Remotely Almost Periodic Solutions of Scalar Differential Equations

Abstract

The aim of this paper is to study the problem of existence of remotely almost periodic solutions for the scalar differential equation $x^{\prime }=f(t,x),$ where $f:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is a continuous, monotone in $x$ and remotely almost periodic in $t$ function. We prove that every solution $\phi$ of this equation bounded on the semi-axis ${\mathbb{R}}_{+}$ is remotely almost periodic. This statement is a generalization of the well-known Opial's theorem for remotely almost periodic scalar differential equations. We also establish a similar statement for scalar difference equations.