On Characterization of Non-Newtonian Superposition Operators in Some Sequence Spaces

Abstract

In this paper, we define a non-Newtonian superposition operator<inf>N</inf> P<inf>f</inf> where f: N × R(N)<inf>α</inf> → R(N)<inf>β</inf> by<inf>N</inf> P<inf>f</inf> (x) =( f (k, x<inf>k</inf> ))∞ k=1for every non-Newtonian real sequencex= (xk). Chew and Lee [4] have characterized P<inf>f</inf>: ℓ<inf>p</inf> → ℓ<inf>1</inf> and P<inf>f</inf>: c<inf>0</inf> → ℓ<inf>1</inf> for 1 ≤ p < ∞. The purpose of this paper is to generalize these works respect to the non-Newtonian calculus. We characterize<inf>N</inf> P<inf>f</inf>: ℓ<inf>∞</inf> (N) → ℓ<inf>1</inf> (N),<inf>N</inf> P<inf>f</inf>: c<inf>0</inf> (N) → ℓ<inf>1</inf> (N),<inf>N</inf> P<inf>f</inf>: c (N) → ℓ<inf>1</inf> (N) and<inf>N</inf> P<inf>f</inf>: ℓ<inf>p</inf> (N) → ℓ<inf>1</inf> (N), respectively. Then we show that such<inf>N</inf> P<inf>f</inf>: ℓ<inf>∞</inf> (N) → ℓ<inf>1</inf> (N) is *-continuous if and only if f (k,.) is *-continuous for every k ∈ N. © University of Nis. All rights reserved.