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Periodic solutions for dynamic equations on time scales. (English) Zbl 1124.34028

The paper is concerned with the existence of periodic solutions for linear and nonlinear dynamic equations on a time scale of the form

x Δ (t)=a(t)x(t)+h(t)(1)

and

x Δ (t)=f(t,x)·(2)

For the ω-periodic linear equation, the analogue of the celebrated Massera’s theorem is proved stating that (1) has an ω-periodic solution if and only if it has a bounded solution. For the ω-periodic nonlinear dynamic equation, it is demonstrated that equi-boundedness and ultimate boundedness of solutions of (2) imply existence of an ω-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.

MSC:
34C25Periodic solutions of ODE
39A12Discrete version of topics in analysis