Open Access

On some spaces of almost lacunary convergent sequences derived by Riesz mean and weighted almost lacunary statistical convergence in a real n-normed space

Journal of Inequalities and Applications20142014:81

https://doi.org/10.1186/1029-242X-2014-81

Received: 4 October 2013

Accepted: 9 January 2014

Published: 18 February 2014

Abstract

In this paper, we introduce some new spaces of almost convergent sequences derived by Riesz mean and the lacunary sequence in a real n-normed space. By combining the definitions of lacunary sequence and Riesz mean, we obtain a new concept of statistical convergence which will be called weighted almost lacunary statistical convergence in a real n-normed space. We examine some connections between this notion with the concept of almost lacunary statistical convergence and weighted almost statistical convergence, where the base space is a real n-normed space.

MSC:40C05, 40A35, 46A45, 40A05, 40F05.

Keywords

Riesz mean weighted lacunary statistical convergence almost convergence lacunary sequence n-norm

1 Introduction

The concept of 2-normed space has been initially introduced by Gähler [1]. Later, this concept was generalized to the concept of n-normed spaces by Misiak [2]. Since then, many others have studied these concepts and obtained various results [310].

The idea of statistical convergence was given by Zygmund [11] in 1935, in order to extend the convergence of sequences. The concept was formally introduced by Fast [12] and Steinhaus [13] and later on by Schoenberg [14], and also independently by Buck [15]. Many years later, it has been discussed in the theory of Fourier analysis, ergodic theory, and number theory under different names. In 1993, Fridy and Orhan [16] introduced the concept of lacunary statistical convergence. Statistical convergence has been generalized to the concept of a 2-normed space by Gürdal and Pehlivan [3] and to the concept of an n-normed space by Reddy [9].

Moricz and Orhan [17] have defined the concept of statistical summability ( R , p r ) . Later on, Karakaya and Chishti [18] have used ( R , p r ) -summability to generalize the concept of statistical convergence and have called this new method weighted statistical convergence. Mursaleen et al. [19] have altered the definition of weighted statistical convergence and have found its relation with the concept of statistical ( R , p r ) -summability. In general, the statistical convergence of weighted mean is studied as a regular matrix transformation. In [18] and [19], the concept of statistical convergence is generalized by using a Riesz summability method and it is called weighted statistical convergence. For more details related to this topic, we may refer to [5, 2023].

In this paper, we introduce some new spaces of almost convergent sequences derived by Riesz mean and lacunary sequence in a real n-normed space. By combining the definitions of lacunary sequence and Riesz mean, we obtain a new concept of statistical convergence, which will be called weighted almost lacunary statistical convergence in a real n-normed space. We examine some connections between this notion with the concept of almost lacunary statistical convergence and weighted almost statistical convergence, where the base space is a real n-normed space.

2 Definitions and preliminaries

Let K be a subset of natural numbers and we denote the set K n = { j K : j n } . The cardinality of K n is denoted by | K n | . The natural density of K is given by δ ( K ) : = lim r 1 r | K r | , if it exists. The sequence x = ( x j ) is statistically convergent to ξ provided that, for every ε > 0 , the set K = K ( ε ) : = { j N : | x j ξ | ε } has natural density zero.

Let ( p k ) be a sequence of non-negative real numbers and P r = p 1 + p 2 + + p r for r N . Then the Riesz transformation of x = ( x k ) is defined as
t r : = 1 P r k = 1 r p k x k .
(2.1)

If the sequence t r has a finite limit ξ, then the sequence x is said to be ( R , p r ) -convergent to ξ. Let us note that if P r as r then the Riesz transformation is a regular summability method, that is, it transforms every convergent sequence to convergent sequence and preserves the limit.

If p k = 1 for all k N in (2.1), then the Riesz mean reduces to the Cesaro mean C 1 of order one.

By a lacunary sequence θ = ( k r ) , where k 0 = 0 , we will 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 ] . We write h r = k r k r 1 . The ratio k r k r 1 will be denoted by q r .

Throughout the paper, we will use the following notations, which have been defined in [24].

Let θ = ( k r ) be a lacunary sequence, ( p k ) be a sequence of positive real numbers such that H r : = k I r p k , P k r : = k ( 0 , k r ] p k , P k r 1 : = k ( 0 , k r 1 ] p k , Q r : = P k r P k r 1 , P 0 = 0 and the intervals determined by θ and ( p k ) are denoted by I r = ( P k r 1 , P k r ] , H r = P k r P k r 1 . If p k = 1 for all k N , then H r , P k r , P k r 1 , Q r and I r reduce to h r , k r , k r 1 , q r and I r , respectively.

If θ = ( k r ) is a lacunary sequence and P r as r , then θ = ( P k r ) is a lacunary sequence, that is, P 0 = 0 , 0 < P k r 1 < P k r and H r = P k r P k r 1 as r .

Throughout the paper, we will take P r as r , unless otherwise stated.

Lorentz [25] has proved that a sequence x is almost convergent to a number ξ if and only if t k m ( x ) ξ as k , uniformly in m, where
t k m ( x ) = x m + x m + 1 + + x m + k 1 k , k N , m 0 .
(2.2)

We write f lim x = ξ if x is almost convergent to ξ. Maddox [26] has defined x = ( x j ) to be strongly almost convergent to a number ξ if and only if t k m ( | x ξ e | ) 0 as k , uniformly in m, where x ξ e = ( x j ξ ) for all j and e = ( 1 , 1 , ) .

Let n N and X be a real vector space of dimension d n 2 . A real-valued function , , : X n R satisfying the following conditions is called an n-norm on X and the pair ( X , , , ) is called a linear n-normed space:
  1. (1)

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

     
  2. (2)

    x 1 , , x n is invariant under permutation,

     
  3. (3)

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

     
  4. (4)

    x 1 , , x n 1 , y + z x 1 , , x n 1 , y + x 1 , , x n 1 , z , for all y , z , x 1 , , x n 1 X .

     

A sequence x = ( x j ) in an n-normed space ( X , , , ) is said to be convergent to some ξ X in the n-norm if for each ε > 0 there exists a positive integer j 0 = j 0 ( ε ) such that x j ξ , z 1 , , z n 1 < ε for all j j 0 and for every nonzero z 1 , , z n 1 X .

A sequence x = ( x j ) is said to be statistically convergent to ξ if for every ε > 0 the set K : = { j N : x j ξ , z 1 , , z n 1 ε } has natural density zero for every nonzero z 1 , , z n 1 X , in other words, x = ( x j ) is statistically convergent to ξ in n-normed space ( X , , , ) if lim j 1 j | { j N : x j ξ , z 1 , , z n 1 ε } | = 0 , for every nonzero z 1 , , z n 1 X . For ξ = 0 , we say this is statistically null.

3 Main results

Throughout the paper w ( X ) , l ( X ) denote the spaces of all and bounded X valued sequence spaces, respectively, where ( X , , , ) is a real n-normed space.

The set of all almost convergent sequences and strongly almost convergent sequences with respect to the n-norm , are denoted by F and [ F ] , respectively, as follows:
F = { x l ( X ) : lim k t k m ( x ξ e ) , z 1 , , z n 1 = 0 ,  uniformly in  m , for every nonzero  z 1 , , z n 1 X } ,
and
[ F ] = { x l ( X ) : lim k t k m ( x ξ e , z 1 , , z n 1 ) = 0 ,  uniformly in  m , for every nonzero  z 1 , , z n 1 X } ,

where t k m ( x ) is defined as in (2.2). We write F lim x = ξ if x is almost convergent to ξ with respect to the n-norm and [ F ] lim x = ξ if x is strongly almost convergent to ξ with respect to the n-norm. It is easy to see that the inclusions [ F ] F l ( X ) hold.

Now, we define some new sequence spaces in a real n-normed space as follows:
[ R ˜ , p r , θ ] n = { x : lim r 1 H r k I r p k t k m ( x ξ e ) , z 1 , , z n 1 = 0 ,  uniformly in  m , for some  ξ  and for every nonzero  z 1 , , z n 1 X } , ( R ˜ , p r , θ ) n = { x : lim r 1 H r k I r p k t k m ( x ξ e ) , z 1 , , z n 1 = 0 ,  uniformly in  m , for some  ξ  and for every nonzero  z 1 , , z n 1 X } , | R ˜ , p r , θ | n = { x : lim r 1 H r k I r p k t k m ( x ξ e , z 1 , , z n 1 ) = 0 ,  uniformly in  m , for some  ξ  and for every nonzero  z 1 , , z n 1 X } .
The following results are obtained for some special cases:
  1. (1)
    If we take m = 0 then the sequence spaces above are reduced to the sequence spaces [ C 1 , θ ] n , ( C 1 , θ ) n , | C 1 , θ | n , respectively as follows:
    [ C 1 , θ ] n = { x : lim r 1 H r k I r p k t k 0 ( x ξ e ) , z 1 , , z n 1 = 0 , for some  ξ  and for every nonzero  z 1 , , z n 1 X } , ( C 1 , θ ) n = { x : lim r 1 H r k I r p k t k 0 ( x ξ e ) , z 1 , , z n 1 = 0 for some  ξ  and for every nonzero  z 1 , , z n 1 X } , | C 1 , θ | n = { x : lim r 1 H r k I r p k t k 0 x ξ e , z 1 , , z n 1 = 0 , for some  ξ  and for every nonzero  z 1 , , z n 1 X } .
     
  2. (2)
    If we take p k = 1 for all k N , then the sequence spaces above are reduced to the following spaces:
    [ w θ ] n = { x : lim r 1 h r k I r t k m ( x ξ e ) , z 1 , , z n 1 = 0 ,  uniformly in  m , for some  ξ  and for every nonzero  z 1 , , z n 1 X } , ( w θ ) n = { x : lim r 1 h r k I r t k m ( x ξ e ) , z 1 , , z n 1 = 0 ,  uniformly in  m , for some  ξ  and for every nonzero  z 1 , , z n 1 X } , | w θ | n = { x : lim r 1 h r k I r t k m ( x ξ e , z 1 , , z n 1 ) = 0 ,  uniformly in  m , for some  ξ  and for every nonzero  z 1 , , z n 1 X } .
     
  3. (3)
    Let us choose θ = ( k r ) = 2 r for r > 0 , then these sequence spaces above are reduced to the following spaces:
    [ R ˜ , p r ] n = { x : lim r 1 P r k = 1 r p k t k m ( x ξ e ) , z 1 , , z n 1 = 0 ,  uniformly in  m ,  for some  ξ  and for every nonzero  z 1 , , z n 1 X } , ( R ˜ , p r ) n = { x : lim r 1 P r k = 1 r p k t k m ( x ξ e ) , z 1 , , z n 1 = 0 ,  uniformly in  m ,  for some  ξ  and for every nonzero  z 1 , , z n 1 X } , | R ˜ , p r | n = { x : lim r 1 P r k = 1 r p k t k m ( x ξ e , z 1 , , z n 1 ) = 0 ,  uniformly in  m ,  for some  ξ  and for every nonzero  z 1 , , z n 1 X } .
     
  4. (4)

    If we select θ = ( k r ) = 2 r for r > 0 and the base space as ( X , , ) then these sequence spaces above are reduced to the sequence spaces which can be seen in [5].

     
  5. (5)

    If we choose p k = 1 for all k N and θ = ( k r ) = 2 r for r > 0 , then these sequence spaces above are reduced to the sequence spaces [ C 1 ] n , ( C 1 ) n , | C 1 | n , respectively.

     

Now, we give the following theorem to demonstrate some inclusion relations among the sequence spaces | R ˜ , p r , θ | n , ( R ˜ , p r , θ ) n , [ R ˜ , p r , θ ] n , | C 1 , θ | n , ( C 1 , θ ) n , [ C 1 , θ ] n with the spaces F and [ F ] .

Theorem 3.1 The following statements are true:
  1. (1)

    [ F ] F ( R ˜ , p r , θ ) n [ R ˜ , p r , θ ] n [ C 1 , θ ] n .

     
  2. (2)

    [ F ] | R ˜ , p r , θ | n ( R ˜ , p r , θ ) n [ R ˜ , p r , θ ] n [ C 1 , θ ] n .

     
  3. (3)

    [ F ] | R ˜ , p r , θ | n | C 1 , θ | ( C 1 , θ ) n [ C 1 , θ ] n .

     
Proof We give the proof only for (2). The proofs of (1) and (3) can be done, similarly. So we omit them. Let x [ F ] and [ F ] lim x = ξ . Then t k m ( x ξ e , z 1 , , z n 1 ) 0 as k , uniformly in m, for every nonzero z 1 , , z n 1 X . Since H r as r , then its weighted lacunary mean also converges to ξ as r uniformly in m. This proves that x | R ˜ , p r , θ | n and [ F ] lim x = | R ˜ , p r , θ | n lim x = ξ . Also since
1 H r k I r p k t k m ( x ξ e ) , z 1 , , z n 1 1 H r k I r p k t k m ( x ξ e ) , z 1 , , z n 1 1 H r k I r p k t k m ( x ξ e , z 1 , , z n 1 ) ,

then it follows that [ F ] | R ˜ , p r , θ | n ( R ˜ , p r , θ ) n [ R ˜ , p r , θ ] n and [ F ] lim x = | R ˜ , p r , θ | n lim x = ( R ˜ , p r , θ ) n lim x = [ R ˜ , p r , θ ] n lim x = ξ . Since uniform convergence of 1 H r k I r p k t k m ( x ξ e ) , z 1 , , z n 1 with respect to m, as r , implies convergence for m = 0 and for every nonzero z 1 , , z n 1 X . It follows that [ R ˜ , p r , θ ] n [ C 1 , θ ] n and [ R ˜ , p r , θ ] n lim x = [ C 1 , θ ] n lim x = ξ . This completes the proof. □

Theorem 3.2 Let θ = ( k r ) be a lacunary sequence and lim inf r Q r > 1 . Then ( R ˜ , p r ) n ( R ˜ , p r , θ ) n with ( R ˜ , p r ) n lim x = ( R ˜ , p r , θ ) n lim x = ξ .

Proof Suppose that lim inf r Q r > 1 , then there exists a δ > 0 such that Q r 1 + δ for sufficiently large values of r, which implies that H r P k r δ 1 + δ . If x ( R ˜ , p r ) n with ( R ˜ , p r ) n lim x = ξ , then for sufficiently large values of r, we have
1 P k r k = 1 k r p k t k m ( x ξ e ) , z 1 , , z n 1 = 1 P k r ( k = 1 k r 1 p k t k m ( x ξ e ) , z 1 , , z n 1 + k = k r 1 + 1 k r p k t k m ( x ξ e ) , z 1 , , z n 1 ) H r P k r ( 1 H r k I r p k t k m ( x ξ e ) , z 1 , , z n 1 ) δ 1 + δ . 1 H r k I r p k t k m ( x ξ e ) , z 1 , , z n 1 ,

for each m 0 and for every nonzero z 1 , , z n 1 X . Then, it follows that x ( R ˜ , p r , θ ) n with ( R ˜ , p r , θ ) n lim x = ξ by taking the limit as r . This completes the proof. □

Theorem 3.3 Let θ = ( k r ) be a lacunary sequence with lim sup r Q r < . Then ( R ˜ , p r , θ ) n ( R ˜ , p r ) n with ( R ˜ , p r , θ ) n lim x = ( R ˜ , p r ) n lim x = ξ .

Proof Let x ( R ˜ , p r , θ ) n with ( R ˜ , p r , θ ) n lim x = ξ . Then for ε > 0 , there exists q 0 such that for every q > q 0
L q = 1 H q k I q p k t k m ( x ξ e ) , z 1 , , z n 1 < ε ,
(3.1)
for each m 0 and for every nonzero z 1 , , z n 1 X , that is, we can find some positive constant M such that
L q M for all  q .
(3.2)
lim sup r Q r < implies that there exists some positive number K such that
Q r K for all  r 1 .
(3.3)
Therefore for k r 1 < r k r , we have by (3.1), (3.2), and (3.3)
1 P r k = 1 r p k t k m ( x ξ e ) , z 1 , , z n 1 1 P k r 1 k = 1 k r p k t k m ( x ξ e ) , z 1 , , z n 1 = 1 P k r 1 ( k I 1 p k t k m ( x ξ e ) , z 1 , , z n 1 + k I 2 p k t k m ( x ξ e ) , z 1 , , z n 1 + + k I q 0 p k t k m ( x ξ e ) , z 1 , , z n 1 + + k I r p k t k m ( x ξ e ) , z 1 , , z n 1 ) = 1 P k r 1 ( L 1 H 1 + L 2 H 2 + + L q 0 H q 0 + L q 0 + 1 H q 0 + 1 + + L r H r ) M P k r 1 ( H 1 + H 2 + + H q 0 ) + ε P k r 1 ( H q 0 + 1 + + H r ) = M P k r 1 ( P k 1 P k 0 + + P k q 0 P k q 0 1 ) + ε P k r 1 ( P k q 0 P k q 0 1 + + P k r P k r 1 ) = M P k q 0 P k r 1 + ε P k r P k q 0 P k r 1 M P k q 0 P k r 1 + ε K ,

for each m 0 and for every nonzero z 1 , , z n 1 X . Since P k r 1 as r , we get x ( R ˜ , p r ) n with ( R ˜ , p r ) n lim x = ξ . This completes the proof. □

Corollary 3.4 Let 1 < lim inf r Q r lim sup r Q r < . Then ( R ˜ , p r , θ ) n = ( R ˜ , p r ) n and ( R ˜ , p r , θ ) n lim x = ( R ˜ , p r ) n lim x = ξ .

Proof It follows from Theorem 3.2 and Theorem 3.3. □

In the following theorem, we give the relations between the sequence spaces ( w θ ) n and ( R ˜ , p r ) n .

Theorem 3.5
  1. (1)

    If p k < 1 for all k N , then ( w θ ) n ( R ˜ , p r ) n and ( w θ ) n lim x = ( R ˜ , p r ) n lim x = ξ .

     
  2. (2)

    If p k > 1 for all k N and ( H r h r ) is upper-bounded, then ( R ˜ , p r ) n ( w θ ) n and ( R ˜ , p r ) n lim x = ( w θ ) n lim x = ξ .

     
Proof
  1. (1)
    If p k < 1 for all k N , then H r < h r for all r N . So, there exists an M 1 , a constant, such that 0 < M 1 H r h r < 1 for all r N . Let x ( w θ ) n with ( w θ ) n lim x = ξ , then for an arbitrary ε > 0 we have
    1 H r k I r p k t k m ( x ξ e ) , z 1 , , z n 1 1 M 1 1 h r k I r t k m ( x ξ e ) , z 1 , , z n 1 ,

    for each m 0 and for every nonzero z 1 , , z n 1 X . Therefore, we get the result by taking the limit as r .

     
  2. (2)
    Let p k > 1 for all k N , then H r > h r for all r N . Suppose that ( H r h r ) is upper-bounded, then there exists an M 2 , a constant, such that 1 < H r h r M 2 < for all r N . Let x ( R ˜ , p r ) n and ( R ˜ , p r ) n lim x = ξ . So the result is obtained by taking the limit as r for each m 0 and for every nonzero z 1 , , z n 1 X , from the following inequality:
    1 h r k I r t k m ( x ξ e ) , z 1 , , z n 1 M 2 1 H r k I r p k t k m ( x ξ e ) , z 1 , , z n 1 .
     

 □

Now, we define a new concept of statistical convergence in n-normed space, which will be called weighted almost lacunary statistical convergence:

Definition 3.6 The weighted almost lacunary density of K N is denoted by δ ( R ˜ , θ ) ( K ) = lim r 1 H r | K r ( ε ) | if the limit exists. We say that the sequence x = ( x j ) is weighted almost lacunary statistically convergent to ξ if for every ε > 0 , the set K r ( ε ) = { k I r : p k t k m ( x ξ e ) , z 1 , , z n 1 ε } has weighted lacunary density zero, i.e.
lim r 1 H r | { k I r : p k t k m ( x ξ e ) , z 1 , , z n 1 ε } | = 0
(3.4)
uniformly in m, for every nonzero z 1 , , z n 1 X . In this case, we write ( S ( R ˜ , θ ) , n ) lim k x k = ξ . By ( S ( R ˜ , θ ) , n ) we denote the set of all weighted almost lacunary statistically convergent sequences in n-normed space.
  1. (1)
    If we take p k = 1 for all k N in (3.4) then we obtain the definition of almost lacunary statistical convergence in n−normed space, that is, x is called almost lacunary statistically convergent to ξ if for every ε > 0 , the set K θ ( ε ) = { k I r : t k m ( x ξ e ) , z 1 , , z n 1 ε } has lacunary density zero, i.e.
    lim r 1 h r | { k I r : t k m ( x ξ e ) , z 1 , , z n 1 ε } | = 0
    (3.5)

    uniformly in m, for every nonzero z 1 , , z n 1 X . In this case, we write ( S θ , n ) lim j x j = ξ . By ( S θ , n ) we denote the set of all weighted almost lacunary statistically convergent sequences in n-normed space.

     
  2. (2)
    Let us choose θ = ( k r ) for r > 0 then the definition of weighted almost lacunary statistical convergence which is given in (3.4) is reduced to the definition of weighted almost statistically convergence, that is, x is called weighted almost statistically convergent to ξ if for every ε > 0 , the set K P r ( ε ) = { k P r : p k t k m ( x ξ e ) , z 1 , , z n 1 ε } has weighted density zero, i.e.
    lim r 1 P r | { k P r : t k m ( x ξ e ) , z 1 , , z n 1 ε } | = 0
    (3.6)

    uniformly in m, for every nonzero z 1 , , z n 1 X . In this case, we write ( S R ˜ , n ) lim j x j = ξ . By ( S R ˜ , n ) we denote the set of all weighted almost lacunary statistically convergent sequences in n-normed space.

     
  3. (3)

    Let us choose θ = ( k r ) for r > 0 and p k = 1 for all k N , then the definition of weighted almost lacunary statistical convergence, which is given in (3.4), is reduced to the definition of almost statistical convergence.

     

Theorem 3.7 If the sequence x is ( R ˜ , p r , θ ) n -convergent to ξ then the sequence x is weighted almost lacunary statistically convergent to ξ.

Proof Let the sequence x be ( R ˜ , p r , θ ) n -convergent to ξ and K r m ( ε ) = { k I r : p k t k m ( x ξ e ) , z 1 , , z n 1 ε } . Then for a given ε > 0 , we have
1 H r k I r p k t k m ( x ξ e ) , z 1 , , z n 1 1 H r k I r k K r m ( ε ) p k t k m ( x ξ e ) , z 1 , , z n 1 ε 1 H r | K r m ( ε ) |

for each m 0 and for every nonzero z X . Hence, we see that the sequence x is weighted almost statistically convergent to ξ by taking the limit as r . □

Theorem 3.8 Let p k t k m ( x ξ e ) , z 1 , , z n 1 M for all k N , for each m 0 and for every nonzero z 1 , , z n 1 X . Then ( S ( R ˜ , θ ) , n ) ( R ˜ , p r , θ ) n with ( S ( R ˜ , θ ) , n ) lim x = ( R ˜ , p r , θ ) n lim x = ξ .

Proof Let x be convergent to ξ in ( S ( R ˜ , θ ) , n ) and let us take
K r m ( ε ) = { k I r : p k t k m ( x ξ e ) , z 1 , , z n 1 ε } .
Since p k t k m ( x ξ e ) , z 1 , , z n 1 M for all k N for each m 0 , for every nonzero z 1 , , z n 1 X and H r as r , then for a given ε > 0 we have
1 H r k I r p k t k m ( x ξ e ) , z 1 , , z n 1 = 1 H r k I r k K r m ( ε ) p k t k m ( x ξ e ) , z 1 , , z n 1 + 1 H r k I r k K r m ( ε ) p k t k m ( x ξ e ) , z 1 , , z n 1 M 1 H r | K r m ( ε ) | + h r H r ε M 1 H r | K r m ( ε ) | + ε ,

for each m 0 and for every nonzero z 1 , , z n 1 X . Since ε is arbitrary, we have x ( R ˜ , p r , θ ) n by taking the limit as r . □

Theorem 3.9 The following statements are true.
  1. (1)

    If p k 1 for all k N , then ( S θ , n ) ( S ( R ˜ , θ ) , n ) .

     
  2. (2)

    Let p k 1 for all k N and ( H r h r ) be upper-bounded, then ( S ( R ˜ , θ ) , n ) ( S θ , n ) .

     
Proof
  1. (1)
    If p k 1 for all k N , then H r h r for all r N . So, there exist M 1 and M 2 , constants, such that 0 < M 1 H r h r M 2 1 for all r N . Let x ( S θ , n ) with ( S θ , n ) lim x = ξ , then for an arbitrary ε > 0 we have
    1 H r | { k I r : p k t k m ( x ξ e ) , z 1 , , z n 1 ε } | = 1 H r | { P k r 1 < k P k r : p k t k m ( x ξ e ) , z 1 , , z n 1 ε } | 1 M 1 1 h r | { P k r 1 k r 1 < k P k r k r : t k m ( x ξ e ) , z 1 , , z n 1 ε } | = 1 M 1 1 h r | { k r 1 < k k r : t k m ( x ξ e ) , z 1 , , z n 1 ε } | = 1 M 1 1 h r | { k I r : t k m ( x ξ e ) , z 1 , , z n 1 ε } | ,
     
for each m 0 and for every nonzero z 1 , , z n 1 X . Hence, we obtain the result by taking the limit as r .
  1. (2)
    Let ( H r h r ) be upper-bounded, then there exist M 1 and M 2 , constants, such that 1 M 1 H r h r M 2 < for all r N . Suppose that p k 1 for all k N , then H r h r for all r N . Let x ( R ˜ , p r ) n and ( R ˜ , p r ) n lim x = ξ , then for an arbitrary ε > 0 we have
    1 h r | { k I r : t k m ( x ξ e ) , z 1 , , z n 1 ε } | = 1 h r | { k r 1 < k k r : t k m ( x ξ e ) , z 1 , , z n 1 ε } | M 2 1 H r | { k r 1 P k r 1 < k k r P k r : p k t k m ( x ξ e ) , z 1 , , z n 1 ε } | = M 2 1 H r | { P k r 1 < k P k r : p k t k m ( x ξ e ) , z 1 , , z n 1 ε } | = M 2 1 H r | { k I r : p k t k m ( x ξ e ) , z 1 , , z n 1 ε } | ,
     

for each m 0 and for every nonzero z 1 , , z n 1 X . Hence, the result is obtained by taking the limit as r . □

Theorem 3.10 For any lacunary sequence θ, if lim inf r Q r > 1 then ( S R ˜ , n ) ( S ( R ˜ , θ ) , n ) and ( S R ˜ , n ) lim x = ( S ( R ˜ , θ ) , n ) lim x = ξ .

Proof Suppose that lim inf r Q r > 1 , then there exists a δ > 0 such that Q r 1 + δ for sufficiently large values of r, which implies that H r P k r δ 1 + δ . If x ( S R ˜ , n ) with ( S R ˜ , n ) lim x = ξ , then for every ε > 0 and for sufficiently large values of r, we have
1 P k r | { k P k r : p k t k m ( x ξ e ) , z 1 , , z n 1 ε } | 1 P k r | { P k r 1 < k P k r : p k t k m ( x ξ e ) , z 1 , , z n 1 ε } | = H r P k r ( 1 H r | { P k r 1 < k P k r : p k t k m ( x ξ e ) , z 1 , , z n 1 ε } | ) δ 1 + δ ( 1 H r | { k I r : p k t k m ( x ξ e ) , z 1 , , z n 1 ε } | ) ,

for each m 0 and for every nonzero z 1 , , z n 1 X . Hence, we get the result by taking the limit as r . □

Theorem 3.11 Let θ = ( k r ) be a lacunary sequence with lim sup r Q r < , then ( S ( R ˜ , θ ) , n ) ( S R ˜ , n ) and ( S R ˜ , n ) lim x = ( S ( R ˜ , θ ) , n ) lim x = ξ .

Proof If lim sup r Q r < , then there is a K > 0 such that Q r K for all r N . Suppose that x ( S ( R ˜ , θ ) , n ) with ( S ( R ˜ , θ ) , n ) lim x = ξ and let
N r : = | { k I r : p k t k m ( x ξ e ) , z 1 , , z n 1 ε } | .
(3.7)
By (3.7), given ε > 0 , there is a r 0 N such that N r H r < ε for all r > r 0 . Now, let M : = max { N r : 1 r r 0 } and let r be any integer satisfying k r 1 < r k r , then for each m 0 and for every nonzero z 1 , , z n 1 X we can write
1 P r | { k P r : p k t k m ( x ξ e ) , z 1 , , z n 1 ε } | 1 P k r 1 | { P k r 1 < k P k r : p k t k m ( x ξ e ) , z 1 , , z n 1 ε } | = 1 P k r 1 ( N 1 + N 2 + + N r 0 + N r 0 + 1 + + N r ) M . r 0 P k r 1 + 1 P k r 1 ε ( H r 0 + 1 + + H r ) = M . r 0 P k r 1 + ε ( P k r P k r 0 ) P k r 1 M . r 0 P k r 1 + ε Q r M . r 0 P k r 1 + ε K ,

which completes the proof by taking the limit as r . □

Corollary 3.12 Let 1 < lim inf r Q r lim sup r Q r < . Then ( S ( R ˜ , θ ) , n ) = ( S R ˜ , n ) and ( S R ˜ , n ) lim x = ( S ( R ˜ , θ ) , n ) lim x = ξ .

Proof It follows from Theorem 3.10 and Theorem 3.11. □

Declarations

Acknowledgements

This paper has been presented in 2nd International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2013) and it was supported by the Research Foundation of Sakarya University (Project Number: 2012-50-02-032).

Authors’ Affiliations

(1)
Department of Mathematics, Sakarya University

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© Konca and Başarır; licensee Springer. 2014

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