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Double almost lacunary statistical convergence of order α

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Abstract

In this paper, we define and study lacunary double almost statistical convergence of order α. Further, some inclusion relations have been examined. We also introduce a new sequence space by combining lacunary double almost statistical convergence and Orlicz function.

MSC:40B05, 40C05.

1 Introduction

The notion of convergence of a real sequence was extended to a statistical convergence by Fast [1] (see also Schoenberg [2]) as follows. If denotes the set of natural numbers and KN, then K(m,n) denotes the cardinality of the set K[m,n]. The upper and lower natural density of the subset K is defined by

d ¯ (K)= lim n sup K ( 1 , n ) n and d ̲ (K)= lim n inf K ( 1 , n ) n .

If d ¯ (K)= d ̲ (K), then we say that the natural density of K exists, and it is denoted simply by d(K). Clearly d(K)= lim n K ( 1 , n ) n .

A sequence x=( x k ) of real numbers is said to be statistically convergent to L if for arbitrary ϵ>0, the set K(ϵ)={kN:| x k L|ϵ} has a natural density zero.

Statistical convergence turned out to be one of the most active areas of research in summability theory after the works of Fridy [3] and Šalát [4]. For some very interesting investigations concerning statistical convergence, one may consult the papers of Cakalli [5], Miller [6], Maddox [7] and many others, where more references on this important summability method can be found.

On the other hand, in [8, 9], a different direction was given to the study of statistical convergence, where the notion of statistical convergence of order α, 0<α<1 was introduced by replacing n by n α in the denominator in the definition of statistical convergence. It was observed in [8] that the behaviour of this new convergence was not exactly parallel to that of statistical convergence, and some basic properties were obtained. One can also see [10] for related works.

In this paper, we define and study lacunary double almost statistical convergence of order α. Also some inclusion relations have been examined.

Let w 2 be the set of all real or complex double sequences. By the convergence of a double sequence, we mean the convergence on the Pringsheim sense, that is, double sequence x=( x i j ) has a Pringsheim limit L, denoted by P-limx=L, provided that given ϵ>0, and there exists NN such that | x i j L|<ϵ whenever i,jN. We shall describe such an x more briefly as ‘P-convergent’ (see, [11]). We denote by c 2 the space of P-convergent sequences. A double sequence x=( x i j ) is bounded if x= sup i , j 0 | x i j |<. Let l 2 and c 2 be the set of all real or complex bounded double sequences and the set bounded and convergent double sequences, respectively. Moricz and Rhoades [12] defined the almost convergence of double sequence as follows: x=( x i j ) is said to be almost convergent to a number L if

P- lim p , q sup m , n | 1 ( p + 1 ) ( q + 1 ) i = m m + p j = n n + q x i j L|=0,

that is, the average value of ( x i j ) taken over any rectangle

D= { ( i , j ) : m i m + p , n j n + q } ,

tends to L as both p and q tend to ∞, and this convergence is uniform in m and n. We denote the space of almost convergent double sequence by c ˆ 2 , as

c ˆ 2 = { x = ( x i j ) : lim k , l | t k l p q ( x ) L | = 0 ,  uniformly in  p , q } ,

where

t k l p q (x)= 1 ( k + 1 ) ( l + 1 ) i = p k + p j = q l + q x i j .

The notion of almost convergence for single sequences was introduced by Lorentz [13] and some others.

A double sequence x is called strongly double almost convergent to a number L if

P- lim k , l 1 ( k + 1 ) ( l + 1 ) i = p k + p j = q l + q | x i j L|=0,uniformly in p,q.

By [ c ˆ 2 ], we denote the space of strongly almost convergent double sequences.

The notion of strong almost convergence for single sequences has been introduced by Maddox [7].

The idea of statistical convergence was extended to double sequences by Mursaleen and Edely [14]. More recent developments on double sequences can be found in [8, 1518]. For the single sequences; statistical convergence of order α and strongly p-Cesàro summability of order α introduced by Çolak [9]. Quite recently, in [10], Çolak and Bektaş generalized this notion by using de la Valée-Poussin mean.

Let KN×N be a two-dimensional set of positive integers, and let K m , n be the numbers of (i,j) in K such that in and jm.

Then the lower asymptotic density of K is defined as

P- lim inf m , n K m , n m n = δ 2 (K).

In the case when the sequence ( K m , n m n ) m , n = 1 , 1 , has a limit, we say that K has a natural density and is defined as

P- lim m , n K m , n m n = δ 2 (K).

For example, let K={( i 2 , j 2 ):i,jN}, where is the set of natural numbers. Then

δ 2 (K)=P- lim m , n K m , n m n P- lim m , n m n m n =0

(i.e., the set K has a double natural density zero).

Mursaleen and Edely [14] presented the notion of a statistical convergence for the double sequence x=( x i j ) as follows: A real double sequence x=( x i j ) is said to be statistically convergent to L, provided that for each ϵ>0

P- lim m , n 1 m n | { ( i , j ) : i m  and  j n , | x i j L | ϵ } | =0.

We now write the following definition.

The double statistical convergence of order α is defined as follows. Let 0<α1 be given. The sequence ( x i j ) is said to be statistically convergent of order α if there is a real number L such that

P- lim m n 1 ( m n ) α | { i m  and  j n : | x i j L | ϵ } |=0

for every ϵ>0, in this, case we say that x is double statistically convergent of order α to L. In this case, we write S 2 α -lim x i j =L. The set of all double statistically convergent sequences of order α will be denoted by S 2 α . If we take α=1 in this definition , we can have the previous definition.

By a lacunary θ=( k r ); r=0,1,2, , where k 0 =0, we shall mean an increasing sequence of nonnegative 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 .

Fridy and Orhan [19] introduced the idea of lacunary statistical convergence for single sequence as follows.

The number sequence x=( x i ) is said to be lacunary statistically convergent to the number if for each ϵ>0,

lim n 1 h r | { k I r : | x i L | ϵ } |=0.

In this case, we write S θ - lim i x i =, and we denote the set of all lacunary statistically convergent sequences by S θ .

Definition 1.1 By a double lacunary θ r s ={( k r l s )}, r,s=0,1,2, , where k 0 =0 and l 0 =0, we shall mean two increasing sequences of nonnegative integers with

h r = k r k k 1 as r

and

h ¯ s = l s l s 1 as s.

Let us denote k r s = k r l s , h r s = h r h ¯ s and the intervals determined by θ r s will be denoted by I r s ={(k,l): k r 1 <k k r  and  l s 1 <l l s }, q r = k r k r 1 , q ¯ s = l s l s 1 , and q r s = q r q ¯ s . We will denote the set of all double lacunary sequences by N θ r s .

Let KN×N have double lacunary density δ 2 θ (K) if

P- lim r s 1 h r s | { ( k , l ) I r s : ( k , l ) K } |

exists.

Example 1 Let θ={( 2 r 1, 3 s 1)} and K={(k,2l):k,lN×N}. Then δ 2 θ (K)=0. But it is obvious that δ 2 (K)=1/2.

In 2005, Patterson and Savaş [17] studied double lacunary statistical convergence by giving the definition for complex sequences as follows.

Definition 1.2 Let θ r s be a double lacunary sequence; the double number sequence x is S θ 2 -convergent to L, provided that for every ϵ>0,

P- lim r s 1 h r s | { ( k , l ) I r s : | x k l L | ϵ } |=0.

In this case, write S θ 2 -limx=L or x k l P L( S θ 2 ).

More investigation in this direction and more applications of double lacunary and double sequences can be found in [2022] and [23].

2 Main results

In this section, we define lacunary double almost statistically convergent sequences of order α. Also we shall prove some inclusion theorems.

We now have the following.

Definition 2.1 Let 0<α1 be given. The sequence x=( x i j ) w 2 is said to be S ˆ θ r s α -statistical convergence of order α if there is a real number L such that

P- lim r s 1 h r s α | { ( k , l ) I r s : | t k l p q ( x ) L | ϵ } |=0,uniformly in p,q,

where h r s α denote the α th power ( h r s ) α of h r s . In case x=( x i j ) is S ˆ θ r s α -statistically convergent of order α to L, we write S ˆ θ r s α -lim x i j =L. We denote the set of all S ˆ θ r s α -statistically convergent sequences of order α by S ˆ θ r s α .

We know that the S ˆ θ r s α -statistical convergence of order α is well defined for 0<α1, but it is not well defined for α>1 in general. It is easy to see by taking x=( x i j ) as fixed.

Definition 2.2 Let 0<α1 be any real number, and let t be a positive real number. A sequence x is said to be strongly w ˆ θ r s α (t)-summable of order α, if there is a real number L such that

P- lim r s 1 h r s α ( k , l ) I r s | t k l p q (x)L | t =0,uniformly in p,q.

If we take α=1, the strong w ˆ θ r s α (t)-summability of order α reduces to the strong w ˆ θ r s (t)-summability.

We denote the set of all strongly w ˆ θ r s α (t)-summable sequence of order α by w ˆ θ r s α (t).

We now state the following theorem.

Theorem 2.1If0<αβ1, then S ˆ θ r s α S ˆ θ r s β .

Proof Let 0<αβ1. Then

1 h r s β | { ( k , l ) I r s : | t k l p q ( x ) L | ϵ } | 1 h r s α | { ( k , l ) I r s : | t k l p q ( x ) L | ϵ } |

for every ϵ>0, and finally, we have that S ˆ θ r s α S ˆ θ r s β . This proves the result. □

Theorem 2.2For any lacunary sequencesθ, S ˆ 2 α S ˆ θ r s α , ifliminf q r >1andliminf q ¯ s >1.

Proof Suppose that liminf q r α >1 and liminf q s α >1, liminf q r α = α 1 and liminf q s α = α 2 , say. Write β 1 =( α 1 1)/2 and β 2 =( α 2 1)/2. Then there exist a positive integer r 0 and s 0 such that q r α 1+ β 1 for r r 0 and q s 1+ β 2 for s s 0 . Hence for r r 0 , and s s 0 ,

h r s α 1 ( k r l s ) α = 1 ( k r 1 α k r α ) × 1 ( l s 1 α l s α ) = ( 1 1 q r α ) × ( 1 1 q s α ) 1 1 ( 1 + β 1 ) × 1 1 ( 1 + β 2 ) = β 1 1 + β 1 × β 2 1 + β 2 .

Take any ( x k l ) S ˆ 2 α , and S ˆ 2 α - lim ( k , l ) x k l =L, say. We prove that S ˆ θ r s α - lim ( k , l ) x k l =L. Then for r r 0 and s s 0 , we have

1 ( k r l s ) α | { k k r , l l s : | t k l p q ( x ) L | ϵ } | 1 ( k r l s ) α | { ( k , l ) I r s : | t k l p q ( x ) L | ϵ } | = h r s α 1 ( k r l s ) α 1 h r s α | { ( k , l ) I r s : | t k l p q ( x ) L | ϵ } | β 1 1 + β 1 × β 2 1 + β 2 1 h r s α | { ( k , l ) I r s : | t k l p q ( x ) L | ϵ } | .

Therefore, S ˆ θ r s α - lim ( k , l ) x(k,l)=L. □

Remark 2.1 The converse of this result is true for α=1. However, for α<1 it is not clear, and we leave it as an open problem.

Theorem 2.3For any double lacunary sequence θ r s , S ˆ θ r s α S ˆ 2 α iflim sup r q r α <andlim sup s q s α <.

Proof Suppose that lim sup r q r α < and lim sup s q s α <. Then there exists H>0 such that q r α <H and q s α <H for all r and s. Suppose that x k l L( S θ r s α ) and

N r s =| { ( k , l ) I r s : | t k l p q ( x ) L | ϵ } |

by the definition of x k l L( S θ r s ) given ϵ>0, there exists r 0 , s 0 N such that N r s h r s α <ϵ for all r> r 0 and s> s 0 . Let

M:=max{ N r s :1r r 0  and 1s s 0 }.

Let n and m be such that k r 1 <m k r and l s 1 <n l s . Therefore, we obtain the following:

1 ( m n ) α | { k m and l n : | t k l p q ( x ) L | ϵ } | 1 ( k r 1 l s 1 ) α | { k k r and l l s : | t k l p q ( x ) L | ϵ } | = 1 ( k r 1 l s 1 ) α { i , j = 1 , 1 r , s N i , j } M r 0 s 0 ( k r 1 l s 1 ) α + 1 ( k r 1 l s 1 ) α { i , j = r 0 + 1 , r 0 + 1 r , s N i , j } M r 0 s 0 ( k r 1 l s 1 ) α + 1 ( k r 1 l s 1 ) α { i , j = r 0 + 1 , r 0 + 1 r , s N i , j h i , j α h i , j α } M r 0 s 0 k r 1 l s 1 + 1 ( k r 1 l s 1 ) α ( sup i , j r 0 , r 0 N i , j h i , j α ) { i , j = r 0 + 1 , r 0 + 1 r , s h i , j α } M r 0 s 0 ( k r 1 l s 1 ) α + ϵ { i , j = r 0 + 1 , r 0 + 1 r , s h i , j α } M r 0 s 0 ( k r 1 l s 1 ) α + ϵ H 2 .

This completes the proof of the theorem. □

Theorem 2.4Let0<αβ1andtbe a positive real number, then w ˆ θ r s α (t) w ˆ θ r s β (t).

Proof Let x=( x i j ) w ˆ θ r s α (t). Then given α and β such that 0<αβ1 and a positive real number t we write

1 h r s β ( k , l ) I r s | t k l p q (x)L | t 1 h r s α ( k , l ) I r s | t k l p q (x)L | t ,

and we get that w ˆ θ r s α (t) w ˆ θ r s β (t). □

As a consequence of Theorem 2.4, we have the following.

Corollary 2.1Let0<αβ1andtbe a positive real number. Then:

  1. (i)

    Ifα=β, then w ˆ θ r s α (t)= w ˆ θ r s β (t).

  2. (ii)

    w ˆ θ r s α (t) w ˆ θ r s (t)for eachα(0,1]and0<t<.

Theorem 2.5Letαandβbe fixed real numbers such that0<αβ1and0<t<. If a sequence is a strongly w ˆ θ r s α (t)-summable sequence of orderα, toL, then it is S ˆ θ r s β -statistically convergent of orderβ, toL, i.e., w ˆ θ r s α (t) S ˆ θ r s β .

Proof For any sequence x=( x i j ) and ϵ>0, we write

( k , l ) I r s | t k l p q ( x ) L | t = ( k , l ) I r s | t k l p q ( x ) L | ϵ | t k l p q ( x ) L | t + ( k , l ) I r s | t k l p q ( x ) L | < ϵ | t k l p q ( x ) L | t ( k , l ) I r s | t k l p q ( x ) L | ϵ | t k l p q ( x ) L | t | { ( k , l ) I r s : | t k l p q ( x ) L | ϵ } | ϵ t

and so that

1 h r s α ( k , l ) I r s | t k l p q ( x ) L | t 1 h r s α | { ( k , l ) I r s : | t k l p q ( x ) L | ϵ } | ϵ t 1 h r s β | { ( k , l ) I r s : | t k l p q ( x ) L | ϵ } | ϵ t ,

this shows that if x=( x i j ) is strongly w ˆ θ r s α (t)-summable sequence of order α to L, then it is S ˆ θ r s β -statistically convergent of order β to L. This completes the proof. □

We have the following.

Corollary 2.2Letαbe fixed real numbers such that0<α1and0<t<.

  1. (i)

    If a sequence is strongly w ˆ θ r s α (t)-summable sequence of orderαtoL, then it is S ˆ θ r s α -statistically convergent of orderαtoL, i.e., w ˆ θ r s α (t) S ˆ θ r s α .

  2. (ii)

    w ˆ θ r s α (t) S ˆ θ r s , for0<α1.

3 New sequence space

In this section, we study the inclusion relations between the set of S ˆ θ r s α -statistical convergent sequences of order α and strongly w ˆ θ r s α [M,t]-summable sequences of order α with respect to an Orlicz function M.

The study of Orlicz sequence spaces was initiated with a certain specific purpose in Banach space theory. Lindenstrauss and Tzafriri [24] investigated Orlicz sequence spaces in more detail, and they proved that every Orlicz sequence space l M contains a subspace isomorphic to l p (1p<). The Orlicz sequence spaces are the special cases of Orlicz spaces studied in [25]. Orlicz spaces find a number of useful applications in the theory of nonlinear integral equations. Whereas the Orlicz sequence spaces are the generalization of l p spaces, the l p -spaces find themselves enveloped in Orlicz spaces [26].

Recall in [25] that an Orlicz function M:[0,)[0,) is continuous, convex, non-decreasing function such that M(0)=0 and M(x)>0 for x>0, and M(x) as x.

An Orlicz function M is said to satisfy Δ 2 -condition for all values of u, if there exists K>0 such that M(2u)KM(u), u0.

In the later stage different classes of Orlicz sequence spaces were introduced and studied by Parashar and Choudhary [27], Savaş [2833] and many others.

Definition 3.1 Let M be an Orlicz function, t=( t k l ) be a sequence of strictly positive real numbers, and let α(0,1] be any real number. Now, we write

w ˆ θ r s α [ M , t ] = { x = ( x k l ) : P - lim r s 1 h r s α ( k , l ) I r s [ M ( | t k l p q ( x ) L | ) ρ ] t k l = 0 , uniformly in  p , q ,  for some  L  and  ρ > 0 } .

If x w ˆ θ r s α [M,t], then we say that x is strongly double almost lacunary summable of order α with respect to the Orlicz function M.

If we consider various assignments of M, θ r s and t in the sequence spaces above, we are granted the following:

  1. (1)

    If M(x)=x, θ= 2 r s , and t k , l =1 for all (k,l) then w ˆ θ r s α [M,t]=[ w ˆ α ].

  2. (2)

    If t k , l =1 for all (k,l), then w ˆ θ r s α [M,t]= w ˆ θ r s α [M].

  3. (3)

    If t k , l =1 for all (k,l) and θ= 2 r s , then w ˆ α θ r s [M,t]= w ˆ α [M].

  4. (4)

    If θ= 2 r s , then w ˆ θ α [M,t]= w ˆ α [M,t].

In the followings theorems, we shall assume that t=( t k l ) is bounded and 0<h= inf k l t k l t k l sup k l t k l =H<.

Theorem 3.1Letα,β(0,1]be real numbers such thatαβ, and letMbe an Orlicz function, then w ˆ α θ r s [M,t] S ˆ β θ r s .

Proof Let x w ˆ θ α [M,t], ϵ>0 be given and ∑1 and ∑2 denote the sums over (k,l) I r s , | t k l p q (x)L|ϵ and (k,l) I r s , | t k l p q (x)L|<ϵ, respectively. Since h r s α h r s β for each r, s we write

1 h r s α ( k , l ) I r s [ M ( | t k l p q ( x ) L | ) ρ ] t k l = 1 h r s α [ 1 [ M ( | t k l p q ( x ) L | ) ρ ] t k l + 2 [ M ( | t k l p q ( x ) L | ) ρ ] t k l ] 1 h r s β [ 1 [ M ( | t k l p q ( x ) L | ) ρ ] t k l + 2 [ M ( | t k l p q ( x ) L | ) ρ ] t k l ] 1 h r s β [ 1 [ M ( ϵ / ρ ) ] ] t k l 1 h r s β 1 min ( [ M ( ϵ 1 ) ] h , [ M ( ϵ 1 ) ] H ) , ϵ 1 = ϵ ρ 1 h r s β | { ( k , l ) I r s : | t k l p q ( x ) L | ϵ } | min ( [ M ( ϵ 1 ) ] h , [ M ( ϵ 1 ) ] H ) .

Since x w ˆ α θ r s [M,t], the left hand side of the inequality above tends to zero as r,s uniformly in p, q. Hence the right hand side tends to zero as r,s uniformly in p, q, and, therefore, x S ˆ θ r s β . This proves the result. □

Corollary 3.1Letα(0,1]andMbe an Orlicz function, then w ˆ θ r s α [M,t] S ˆ θ r s α .

We finally prove the following theorem.

Theorem 3.2LetMbe an Orlicz function, and letx=( x i j )be a bounded sequence, then S ˆ θ r s α w ˆ θ r s α [M,t].

Proof Suppose that x 2 and S ˆ θ r s α -lim x i j =L. Since x 2 , then there is a constant K>0 such that | t k l p q (x)|K. Given ϵ>0, we write for all p, q

1 h r s α ( k , l ) I r s [ M ( | t k l p q ( x ) L | ρ ) ] r k l = 1 h r s α 1 [ M ( | t k l p q ( x ) L | ρ ) ] r k l + 1 h r s α 2 [ M ( | t k l p q ( x ) L | ρ ) ] r k l 1 h r s α 1 max { [ M ( K ρ ) ] h , [ M ( K ρ ) ] H } + 1 h r s α 2 [ M ( ϵ ρ ) ] t k l max { [ M ( T ) ] h , [ M ( T ) ] H } × 1 h r s α | { ( k , l ) I r s : | t k l p q ( x ) L | ϵ } | + max { [ M ( ϵ 1 ) ] h , [ M ( ϵ 1 ) ] H } , K ρ = T , ϵ ρ = ϵ 1 .

Therefore, xx w ˆ θ r s α [M,t]. This proves the result. □

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Acknowledgements

The author would like to thank the referees for their help and useful discussions.

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Correspondence to Ekrem Savaş.

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The author declares that they have no competing interests.

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Keywords

  • statistical convergence
  • Orlicz function
  • double statistical convergence of order α
  • lacunary statistical convergence
  • double almost statistical convergence