Open Access

Two new sequence spaces generated by the composition of m th order generalized difference matrix and lambda matrix

Journal of Inequalities and Applications20142014:274

https://doi.org/10.1186/1029-242X-2014-274

Received: 10 January 2014

Accepted: 4 July 2014

Published: 23 July 2014

Abstract

In this work, we introduce two sequence spaces c 0 λ ( G m ) and c λ ( G m ) generated by the composition of m th order generalized difference matrix and lambda matrix and define an isomorphism between new sequence spaces and classical sequence spaces. Afterward, we investigate inclusion relations and obtain the Schauder basis of those spaces. Furthermore, we determine their α-, β- and γ-duals. Lastly, we characterize some matrix classes related to those spaces.

MSC:40C05, 40H05, 46B45.

Keywords

matrix transformationsmatrix domainSchauder basisα-, β- and γ-dualsmatrix classes

1 Introduction

The family of all real (or complex) valued sequences is denoted by w. w is a vector space under point-wise addition and scalar multiplication. Any vector subspace of w is called a sequence space. In the literature, the classical sequence spaces are symbolized with , c 0 , c and p which are called all bounded, null, convergent and absolutely p-summable sequence spaces, respectively, where 1 p < .

A sequence space X with a linear topology is called a K-space provided each of the maps p i : X C defined by p i ( x ) = x i is continuous for all i N . It is assumed that w is always endowed with its locally convex topology generated by the sequence { p n } n = 0 of semi-norms on w, where p n ( x ) = | x n | , n = 0 , 1 , 2 ,  . A K-space X is called an FK-space provided X is a complete linear metric space. An FK-space whose topology is normable is called a BK-space [1].

The classical sequence spaces , c 0 and c are BK-spaces with their usual sup - norm defined by x = sup k N | x k | and p is a BK-space with its p - norm defined by
x p = ( k = 0 | x k | p ) 1 p ,

where p [ 1 , ) [2].

Given an infinite matrix A = ( a n k ) with a n k C , for all n , k N and a sequence x w , the A-transform of x is defined by
( A x ) n = k = 0 a n k x k
(1.1)

and is assumed to be convergent for all n N [3]. For using simple notations here and in what follows, the summation without limits runs from 0 to ∞. If x X implies that A x Y , then we say that A defines a matrix mapping from X into Y and denote it by A : X Y . By using the notation ( X : Y ) , we mean the class of all infinite matrices A such that A : X Y .

For an arbitrary sequence space X, X A is called the domain of an infinite matrix A and is defined by
X A = { x = ( x k ) w : A x X } ,
(1.2)
which is also a sequence space. By bs and cs we denote the spaces of all bounded and convergent series, and define them by means of the matrix domain of the summation matrix S = ( s n k ) such that b s = ( ) S and c s = c S , respectively, where S = ( s n k ) is defined by
s n k = { 1 , 0 k n , 0 , k > n ,

which is a triangle matrix too. A matrix A is called a triangle if a n k = 0 for k > n and a n n 0 for all n , k N . Moreover, a triangle matrix A uniquely has an inverse A 1 = B which is also a triangle matrix.

Unless stated otherwise, any term with negative subscript is assumed to be zero. The theory of matrix transformation was prompted by summability theory which was obtained by Cesàro, Riesz and others. The Cesàro mean of order one and the Riesz mean according to the sequence p = ( p n ) are defined by using the matrices C = ( c n k ) and R p = ( r n k p ) such that
c n k = { 1 n + 1 , 0 k n , 0 , k > n , and r n k p = { p k P n , 0 k n , 0 , k > n ,

respectively, where p 0 > 0 , p n 0 ( n 1 ) and P n = k = 0 n p k .

Moreover, the theory of matrix transformation has been continued until nowadays. Many authors have constructed new sequence spaces by using matrix domain of infinite matrices. For example, ( ) N q and c N q in [4], a p r and a r in [5], a 0 r and a c r in [6], Z ( u , v ; p ) in [7], X p and X in [8], e p r and e r in [9], r t , r 0 t and r c t in [10], r p t in [11], e 0 r and e c r in [12], c ˜ and c ˜ 0 in [13]. Also, many authors introduced new sequence spaces by using especially difference matrices. For instance, c 0 ( Δ ) , c ( Δ ) and ( Δ ) in [14], c 0 ( Δ 2 ) , c ( Δ 2 ) and ( Δ 2 ) in [15], a 0 r ( Δ ) and a c r ( Δ ) in [16], c 0 ( Δ m ) , c ( Δ m ) and ( Δ m ) in [17], Δ c 0 ( p ) , Δ c ( p ) and Δ ( p ) in [18], c 0 ( u : Δ 2 ) , c ( u : Δ 2 ) and ( u : Δ 2 ) in [19], c 0 ( u , Δ , p ) , c ( u , Δ , p ) and ( u , Δ , p ) in [20], c 0 ( u , Δ 2 , p ) , c ( u , Δ 2 , p ) and ( u , Δ 2 , p ) in [21], Δ m c 0 ( p ) , Δ m c ( p ) and Δ m ( p ) in [22], r q ( p , B m ) in [23], ˆ , c ˆ , c ˆ 0 and ˆ p in [24], c 0 ( B ) , c ( B ) , ( B ) and p ( B ) in [25], c 0 ( Δ ( m ) ) , c ( Δ ( m ) ) and ( Δ ( m ) ) in [26], Δ u ( m ) X in [27].

In this work, we introduce two sequence spaces c 0 λ ( G m ) and c λ ( G m ) generated by the composition of m th order generalized difference matrix and lambda matrix and define an isomorphism between new sequence spaces and classical sequence spaces. Afterward, we investigate inclusion relations and obtain the Schauder basis of those spaces. Furthermore, we determine their α-, β- and γ-duals. Lastly, we characterize some matrix classes related to those spaces.

2 Two new sequence spaces

In this section, we give some historical information and define the sequence spaces c 0 λ ( G m ) and c λ ( G m ) generated by the composition of m th order generalized difference matrix and lambda matrix. Moreover, we speak of some inclusion relations.

The idea of using the notion of λ-convergent was first motivated by Mursaleen and Noman in [28]. They defined the sequence spaces c 0 λ , λ and c λ by means of the Lambda matrix Λ = ( λ n k ) such that
c 0 λ = { x = ( x k ) w : lim n 1 λ n k = 0 n ( λ k λ k 1 ) x k = 0 } , c λ = { x = ( x k ) w : lim n 1 λ n k = 0 n ( λ k λ k 1 ) x k  exists }
and
λ = { x = ( x k ) w : sup n N | 1 λ n k = 0 n ( λ k λ k 1 ) x k | < } ,
where λ = ( λ k ) consists of positive reals such that
0 < λ 0 < λ 1 < and lim k λ k =
and the lambda matrix Λ = ( λ n k ) is defined by
λ n k = { λ k λ k 1 λ n , 0 k n , 0 , k > n

for all k , n N . Here, we would like to touch on a point, if we take λ n = n + 1 and λ n = P n , for all n N , we obtain the Cesàro mean of order one and the Riesz mean matrix, respectively. So, the Λ = ( λ n k ) matrix generalizes the C = ( c n k ) and R p = ( r n k p ) matrices.

Also, they improved their work by constructing the spaces c 0 λ ( Δ ) and c λ ( Δ ) in [29]. The sequence spaces c 0 λ ( Δ ) and c λ ( Δ ) are defined by
c 0 λ ( Δ ) = { x = ( x k ) w : lim n 1 λ n k = 0 n ( λ k λ k 1 ) ( x k x k 1 ) = 0 }
and
c λ ( Δ ) = { x = ( x k ) w : lim n 1 λ n k = 0 n ( λ k λ k 1 ) ( x k x k 1 )  exists } ,

where Δ is a difference matrix.

Afterward, Sönmez and Başar defined the sequence spaces c 0 λ ( B ) and c λ ( B ) in [30] and improved Mursaleen and Noman’s work as follows:
c 0 λ ( B ) = { x = ( x k ) w : lim n 1 λ n k = 0 n ( λ k λ k 1 ) ( b 1 x k + b 2 x k 1 ) = 0 }
and
c λ ( B ) = { x = ( x k ) w : lim n 1 λ n k = 0 n ( λ k λ k 1 ) ( b 1 x k + b 2 x k 1 )  exists } ,
where B = B ( b 1 , b 2 ) is called a double band (generalized difference) matrix defined by
b n k = { b 1 , k = n , b 2 , k = n 1 , 0 , otherwise .
Let r and s be non-zero real numbers, then the m th order generalized difference matrix G m ( r , s ) = ( g n k m ( r , s ) ) is defined by
g n k m ( r , s ) = { ( m 1 n k ) r m n + k 1 s n k , max { 0 , n m + 1 } k n , 0 , otherwise

for all n , k N and m N 2 = { 2 , 3 , 4 , } . We want to recall that G 2 ( r , s ) = B ( b 1 , b 2 ) , G 3 ( r , s ) = B ( b 1 , b 2 , b 3 ) , G 4 ( r , s ) = B ( b 1 , b 2 , b 3 , b 4 ) , G 5 ( r , s ) = B ( b 1 , b 2 , b 3 , b 4 , b 5 ) , … where B ( b 1 , b 2 ) , B ( b 1 , b 2 , b 3 ) , B ( b 1 , b 2 , b 3 , b 4 ) , B ( b 1 , b 2 , b 3 , b 4 , b 5 ) , … are double band (that is, the generalized difference matrix), triple band, quadruple band, quinary band, … matrix, respectively. Moreover, G m ( 1 , 1 ) = Δ m , G 2 ( 1 , 1 ) = Δ , G 3 ( 1 , 1 ) = Δ 2 . So, our results obtained from the matrix domain of the m th order generalized difference matrix G m are more general and more extensive than the results on the matrix domain of B ( b 1 , b 2 ) , B ( b 1 , b 2 , b 3 ) , B ( b 1 , b 2 , b 3 , b 4 ) , B ( b 1 , b 2 , b 3 , b 4 , b 5 ) , … , Δ m , Δ 2 and Δ.

By considering the definition of m th order generalized difference matrix G m , we define the sequence spaces c 0 λ ( G m ) and c λ ( G m ) as follows:
c 0 λ ( G m ) = { x = ( x k ) w : lim n 1 λ n k = 0 n ( λ k λ k 1 ) ϑ = 0 m 1 ( m 1 ϑ ) r m ϑ 1 s ϑ x k ϑ = 0 }
and
c λ ( G m ) = { x = ( x k ) w : lim n 1 λ n k = 0 n ( λ k λ k 1 ) ϑ = 0 m 1 ( m 1 ϑ ) r m ϑ 1 s ϑ x k ϑ  exists } .
If we recall the notation of (1.2), the sequence spaces c 0 λ ( G m ) and c λ ( G m ) can be redefined by the matrix domain of G m as follows:
c 0 λ ( G m ) = ( c 0 λ ) G m and c λ ( G m ) = c G m λ .
(2.1)
Also, by constructing a triangle matrix T m λ = ( t n k m λ ) so that
t n k m λ = { 1 λ n ϑ = 0 m 1 ( m 1 ϑ ) r m ϑ 1 s ϑ ( λ k + ϑ λ k + ϑ 1 ) , k < n m + 2 , 1 λ n ϑ = 1 m 1 ( m 1 ϑ 1 ) r m ϑ s ϑ 1 ( λ n m + ϑ + 1 λ n m + ϑ ) , k = n m + 2 , 1 λ n ϑ = 2 m 1 ( m 1 ϑ 2 ) r m ϑ + 1 s ϑ 2 ( λ n m + ϑ + 1 λ n m + ϑ ) , k = n m + 3 , r m 1 ( λ n 1 λ n 2 ) + ( m 1 ) r m 2 s ( λ n λ n 1 ) λ n , k = n 1 , r m 1 ( λ n λ n 1 ) λ n , k = n , 0 , k > n
for all n , k N and m N 2 , we rearrange the sequence spaces c 0 λ ( G m ) and c λ ( G m ) by means of the T m λ = ( t n k m λ ) matrix as follows:
c 0 λ ( G m ) = ( c 0 ) T m λ and c λ ( G m ) = c T m λ .
(2.2)
So, for a given arbitrary sequence x = ( x k ) , the T m λ -transform of x is denoted by
y k = ( T m λ x ) k = 1 λ k j = 0 k ( λ j λ j 1 ) ϑ = 0 m 1 ( m 1 ϑ ) r m ϑ 1 s ϑ x j ϑ
(2.3)
for all k N , or, by using another representation, we can rewrite the sequence y = ( y k ) as follows:
y k = 1 λ k j = 0 k m + 1 ϑ = 0 m 1 ( m 1 ϑ ) r m ϑ 1 s ϑ ( λ j + ϑ λ j + ϑ 1 ) x j + + r m 1 ( λ k λ k 1 ) λ k x k
(2.4)

for all k N .

Theorem 2.1 The sequence spaces c 0 λ ( G m ) and c λ ( G m ) are BK-spaces according to their norms defined by
x c 0 λ ( G m ) = x c λ ( G m ) = ( T m λ x ) n = sup n N | ( T m λ x ) n | .

Proof It is known that c and c 0 are BK-spaces with their sup - norm [2]. Also (2.2) holds and T m λ = ( t n k m λ ) is a triangle matrix. If we consider these three facts and Theorem 4.3.12 of Wilansky [3], we conclude that c 0 λ ( G m ) and c λ ( G m ) are BK-spaces. This step completes the proof. □

Theorem 2.2 The sequence spaces c 0 λ ( G m ) and c λ ( G m ) are linearly isomorphic to the sequence spaces c 0 and c, respectively.

Proof To avoid the repetition of similar statements, we give the proof of the theorem for only the sequence space c λ ( G m ) . For the proof, the existence of a linear bijection between the spaces c λ ( G m ) and c should be shown. Let us define a transformation L such that L : c λ ( G m ) c , L ( x ) = T m λ x . Then it is clear that for all x c λ ( G m ) , L ( x ) = T m λ x c . Also, it is trivial that L is a linear transformation and x = 0 whenever L ( x ) = 0 . On account of this, L is injective.

Furthermore, for a given sequence y = ( y k ) c , we define the sequence x = ( x k ) as follows:
x k = 1 r m 1 j = 0 k ( m + k j 2 m 2 ) ( s r ) k j i = j 1 j ( 1 ) j i λ i λ j λ j 1 y i
for all k N . Then, for every n N , we obtain
ϑ = 0 m 1 ( m 1 ϑ ) r m ϑ 1 s ϑ x k ϑ = i = k 1 k ( 1 ) k i λ i λ k λ k 1 y i .
If we consider the equality above, for all n N , we conclude that
( T m λ x ) n = 1 λ n k = 0 n ( λ k λ k 1 ) ϑ = 0 m 1 ( m 1 ϑ ) r m ϑ 1 s ϑ x k ϑ = 1 λ n k = 0 n ( λ k λ k 1 ) i = k 1 k ( 1 ) k i λ i λ k λ k 1 y i = 1 λ n k = 0 n i = k 1 k ( 1 ) k i λ i y i = y n .

So, T m λ x = y and since y c , we bring to a conclusion that T m λ x c . Hence, we conclude that x c λ ( G m ) and L ( x ) = y . Thus L is surjective.

Furthermore, we have for every x c λ ( G m ) that
L ( x ) = T m λ x = x c λ ( G m ) .

So, L is norm preserving. As a consequence, L is a linear bijection. This step shows that the spaces c λ ( G m ) and c are linearly isomorphic, namely c λ ( G m ) c . □

Lemma 2.3 [28]

The inclusions c 0 c 0 λ and c c λ hold.

Theorem 2.4 The inclusion c 0 λ ( G m ) c λ ( G m ) strictly holds.

Proof It is well known that every null sequence is also convergent. So, the inclusion c 0 λ ( G m ) c λ ( G m ) holds. Now we define a sequence x = ( x k ) such that
x k = 1 r m 1 j = 0 k ( m + j 2 m 2 ) ( s r ) j
for all k N . Then we obtain by (2.3) that
( T m λ x ) n = 1 λ n k = 0 n ( λ k λ k 1 ) = 1

for all n N , which gives us that T m λ x = e , where e = ( 1 , 1 , ) . Then T m λ x = e c c 0 , namely x c λ ( G m ) c 0 λ ( G m ) . This shows that the inclusion c 0 λ ( G m ) c λ ( G m ) strictly holds. This step completes the proof. □

Theorem 2.5 If r + s = 0 , the inclusion c c 0 λ ( G m ) is strict.

Proof It is clear that Δ x , Δ 2 x , Δ 3 x , , Δ m x c 0 whenever x c . Assume that r + s = 0 and x c . Then G m x = r m 1 Δ m x and because of Δ m x c 0 , G m x c 0 . If we consider this fact and Lemma 2.3, we deduce that G m x c 0 λ . This shows that x c 0 λ ( G m ) . As a consequence, c c 0 λ ( G m ) holds. Now we define a sequence y = ( y k ) such that y k = l n k for all k N and k > m . Then it is obvious that G m y c 0 but y c . Because of c 0 c 0 λ , we conclude that G m y c 0 λ and thereby y c 0 λ ( G m ) . This shows that c c 0 λ ( G m ) strictly holds if r + s = 0 . This step completes the proof. □

If we combine Theorem 2.4 and Theorem 2.5, we give the following result.

Corollary 2.6 If r + s = 0 , the inclusions c 0 c 0 λ ( G m ) and c c λ ( G m ) are strict.

Now we define two sequences x = ( x k ) and y = ( y k ) such that x k = k k + 1 and y k = k + m for all k N and m N 2 . Then we can see that x and G m x c 0 c 0 λ , namely x c 0 λ ( G m ) . Also, it is clear that y c 0 λ ( G m ) . These two facts give us the following corollary.

Corollary 2.7 The spaces and c 0 λ ( G m ) overlap but the space does not include the space c 0 λ ( G m ) .

Now we give the following lemma which is needed in the next theorem.

Lemma 2.8 [31]

A ( : c 0 ) lim n k | a n k | = 0 .

Theorem 2.9 Let a sequence z = ( z k ) be as follows:
z k = | 1 λ k λ k 1 ϑ = 0 m 1 ( m 1 ϑ ) r m ϑ 1 s ϑ ( λ k + ϑ λ k + ϑ 1 ) |

for all k N . Then the inclusion c 0 λ ( G m ) is strict if and only if z c 0 λ .

Proof We assume that the inclusion c 0 λ ( G m ) holds. Then it is obvious that for every x , x c 0 λ ( G m ) , namely T m λ c 0 . Thus T m λ ( : c 0 ) . If we consider the last result and Lemma 2.8, we deduce that
lim n k | t n k m λ | = 0 .
Also, by using the definition of the matrix T m λ = ( t n k m λ ) , we obtain by the equality above that
By considering equalities (2.5), (2.6), …, (2.3+m) and (2.4+m), we conclude that
lim n λ n 1 λ n = 1 , lim n λ n 2 λ n = 1 , , lim n λ n m + 1 λ n = 1 .
For all n m 1 , we write the equality as follows:
By combining lim n λ n m + 1 λ n = 1 , (2.5) and (2.5+m), we deduce that

This means that z c 0 λ .

On the contrary, assume that z c 0 λ . Then we have (2.6+m). Also, for all n m 1 , we write the inequality as follows:
0 | r m 1 λ n m + 1 + ( m 1 ) r m 2 s ( λ n m + 2 λ 0 ) + + s m 1 ( λ n λ m 2 ) λ n | = | 1 λ n k = 0 n m + 1 ϑ = 0 m 1 ( m 1 ϑ ) r m ϑ 1 s ϑ ( λ k + ϑ λ k + ϑ 1 ) | 1 λ n k = 0 n m + 1 | ϑ = 0 m 1 ( m 1 ϑ ) r m ϑ 1 s ϑ ( λ k + ϑ λ k + ϑ 1 ) | = 1 λ n k = 0 n m + 1 ( λ k λ k 1 ) z k 1 λ n m + 1 k = 0 n m + 1 ( λ k λ k 1 ) z k .
By combining the last inequality and (2.6+m), we conclude that
lim n r m 1 λ n m + 1 + ( m 1 ) r m 2 s ( λ n m + 2 λ 0 ) + + s m 1 ( λ n λ m 2 ) λ n = 0 .

Specially, if we take ( r = 1 , s = 1 and m = 2 ), ( r = 1 , s = 1 and m = 3 ), … then we obtain lim n λ n λ n 1 λ n = 0 , lim n λ n 1 λ n 2 λ n = 0 , …, respectively. These equalities show that (2.4+m), (2.3+m), …, (2.6) and (2.5) hold, respectively. If we take into account the last result and Lemma 2.8, we conclude that T m λ ( : c 0 ) . Thus, the inclusion c 0 λ ( G m ) holds and is strict by Corollary 2.7. This step completes the proof. □

3 The Schauder basis and α-, β- and γ-duals

In the present section, we give the Schauder basis and determine α-, β- and γ-duals of the sequence spaces c 0 λ ( G m ) and c λ ( G m ) .

Let ( X , x X ) be a normed space. A set { x k : x k X , k N } is called a Schauder basis for X if for every x X there exist unique scalars μ k , k N , such that x = k μ k x k ; i.e.,
x k = 0 n μ k x k X 0

as n .

Note that the Hamel basis is free from topology, whereas the Schauder basis involves convergence and hence topology (see [1]).

For example, let e ( k ) be a sequence with 1 in the k th place and zeros elsewhere, and let e = ( 1 , 1 , 1 , ) . Then the sequence ( e ( k ) ) is a Schauder basis for c 0 . Moreover, { e , e 0 , e 1 , } is a Schauder basis for c.

Due to the transformation, L defined in the proof of Theorem 2.2 is an isomorphism; the inverse image of ( e ( k ) ) is a Schauder basis for c 0 λ ( G m ) .

Now we give the following results.

Theorem 3.1 Let σ k = { T m λ x } k for all k N . For every fixed k N , we define the sequences h = ( h n ) and h ( k ) m λ ( r , s ) = { h n ( k ) m λ ( r , s ) } n N such that
h n = 1 r m 1 k = 0 n ( m + k 2 m 2 ) ( s r ) k , h n ( k ) m λ ( r , s ) = { 1 r m 2 ( s r ) n k [ ( m + n k 2 m 2 ) λ k r ( λ k λ k 1 ) + ( m + n k 3 m 2 ) λ k s ( λ k + 1 λ k ) ] , k < n , λ k r m 1 ( λ k λ k 1 ) , k = n , 0 , k > n .
Then
  1. (a)
    The sequence { h ( k ) m λ ( r , s ) } k N is a Schauder basis for the space c 0 λ ( G m ) , and every x c 0 λ ( G m ) has a unique representation of the form
    x = k σ k h ( k ) m λ ( r , s ) .
     
  2. (b)
    The sequence { h , h ( 0 ) m λ ( r , s ) , h ( 1 ) m λ ( r , s ) , } is a Schauder basis for the space c λ ( G m ) , and every x c λ ( G m ) has a unique representation of the form
    x = l h + k [ σ k l ] h ( k ) m λ ( r , s ) ,
     

where l = lim k σ k .

If we consider the results of Theorem 2.1 and Theorem 3.1, we give the following result.

Corollary 3.2 The sequence spaces c 0 λ ( G m ) and c λ ( G m ) are separable.

Given arbitrary sequence spaces X and Y, the set M ( X , Y ) defined by
M ( X , Y ) = { y = ( y k ) w : x y = ( x k y k ) Y  for all  x = ( x k ) X }
(3.1)

is called the multiplier space of X and Y. For a sequence space Z with Y Z X , one can easily observe that M ( X , Y ) M ( Z , Y ) and M ( X , Y ) M ( X , Z ) hold, in turn.

By using the sequence spaces 1 , cs and bs and notation (3.1), the α-, β- and γ-duals of a sequence space X are defined by
X α = M ( X , 1 ) , X β = M ( X , c s ) and X γ = M ( X , b s ) ,

respectively.

Now we give some properties which are needed in the next lemma
sup K F n | k K a n k | p < ,
(3.2)
sup n N k | a n k | < ,
(3.3)
lim n a n k = α k for each  k N ,
(3.4)
lim n k a n k = α ,
(3.5)

where is the collection of all finite subsets of and p [ 1 , ) .

Lemma 3.3 [31]

Let A = ( a n k ) be an infinite matrix, then the following hold:
  1. (i)

    A = ( a n k ) ( c 0 : 1 ) = ( c : 1 ) (3.2) holds with p = 1 ;

     
  2. (ii)

    A = ( a n k ) ( c 0 : c ) (3.3) and (3.4) hold ;

     
  3. (iii)

    A = ( a n k ) ( c : c ) (3.3) , (3.4) and (3.5) hold ;

     
  4. (iv)

    A = ( a n k ) ( c 0 : ) = ( c : ) (3.3) holds ;

     
  5. (v)

    A = ( a n k ) ( c 0 : p ) = ( c : p ) (3.2) holds with 1 p < ;

     
  6. (vi)

    A = ( a n k ) ( c : c 0 ) (3.3) , (3.4) and (3.5) hold with α k = 0 , k N and α = 0 ;

     
  7. (vii)

    A = ( a n k ) ( c 0 : c 0 ) (3.3) and (3.4) hold with α k = 0 , k N .

     
Theorem 3.4 Define the set u 1 m λ ( r , s ) by
u 1 m λ ( r , s ) = { a = ( a n ) w : sup K F n | k K d n k m λ | < } ,
where the matrix D m λ = ( d n k m λ ) is defined by means of the sequence a = ( a n ) by
d n k m λ ( r , s ) = { 1 r m 2 ( s r ) n k [ ( m + n k 2 m 2 ) λ k r ( λ k λ k 1 ) + ( m + n k 3 m 2 ) λ k s ( λ k + 1 λ k ) ] a n , k < n , λ n r m 1 ( λ n λ n 1 ) a n , k = n , 0 , k > n .

Then { c 0 λ ( G m ) } α = { c λ ( G m ) } α = u 1 m λ ( r , s ) .

Proof For given a = ( a n ) w , by taking into account the sequence x = ( x n ) that is defined in the proof of Theorem 2.2, we obtain
a n x n = 1 r m 1 k = 0 n ( m + n k 2 m 2 ) ( s r ) n k i = k 1 k ( 1 ) k i λ i λ k λ k 1 a n y i = D n m λ ( y )
for all n N . If we consider the equality above, we conclude that a x = ( a n x n ) 1 whenever x = ( x k ) c 0 λ ( G m ) or c λ ( G m ) if and only if D m λ y 1 whenever y = ( y k ) c 0 or c. This means that a = ( a n ) { c 0 λ ( G m ) } α = { c λ ( G m ) } α if and only if D m λ ( c 0 : 1 ) = ( c : 1 ) . If we consider this and Lemma 3.3(i), we write
a = ( a n ) { c 0 λ ( G m ) } α = { c λ ( G m ) } α sup K F n | k K d n k m λ | <

and conclude { c 0 λ ( G m ) } α = { c λ ( G m ) } α = u 1 m λ ( r , s ) . This step completes the proof. □

Theorem 3.5 Given the sets u 2 m λ ( r , s ) , u 3 m λ ( r , s ) , u 4 m λ ( r , s ) and u 5 m λ ( r , s ) as follows:
u 2 m λ ( r , s ) = { a = ( a k ) w : j = k ( m + n j 2 m 2 ) ( s r ) n j a j exists for all k N } , u 3 m λ ( r , s ) = { a = ( a k ) w : sup n N k = 0 n 1 | b k m λ ( n ) | < } , u 4 m λ ( r , s ) = { a = ( a k ) w : sup n N | λ n r m 1 ( λ n λ n 1 ) a n | < }
and
u 5 m λ ( r , s ) = { a = ( a k ) w : k 1 r m 1 j = 0 k ( m + j 2 m 2 ) ( s r ) j a k converges } ,
where
b k m λ ( n ) = λ k [ a k r m 1 ( λ k λ k 1 ) + 1 r m 2 j = k + 1 n ( s r ) n j ( ( m + n j 2 m 2 ) r ( λ k λ k 1 ) + ( m + n j 3 m 2 ) s ( λ k + 1 λ k ) ) a j ] , k < n .

Then { c 0 λ ( G m ) } β = u 2 m λ ( r , s ) u 3 m λ ( r , s ) u 4 m λ ( r , s ) and { c λ ( G m ) } β = u 3 m λ ( r , s ) u 4 m λ ( r , s ) u 5 m λ ( r , s ) .

Proof Given a = ( a k ) w , by considering the sequence x = ( x k ) that is defined in the proof of Theorem 2.2, we obtain
k = 0 n a k x k = k = 0 n { 1 r m 1 j = 0 k ( m + k j 2 m 2 ) ( s r ) k j i = j 1 j ( 1 ) j i λ i λ j λ j 1 y i } a k = k = 0 n 1 λ k [ a k r m 1 ( λ k λ k 1 ) + 1 r m 2 j = k + 1 n ( s r ) n j ( ( m + n j 2 m 2 ) r ( λ k λ k 1 ) + ( m + n j 3 m 2 ) s ( λ k + 1 λ k ) ) a j ] y k + λ n r m 1 ( λ n λ n 1 ) a n y n = k = 0 n 1 b k m λ ( n ) y k + λ n r m 1 ( λ n λ n 1 ) a n y n = V n m λ ( y )
n N , where the matrix V m λ = ( v n k m λ ) is defined as follows:
v n k m λ ( r , s ) = { b k m λ ( n ) , k < n , λ n r m 1 ( λ n λ n 1 ) a n , k = n , 0 , k > n
for all n , k N . Then a x = ( a k x k ) c s whenever x = ( x k ) c 0 λ ( G m ) if and only if V m λ y c whenever y = ( y k ) c 0 . This shows that a = ( a k ) { c 0 λ ( G m ) } β if and only if V m λ ( c 0 : c ) . If we consider this and Lemma 3.3(ii), we obtain
j = k ( m + n j 2 m 2 ) ( s r ) n j a j  exists k N ,
(3.6)
sup n N k = 0 n 1 | b k m λ ( n ) | <
(3.7)
and
sup n N | λ n r m 1 ( λ n λ n 1 ) a n | < .
(3.8)

These results show that { c 0 λ ( G m ) } β = u 2 m λ ( r , s ) u 3 m λ ( r , s ) u 4 m λ ( r , s ) .

By using a similar way, we obtain a = ( a k ) { c λ ( G m ) } β if and only if V m λ ( c : c ) . If we consider this and Lemma 3.3(iii), we conclude that (3.6), (3.7) and (3.8) hold.

Moreover, one can easily see that
1 r m 1 k = 0 n j = 0 k ( m + j 2 m 2 ) ( s r ) j a k = k = 0 n 1 b k m λ ( n ) + λ n r m 1 ( λ n λ n 1 ) a n = k v n k m λ .
As a consequence, we derive from (3.5) that
{ 1 r m 1 j = 0 k ( m + j 2 m 2 ) ( s r ) j a k } c s .

Since condition (3.6) is weaker, it can be omitted.

Therefore we conclude that { c λ ( G m ) } β = u 3 m λ ( r , s ) u 4 m λ ( r , s ) u 5 m λ ( r , s ) . This step completes the proof. □

Theorem 3.6 { c 0 λ ( G m ) } γ = { c λ ( G m ) } γ = u 3 m λ ( r , s ) u 4 m λ ( r , s ) .

Proof It can be proved by combining the proof method of Theorem 3.5 and Lemma 3.3(iv). □

4 Matrix transformations

In the present section, we determine some matrix classes related to the sequence spaces c 0 λ ( G m ) and c λ ( G m ) . Let us begin with two lemmas which are needed in the proof of theorems.

Lemma 4.1 [3]

Any matrix map between BK-spaces is continuous.

Lemma 4.2 [32]

Let X, Y be any two sequence spaces, A be an infinite matrix and U be a triangle matrix. Then A ( X : Y U ) U A ( X : Y ) .

For simplicity of notation, in what follows, we use the following equalities.
b n k m λ ( i ) = λ k [ a n k r m 1 ( λ k λ k 1 ) + 1 r m 2 j = k + 1 i ( s r ) n j ( ( m + n j 2 m 2 ) r ( λ k λ k 1 ) + ( m + n j 3 m 2 ) s ( λ k + 1 λ k ) ) a n j ] , k < i
and
b n k m λ = λ k [ a n k r m 1 ( λ k λ k 1 ) + 1 r m 2 j = k + 1 ( s r ) n j ( ( m + n j 2 m 2 ) r ( λ k λ k 1 ) + ( m + n j 3 m 2 ) s ( λ k + 1 λ k ) ) a n j ]
for all n , k , i N provided the convergence of the series. Also, unless stated otherwise, we assume throughout Section 4 that the sequence y = ( y k ) is connected with the sequence x = ( x k ) as follows:
x k = 1 r m 1 j = 0 k ( m + k j 2 m 2 ) ( s r ) k j φ = j 1 j ( 1 ) j φ λ φ λ j λ j 1 y φ

for all k N .

Theorem 4.3 Given an infinite matrix A = ( a n k ) of complex numbers, the following statements hold.
  1. (1)
    Let 1 p < . Then A ( c λ ( G m ) : p ) if and only if
    sup K F n | k K b n k m λ | p < ,
    (4.1)
    sup i N k = 0 i 1 | b n k m λ ( i ) | < ( n N ) ,
    (4.2)
    { 1 r m 1 j = 0 k ( m + j 2 m 2 ) ( s r ) j a n k } k = 0 c s ( n N ) ,
    (4.3)
    lim k λ k r m 1 ( λ k λ k 1 ) a n k = a n ( n N ) ,
    (4.4)
    ( a n ) p .
    (4.5)
     
  2. (2)
    A ( c λ ( G m ) : ) if and only if (4.3) and (4.4) hold, and
    sup n N k | b n k m λ | < ,
    (4.6)
    ( a n ) .
    (4.7)
     

Proof For a given sequence x = ( x k ) c λ ( G m ) , we assume that conditions (4.1)-(4.5) hold. Then, by remembering Theorem 3.5, we deduce that { a n k } k N { c λ ( G m ) } β for all n N . Therefore the A-transform of x exists. Moreover, it is trivial that y c , namely l C lim k | y k l | = 0 . Furthermore, if we consider Lemmas 3.3 and 4.1, we conclude that the matrix B m λ ( c : p ) , where 1 p < .

Now, we consider the following equality:
k = 0 i a n k x k = k = 0 i 1 b n k m λ ( i ) y k + λ i r m 1 ( λ i λ i 1 ) a n i y i ( n , i N ) .
(4.8)
Then B m λ y exists and the series k b n k m λ y k converges for all n N . Moreover, we derive from (4.3) that the series j = k ( s r ) n j [ ( m + n j 2 m 2 ) r ( λ k λ k 1 ) + ( m + n j 3 m 2 ) s ( λ k + 1 λ k ) ] a n j converges for all n , k N ; and therefore lim i b n k m λ ( i ) = b n k m λ . Hence, if we take limit (4.8) side by side as i , we obtain by (4.4) that
k a n k x k = k b n k m λ y k + l a n
(4.9)
for all n N . Then we write the equality above as follows:
A n ( x ) = B n m λ ( y ) + l a n
(4.10)
for all n N . Also, we know ( B m λ y ) n p and a = ( a n ) p . Then we have B m λ y p < and a n p < . By taking p -norm (4.10) side by side, we obtain that
A x p B m λ y p + | l | a n p < .

Therefore A x p and so A ( c λ ( G m ) : p ) .

On the contrary, assume that A ( c λ ( G m ) : p ) , where 1 p < . This leads us to { a n k } k N { c λ ( G m ) } β for all n N . Then, if we consider Theorem 3.5, conditions (4.2) and (4.3) hold.

We know that c λ ( G m ) and p are BK-spaces. If we combine this fact and Lemma 4.1, we conclude that there is a constant M > 0 such that
A x p M x c λ ( G m )
(4.11)

holds for all x c λ ( G m ) . Let us define a sequence z = ( z k ) such that z = k K h ( k ) m λ ( r , s ) for every fixed k N , where the sequence h ( k ) m λ ( r , s ) = { h n ( k ) m λ ( r , s ) } n N and K F .

We know from Theorem 3.1 that z c λ ( G m ) and T m λ ( h ( k ) m λ ( r , s ) ) = e ( k ) , k N . Then we obtain
z c λ ( G m ) = T m λ ( z ) = k K T m λ ( h ( k ) m λ ( r , s ) ) = k K e ( k ) = 1
and
A n ( z ) = k K A n ( h ( k ) m λ ( r , s ) ) = k K j a n j h j ( k ) m λ ( r , s ) = k K b n k m λ
for all n N . Since inequality (4.11) holds for every x c λ ( G m ) , the inequality is satisfied also for z c λ ( G m ) . Then we have
( n | k K b n k m λ | p ) 1 p M
for all K F . Therefore (4.1) holds. If we consider this and Lemma 3.3(v), we conclude that B m λ = ( b n k m λ ) ( c : p ) . Given y = ( y k ) c c 0 . Then x c λ ( G m ) so that y = T m λ ( x ) . Hence Ax and B m λ y exist. So one can easily see that the series k a n k x k and k b n k m λ y k are convergent for all n N . Thus, we conclude that
lim i k = 0 i 1 b n k m λ ( i ) y k = k b n k m λ y k
(4.12)
or all n N . As a consequence, if we pass to the limit in (4.8) as i , we obtain
lim i λ i r m 1 ( λ i λ i 1 ) a n i y i  exists
for all n N . Because of y = ( y k ) c c 0 , this leads us
lim i λ i r m 1 ( λ i λ i 1 ) a n i  exists

for all n N . Therefore, (4.4) holds. If we take l = lim k y k , also relation (4.10) holds. Because of A x , B m λ y p , we conclude a = ( a n ) p . The last result is the necessity of (4.5). This step completes the proof of part (1).

If we take Lemma 3.3(iv) instead of Lemma 3.3(v), the second part of theorem can be proved similarly. □

Moreover, from (4.12) we derive
lim i k = 0 i 1 | b n k m λ ( i ) | = k | b n k m λ | sup n N k | b n k m λ | .
(4.13)
If we combine (4.13) and (4.6), we conclude that
lim i k = 0 i 1 | b n k m λ ( i ) |

exists for each n N . So, condition (4.6) implies condition (4.2).

Theorem 4.4 Given an infinite matrix A = ( a n k ) of complex numbers, the following statements hold.
  1. (1)
    Let 1 p < . Then A ( c 0 λ ( G m ) : p ) if and only if (4.1) and (4.2) hold, and
    j = k ( s r ) n j [ ( m + n j 2 m 2 ) r ( λ k λ k 1 ) + ( m + n j 3 m 2 ) s ( λ k + 1 λ k ) ] a n j exists for all n , k N ,
    (4.14)
    { λ k r m 1 ( λ k λ k 1 ) a n k } k = 0 for all n N .
    (4.15)
     
  2. (2)

    A ( c 0 λ ( G m ) : ) if and only if (4.6), (4.14) and (4.15) hold.

     

Proof From Lemma 3.3(iv) and (v), we know ( c 0 : p ) = ( c : p ) and ( c 0 : ) = ( c : ) . Therefore, the theorem can be proved similarly. □

Theorem 4.5 A ( c λ ( G m ) : c ) if and only if (4.3), (4.4) and (4.6) hold, and the conditions
lim n a n = a ,
(4.16)
lim n b n k m λ = α k , k N ,
(4.17)
lim n k b n k m λ = α
(4.18)

hold.

Proof Given arbitrary x c λ ( G m ) , we assume that conditions (4.3), (4.4), (4.6), (4.16), (4.17) and (4.18) hold for an infinite matrix A = ( a n k ) . We consider Theorem 3.5, and condition (4.6) implies condition (4.2). Then we conclude that { a n k } k N { c λ ( G m ) } β for all n N , and so Ax exists. From (4.6) and (4.17) we have
j = 0 k | α j | = lim n j = 0 k | b n j m λ | sup n N j | b n j m λ | <
for all k N . This leads us to ( α k ) 1 , and therefore the series k α k ( y k l ) converges, where lim k y k = l and so y c . If we combine Lemma 3.3(iii) with conditions (4.6), (4.17) and (4.18), we deduce that B m λ = ( b n k m λ ) ( c : c ) . Also from condition (4.9) we have
A n ( x ) = k a n k x k = k b n k m λ y k + l a n .
With a basic calculation, we obtain
k a n k x k = k b n k m λ ( y k l ) + l k b n k m λ + l a n
(4.19)
for all n N . If we pass to the limit in (4.19), we write
lim n A n ( x ) = k α k ( y k l ) + l ( α + a ) .

This shows that A x c and so A ( c λ ( G m ) : c ) .

On the contrary, we assume that A ( c λ ( G m ) : c ) . Since every convergent sequence is also bounded, we deduce that A ( c λ ( G m ) : ) . If we consider this fact and Theorem 4.3, we conclude that conditions (4.3), (4.4) and (4.6) hold. Let us take the sequences h ( k ) m λ ( r , s ) = { h n ( k ) m λ ( r , s ) } n N c λ ( G m ) and z = k h ( k ) m λ ( r , s ) defined in Theorem 3.1 and the proof of Theorem 4.3, respectively. Then it is clear that A h ( k ) m λ ( r , s ) = { b n k m λ } n N c for every k N . Hence condition (4.17) holds. Moreover, from Theorem 2.2 we know that the transformation L : c λ ( G m ) c , L ( x ) = T m λ x is continuous. So, we write
T n m λ ( z ) = k T n m λ ( h ( k ) m λ ( r , s ) ) = k δ n k = 1
for all n N , where
δ n k = { 1 , k = n , 0 , k n .

This leads us to T m λ z = e c and so z c λ ( G m ) .

It is well known that c is a BK-space. If we combine Theorem 2.1 and Lemma 4.1, we conclude that the matrix transformation A : c λ ( G m ) c is continuous. Therefore the equality
A n ( z ) = k A n ( h ( k ) m λ ( r , s ) ) = k b n k m λ

holds for all n N . This shows that (4.18) holds.

By considering conditions (4.6), (4.17), (4.18) and Lemma 3.3(iii), we deduce that B m λ = ( b n k m λ ) ( c : c ) . Hence, (4.3), (4.4) and the last result give us that condition (4.10) holds for all x c λ ( G m ) and y c . Finally, if we consider A x , B m λ y c and (4.10), we conclude that condition (4.16) holds. This step completes the proof. □

Theorem 4.6 A ( c λ ( G m ) : c 0 ) if and only if (4.3), (4.4), (4.6) and the following conditions hold:
lim n a n = 0 ,
(4.20)
lim n b n k m λ = 0 , k N ,
(4.21)
lim n k b n k m λ = 0 .
(4.22)

Proof In Theorem 4.5, if we take Lemma 3.3(vi) instead of Lemma 3.3(iii), the present theorem can be proved by using a similar way. □

Theorem 4.7 A ( c 0 λ ( G m ) : c ) if and only if (4.6), (4.14), (4.15) and (4.17) hold.

Proof If we combine Lemma 3.3(ii) Theorem 3.5 and Theorem 4.4(2), the present theorem can be proved by using a similar way. □

Theorem 4.8 A ( c 0 λ ( G m ) : c 0 ) if and only if (4.6), (4.14), (4.15), (4.20), (4.21) and (4.22) hold.

Proof If we combine Lemma 3.3(vii), Theorem 3.5 and Theorem 4.7, the present theorem can be proved by using a similar way. □

Now, by using Lemma 4.2, we give one more result.

Corollary 4.9 Given an infinite matrix A = ( a n k ) of complex numbers, we define a matrix E = ( e n k ) as follows:
e n k = 1 λ n j = 0 n ( λ j λ j 1 ) ϑ = 0 m 1 ( m 1 ϑ ) r m ϑ 1 s ϑ a ( j ϑ ) k

for all n , k N . Then A belongs to matrix classes ( c 0 : c 0 λ ( G m ) ) , ( c : c 0 λ ( G m ) ) , ( p : c 0 λ ( G m ) ) , ( c 0 : c λ ( G m ) ) , ( c : c λ ( G m ) ) and ( p : c λ ( G m ) ) if and only if E belongs to matrix classes ( c 0 : c 0 ) , ( c : c 0 ) , ( p : c 0 ) , ( c 0 : c ) , ( c : c ) and ( p : c ) .

Finally, we put a period to our work by mentioning as of now that the sequence space f ( G m ) of almost convergent sequences derived by the domain of m th order generalized difference matrix will be defined and studied analogously in the next paper.

Declarations

Acknowledgements

We would like to express our thanks to the anonymous reviewers for their valuable comments.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Erciyes University

References

  1. Choudhary B, Nanda S: Functional Analysis with Applications. Wiley, New Delhi; 1989.MATHGoogle Scholar
  2. Musayev B, Alp M: Fonksiyonel Analiz. Balcı Yayınları, Ankara; 2000.Google Scholar
  3. Wilansky A North-Holland Mathematics Studies 85. In Summability Through Functional Analysis. Elsevier, Amsterdam; 1984.Google Scholar
  4. Wang C-S: On Nörlund sequence spaces. Tamkang J. Math. 1978, 9: 269-274.MathSciNetGoogle Scholar
  5. Aydın C, Başar F:Some new sequence spaces which include the spaces p and . Demonstratio Math. 2005,38(3):641-656.MathSciNetMATHGoogle Scholar
  6. Aydın C, Başar F: On the new sequence spaces which include the spaces c 0 and c . Hokkaido Math. J. 2004,33(2):383-398. 10.14492/hokmj/1285766172MathSciNetView ArticleMATHGoogle Scholar
  7. Malkowsky E, Savaş E: Matrix transformations between sequence spaces of generalized weighted means. Appl. Math. Comput. 2004, 147: 333-345. 10.1016/S0096-3003(02)00670-7MathSciNetView ArticleMATHGoogle Scholar
  8. Ng P-N, Lee P-Y: Cesàro sequence spaces of non-absolute type. Comment. Math. (Prace Mat.) 1978,20(2):429-433.MathSciNetMATHGoogle Scholar
  9. Altay B, Başar F, Mursaleen M:On the Euler sequence spaces which include the spaces p and . I. Inform. Sci. 2006,176(10):1450-1462. 10.1016/j.ins.2005.05.008MathSciNetView ArticleMATHGoogle Scholar
  10. Malkowsky E: Recent results in the theory of matrix transformations in sequence spaces. Mat. Vesnik 1997, 49: 187-196.MathSciNetMATHGoogle Scholar
  11. Altay B, Başar F: On the paranormed Riesz sequence spaces of non-absolute type. Southeast Asian Bull. Math. 2002,26(5):701-715.Google Scholar
  12. Altay B, Başar F: Some Euler sequence spaces of non-absolute type. Ukrainian Math. J. 2005,57(1):1-17. 10.1007/s11253-005-0168-9MathSciNetView ArticleGoogle Scholar
  13. Şengönül M, Başar F: Some new Cesàro sequence spaces of non-absolute type which include the spaces c 0 and c . Soochow J. Math. 2005,31(1):107-119.MathSciNetMATHGoogle Scholar
  14. Kızmaz H: On certain sequence spaces. Canad. Math. Bull. 1981,24(2):169-176. 10.4153/CMB-1981-027-5MathSciNetView ArticleMATHGoogle Scholar
  15. Et M: On some difference sequence spaces. Turkish J. Math. 1993, 17: 18-24.MathSciNetMATHGoogle Scholar
  16. Aydın C, Başar F: Some new difference sequence spaces. Appl. Math. Comput. 2004,157(3):677-693. 10.1016/j.amc.2003.08.055MathSciNetView ArticleMATHGoogle Scholar
  17. Et M, Çolak R: On some generalized difference sequence spaces. Soochow J. Math. 1995,21(4):377-386.MathSciNetMATHGoogle Scholar
  18. Ahmad ZU, Mursaleen : Köthe-Toeplitz duals of some new sequence spaces and their matrix maps. Publ. Inst. Math. (Beograd) 1987, 42: 57-61.MathSciNetGoogle Scholar
  19. Mursaleen M: Generalized spaces of difference sequences. J. Math. Anal. Appl. 1996,203(3):738-745. 10.1006/jmaa.1996.0409MathSciNetView ArticleMATHGoogle Scholar
  20. Asma Ç, Çolak R: On the Köthe-Toeplitz duals of some generalized sets of difference sequences. Demonstratio Math. 2000, 33: 797-803.MathSciNetMATHGoogle Scholar
  21. Bektaş Ç: On some new generalized sequence spaces. J. Math. Anal. Appl. 2003, 277: 681-688. 10.1016/S0022-247X(02)00619-4MathSciNetView ArticleMATHGoogle Scholar
  22. Et M, Başarır M: On some new generalized difference sequence spaces. Period. Math. Hungar. 1997,35(3):169-175. 10.1023/A:1004597132128MathSciNetView ArticleMATHGoogle Scholar
  23. Başarır M, Kayıkçı M: On generalized B m -Riesz difference sequence space and β -property. J. Inequal. Appl. 2009. Article ID 385029, 2009: Article ID 385029Google Scholar
  24. Kirişçi M, Başar F: Some new sequence spaces derived by the domain of generalized difference matrix. Comput. Math. Appl. 2010,60(5):1299-1309. 10.1016/j.camwa.2010.06.010MathSciNetView ArticleMATHGoogle Scholar
  25. Sönmez A: Some new sequence spaces derived by the domain of the triple band matrix. Comput. Math. Appl. 2011,62(2):641-650. 10.1016/j.camwa.2011.05.045MathSciNetView ArticleMATHGoogle Scholar
  26. Malkowsky E, Parashar SD: Matrix transformations in space of bounded and convergent sequences of order m . Analysis. 1997, 17: 87-97.MathSciNetView ArticleMATHGoogle Scholar
  27. Malkowsky E, Mursaleen M, Suantai S: The dual spaces of sets of difference sequences of order m and matrix transformations. Acta Math. Sinica (Engl. Ser.) 2007,23(3):521-532. 10.1007/s10114-005-0719-xMathSciNetView ArticleMATHGoogle Scholar
  28. Mursaleen M, Noman AK: On the spaces of λ -convergent and bounded sequences. Thai J. Math. 2010,8(2):311-329.MathSciNetMATHGoogle Scholar
  29. Mursaleen M, Noman AK: On some new difference sequence spaces of non-absolute type. Math. Comput. Modelling. 2010, 52: 603-617. 10.1016/j.mcm.2010.04.006MathSciNetView ArticleMATHGoogle Scholar
  30. Sönmez A, Başar F: Generalized difference spaces of non-absolute type of convergent and null sequences. Abstr. Appl. Anal. 2012. Article ID 435076, 2012: Article ID 435076Google Scholar
  31. Stieglitz M, Tietz H: Matrix transformationen von folgenräumen eine ergebnisübersicht. Math. Z. 1977, 154: 1-16. 10.1007/BF01215107MathSciNetView ArticleMATHGoogle Scholar
  32. Başar F, Altay B: On the space of sequences of p -bounded variation and related matrix mappings. Ukrainian Math. J. 2003,55(1):136-147. 10.1023/A:1025080820961MathSciNetView ArticleGoogle Scholar

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© Bi¿gin and Sönmez; licensee Springer. 2014

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