IJANAL International Journal of Analysis 2314-4998 2314-498X Hindawi Publishing Corporation 10.1155/2014/945902 945902 Research Article ( σ , f ) -Asymptotically Lacunary Equivalent Sequences Bilgin Tunay Qamar Shamsul Department of Mathematics, Faculty of Education Yüzüncü Yıl University 65080 Van Turkey yyu.edu.tr 2014 3122014 2014 02 07 2014 22 09 2014 3 12 2014 2014 Copyright © 2014 Tunay Bilgin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We introduce the strong ( σ , f ) -asymptotically equivalent and strong ( σ , f ) -asymptotically lacunary equivalent sequences which are some combinations of the definitions for asymptotically equivalent, statistical limit, modulus function, σ -convergence, and lacunary sequences. Then we use these definitions to prove strong ( σ , f ) -asymptotically equivalent and strong ( σ , f ) -asymptotically lacunary equivalent analogues of Connor’s results in Connor, 1988, Fridy and Orhan’s results in Fridy and Orhan, 1993, and Das and Patel’s results in Das and Patel, 1989.

1. Introduction

Let s , l , and c denote the spaces of all real sequences, bounded and convergent sequences, respectively. Any subspace of s is called a sequence space.

Let σ be a mapping of the set of positive integers into itself. A continuous linear functional ϕ on l , the space of real bounded sequences, is said to be an invariant mean or σ -mean if and only if

ϕ ( x ) 0 when the sequence x = ( x k ) has x k 0 for all k ;

ϕ ( e ) = 1 , where e = ( 1,1 , 1 , ) ;

ϕ ( x ) = ϕ ( σ x ) for all x l , where σ x = ( x σ ( n ) ) . The mapping σ is one to one with σ k ( n ) n for all positive integers n and k , where σ k denotes the k th iterate of the mapping σ at n . Thus ϕ extends the limit functional on c in the sense that ϕ ( x ) = lim x for all x c . If x = ( x n ) , write T x = T x n = ( x σ ( n ) ) .

Several authors including Bilgin , Mursaleen , Savas , Schaefer , and others have studied invariant convergent sequences.

The notion of modulus function was introduced by Nakano . We recall that a modulus 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 x , y 0 , (iii) f is increasing, and (iv) f is continuous from the right at 0. Hence f must be continuous everywhere on [ 0 , ) . Kolk , Maddox , Öztürk and Bilgin , Pehlivan and Fisher , Ruckle , and others used a modulus function to construct sequence spaces.

Following Freedman et al. , we call the sequence θ = ( k r ) lacunary if it is an increasing sequence of integers such that k 0 = 0 , h r = k r - k r - 1 as r . The intervals determined by θ will be denoted by I r = ( k r - 1 , k r ] and q r = k r / k r - 1 . These notations will be used throughout the paper. The sequence space of lacunary strongly convergent sequences N θ was defined by Freedman et al.  as follows: (1) N θ = x = x i s : h r - 1 i I r x i - s = 0    for    some    s .

Lacunary convergent sequences have been studied by Bilgin , Das and Mishra , Das and Patel , Savas and Patterson , and others. Marouf presented definitions for asymptotically equivalent sequences and asymptotic regular matrices in . Patterson extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices in .

Recently, the concept of asymptotically equivalent was generalized by Bilgin , Kumar and Sharma , Patterson and Savas , Savas and Basarr , and Patterson and Savas . In this paper we introduce the strong ( σ , f ) -asymptotically equivalent and strong ( σ , f ) -asymptotically lacunary equivalent sequences which are some combinations of the definitions for asymptotically equivalent, statistical limit, modulus function, σ -convergence, and lacunary sequences.

In addition to these definitions, natural inclusion theorems will also be presented.

2. Definitions and Notations

Now we recall some definitions of sequence spaces (see [7, 12, 1517, 2125]).

Definition 1.

A sequence [ x ] is statistically convergent to L if (2) lim n 1 n the    number    of    k n : x k - L ɛ = 0 for    every    ɛ > 0 , denoted    by    s t - lim x = L .

Definition 2.

A sequence [ x ] is strongly (Cesàro) summable to L if (3) lim n 1 n k = 1 n x k - L = 0 , denoted    by    w - lim x = L .

Definition 3.

Let f be any modulus; the sequence [ x ] is strongly (Cesàro) summable to L with respect to a modulus if (4) lim n 1 n k = 1 n f x k - L = 0 , ( denoted    by    w f - lim x = L ) .

Definition 4.

Two nonnegative sequences [ x ] and [ y ] are said to be asymptotically equivalent if lim k ( x k / y k ) = 1 , (denoted by x y ).

Definition 5.

Two nonnegative sequences [ x ] and [ y ] are said to be asymptotically statistical equivalent of multiple L provided that for every ɛ > 0 (5) lim n 1 n the    number    of    k n : x k y k - L ɛ = 0 , hhhhhhhhhhhhhhhhhhhh denoted    by    x S y and simply asymptotically statistical equivalent, if L = 1 .

Definition 6.

Two nonnegative sequences [ x ] and [ y ] are said to be strong asymptotically equivalent of multiple L provided that (6) lim n 1 n k = 1 n x k y k - L = 0 ( denoted    by    x w y ) and simply strong asymptotically equivalent, if L = 1 .

Definition 7.

Let θ be a lacunary sequence; the two nonnegative sequences [ x ] and [ y ] are said to be asymptotically lacunary statistical equivalent of multiple L provided that for every ɛ > 0 (7) lim r 1 h r the    number    of    k I r : x k y k - L ɛ = 0 , hhhhhhhhhhhhhhhhhhhh denoted    by    x S θ y and simply asymptotically lacunary statistical equivalent, if L = 1 .

Definition 8.

Let θ be a lacunary sequence; the two nonnegative sequences [ x ] and [ y ] are said to be strong asymptotically lacunary equivalent of multiple L provided that (8) lim r 1 h r k I r x k y k - L = 0 denoted    by    x N θ y and simply strong asymptotically lacunary equivalent, if L = 1 .

Definition 9.

Let f be any modulus; the two nonnegative sequences [ x ] and [ y ] are said to be f -asymptotically equivalent of multiple L provided that (9) lim k f x k y k - L = 0 denoted    by    x f y and simply strong f -asymptotically equivalent, if L = 1 .

Definition 10.

Let f be any modulus; the two nonnegative sequences [ x ] and [ y ] are said to be strong f -asymptotically equivalent of multiple L provided that (10) lim n 1 n k = 1 n f x k y k - L = 0 denoted    by    x w f y and simply strong f -asymptotically equivalent, if L = 1 .

Definition 11.

Two nonnegative sequences [ x ] and [ y ] are said to be S σ -asymptotically statistical equivalent of multiple L provided that for every ɛ > 0 (11) lim n 1 n the    number    of    k n : x σ k ( i ) y σ k ( i ) - L ɛ = 0 , hh uniformly    in    i = 1,2 , 3 , , ( denoted    by    x S σ y ) and simply σ -asymptotically statistical equivalent, if L = 1 .

Definition 12.

Let θ be a lacunary sequence; two nonnegative sequences [ x ] and [ y ] are said to be S σ , θ -asymptotically lacunary statistical equivalent of multiple L provided that for every ɛ > 0 (12) lim r 1 h r the    number    of    k I r : x σ k ( i ) y σ k ( i ) - L ɛ = 0 , hh uniformly    in    i = 1,2 , 3 , , ( denoted    by    x S σ , θ y ) and simply S σ , θ -asymptotically lacunary statistical equivalent, if L = 1 .

Definition 13.

Let θ be a lacunary sequence; two nonnegative sequences [ x ] and [ y ] are said to be strong σ -asymptotically lacunary equivalent of multiple L provided that (13) lim r 1 h r k I r x σ k ( i ) y σ k ( i ) - L = 0 , uniformly    in    i = 1,2 , 3 , , denoted    by    x N σ , θ y and simply σ -asymptotically lacunary statistical equivalent, if L = 1 .

Definition 14.

Let f be any modulus and let θ be a lacunary sequence; two nonnegative sequences [ x ] and [ y ] are said to be strong f -asymptotically lacunary equivalent of multiple L provided that (14) lim r 1 h r k I r f x k y k - L = 0 , denoted    by    x N θ , f y and simply f -asymptotically lacunary equivalent, if L = 1 .

Definition 15.

Let f be any modulus and let θ be a lacunary sequence; two nonnegative sequences [ x ] and [ y ] are said to be strong ( σ , f ) -asymptotically equivalent of multiple L provided that (15) lim n 1 n k = 1 n f x σ k ( i ) y σ k ( i ) - L = 0 , uniformly    in    i = 1,2 , 3 , , denoted    by    x w σ , f y and simply ( σ , f ) -asymptotically equivalent, if L = 1 .

For ( i ) = i + 1 we write [ w f ] for w σ , f . Hence the two nonnegative sequences [ x ] and y are said to be strong almost f -asymptotically equivalent of multiple L provided that (16) lim n 1 n k = 1 n f x k + i y k + i - L = 0 , uniformly    in    i = 1,2 , 3 , , denoted    by    x [ w f ] y and simply f -asymptotically equivalent, if L = 1 .

For f ( x ) = x for all x [ 0 , ) we write [ w ] for [ w f ] . Hence the two nonnegative sequences [ x ] and [ y ] are said to be strong almost asymptotically equivalent of multiple L provided that (17) lim n 1 n k = 1 n x k + i y k + i - L = 0 uniformly    in    i = 1,2 , 3 , , denoted    by    x [ w f ] y and simply strong almost asymptotically equivalent, if L = 1 .

Definition 16.

Let f be any modulus and let θ be a lacunary sequence; two nonnegative sequences [ x ] and [ y ] are said to be strong ( σ , f ) -asymptotically lacunary equivalent of multiple L provided that (18) lim r 1 h r k I r f x σ k ( i ) y σ k ( i ) - L = 0 , uniformly    in    i = 1,2 , 3 , , denoted    by    x N σ , θ , f y and simply ( σ , f ) -asymptotically lacunary equivalent, if L = 1 .

For ( i ) = i + 1 we write M θ , f for N σ , θ , f . Hence the two nonnegative sequences [ x ] and [ y ] are said to be strong almost f -asymptotically lacunary equivalent of multiple L provided that (19) lim r 1 h r k I r f x k + i y k + i - L = 0 , uniformly    in    i = 1,2 , 3 , , denoted    by    x M θ , f y and simply strong almost f -asymptotically lacunary equivalent, if L = 1 .

For f ( x ) = x for all x [ 0 , ) we write [ w θ ] for w θ , f . Hence the two nonnegative sequences [ x ] and [ y ] are said to be strong almost asymptotically lacunary equivalent of multiple L provided that (20) lim r 1 h r k I r x k + i y k + i - L = 0 , uniformly    in    i = 1,2 , 3 , , denoted    by    x M θ y and simply strong almost asymptotically lacunary equivalent, if L = 1 .

3. Main Theorems

We now give lemma to be used later.

Lemma 17.

Let f be any modulus. Suppose for given ɛ > 0 , there exist n o and i o such that (21) 1 n k = 0 n - 1 f x σ k ( i ) y σ k ( i ) - L < ɛ n n o , i i o . Then x w σ , f y .

Proof.

Let ɛ > 0 be given. Chose n 1 and i o such that (22) 1 n k = 0 n - 1 f x σ k ( i ) y σ k ( i ) - L < ɛ 2 n n 1 , i i o . It is sufficient to prove that there exists n 2 such that for n n 2 , i o i 0 (23) 1 n k = 0 n - 1 f x σ k ( i ) y σ k ( i ) - L < ɛ since, taking n o = max ( n 1 , n 2 ) , (23) holds for n > n o and for all i , which gives the result. Once i o has been chosen, i o is fixed, so (24) k = 0 i o f x σ k ( i ) y σ k ( i ) - L = R say . Now, taking i o i 0 and n > i o , we have (25) 1 n k = 0 n - 1 f x σ k ( i ) y σ k ( i ) - L = 1 n k = 0 i o - 1 k = i o n - 1 f x σ k ( i ) y σ k ( i ) - L R n + ɛ 2 . Taking n sufficiently large, we can make R / n + ɛ / 2 < ɛ which gives (23) and hence the result.

The next theorems show the relationship between the strong ( σ , f ) -asymptotically equivalence and the strong ( σ , f ) -asymptotically lacunary equivalence.

Theorem 18.

Let f be any modulus. Then x N σ , θ , f y x w σ , f y for every lacunary sequence θ .

Proof.

Let x N σ , θ , f y . Then, given ɛ > 0 , there exist r o and L such that (26) 1 h r k I r f x σ k ( i ) y σ k ( i ) - L < ɛ for r r o and i = k r - 1 + 1 + v , v 0 . Let n h r ; write n = m h r + p , where 0 p h r ; m is an integer. Since n h r , m 0 . We have (27) 1 n k = 0 n - 1 f x σ k ( i ) y σ k ( i ) - L 1 n k = 0 ( m + 1 ) h r - 1 f x σ k ( i ) y σ k ( i ) - L = 1 n j = 0 m k = j h r ( j + 1 ) h r - 1 f x σ k i y σ k i - L m + 1 h r ɛ n 2 m h r ɛ n for    m 0 . For h r / n 1 , since m h r / n 1 , therefore, (28) 1 n k = 0 n - 1 f x σ k ( i ) y σ k ( i ) - L 2 ɛ .

Then by lemma x N σ , θ , f y implies x w σ , f y . It is easy to see that x w σ , f y implies x N σ , θ , f y for every θ .

Proposition 19.

Let f be any modulus. Then x M θ , f y x [ w f ] y for every lacunary sequence θ .

Proof.

It follows from Theorem 18 for σ ( i ) = i + 1 for all i = 1,2 , 3 , .

Proposition 20.

x M θ y x [ w ] y for every lacunary sequence θ .

Proof.

It follows from Proposition 19 for f ( x ) = x for all x [ 0 , ) .

Theorem 21.

Let f be any modulus. Then

if liminf r q r > 1 , then x w σ , f y implies x N σ , θ , f y ;

if limsup r q r < , then x N σ , θ , f y implies x w σ , f y ;

if 1 < liminf r q r limsup r q r < , then x w σ , f y x N σ , θ , f y .

Proof.

Part (1): let x w σ , f y and liminf r q r > 1 . There exists δ > 0 such that q r = ( k r / k r - 1 ) 1 + δ for sufficiently large r . We have, for sufficiently large r , that ( h r / k r ) δ / ( 1 + δ ) . Then (29) 1 k r - 1 k = 1 k r f x σ k ( i ) y σ k ( i ) - L 1 k r - 1 k I r f x σ k ( i ) y σ k ( i ) - L = h r k r 1 h r k I r f x σ k ( i ) y σ k ( i ) - L δ 1 + δ 1 h r k I r f x σ k ( i ) y σ k ( i ) - L which yields that x N σ , θ , f y .

Part (2): if limsup r q r < , then there exists K > 0 such that q r < K for every r .

Now suppose that x N σ , θ , f y and ɛ > 0 . There exists m 0 such that for every m m 0 , (30) H m = 1 h m k I m f x σ k ( i ) y σ k ( i ) - L < ɛ , i .

We can also find R > 0 such that H m R for all m . Let n be any integer with k r n > k r - 1 where r > m 0 . Now write (31) 1 n k = 0 n - 1 f x σ k i y σ k i - L 1 k r - 1 k = 1 k r f x σ k ( i ) y σ k ( i ) - L = 1 k r - 1 k I 1 f x σ k i y σ k i - L + k I 2 f x σ k i y σ k i - L + + k I r f x σ k i y σ k i - L = 1 k r - 1 k 1 k 1 k I 1 f x σ k i y σ k i - L + k 2 - k 1 k 2 - k 1 × k I 2 f x σ k i y σ k i - L + + k m 0 - k m 0 - 1 k m 0 - k m 0 - 1 k I m 0 f x σ k ( i ) y σ k ( i ) - L + + k r - k r - 1 k r - k r - 1 k I r f x σ k ( i ) y σ k ( i ) - L = k 1 k r - 1 H 1 + k 2 - k 1 k r - 1 H 2 + + k m 0 - k m 0 - 1 k r - 1 H m 0 + + k r - k r - 1 k r - 1 H r k m 0 k r - 1 sup 1 < k < k m 0 H k + sup k > k m 0 H k ( k r - k m 0 ) k r - 1 < R k m 0 k r - 1 + ɛ K from which we deduce that x w σ , f y .

Part (3): this immediately follows from ( 1 ) and ( 2 ) .

Proposition 22.

Let f be any modulus. Then

if liminf r q r > 1 , then x [ w f ] y implies x M θ , f y ;

if limsup r q r < , then x M θ , f y implies x [ w f ] y ;

if 1 < liminf r q r limsup r q r < , then x [ w f ] y x M θ , f y .

Proof.

It follows from Theorem 21 for σ ( i ) = i + 1 for all i = 1,2 , 3 , .

Proposition 23.

Consider the following:

if liminf r q r > 1 , then x [ w ] y implies x M θ y ;

if limsup r q r < , then x M θ y implies x [ w ] y ;

if 1 < liminf r q r limsup r q r < , then x [ w ] y x M θ y .

Proof.

It follows from Proposition 22 for f ( x ) = x for all x [ 0 , ) .

In the following theorem we study the relationship between the strong ( σ , f ) -asymptotically lacunary equivalence and the strong σ -asymptotically lacunary equivalence.

Theorem 24.

Let f be any modulus. Then

if x N σ , θ y , then x N σ , θ , f y ;

if lim t ( f t / t ) = β > 0 , then x N σ , θ y x N σ , θ , f y .

Proof.

Part ( 1 ) : let x N σ , θ y and ɛ > 0 . We choose 0 < δ < 1 such that f ( u ) < ɛ for every u with 0 u δ . We can write (32) 1 h r k I r f x σ k ( i ) y σ k ( i ) - L = 1 h r 1 f x σ k ( i ) y σ k ( i ) - L + 1 h r 2 f x σ k i y σ k i - L , where the first summation is over x σ k ( i ) / y σ k ( i ) - L δ and the second summation over x σ k ( i ) / y σ k ( i ) - L > δ with k I r . By definition of f , we have (33) 1 h r k I r f x σ k ( i ) y σ k ( i ) - L ɛ + 2 f 1 δ - 1 1 h r k I r f x σ k i y σ k i - L .

Therefore, x N σ , θ , f y .

Part ( 2 ) : if lim t ( f ( t ) / t ) = β > 0 , then f ( t ) β t for all t > 0 . Let x N σ , θ , f y ; clearly (34) 1 h r k I r f x σ k ( i ) y σ k ( i ) - L 1 h r k I r β x σ k ( i ) y σ k ( i ) - L = β 1 h r k I r x σ k i y σ k i - L . Therefore, x N σ , θ y . By using (23) the proof is complete.

Proposition 25.

Let f be any modulus. Then

if x M θ y , then x M θ , f y ;

if lim t ( f ( t ) / t ) = β > 0 , then x M θ y x M θ , f y .

Proof.

It follows from Theorem 24 for σ ( i ) = i + 1 for all i = 1,2 , 3 , .

Finally we give the relation between S σ , θ -asymptotically lacunary statistical equivalence and strong ( σ , f ) -asymptotically lacunary equivalence. Also we give relation between S σ , θ -asymptotically lacunary statistical equivalence and strong ( σ , f ) -asymptotically equivalence.

Theorem 26.

Let f be any modulus. Then

if x N σ , θ , f y , then x S σ , θ y ;

if f is bounded and x S σ , θ y , then x N σ , θ , f y ;

if f is bounded, then x N σ , θ , f y x S σ , θ y .

Proof.

Part ( 1 ) : take ɛ > 0 and x N σ , θ , f y . Let 1 denote the sum over k I r and x σ k ( i ) / y σ k ( i ) - L ɛ .

Then (35) 1 h r k I r f x σ k ( i ) y σ k ( i ) - L 1 h r 1 f x σ k ( i ) y σ k ( i ) - L f ɛ the    number    of    k I r : x σ k i y σ k i - L ɛ .

Therefore, x S σ , θ y .

Part ( 2 ) : suppose that f is bounded and x S σ , θ y . Since f is bounded, there exists an integer T such that f ( x ) T for all x 0 . Given ɛ > 0 , (36) 1 h r k I r f x σ k ( i ) y σ k ( i ) - L T 1 h r the    number    of    k I r : x σ k i y σ k i - L ɛ + f ( ɛ ) .

Therefore, x N σ , θ , f y .

Part ( 3 ) : follows from ( 1 ) and ( 2 ) .

For ( i ) = i + 1 we write [ S θ ] for S σ , θ . Hence the two nonnegative sequences [ x ] and [ y ] are said to be [ S θ ] -asymptotically lacunary statistical equivalent of multiple L provided that for every ɛ > 0 (37) lim r 1 h r the    number    of    k I r : x σ k i y σ k i - L ɛ = 0 , uniformly    in    i = 1,2 , 3 , , ( denoted    by    x [ S θ ] y ) and [ S θ ] -asymptotically lacunary statistical equivalent, if L = 1 . Hence we have the following.

Proposition 27.

Let f be any modulus. Then

if x M θ , f y , then x [ S θ ] y ;

if f is bounded and x [ S θ ] y , then x M θ , f y ;

if f is bounded, then x M θ , f y x [ S θ ] y .

Proof.

It follows from Theorem 26 for σ ( i ) = i + 1 for all i = 1,2 , 3 , .

Proposition 28.

x M θ y implies x [ S θ ] y .

Proof.

It follows from Proposition 27 ( 1 ) for f ( x ) = x for all x [ 0 , ) .

Theorem 29.

Let f be bounded; then x S σ , θ y implies x w σ , f y for every lacunary sequence θ .

Proof.

Let n be any integer with n I r ; then (38) 1 n k = 1 n f x σ k ( i ) y σ k ( i ) - L = 1 n p = 1 r - 1 k I p f x σ k ( i ) y σ k ( i ) - L + 1 n k = 1 + k r - 1 n f x σ k i y σ k i - L .

Consider the first term on the right in (38): (39) 1 n p = 1 r - 1 k I p f x σ k i y σ k i - L 1 k r - 1 p = 1 r - 1 k I p f x σ k ( i ) y σ k ( i ) - L = 1 k r - 1 p = 1 r - 1 h p 1 h p k I p f x σ k i y σ k i - L .

Since f is bounded and x S σ , θ y , it follows from Theorem 26 ( 2 ) that (40) 1 h p k I p f x σ k ( i ) y σ k ( i ) - L 0 .

Hence (41) 1 k r - 1 p = 1 r - 1 h p 1 h p k I p f x σ k i y σ k i - L 0 .

Consider the second term on the right in (38); since f is bounded, there exists an integer T such that f ( x ) T for all x 0 . We split the sum for k r - 1 < k n into sums over x σ k ( i ) / y σ k ( i ) - L ɛ and x σ k i / y σ k i - L < ɛ . Therefore, we have for every ɛ > 0 that (42) 1 n k = 1 + k r - 1 n f x σ k ( i ) y σ k ( i ) - L T 1 h r the    number    of    k I r : x σ k i y σ k i - L ɛ + f ( ɛ ) .

Since x S σ , θ y , f is continuous from the right at 0 and ɛ is arbitrary; the expression on left side of (42) tends to zero as r , uniformly in i . Hence (38), (41), and (42) imply that x w σ , f y .

Proposition 30.

Let f be bounded; then x [ S θ ] y implies x [ w f ] y for every lacunary sequence θ .

Proof.

It follows from Theorem 29 for σ ( i ) = i + 1 for all i = 1,2 , 3 , .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work is supported by the Presidency of Scientific Research Projects of Yüzüncü Yıl University (IECMSA-2013). The author is thankful to the anonymous referees for careful reading and useful suggestions.

Bilgin T. Lacunary strong ( A σ ,   p ) -convergence Czechoslovak Mathematical Journal 2005 55 3 691 697 10.1007/s10587-005-0056-3 2-s2.0-25444500669 Mursaleen Some new spaces of lacunary sequences and invariant means Italian Journal of Pure and Applied Mathematics 2002 11 175 181 Savas E. Lacunary strong σ-convergence Indian Journal of Pure and Applied Mathematics 1990 21 4 359 365 Schaefer P. Infinite matrices and invariant means Proceedings of the American Mathematical Society 1972 36 104 110 10.1090/S0002-9939-1972-0306763-0 MR0306763 ZBL0255.40003 Nakano Some new spaces of lacunary sequences and invariant means Italian Journal of Pure and Applied Mathematics 2002 11 175 181 MR1994291 Kolk E. On strong boundedness and summability with respect to a sequence of moduli Tartu Ülikooli Toimetised 1993 960 41 50 MR1231936 Maddox I. J. Sequence spaces defined by a modulus Mathematical Proceedings of the Cambridge Philosophical Society 1986 100 1 161 166 10.1017/S0305004100065968 MR838663 ZBL0631.46010 Öztürk E. Bilgin T. Strongly summable sequence spaces defined by a modulus Indian Journal of Pure and Applied Mathematics 1994 25 6 621 625 MR1285224 Pehlivan S. Fisher B. On some sequence spaces Indian Journal of Pure and Applied Mathematics 1994 25 10 1067 1071 MR1301201 ZBL0829.46002 Ruckle W. H. FK spaces in which the sequence of coordinate vectors is bounded Canadian Journal of Mathematics 1973 25 973 978 MR0338731 Freedman A. R. Sember J. J. Raphael M. Some Cesàro-type summability spaces Proceedings of the London Mathematical Society 1978 37 3 508 520 10.1112/plms/s3-37.3.508 MR512023 Bilgin T. f -asymptotically lacunary equivalent sequences Acta Universitatis Apulensis: Mathematics—Informatics 2011 28 271 278 MR2933999 Das G. Mishra S. K. Sublinear functional and a class of conservative matrices Journal of Orissa Mathematical Society 1989 20 64 67 Das G. Patel B. K. Lacunary distribution of sequences Indian Journal of Pure and Applied Mathematics 1989 20 1 64 74 MR977401 ZBL0726.40002 Savas E. Patterson R. F. σ -asymptotically lacunary statistical equivalent sequences Central European Journal of Mathematics 2006 4 4 648 655 10.2478/s11533-006-0031-8 MR2257413 2-s2.0-33749246523 Marouf M. S. Asymptotic equivalence and summability International Journal of Mathematics and Mathematical Sciences 1993 16 4 755 762 10.1155/S0161171293000948 MR1234822 ZBL0788.40001 Patterson R. F. On asymptotically statistically equivalent sequences Demonstratio Mathematica 2003 36 1 149 153 MR1968498 Kumar V. Sharma A. On asymptotically λ-statistically equivalent sequences Pure and Applied Mathematics Letters 2013 1 7 10 Savaş E. Patterson R. F. On asymptotically lacunary statistical equivalent sequences Thai Journal of Mathematics 2006 4 2 267 272 MR2407260 Savas R. Başarir M. ( σ , λ ) asymptotically statistically equivalent sequences Flomat 2006 20 1 35 42 Connor J. S. The statistical and strong p -Cesàro convergence of sequences Analysis 1988 8 1-2 47 63 MR954458 Fast H. Sur la convergence statistique College Mathematics Journal 1951 2 241 244 MR0048548 Fridy J. A. On statistical convergence Analysis 1985 5 4 301 313 10.1524/anly.1985.5.4.301 MR816582 Fridy J. A. Orhan C. Lacunary statistical convergence Pacific Journal of Mathematics 1993 160 1 43 51 10.2140/pjm.1993.160.43 MR1227502 ZBL0794.60012 Fridy J. A. Orhan C. Lacunary statistical summability Journal of Mathematical Analysis and Applications 1993 173 2 497 504 10.1006/jmaa.1993.1082 MR1209334 ZBL0786.40004 2-s2.0-0000986307