Open Access

Double sequence spaces over n-normed spaces defined by a sequence of Orlicz functions

  • Abdullah Alotaibi1,
  • Mohammad Mursaleen2Email author and
  • Sunil K Sharma3
Journal of Inequalities and Applications20142014:216

https://doi.org/10.1186/1029-242X-2014-216

Received: 21 February 2014

Accepted: 9 May 2014

Published: 29 May 2014

Abstract

In the present paper we introduce double sequence space m 2 ( M , A , ϕ , p , , , ) defined by a sequence of Orlicz functions over n-normed space. We examine some of its topological properties and establish some inclusion relations.

MSC:40A05, 46A45.

Keywords

double sequence spacesparanormed spaceOrlicz functionn-normed space

1 Introduction and preliminaries

The initial works on double sequences is found in Bromwich [1]. Later on, it was studied by Hardy [2], Moricz [3], Moricz and Rhoades [4], Başarır and Sonalcan [5] and many others. Hardy [2] introduced the notion of regular convergence for double sequences. Quite recently, Zeltser [6] in her PhD thesis has essentially studied both the theory of topological double sequence spaces and the theory of summability of double sequences. Mursaleen and Edely [7] have recently introduced the statistical convergence which was further studied in locally solid Riesz spaces [8]. Nextly, Mursaleen [9] and Mursaleen and Savas [10] have defined the almost regularity and almost strong regularity of matrices for double sequences and applied these matrices to establish core theorems and introduced the M-core for double sequences and determined those four dimensional matrices transforming every bounded double sequences x = ( x k , l ) into one whose core is a subset of the M-core of x. More recently, Altay and Başar [11] have defined the spaces B S , B S ( t ) , C S p , C S b p , C S r and B V of double sequences consisting of all double series whose sequence of partial sums are in the spaces M u , M u ( t ) , C p , C b p , C r and L u , respectively and also examined some properties of these sequence spaces and determined the α-duals of the spaces B S , B V , C S b p and the β ( v ) -duals of the spaces C S b p and C S r of double series. Recently Başar and Sever [12] have introduced the Banach space L q of double sequences corresponding to the well known space q of single sequences and examined some properties of the space  L q . Now, recently Raj and Sharma [13] have introduced entire double sequence spaces. By the convergence of a double sequence we mean the convergence in the Pringsheim sense i.e. a double sequence x = ( x k , l ) has Pringsheim limit L (denoted by P - lim x = L ) provided that given ϵ > 0 there exists n N such that | x k , l L | < ϵ whenever k , l > n , see [14]. The double sequence x = ( x k , l ) is bounded if there exists a positive number M such that | x k , l | < M for all k and l.

Throughout this paper, and denote the set of positive integers and complex numbers, respectively. A complex double sequence is a function x from N × N into and briefly denoted by { x k , l } . If for all ϵ > 0 , there is n ϵ N such that | x k , l a | < ϵ where k > n ϵ and l > n ϵ , then a double sequence { x k , l } is said to be convergent to a C . A real double sequence { x k , l } is non-decreasing, if x k , l x p , q for ( k , l ) < ( p , q ) . A double series is infinite sum k , l = 1 x k , l and its convergence implies the convergence of partial sums sequence { S n , m } , where S n , m = k = 1 m l = 1 n x k , l (see [15]). For recent development on double sequences, we refer to [1620] and [2123].

A double sequence space E is said to be solid if { x k , l y k , l } E for all double sequences { y k , l } of scalars such that | y k , l | < 1 for all k , l N whenever { x k , l } E .

Let x = { x k , l } be a double sequence. A set S ( x ) is defined by
S ( x ) = { { X π 1 ( k ) , π 2 ( k ) } : π 1  and  π 2  are permutation of  N } .
If S ( x ) E for all x E , then E is said to be symmetric. Now let P s be a family of subsets σ having at most elements s in . Also P s , t denotes the class of subsets σ = σ 1 × σ 2 in N × N such that the element numbers of σ 1 and σ 2 are at most s and t, respectively. Besides { ϕ k , l } is taken as a non-decreasing double sequence of the positive real numbers such that
k ϕ k + 1 , l ( k + 1 ) ϕ k , l , l ϕ k , l + 1 ( l + 1 ) ϕ k , l .

An Orlicz function M : [ 0 , ) [ 0 , ) is a continuous, non-decreasing, and convex function such that M ( 0 ) = 0 , M ( x ) > 0 for x > 0 and M ( x ) as x .

Lindenstrauss and Tzafriri [24] used the idea of Orlicz function to define the following sequence space:
M = { x w : k = 1 M ( | x k | ρ ) < } ,
which is called an Orlicz sequence space. Also M is a Banach space with the norm
x = inf { ρ > 0 : k = 1 M ( | x k | ρ ) 1 } .
Also, it was shown that every Orlicz sequence space M contains a subspace isomorphic to p ( p 1 ). The Δ 2 -condition is equivalent to M ( L x ) L M ( x ) , for all L with 0 < L < 1 . An Orlicz function M can always be represented in the following integral form:
M ( x ) = 0 x η ( t ) d t ,

where η, known as the kernel of M, is right differentiable for t 0 , η ( 0 ) = 0 , η ( t ) > 0 , η is non-decreasing and η ( t ) as t .

For further reading on Orlicz spaces, we refer to [2529].

Let X be a linear metric space. A function p : X R is called a paranorm if
  1. (1)

    p ( x ) 0 for all x X ,

     
  2. (2)

    p ( x ) = p ( x ) for all x X ,

     
  3. (3)

    p ( x + y ) p ( x ) + p ( y ) for all x , y X ,

     
  4. (4)

    if ( λ n ) is a sequence of scalars with λ n λ as n and ( x n ) is a sequence of vectors with p ( x n x ) 0 as n , then p ( λ n x n λ x ) 0 as n .

     

A paranorm p for which p ( x ) = 0 implies x = 0 is called a total paranorm and the pair ( X , p ) is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see [30], Theorem 10.4.2, p.183).

The concept of 2-normed spaces was initially developed by Gähler [31] in the mid-1960s, while that of n-normed spaces one can see in Misiak [32]. Since then, many others have studied this concept and obtained various results; see Gunawan [33, 34] and Gunawan and Mashadi [35] and references therein. Let n N and X be a linear space over the field , where is the field of real or complex numbers of dimension d, where d n 2 . A real valued function , , on X n satisfying the following four conditions:
  1. (1)

    x 1 , x 2 , , x n = 0 if and only if x 1 , x 2 , , x n are linearly dependent in X;

     
  2. (2)

    x 1 , x 2 , , x n is invariant under permutation;

     
  3. (3)

    α x 1 , x 2 , , x n = | α | x 1 , x 2 , , x n for any α K , and

     
  4. (4)

    x + x , x 2 , , x n x , x 2 , , x n + x , x 2 , , x n

     
is called a n-norm on X, and the pair ( X , , , ) is called a n-normed space over the field . For example, we may take X = R n being equipped with the Euclidean n-norm x 1 , x 2 , , x n E , the volume of the n-dimensional parallelepiped spanned by the vectors x 1 , x 2 , , x n which may be given explicitly by the formula
x 1 , x 2 , , x n E = | det ( x i j ) | ,
where x i = ( x i 1 , x i 2 , , x i n ) R n for each i = 1 , 2 , , n . Let ( X , , , ) be a n-normed space of dimension d n 2 and { a 1 , a 2 , , a n } be linearly independent set in X. Then the function , , on X n 1 defined by
x 1 , x 2 , , x n 1 = max { x 1 , x 2 , , x n 1 , a i : i = 1 , 2 , , n }

defines an ( n 1 ) -norm on X with respect to { a 1 , a 2 , , a n } .

A sequence ( x k ) in a n-normed space ( X , , , ) is said to converge to some L X if
lim k x k L , z 1 , , z n 1 = 0 for every  z 1 , , z n 1 X .
A sequence ( x k ) in a n-normed space ( X , , , ) is said to be Cauchy if
lim k p x k x p , z 1 , , z n 1 = 0 for every  z 1 , , z n 1 X .

If every Cauchy sequence in X converges to some L X , then X is said to be complete with respect to the n-norm. Any complete n-normed space is said to be n-Banach space.

The space m ( ϕ ) was introduced by Sargent [36]:
m ( ϕ ) = { x = ( x k ) w : x m ( ϕ ) = sup s 1 , σ P s 1 ϕ s k σ | x k | < } ,

which was further studied in [37, 38] and [39]. Recently, Duyar and Oǧur [40] introduced the sequence space m 2 ( M , A , ϕ , p ) and studied some of its properties.

Let A = ( a i j k l ) be an infinite double matrix of complex numbers, M = ( M k , l ) be a sequence of Orlicz functions, and p = ( p k , l ) be a bounded double sequence of positive real numbers. In the present paper we define the following sequence space:
m 2 ( M , A , ϕ , p , , , ) = { x = ( x k , l ) w 2 ( X ) : sup { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( x ) ρ , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } <  for some  ρ > 0 } ,

where A ( x ) = ( A i j ( x ) ) if A i j ( x ) = k , l = 1 a i j k l x k , l converges for each ( i , j ) N × N .

If p = ( p i j ) = 1 , we have
m 2 ( M , A , ϕ , , , ) = { x = ( x k , l ) w 2 ( X ) : sup { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( x ) ρ , z 1 , , z n 1 ) : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } <  for some  ρ > 0 } .
The following inequality will be used throughout the paper:
| a + b | p i j max ( 1 , 2 H 1 ) ( | a | p i j + | b | p i j ) ,
(1.1)

where a , b C and H = sup { p i j : ( i , j ) N × N } .

We examine some topological properties of m 2 ( M , A , ϕ , p , , , ) and establish some inclusion relations.

2 Main results

Theorem 2.1 Let M = ( M k , l ) be a sequence of Orlicz functions and p = ( p k , l ) be a bounded sequence of positive real numbers, then the space m 2 ( M , A , ϕ , p , , , ) is linear space over the field of complex number .

Proof Let x = ( x k , l ) , y = ( y k , l ) m 2 ( M , A , ϕ , p , , , ) and α , β C . Then there exist positive numbers ρ 1 and ρ 2 such that
sup { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( x ) ρ 1 , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } <
and
sup { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( y ) ρ 2 , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } < .
Let ρ 3 = max ( 2 | α | ρ 1 , 2 | β | ρ 2 ) . Since is a non-decreasing and convex function, we have
i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( α x + β y ) ρ 3 , z 1 , , z n 1 ) p i j i σ 1 j σ 2 k , l = 1 M k , l ( α a i j k l x k , l + β a i j k l y k , l ρ 3 , z 1 , , z n 1 ) p i j i σ 1 j σ 2 k , l = 1 M k , l ( α a i j k l x k , l 2 | α | ρ 1 , z 1 , , z n 1 + β a i j k l y k , l 2 | β | ρ 2 , z 1 , , z n 1 ) p i j = i σ 1 j σ 2 k , l = 1 M k , l ( a i j k l x k , l 2 ρ 1 , z 1 , , z n 1 + a i j k l y k , l 2 ρ 2 , z 1 , , z n 1 ) p i j max ( 1 , 2 H 1 ) ( i σ 1 j σ 2 k , l = 1 M k , l ( a i j k l x k , l 2 ρ 1 , z 1 , , z n 1 ) p i j + i σ 1 j σ 2 k , l = 1 M k , l ( a i j k l y k , l 2 ρ 2 , z 1 , , z n 1 ) p i j ) .
Thus, we have
sup { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( α x + β y ) ρ 3 , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } max ( 1 , 2 H 1 ) { sup { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( a i j k l x k , l 2 ρ 1 , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } + sup { 1 ϕ s , t i σ 1 j σ 2 M k , l ( a i j k l y k , l 2 ρ 2 , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } } .

This proves that α x + β y m 2 ( M , A , ϕ , p , , , ) . Hence m 2 ( M , A , ϕ , p , , , ) is a linear space. This completes the proof of the theorem. □

Theorem 2.2 M = ( M k , l ) be a sequence of Orlicz functions and p = ( p k , l ) be a bounded sequence of positive real numbers, then the space m 2 ( M , A , ϕ , p , , , ) is a paranormed space with the paranorm defined by
g ( x ) = inf { ρ p q r H : [ sup { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( x ) ρ , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } 1 ] 1 / H , q N , r N } .
Proof It is clear that g ( x ) = g ( x ) and g ( x ) = 0 if x = 0 . Then there exist positive numbers ρ 1 and ρ 2 such that
sup { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( x ) ρ 1 , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } < 1
and
sup { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( y ) ρ 2 , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } < 1 .
Then, by using Minkowski’s inequality, we have
sup { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( x + y ) ρ 1 + ρ 2 , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } sup { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( x ) ρ 1 + ρ 2 , z 1 , , z n 1 + A i j ( y ) ρ 1 + ρ 2 , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } ( ρ 1 ρ 1 + ρ 2 ) h { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( x ) ρ 1 , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } + ( ρ 2 ρ 1 + ρ 2 ) h { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( y ) ρ 2 , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } ,
where h = inf p i j . This shows that g ( x + y ) g ( x ) + g ( y ) . Using this triangle inequality we can write
g ( λ n x n λ x ) g ( λ n x n λ n x ) + g ( λ n x λ x ) .
Thus we have
g ( λ n x n λ n x ) = inf { ρ n p q r H : [ sup { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( λ n x n λ n x ) ρ n , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 σ 2 P s , t } 1 ] 1 / H 1 , q N , r N } = inf { ρ n p q r / H : [ sup { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( x n x ) ( ρ n / | λ n | ) , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } 1 ] 1 / H 1 , q N , r N } = inf { ( λ n ρ n ) p q r / H : [ sup { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( x n x ) ρ n , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } ] 1 / H 1 , q N , r N } max { | λ n | h / H , | λ n | } × inf { ( | λ n | ρ n ) p q r / H : [ sup { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( x n x ) ρ n , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } ] 1 / H 1 , q N , r N } = max { | λ n | h / H , | λ n | } g ( x n x ) .
Thus
g ( λ n x λ x ) = inf { ρ n p q r / H : [ sup { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( ( λ n λ ) x ) ρ n , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } ] 1 / H 1 , q N , r N } = inf { ρ n p q r / H : [ sup { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( x ) ρ n / | λ n λ | , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } ] 1 / H 1 , q N , r N } = inf { ( | λ n λ | ρ n ) p q r / H : [ sup { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( x ) ρ n , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } ] 1 / H 1 , q N , r N } max { | λ n λ | h / H , | λ n λ | } g ( x ) .

Hence g ( λ n x n λ x ) 0 where λ n λ and x n x as n . This proves that m 2 ( M , A , ϕ , p , , , ) is a paranormed space with the paranorm defined by g. This completes the proof of the theorem. □

Theorem 2.3 Let ϕ and ψ be two double sequences then m 2 ( M , A , ϕ , p , , , ) m 2 ( M , A , ψ , p , , , ) if and only if sup ( s , t ) ( 1 , 1 ) ( ϕ s , t / ψ s , t ) < .

Proof Let K = sup ( s , t ) ( 1 , 1 ) ( ϕ s , t / ψ s , t ) < . Then ϕ s , t K ψ s , t for all ( s , t ) ( 1 , 1 ) . If x = { x k , l } m 2 ( M , A , ϕ , p , , , ) , then
sup { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( x ) ρ , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } < for some  ρ > 0 .
Thus
sup { 1 K ψ s , t i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( x ) ρ , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } < for some  ρ > 0 ,
and hence x = { x k , l } m 2 ( M , A , ψ , p , , , ) . This shows that
m 2 ( M , A , ϕ , p , , , ) m 2 ( M , A , ψ , p , , , ) .
Conversely, let m 2 ( M , A , ϕ , p , , , ) m 2 ( M , A , ψ , p , , , ) and α s , t = ϕ s , t ψ s , t for all ( s , t ) ( 1 , 1 ) , and suppose sup ( s , t ) ( 1 , 1 ) α s , t = . Then there exists a subsequence { α s i , t i } of { α s , t } such that lim i α s i , t i = . If x = { x k , l } m 2 ( M , A , ϕ , p , , , ) , then we have
sup { 1 ψ s , t i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( x ) ρ , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } = sup { α s , t 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( x ) ρ , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } { sup m 1 α s m , t m } sup { 1 ϕ s m , t m i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( x ) ρ , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } = .

This is a contradiction as x = { x k , l } m 2 ( M , A , ϕ , p , , , ) . This completes the proof of the theorem. □

Corollary 2.4 Let ϕ and ψ be two double sequences then m 2 ( M , A , ϕ , p , , , ) = m 2 ( M , A , ψ , p , , , ) if and only if sup ( s , t ) ( 1 , 1 ) α s , t < and sup ( s , t ) ( 1 , 1 ) α s , t 1 < .

Proof It is easy to prove so we omit the details. □

Theorem 2.5 Let M = ( M k , l ) , M = ( M k , l ) and M = ( M k , l ′′ ) be sequences of Orlicz functions satisfying Δ 2 -condition. Then
  1. (i)

    m 2 ( M , ϕ , , , ) m 2 ( M M , ϕ , , , ) ,

     
  2. (ii)

    m 2 ( M , A , ϕ , p , , , ) m 2 ( M , A , ϕ , p , , , ) m 2 ( M + M , A , ϕ , p , , , ) .

     
Proof (i) Let x = { x k , l } m 2 ( M , A , ϕ , p , , , ) . Then there exists ρ > 0 such that
sup { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( A i j ( x ) ρ , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } < .
By the continuity of , we can take a number δ with 0 < δ < 1 such that M k , l ( t ) < ϵ , whenever 0 t < δ , for arbitrary 0 < ϵ < 1 . Now let
y i , j = ( A i , j ( x ) ρ , z 1 , , z n 1 ) .
Thus we have
i σ 1 j σ 2 k , l = 1 M k , l ( y i , j ) p i , j = y i , j δ k , l = 1 M k , l ( y i , j ) p i , j + y i , j > δ k , l = 1 M k , l ( y i , j ) p i , j .
By the properties of the Orlicz function we have
y i , j k , l = 1 M k , l ( y i , j ) p i , j max { 1 , M k , l ( 1 ) H } y i , j ( y i , j ) p i , j .
Again, we have
M k , l ( y i , j ) < M k , l ( 1 + y i , j δ ) < 1 2 M k , l ( 2 ) + 1 2 M k , l ( 2 y i , j δ )
for y i , j > δ . If satisfies the Δ 2 -condition, then we have
M k , l ( y i , j ) < 1 2 T y i , j δ M k , l ( 2 ) + 1 2 T y i , j δ M k , l ( 2 ) ,
and so
y i , j > δ M k , l ( y i , j ) p i , j max ( 1 , ( T δ M k , l ( 2 ) ) H ) y i , j > δ y i , j .
Hence, we have
sup { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( M k , l ( A i j ( x ) ρ , z 1 , , z n 1 ) ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } < max { 1 , M k , l ( 1 ) H } sup { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( y i , j ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } < + max ( 1 , ( T δ M k , l ( 2 ) ) H ) × sup { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( y i , j ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } < .
Thus, we have x = { x k , l } m 2 ( M M , ϕ , , , ) and hence m 2 ( M , ϕ , , , ) m 2 ( M M , ϕ , , , ) .
  1. (ii)
    Let x = { x k , l } m 2 ( M , A , ϕ , p , , , ) m 2 ( M , A , ϕ , p , , , ) . Then there exists a ρ > 0 such that
    sup { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ( A i , j ( x ) ρ , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } <
     
and
sup { 1 ϕ s , t i σ 1 j σ 2 k , l = 1 M k , l ′′ ( A i , j ( x ) ρ , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } < .
By the inequality, we have
i σ 1 j σ 2 k , l = 1 ( M k , l + M k , l ′′ ) ( A i , j ( x ) ρ , z 1 , , z n 1 ) p i j max ( 1 , 2 H 1 ) i σ 1 j σ 2 k , l = 1 M k , l ( A i , j ( x ) ρ , z 1 , , z n 1 ) p i j + max ( 1 , 2 H 1 ) i σ 1 j σ 2 k , l = 1 M k , l ′′ ( A i , j ( x ) ρ , z 1 , , z n 1 ) p i j .
Hence
m 2 ( M , A , ϕ , p , , , ) m 2 ( M , A , ϕ , p , , , ) m 2 ( M + M , A , ϕ , p , , , ) .

This completes the proof of the theorem. □

Theorem 2.6 The sequence space m 2 ( M , ϕ , p , , , ) is solid.

Proof Let α = { α k , l } be a double sequence of scalars such that | α k , l | 1 and y = { y k , l } m 2 ( M , ϕ , p , , , ) . Then we have
sup { 1 ϕ s , t k σ 1 l σ 2 M k , l ( α k , l x k , l ρ , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } sup { 1 ϕ s , t k σ 1 l σ 2 M k , l ( α k , l y k , l ρ , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } sup { 1 ϕ s , t k σ 1 l σ 2 M k , l ( y k , l ρ , z 1 , , z n 1 ) p i j : ( s , t ) ( 1 , 1 ) , σ 1 × σ 2 P s , t } .

This implies that { α k , l y k , l } m 2 ( M , ϕ , p , , , ) . This proves that the space m 2 ( M , ϕ , p , , , ) is a solid. □

Corollary 2.7 The sequence space m 2 ( M , ϕ , p , , , ) is monotone.

Proof It is trivial so we omit the details. □

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Abdulaziz University
(2)
Department of Mathematics, Aligarh Muslim University
(3)
Department of Mathematics, Model Institute of Engineering and Technology

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© Alotaibi et al.; licensee Springer. 2014

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