Global Journal of Science Frontier Research, F: Mathematics and Decision Science, Volume 21 Issue 4

Difference Sequence Spaces of Second order Defined by a Sequence of Moduli Khalid Ebadullah α & Kibreab Gebreselassie σ I. I ntroduction 1 Global Journal of Science Frontier Research Volume XXI Issue IV Year 2021 27 ( F ) © 2021 Global Journals Version I Let N, R and C be the sets of all natural, real and complex numbers respectively. We write ω = { x = ( x k ) : x k ∈ R or C } , the space of all real or complex sequences. Let ` ∞ , c and c 0 denote the Banach spaces of bounded, convergent and null sequences respectively. The following subspaces of ω were first introduced and discussed by Maddox [10-12]. l ( p ) = { x ∈ ω : P k | x k | p k < ∞} ` ∞ ( p ) = { x ∈ ω : sup k | x k | p k < ∞} c ( p ) = { x ∈ ω : lim k | x k − l | p k = 0 , for some l ∈ C } c 0 ( p ) = { x ∈ ω : lim k | x k | p k = 0 } where p = ( p k ) is a sequence of strictly positive real numbers. The idea of difference sequence sets X 4 = { x = ( x k ) ∈ ω : 4 x = ( x k − x k +1 ) ∈ X } where X = ` ∞ , c or c 0 was introduced by Kizmaz [6]. Kizmaz [6] defined the following sequence spaces, Author α σ : Department of Mathematics, Mai Nefhi College of Science, Eritrea. e-mails: khalidebadullah@gmail.com , kgebreselassie@gmail.com Abstract- In this article we introduce the sequence spaces and for a sequence of moduli, sequence of positive reals and the set of all sequences and establish some inclusion relations. Keywords: paranorm, sequence of moduli, difference sequence spaces. c 0 ( u, 4 2 , F, p ), c ( u, 4 2 , F, p ) ` ∞ ( u, 4 2 , F, p F = ( f k p = ( p k u ∈ U ) ) ) 6. Kizmaz, H. On Certain sequence spaces, Canad.Math.Bull. 24(1981) 169-176. R ef