We would like to generalize to non-Newtonian real numbers the usual Lebesgue measure in real numbers. For this purpose, we introduce the Lebesgue measure on open and closed sets in non-Newtonian sense and examine their basic properties.
In this paper, we define the superposition operator Pg where g : N2 x R ? R
by Pg((xks))=g(k,s,xks) for all real double sequence (xks). Chew & Lee
[4] and Petranuarat & Kemprasit [7] have characterized Pg : lp ? l1 and Pg :
lp ? lq where 1 ? p,q < ?, respectively. The main goal of this paper is to
construct the necessary and sufficient conditions for the continuity of Pg:
Lp ? L1 and Pg : Lp ? Lq where 1 ? p,q < ?.
In this paper, we define a non-Newtonian superposition operator N P f where f : N × R(N) α → R(N) β by N P f (x) = f (k, x k) ∞ k=1 for every non-Newtonian real sequence x = (x k). Chew and Lee [4] have characterized P f : p → 1 and P f : c 0 → 1 for 1 ≤ p < ∞. The purpose of this paper is to generalize these works respect to the non-Newtonian calculus. We characterize N P f : ∞ (N) → 1 (N) , N P f : c 0 (N) → 1 (N) , N P f : c (N) → 1 (N) and N P f : p (N) → 1 (N), respectively. Then we show that such N P f : ∞ (N) → 1 (N) is *-continuous if and only if f (k, .) is *-continuous for every k ∈ N.
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