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Zbl 1124.34028
Liu, Xi-Lan; Li, Wan-Tong
Periodic solutions for dynamic equations on time scales. (English)
[J] Nonlinear Anal., Theory Methods Appl. 67, No. 5, A, 1457-1463 (2007). ISSN 0362-546X

The paper is concerned with the existence of periodic solutions for linear and nonlinear dynamic equations on a time scale of the form $$x^{\Delta}(t)=a(t)x(t)+h(t)\tag1$$ and $$x^{\Delta}(t)=f(t,x).\tag2$$ 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.
[Svitlana P. Rogovchenko (Famagusta)]

MSC 2000:: *34C25 Periodic solutions of ODE
39A12 Discrete version of topics in analysis

Keywords: dynamic equations on a time scale; bounded solutions; periodic solutions; existence; Lyapunov functions

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