The superposition operator in the space of functions continuous and converging at infinity on the real half-axis

Abstract

Abstract We will consider the so-called superposition operator in the space CC(ℝ+) of real functions defined, continuous on the real half-axis ℝ+ and converging to finite limits at infinity. We will assume that the function f = f(t, x) generating the mentioned superposition operator is locally uniformly continuous with respect to the variable x uniformly for t ∈ ℝ+. Moreover, we require that the function t → f(t, x) satisfies the Cauchy condition at infinity uniformly with respect to the variable x. Under the above indicated assumptions a few properties of the superposition operator in question are derived. Examples illustrating our considerations will be included.