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 $x'=f(t,x),$ 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 $\varphi$ of this equation is S-asymptotically $\tau$-periodic, i.e., $\lim\limits_{t\to +\infty}|\varphi(t+\tau)-\varphi(t)|=0$. 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.