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Existence results for periodic solutions of integro-dynamic equations on time scales. (English) Zbl 1176.45008
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)].

45G15Systems of nonlinear integral equations
34N05Dynamic equations on time scales or measure chains
45M15Periodic solutions of integral equations
Full Text: DOI
[1] Akin-Bohner, E., Raffoul, Y.N.: Boundedness in functional dynamic equations on time scales. Adv. Difference Equ., pages Art. ID 79689, 18 (2006) · Zbl 1139.39005
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[9] Islam M.N., Raffoul Y.N.: Periodic solutions of neutral nonlinear system of differential equations with functional delay. J. Math. Anal. Appl. 331(2), 1175--1186 (2007) · Zbl 1118.34057 · doi:10.1016/j.jmaa.2006.09.030
[10] Kaufmann E.R., Raffoul Y.N.: Periodic solutions for a neutral nonlinear dynamical equation on a time scale. J. Math. Anal. Appl. 319(1), 315--325 (2006) · Zbl 1096.34057 · doi:10.1016/j.jmaa.2006.01.063
[11] Kaufmann, E.R., Raffoul, Y.N.: Periodicity and stability in neutral nonlinear dynamic equations with functional delay on a time scale. Electron. J. Differ. Equ., No. 27, 12 pp. (electronic) (2007) · Zbl 1118.34058
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