Annali di Matematica Pura ed Applicata

, Volume 188, Issue 4, pp 543-559

First online:

Existence results for periodic solutions of integro-dynamic equations on time scales

  • Murat AdıvarAffiliated withDepartment of Mathematics, Izmir University of Economics Email author 
  • , Youssef N. RaffoulAffiliated withDepartment of Mathematics, University of Dayton

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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)


Periodic time scale Dynamic equation Volterra integral equation Sobolev’s inequality Schaefer Lyapunov Periodic solution

Mathematics Subject Classification (2000)

45G15 45D05 34A12 34A34