Massera's theorems for various types of equations with discontinuous solutions (Q2003975)

In the article under review, the authors present new Massera's type results for generalized ordinary differential equations in finite dimensional spaces \(\mathbb{R}^n\). For \(n=1\) it is proven that, under appropriate conditions, each bounded solution of the equation \[ \frac{d}{dt}x=D_t F(x,t) \] is asymptotic to a periodic solution. For the general case of a finite-dimensional space, for linear problems \[ \frac{d}{dt}x=D_t[A(t)x+f(t)] \] the existence of a solution implies the existence of a periodic solution. These results are then used to get new Massera's type theorems for measure differential equations, for dynamic equations on time scales and for impulsive differential equations with a countable number of impulses.