Remotely Almost Periodic Solutions of Scalar Differential Equations
David Cheban
TL;DR
The paper addresses when bounded solutions of scalar nonautonomous systems with time-dependent forcing that is remotely almost periodic inherit remote almost periodicity themselves. It develops a one-dimensional monotone cocycle/skew-product framework and analyzes omega-limit sets to establish remotely almost periodic behavior under monotone, regular forcing, extending Opial's theorem to remotely almost periodic coefficients for both differential and difference equations. The main contribution is a general theorem showing that, under asymptotically almost periodic (or tau-periodic) time dependence, monotonicity in the state, and a minimal omega-limit set, bounded solutions are remotely almost periodic; the work also provides discrete analogs (e.g., Beverton–Holt) and clarifies the necessity of monotonicity through counterexamples. The results offer robust criteria for the asymptotic structure of solutions to nonautonomous scalar equations with remote forcing, with implications for stability analysis and applications in continuous and discrete settings.
Abstract
The aim of this paper is to study the problem of existence of remotely almost periodic solutions for the scalar differential equation $x'=f(t,x),$ 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 $\varphi$ 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.