Continuity of superposition operators on the double sequence spaces Lp

Abstract

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 < ?.

On characterization of boundedness of superposition operators on the Maddox space C_{r0}( p) of double sequences

“…If P g (x) ∈ Y for all x ∈ X, we say that P g acts from X into Y and write P g : X → Y [13]. Also, we shall use some of the following conditions:…”

“…Boundedness of the superposition operators on some sequence spaces was studied by Samae [16], Sagır and Güngör [14] and Chew [4], [5], [6], [8], [10], [11], [18]. Sagır and Güngör [15] characterized the superposition operators P g on C r0 (p) as follows …”

“…Let x = (x ks ) ∈ C r0 (p) such that z − x C r0(p) ≤ (13) for each k, s ∈ N. By (12) and (13), we obtain…”

ON *-BOUNDEDNESS AND *-LOCAL BOUNDEDNESS OF NON-NEWTONIAN SUPERPOSITION OPERATORS IN c_(0,α) AND c_α TO l_(1,β)

“…Dedagich and Zabreiko [2] studied on the superposition operators in the space p . After, some properties of superposition operator, such as boundedness, continuity, were studied by Tainchai [3], Sama-ae [4], Sagır and Güngör [6] and many others. Non-Newtonian superposition operator was defined and characterized in some non-Newtonian sequence spaces by Sagır and Erdogan in [9].…”

On characterization of non-Newtonian superposition operators in some sequence spaces

“…Dedagich and Zabreiko [7] have given the necessary and sufficient conditions for the superposition operators on the sequence spaces p , ∞ and c 0 . After, some properties of superposition operator, such as boundedness, compactness, were studied by Sama-ae [19], Sagır and Güngör [11,16], Kolk and Raidjoe [12], Ogur [14,15] and many others. The purpose of this paper is to generalize these works respect to the non-Newtonian calculus.…”