Locally Boundedness and Continuity of Superposition Operators on Double Sequence Spaces C-r0

Abstract

Let R be set of all real numbers, N be the set of all natural numbers and N-2 = N x N. In this paper, we define the superposition operator P-g where g : N-2 x R -> R by P-g ((x(ks))) = g (k, s, x(ks)) for all real double sequence (x(ks)). Chew & Lee [4] and Petranuarat & Kemprasit [11] have characterized P-g : c(0) -> l(1) and P-g : c(0) -> l(q) where 1 <= q < infinity, respectively. The main aim of this paper is to construct the necessary and sufficient conditions for the boundedness and continuity of P-g : C-r0 -> L-1 and P-g : C-r0 -> L-p where 1 <= p < infinity.