Generalized spaces of double sequences for Orlicz functions and bounded-regular matrices over n-normed spaces

Mohiuddine et al. Journal of Inequalities and Applications 2014, 2014:332 http://www.journalofinequalitiesandapplications.com/content/2014/1/332 RESEARCH Open Access Generalized spaces of double sequences for Orlicz functions and bounded-regular matrices over n-normed spaces Syed Abdul Mohiuddine1* , Kuldip Raj2 and Abdullah Alotaibi1 * Correspondence: 1 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia Full list of author information is available at the end of the article Abstract The aim of this paper is to introduce some generalized spaces of double sequences with the help of the Musielak-Orlicz function M = (Mjk ) and four-dimensional bounded-regular (shortly, RH-regular) matrices A = (anmjk ) over n-normed spaces. Some topological properties and inclusion relations between these spaces are investigated. MSC: 40A05; 40D25 Keywords: double sequence; Orlicz function; difference sequence; paranormed space; over n-normed spaces; bounded-regular matrices 1 Introduction, notations, and preliminaries The concept of -normed spaces was first introduced by Gähler [] in the mid-s, while that of n-normed spaces one can find in Misiak []. Since then, many others have studied this concept and obtained various results; see Gunawan [, ] and Gunawan and Mashadi []. Let n ∈ N and X be a linear space over the field of real numbers R of dimension d, where d ≥ n ≥ . A real valued function ·, . . . , · on X n satisfying the following four conditions: () x , x , . . . , xn  =  if and only if x , x , . . . , xn are linearly dependent in X, () x , x , . . . , xn  is invariant under permutation, () αx , x , . . . , xn  = |α|x , x , . . . , xn  for any α ∈ R, and () x + x , x , . . . , xn  ≤ x, x , . . . , xn  + x , x , . . . , xn  is called an n-norm on X, and the pair (X, ·, . . . , ·) is called a n-normed space over the field R. For example, we may take X = Rn being equipped with the n-norm x , x , . . . , xn E = the volume of the n-dimensional parallelepiped spanned by the vectors x , x , . . . , xn which may be given explicitly by the formula   x , x , . . . , xn E = det(xij ), where xi = (xi , xi , . . . , xin ) ∈ Rn for each i = , , . . . , n. Let (X, ·, . . . , ·) be an n-normed space of dimension d ≥ n ≥  and {a , a , . . . , an } be a linearly independent set in X. Then © 2014 Mohiuddine 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. Mohiuddine et al. Journal of Inequalities and Applications 2014, 2014:332 http://www.journalofinequalitiesandapplications.com/content/2014/1/332 the function ·, . . . , ·∞ on X n– defined by   x , x , . . . , xn– ∞ = max x , x , . . . , xn– , ai  : i = , , . . . , n defines an (n – )-norm on X with respect to {a , a , . . . , an }. A sequence (xk ) in a n-normed space (X, ·, . . . , ·) is said to converge to some L ∈ X if lim xk – L, z , . . . , zn–  =  for every z , . . . , zn– ∈ X. k→∞ A sequence (xk ) in a n-normed space (X, ·, . . . , ·) is said to be Cauchy if lim xk – xp , z , . . . , zn–  =  for every z , . . . , zn– ∈ X. k,p→∞ If every Cauchy sequence in X converges to some L ∈ X, then X is said to be complete with respect to the n-norm. A complete n-normed space is called n-Banach space. Of the definitions of convergence commonly employed for double series, only that due to Pringsheim permits a series to converge conditionally. Therefore, in spite of any disadvantages which it may possess, this definition is better adapted than others to the study of many problems in double sequences and series. Chief among the reasons why the theory of double sequences, under the Pringsheim definition of convergence, presents difficulties not encountered in the theory of simple sequences is the fact that a double sequence {xij } may converge without xij being a bounded function of i and j. Thus it is not surprising that many authors in dealing with the convergence of double sequences should have restricted themselves to the class of bounded sequences or, in dealing with the summability of double series, to the class of series for which the function whose limit is the sum of the series is a bounded function of i and j. Without such a restriction, peculiar things may sometimes happen; for example, a double power series may converge with partial sum {Sij } unbounded at a place exterior to its associated circles of convergence. Nevertheless there are problems in the theory of double sequences and series where this restriction of boundedness as it has been applied is considerably more stringent than need be. In [], Hardy introduced the concept of regular convergence for double sequences. Some important work on double sequences has also been done by Bromwich []. Later on, it was studied by various authors, e.g. Móricz [], Móricz and Rhoades [], Başarır and Sonalcan [], Mursaleen and Mohiuddine [, ], and many others. Mursaleen [] has defined and characterized the notion of almost strong regularity of four-dimensional matrices and applied these matrices to establish a core theorem (also see []). Altay and Başar [] have recently introduced the double sequence spaces BS , BS (t), CSp , CSbp , CSr , and BV consisting of all double series whose sequence of partial sums are in the spaces Mu , Mu (t), Cp , Cbp , Cr , and Lu , respectively. Başar and Sever [] extended the well known space q from single sequence to double sequences, denoted by Lq , and established its interesting properties. The authors of [] defined some convex and paranormed sequences spaces and presented some interesting characterization. Most recently, Mohiuddine and Alotaibi [] introduced some new double sequences spaces for σ -convergence of double sequences and invariant mean, and also determined some inclusion results for these spaces. For more details on these concepts, one is referred to [–]. Page 2 of 16 Mohiuddine et al. Journal of Inequalities and Applications 2014, 2014:332 http://www.journalofinequalitiesandapplications.com/content/2014/1/332 The notion of difference sequence spaces was introduced by Kızmaz [], who studied the difference sequence spaces l∞ (), c(), and c (). The notion was further generalized by Et and Çolak [] by introducing the spaces l∞ (r ), c(r ), and c (r ). Let w be the space of all complex or real sequences x = (xk ) and let r and s be two nonnegative integers. Then for Z = l∞ , c, c , we have the following sequence spaces:       Z rs = x = (xk ) ∈ w : rs xk ∈ Z , r–  where rs x = (rs xk ) = (r– s xk – s xk+ ) and  xk = xk for all k ∈ N, which is equivalent to the following binomial represe (...truncated)