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

Alotaibi et al. Journal of Inequalities and Applications 2014, 2014:216 http://www.journalofinequalitiesandapplications.com/content/2014/1/216 RESEARCH Open Access Double sequence spaces over n-normed spaces defined by a sequence of Orlicz functions Abdullah Alotaibi1 , Mohammad Mursaleen2* and Sunil K Sharma3 * Correspondence: 2 Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India Full list of author information is available at the end of the article Abstract In the present paper we introduce double sequence space m2 (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 spaces; paranormed space; Orlicz function; n-normed space 1 Introduction and preliminaries The initial works on double sequences is found in Bromwich []. Later on, it was studied by Hardy [], Moricz [], Moricz and Rhoades [], Başarır and Sonalcan [] and many others. Hardy [] introduced the notion of regular convergence for double sequences. Quite recently, Zeltser [] 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 [] have recently introduced the statistical convergence which was further studied in locally solid Riesz spaces []. Nextly, Mursaleen [] and Mursaleen and Savas [] 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 = (xk,l ) into one whose core is a subset of the M-core of x. More recently, Altay and Başar [] have defined the spaces BS , BS (t), CSp , CSbp , CSr and BV of double sequences consisting of all double series whose sequence of partial sums are in the spaces Mu , Mu (t), Cp , Cbp , Cr and Lu , respectively and also examined some properties of these sequence spaces and determined the α-duals of the spaces BS , BV , CSbp and the β(v)-duals of the spaces CSbp and CSr of double series. Recently Başar and Sever [] have introduced the Banach space Lq of double sequences corresponding to the well known space q of single sequences and examined some properties of the space Lq . Now, recently Raj and Sharma [] 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 = (xk,l ) has Pringsheim limit L (denoted by P-lim x = L) provided that given  >  there exists n ∈ N such that |xk,l – L| <  whenever k, l > n, see []. The double ©2014 Alotaibi et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Alotaibi et al. Journal of Inequalities and Applications 2014, 2014:216 http://www.journalofinequalitiesandapplications.com/content/2014/1/216 sequence x = (xk,l ) is bounded if there exists a positive number M such that |xk,l | < M for all k and l. Throughout this paper, N and C denote the set of positive integers and complex numbers, respectively. A complex double sequence is a function x from N × N into C and briefly denoted by {xk,l }. If for all  > , there is n ∈ N such that |xk,l – a| <  where k > n and l > n , then a double sequence {xk,l } is said to be convergent to a ∈ C. A real double sequence {xk,l } is non-decreasing, if xk,l ≤ xp,q for (k, l) < (p, q). A double series is infinite sum ∞ k,l= xk,l and its convergence implies the convergence of partial sums sequence {Sn,m },  n where Sn,m = m k= l= xk,l (see []). For recent development on double sequences, we refer to [–] and [–]. A double sequence space E is said to be solid if {xk,l yk,l } ∈ E for all double sequences {yk,l } of scalars such that |yk,l | <  for all k, l ∈ N whenever {xk,l } ∈ E. Let x = {xk,l } be a double sequence. A set S(x) is defined by   S(x) = {Xπ (k),π (k) } : π and π are permutation of N . If S(x) ⊆ E for all x ∈ E, then E is said to be symmetric. Now let Ps be a family of subsets σ having at most elements s in N. Also Ps,t denotes the class of subsets σ = σ × σ in N × N such that the element numbers of σ and σ are at most s and t, respectively. Besides {φk,l } is taken as a non-decreasing double sequence of the positive real numbers such that kφk+,l ≤ (k + )φk,l , lφk,l+ ≤ (l + )φk,l . An Orlicz function M : [, ∞) → [, ∞) is a continuous, non-decreasing, and convex function such that M() = , M(x) >  for x >  and M(x) → ∞ as x → ∞. Lindenstrauss and Tzafriri [] used the idea of Orlicz function to define the following sequence space:  M = x ∈ w : ∞   M k= |xk | ρ  <∞ , which is called an Orlicz sequence space. Also M is a Banach space with the norm  x = inf ρ >  : ∞  k=  |xk | M ρ  ≤ . Also, it was shown that every Orlicz sequence space M contains a subspace isomorphic to p (p ≥ ). The  -condition is equivalent to M(Lx) ≤ LM(x), for all L with  < L < . An Orlicz function M can always be represented in the following integral form: x M(x) = η(t) dt,  where η, known as the kernel of M, is right differentiable for t ≥ , η() = , η(t) > , η is non-decreasing and η(t) → ∞ as t → ∞. Page 2 of 12 Alotaibi et al. Journal of Inequalities and Applications 2014, 2014:216 http://www.journalofinequalitiesandapplications.com/content/2014/1/216 For further reading on Orlicz spaces, we refer to [–]. Let X be a linear metric space. A function p : X → R is called a paranorm if () p(x) ≥  for all x ∈ X, () p(–x) = p(x) for all x ∈ X, () p(x + y) ≤ p(x) + p(y) for all x, y ∈ X, () if (λn ) is a sequence of scalars with λn → λ as n → ∞ and (xn ) is a sequence of vectors with p(xn – x) →  as n → ∞, then p(λn xn – λx) →  as n → ∞. A paranorm p for which p(x) =  implies x =  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 [], Theorem .., p.). The concept of -normed spaces was initially developed by Gähler [] in the mid-s, while that of n-normed spaces one can see in Misiak []. Since then, many others have studied this concept and obtained various results; see Gunawan [, ] and Gunawan and Mashadi [] and references therein. Let n ∈ N and X be a linear space over the field K, where K is the field of real or complex numbers of dimension d, where d ≥ n ≥ . A real valued function ·, . . . , · on X n satisfying the following four conditions: () x , x , . . . , xn  =  if and only i (...truncated)