Massera criterion for periodic solutions of differential equations with piecewise constant argu\-ment (Q1763792)

This paper deals with differential equations with piecewise constant argument of the form \[ \dot x=Ax([t])+f([t]), \quad x(t)\in C^n, \] where \(A\) is a linear operator on \(C^n\), \(f\) is a bounded continuous function from \(\mathbb{R}\) to \(C^n\), [\(\cdot\)] is the largest integer function. The main purpose of this paper is to show a spectral condition for almost-periodicity of bounded solutions and the existence of periodic solutions of the above equation via J. L. Massera's criterion [Duke Math. J. 17, 457--475 (1950; Zbl 0038.25002)].