Periodic solutions for dynamic equations on time scales (Q885317)

The paper is concerned with the existence of periodic solutions for linear and nonlinear dynamic equations on a time scale of the form \[ x^{\Delta}(t)=a(t)x(t)+h(t)\tag{1} \] and \[ x^{\Delta}(t)=f(t,x).\tag{2} \] For the \(\omega\)-periodic linear equation, the analogue of the celebrated Massera's theorem is proved stating that (1) has an \(\omega\)-periodic solution if and only if it has a bounded solution. For the \(\omega\)-periodic nonlinear dynamic equation, it is demonstrated that equi-boundedness and ultimate boundedness of solutions of (2) imply existence of an \(\omega \)-periodic solution. In the final part of the paper, a criterion for the ultimate boundedness of solutions of (2) in terms of Lyapunov functions is established.