Massera's theorem for Asymptotically Periodic Scalar Differential Equations

The aim of this paper is studying the problem of existence of asymptotically periodic solutions of the scalar differential equation $x^{\mathrm{\prime }}=f(t,x),$ where $f:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is a continuous asymptotically $\tau$-periodic function. We prove that every bounded on semi-axis ${\mathbb{R}}_{+}$ solution $\phi$ of this equation is S-asymptotically $\tau$-periodic, i.e., $\underset{t\to +\mathrm{\infty }}{\lim }\mathrm{\mid }\phi (t+\tau )-\phi (t)\mathrm{\mid }=0$. This statement is a generalization of the well-known Massera’s theorem for asymptotically periodic scalar differential equations. We also establish a similar statement for scalar difference equations.