Massera's theorems for various types of equations with discontinuous solutions

Abstract

We present new Massera-type theorems for various types of equations with periodic right-hand sides. We deal with generalized ordinary differential equations, measure differential equations, impulsive equations (all of which might have discontinuous solutions), as well as dynamic equations on time scales.

For scalar nonlinear equations, we find sufficient conditions guaranteeing that each bounded solution is asymptotic to a periodic solution. For linear systems, we show that the existence of a bounded solution implies the existence of a periodic solution.

We include some examples to illustrate our results.

Keywords

• Massera theorem
• Periodic solution
• Generalized ordinary differential equations
• Measure differential equations
• Impulsive equations
• Dynamic equations on time scales