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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 T. A. Burton, P. W. Eloe and M. N. Islam [Ann. Mat. Pura Appl., IV. Ser. 161, 271–283 (1992; Zbl 0756.45012)].
MSC:
45G15Systems of nonlinear integral equations
34N05Dynamic equations on time scales or measure chains
45M15Periodic solutions of integral equations
References:
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[4]Bohner M., Peterson A.: Dynamic equations on time scales: an introduction with applications. Birkhäuser Boston Inc., Boston (2001)
[5]Bohner, M., Raffoul, Y.N.: Volterra dynamic equations on time scales. Preprint
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[7]Burton T.A., Eloe P.W., Islam M.N.: Nonlinear integrodifferential equations and a priori bounds on periodic solutions. Ann. Mat. Pura Appl. (4) 161, 271–283 (1992) · Zbl 0756.45012 · doi:10.1007/BF01759641
[8]Elaydi S.: Periodicity and stability of linear Volterra difference systems. J. Math. Anal. Appl. 181(2), 483–492 (1994) · Zbl 0796.39004 · doi:10.1006/jmaa.1994.1037
[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)
[12]Peterson A.C., Tisdell C.C.: Boundedness and uniqueness of solutions to dynamic equations on time scales. J. Difference Equ. Appl. 10(13–15), 1295–1306 (2004) · Zbl 1072.39017 · doi:10.1080/10236190410001652793
[13]Raffoul, Y.N.: Periodicity in nonlinear systems with infinite delay. Submitted
[14]Schaefer H.: Über die Methode der a priori-Schranken. Math. Ann. 129, 415–416 (1955) · Zbl 0064.35703 · doi:10.1007/BF01362380