Locally Boundedness and Continuity of Superposition Operators on Double Sequence Spaces Cr0

Abstract

Let R be set of all real numbers, N be the set of all natural numbers and N2 = N×N. In this paper, we define the superposition operator P<inf>g</inf> where g: N2×R → R by P<inf>g</inf> ((x<inf>ks</inf>)) = g(k,s,x<inf>ks</inf>) for all real double sequence (x<inf>ks</inf>). Chew & Lee [4] and Petranuarat & Kemprasit [11] have characterized P<inf>g</inf>: c<inf>0</inf> → l<inf>1</inf> and P<inf>g</inf>: c<inf>0</inf> → l<inf>q</inf> where 1 ≤ q < ∞, respectively. The main aim of this paper is to construct the necessary and sufficient conditions for the boundedness and continuity of P<inf>g</inf>: C<inf>r0</inf> → L<inf>1</inf> and Pg: C<inf>r0</inf> → L<inf>p</inf> where 1≤ p< ∞. © 2015 by Eudoxus Press,LLC,all rights reserved.