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Zbl 1176.45008
Ad ivar, Murat; Raffoul, Youssef N.
Existence results for periodic solutions of integro-dynamic equations on time scales. (English)
[J] Ann. Mat. Pura Appl. (4) 188, No. 4, 543-559 (2009). ISSN 0373-3114; ISSN 1618-1891/e

Authors' abstract: Using the topological degree method and Schaefer's fixed point theorem, we deduce the existence of periodic solutions of a system of nonlinear 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 {\it T. A. Burton, P. W. Eloe} and {\it M. N. Islam} [Ann. Mat. Pura Appl., IV. Ser. 161, 271--283 (1992; Zbl 0756.45012)].
[Ahmed Hegazi (Mansoura)]

MSC 2000:: *45G15 Systems of nonlinear integral equations
34N05
45M15 Periodic solutions of integral equations

Keywords: periodic time scale; dynamic equation; Volterra integral equation; Sobolev's inequality; periodic solution; Schaefer's fixed point theorem; Lyapunov's direct method

Citations: Zbl 0756.45012

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