Remotely Almost Periodic Solutions of Scalar Differential Equations

The aim of this paper is to study the problem of existence of remotely almost periodic solutions for the scalar differential equation $x^{\mathrm{\prime }}=f(t,x),$ 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.