k = Δ m 1 x k Δ m 1 x k + 1 , Δ 0 x k = x k .

The sequence x is said to be Δ m -statistically convergent to the number L provided that for each ε > 0 ,
lim n 1 n | { k n : | Δ m x k L | ε } | = 0 .

The set of all Δ m -statistically convergent sequences was denoted by S ( Δ m ) in [15].

Furthermore, this notion was studied in [16, 17].

A paranorm is a function g : X R defined on a linear space X such that for all x , y , z X ,

(i) g ( x ) = 0 if x = θ ;

(ii) g ( x ) = g ( x ) ;

(iii) g ( x + y ) g ( x ) + g ( y ) ;

(iv) If ( α n ) is a sequence of scalars with α n α 0 ( n ) and x n , a X with x n a ( n ) in the sense that g ( x n a ) 0 ( n ), then α n x n α 0 a ( n ), in the sense that g ( α n x n α 0 a ) 0 ( n ).

A paranorm g for which g ( x ) = 0 implies x = θ is called a total paranorm on X and the pair ( X , g ) is called a total paranormed space.

Note that each seminorm (norm) p on X is a paranorm (total) but converse need not be true.

The concept of paranorm is a generalization of absolute value [18].

A modulus function f is a function from [ 0 , ) to [ 0 , ) such that

(i) f ( x ) = 0 if and only if x = 0 ;

(ii) f ( x + y ) f ( x ) + f ( y ) for all x , y 0 ;

(iii) f increasing;

(iv) f is continuous from at the right zero.

Since | f ( x ) f ( y ) | f ( | x y | ) , it follows from condition (iv) that f is continuous on [ 0 , ) . Furthermore, we have f ( n x ) n f ( x ) for all n N from condition (ii) and so
f ( x ) = f ( n x 1 n ) n f ( x n ) .
Hence, for all n N ,
1 n f ( x ) f ( x n ) .
A modulus may be bounded or unbounded. For example, f ( x ) = x p for 0 < p 1 is unbounded, but f ( x ) = x 1 + x is bounded. Ruckle [19] used the idea of a modulus function f to construct a class of FK spaces
L ( f ) = { x = ( x k ) : f ( | x k | ) < } .

In [9], the notion of statistical convergence was defined in a paranormed space.

Definition 2.1 A sequence x = ( x k ) is said to be statistically convergent to the number L in ( X , g ) if for each ε > 0 ,
lim n 1 n | { k n : g ( x k L ) ε } | = 0 .

In this case, we write g ( s t ) - lim x = L . We denote the set of all g ( s t ) -convergent sequences by S g [9].

Definition 2.2 A sequence x = ( x k ) is said to be strongly p-Cesaro summable ( 0 < p < ) to the limit L in ( X , g ) if
lim n 1 n j = 1 n ( g ( x j L ) ) p = 0 ,

and we write it as x k L ( [ C 1 , g ] p ) . In this case, L is called the [ C 1 , g ] p - lim it of x [9].

In this article, we shall study the concept of Δ m -lacunary statistical convergence, Δ m -lacunary strong convergence and Δ m -lacunary strong convergence with respect to a modulus function in a paranormed space.

3 Generalized difference statistical convergence in a paranormed space

Definition 3.1 A sequence x = ( x k ) is said to be Δ m -statistically convergent to the number L in ( X , g ) if for each ε > 0 ,
lim n 1 n | { k n : g ( Δ m x k L ) ε } | = 0 .

In this case, we write S g ( Δ m ) - lim x = L . We denote the set of all Δ m -statistically convergent sequences in ( X , g ) by S g ( Δ m ) .

Definition 3.2 Let θ be a lacunary sequence. A sequence x = ( x k ) is said to be Δ m -lacunary statistically convergent to the number L in ( X , g ) if for each ε > 0 ,
lim r 1 h r | { k I r : g ( Δ m x k L ) ε } | = 0 .

In this case, we write S g θ ( Δ m ) - lim x = L . We denote the set of all Δ m -lacunary statistically convergent sequences in ( X , g ) by S g θ ( Δ m ) .

Definition 3.3 A sequence x = ( x k ) is said to be strongly Δ m -Cesaro summable to the limit L in ( X , g ) if
lim n 1 n k = 1 n ( g ( Δ m x k L ) ) = 0 ,

and we write it as x k L ( | σ 1 | g ( Δ m ) ) . In this case L is called the | σ 1 | g ( Δ m ) - lim of x.

Definition 3.4 A sequence x = ( x k ) is said to be strongly Δ m -lacunary strongly summable to the limit L in ( X , g ) if
lim r 1 h r k I r ( g ( Δ m x k L ) ) = 0 ,

and we write it as x k L ( N g θ ( Δ m ) ) . In this case L is called the N g θ ( Δ m ) - lim of x.

Theorem 3.1 Let θ be a lacunary sequence and ( X , g ) be a paranormed space. Then

(i) If x k L ( N g θ ( Δ m ) ) , then x k L ( S g θ ( Δ m ) ) and the inclusion is strict;

(ii) If x is a Δ m -bounded sequence and x k L ( S g θ ( Δ m ) ) , then x k L ( N g θ ( Δ m ) ) ;

(iii) l g ( Δ m ) S g θ ( Δ m ) = l g ( Δ m ) N g θ ( Δ m ) .

Proof (i) If ε > 0 and x k L ( N g θ ( Δ m ) ) , we can write
k I r g ( Δ m x k L ) k I r g ( Δ m x k L ) ε g ( Δ m x k L ) ε | { k I r : g ( Δ m x k L ) ε } | ,

which yields the result.

In order to prove that the inclusion N g θ ( Δ m ) S g θ ( Δ m ) is proper, let θ be given and X = N 0 θ ( Δ , 1 h r ) = { x = ( x k ) : | 1 h r k I r Δ x k | 1 h r 0 , r } with the paranorm g ( x ) = | x 1 | + sup r | 1 h r k I r Δ x k | 1 h r . Define x = ( x k ) to be 2 h r 1 h r at the first term in I r for every r 1 , x k = h r ( 1 h r 2 h r ( k 1 ) h r ) between the second term and ( [ h r ] + 1 ) th term in I r , x k = h r ( 1 h r 2 h r ( [ h r ] ) h r ) at the ( [ h r ] + 2 ) th term in I r and x k = 0 otherwise.

We see that
Δ x k = { h r 1 h r , h r 2 h r , , h r [ h r ] h r , at the first  [ h r ]  integers in  I r , 0 , otherwise .
and
g ( Δ x k ) = { 1 , 2 , , [ h r ] , at the first  [ h r ]  integers in  I r , 0 , otherwise .
Note that x is not Δ-bounded in ( X , g ) . We have, for every ε > 0 ,
1 h r | { k I r : g ( Δ x k ) ε ) } | = [ h r ] h r 0
as r , i.e., x k 0 ( S g θ ( Δ ) ) . On the other hand,
1 h r k I r g ( Δ x k ) = 1 h r [ h r ] ( [ h r ] + 1 ) 2 1 2 0 ;

hence x k 0 ( N g θ ( Δ ) ) .

(ii) Suppose that x k L ( S g θ ( Δ m ) ) and say g ( Δ m x k L ) M for all k. Given ε > 0 , we get
1 h r k I r g ( Δ m x k L ) = 1 h r k I r g ( Δ m x k L ) ε g ( Δ m x k L ) + 1 h r k I r g ( Δ m x k L ) < ε g ( Δ m x k L ) M h r | { k I r : g ( Δ m x k L ) ε } | + ε ,

from which the result follows.

(iii) This is an immediate consequence of (i) and (ii). □

Corollary 3.1 If x k L ( | σ 1 | g ( Δ m ) ) , then x k L ( S g ( Δ m ) ) . If x l g ( Δ m ) and if x k L ( S g ( Δ m ) ) , then x k L ( | σ 1 | g ( Δ m ) ) .

Theorem 3.2 Let θ be a lacunary sequence and ( X , g ) be a paranormed space, then S g θ ( Δ m ) = S g ( Δ m ) if and only if 1 < lim inf r q r lim  sup r q r < .

Proof Suppose that lim inf q r > 1 , then there exists a δ > 0 such that q r 1 + δ for sufficiently large r, which implies that
h r k r δ 1 + δ .
If x k L ( S g ( Δ m ) ) , then for every ε > 0 and sufficiently large r, we have
1 k r | { k k r : g ( Δ m x k L ) ε } | 1 k r | { k I r : g ( Δ m x k L ) ε } | δ 1 + δ 1 h r | { k I r : g ( Δ m x k L ) ε } | ,

which proves the S g ( Δ m ) S g θ ( Δ m ) .

Conversely, suppose that lim inf q r = 1 . Since θ is lacunary, we can select a subsequence ( k r j ) of θ satisfying k r j k r j 1 < 1 + 1 j and k r j 1 k r ( j 1 ) > j , where r j r j 1 + 2 and X = N 0 θ ( Δ , 1 h r ) = { x = ( x k ) : | 1 h r k I r Δ x k | 1 h r 0 , r } with the paranorm g ( x ) = | x 1 | + sup r | 1 h r k I r Δ x k | 1 h r .

Now define a sequence by
Δ x k = { h r + k , k I r ( j ) , j = 1 , 2 , , 0 , otherwise .
We can see that
g ( Δ x k ) = { 1 , k I r ( j ) , j = 1 , 2 , , 0 , otherwise .

and hence x is Δ-bounded in ( X , g ) .

We can see that x N g θ ( Δ m ) , but x σ g 1 ( Δ m ) . Theorem 3.1(ii) implies that x S g θ ( Δ m ) , but it follows from Corollary 3.1 that x S g ( Δ m ) . Hence S g ( Δ m ) S g θ ( Δ m ) and S g ( Δ m ) S g θ ( Δ m ) implies that lim inf q r > 1 .

To show for any lacunary sequence θ, S g θ ( Δ m ) S g ( Δ m ) implies lim sup q r < , the same technique of Lemma 3 of [20] can be used. Now suppose that lim sup q r = . Consider the same space defined above and the sequence defined by
Δ x i = { h r + i , k r j 1 < i 2 k r j 1 , j = 1 , 2 , , 0 , otherwise .
Then we get
g ( Δ x i ) = { 1 , k r j 1 < i 2 k r j 1 , j = 1 , 2 , , 0 , otherwise .

Then x N g θ ( Δ m ) , but x σ g 1 ( Δ m ) . By Theorem 3.1(i) we conclude that x S g θ ( Δ m ) , but by Corollary 3.1 that x S g ( Δ m ) . Hence, S g θ ( Δ m ) S g ( Δ m ) . This completes the proof. □

Definition 3.5 Let f be a modulus function. Then a sequence x = ( x k ) is lacunary strongly p-Cesaro summable to L with respect to f in ( X , g ) if
lim r 1 h r k ϵ I r [ f ( g ( Δ m x k L ) ) ] p k = 0 .

In this case, we write x k L ( N g θ ( f , Δ m , p ) ) . If we take p k = 1 for all k N , we say x k L ( N g θ ( f , Δ m ) ) .

Lemma 3.1 Let f be a modulus function and let 0 < δ < 1 . Then for each x > δ we have f ( x ) 2 f ( 1 ) δ 1 x [21].

Theorem 3.3 Let f be a modulus function and ( X , g ) be a paranormed space. Then N g θ ( Δ m ) N g θ ( f , Δ m ) .

Proof Let x N g θ ( Δ m ) . Then we have τ r = 1 h r k I r g ( Δ m x k L ) 0 as r for some L.

Let ε > 0 and choose δ with 0 < δ < 1 such that f ( u ) < ε for u with 0 u δ . Then we can write
1 h r k I r f ( g ( Δ m x k L ) ) = 1 h r k I r g ( Δ m x k L ) δ f ( g ( Δ m x k L ) ) + 1 h r k I r g ( Δ m x k L ) > δ f ( g ( Δ m x k L ) ) 1 h r ( h r δ ) + 1 h r 2 f ( 1 ) δ 1 h r τ r

from Lemma 3.1. Therefore x N g θ ( f , Δ m ) . □

Theorem 3.4 Let 0 < inf k p k p k sup k p k < . Then S g θ ( Δ m ) = N g θ ( f , Δ m , p ) if and only if f is bounded.

Proof Following the technique applied for establishing Theorem 3.16 of [22], we can prove the theorem. □

Declarations

Acknowledgements

The author would like to thank the referees for their careful reading of the manuscript and for their helpful suggestions.

Authors’ Affiliations

(1)
Mathematics Department, Faculty of Arts and Science, Sakarya University

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© Altundağ; licensee Springer. 2013

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