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

1 Global Journal of Science Frontier Research Volume XXI Issue IV Year 2021 3 ( F ) © 2021 Global Journals Version I Difference Sequence Spaces of Second order Defined by a Sequence of Moduli where u ∈ U. Asma and Colak[1] defined the following sequence spaces ` ∞ ( u, 4 , p ) = { x = ( x k ) ∈ ω : ( | u k 4 x k | ) p k ∈ ` ∞ ( p ) } c ( u, 4 , p ) = { x = ( x k ) ∈ ω : ( | u k 4 x k | ) p k ∈ c ( p ) } c 0 ( u, 4 , p ) = { x = ( x k ) ∈ ω : ( | u k 4 x k | ) p k ∈ c 0 ( p ) } where u ∈ U , p = ( p k ) be any sequence of positive reals. Khan and Lohani [5] defined the following sequence spaces ` ∞ ( u, 4 , F, p ) = { x = ( x k ) ∈ ω : sup k ≥ 0 ( f k ( | u k 4 x k | )) p k < ∞} c ( u, 4 , F, p ) = { x = ( x k ) ∈ ω : lim k →∞ ( f k ( | u k 4 x k − l | )) p k = 0 , l ∈ C } c 0 ( u, 4 , F, p ) = { x = ( x k ) ∈ ω : lim k →∞ ( f k ( | u k 4 x k | )) p k = 0 } which are paranormed spaces paranormed with Q ( x ) = sup k ≥ 0 ( f k ( | u k 4 x k | )) p k ) 1 H ≤ a where H = max (1 , sup k ≥ 0 p k ) and a = f k ( l ), l = sup k ≥ 0 ( | u k 4 x k | ) . In this article we introduce the following class of sequence spaces. ` ∞ ( u, 4 2 , F, p ) = { x = ( x k ) ∈ ω : sup k ≥ 0 ( f k ( | u k 4 2 x k | )) p k < ∞} c ( u, 4 2 , F, p ) = { x = ( x k ) ∈ ω : lim k →∞ ( f k ( | u k 4 2 x k − l | )) p k = 0 , l ∈ C } c 0 ( u, 4 2 , F, p ) = { x = ( x k ) ∈ ω : lim k →∞ ( f k ( | u k 4 2 x k | )) p k = 0 } ` ∞ ( u, 4 2 , F ) is a Banach space with norm || x || 4 2 = sup k ≥ 0 ( f k ( | u k 4 2 x k | )) ≤ α, where α = f k ( l ) and l = sup k ≥ 0 ( | u k 4 2 x k | ) . II. M ain R esults Theorem 2.1. 1. Asma, C., Colak, R. On the K Ö the-Toeplitz duals of generalised sets of difference sequences. Demonstratio Math; 33(4), (2000)797-803. R ef