Abstract
Using the topological degree method and Schaefer’s fixed point theorem, we deduce the existence of periodic solutions of nonlinear system of integro-dynamic equations on periodic time scales. Furthermore, we provide several applications to scalar equations, in which we develop a time scale analog of Lyapunov’s direct method and prove an analog of Sobolev’s inequality on time scales to arrive at a priori bound on all periodic solutions. Therefore, we improve and generalize the corresponding results in Burton et al. (Ann Mat Pura Appl 161:271–283, 1992)
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Adıvar, M., Raffoul, Y.N. Existence results for periodic solutions of integro-dynamic equations on time scales. Annali di Matematica 188, 543–559 (2009). https://doi.org/10.1007/s10231-008-0088-z
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DOI: https://doi.org/10.1007/s10231-008-0088-z
Keywords
- Periodic time scale
- Dynamic equation
- Volterra integral equation
- Sobolev’s inequality
- Schaefer
- Lyapunov
- Periodic solution