Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-19T04:32:48.318Z Has data issue: false hasContentIssue false
  • English
  • Français

Periodic Solutions of Almost Linear Volterra Integro-dynamic Equations on Periodic Time Scales

Published online by Cambridge University Press:  20 November 2018

Youssef N. Raffoul
Youssef N. Raffoul*
Affiliation:
Department of Mathematics, University of Dayton, Dayton, OH 45469-2316 USA. e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Using Krasnoselskii’s fixed point theorem, we deduce the existence of periodic solutions of nonlinear system of integro-dynamic equations on periodic time scales. These equations are studied under a set of assumptions on the functions involved in the equations. The equations will be called almost linear when these assumptions hold. The results of this paper are new for the continuous and discrete time scales.

Keywords

45J0545D05Volterra integro-dynamic equationtime scalesKrasnoselsii’s fixed point theoremperiodic solution
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 1 , 01 March 2015 , pp. 174 - 181
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Adivar, M. and Raffoul, Y. N., Existence results for periodic solutions of integro-dynamic equations on time scales. Ann. Mat. Pura Appl. 188 (2009), no. 4, 543–559. http://dx.doi.org/10.1007/s10231-008-0088-Z http://dx.doi.org/10.1007/s10231-008-0088-z Google Scholar
[2] Bohner, M. and Guseinov, G. Sh., Double integral calculus of variations on time scales. Comput. Math. Appl. 54 (2007), no. 1, 45–57. http://dx.doi.org/10.1016/j.camwa.2006.10.032 Google Scholar
[3] Akin-Bohner, E. and Raffoul, Y. N., Boundedness in functional dynamic equations on time scales. Adv. Difference Equ. 2006, Art. ID 79689.Google Scholar
[4] Bohner, M. and Peterson, A., Dynamic equations on time scales. An introduction with applications. Birkhäuser Boston, Inc., Boston, MA, 2001.Google Scholar
[5] Elaydi, S., Periodicity and stability of linear Volterra difference systems. J. Math. Anal. Appl. 181 (1994), no. 2, 483–492. http://dx.doi.org/10.1006/jmaa.1994.1037 Google Scholar
[6] Grossman, S. I. and Miller, R. K., Perturbation theory for Volterra integrodifferential systems. J. Differential Equations 8 (1970), 457–474. http://dx.doi.org/10.1016/0022-0396(70)90018-5 Google Scholar
[7] Hino, Y. and Murakami, S., Stabilities in linear integrodifferential equations. In: Finite- and infinite-dimensional dynamics (Kyoto, 1988), Lecture Notes Numer. Appl. Anal. 15, Kinokuniya, Tokyo, 1996, pp. 31–46..Google Scholar
[8] Islam, M. N. and Neugebauer, J., Qualitative properties of nonlinear Volterra integral equations. Electron. J. Qual. Theory Differ. Equ. 2008, no. 12, 1–16. Google Scholar
[9] Islam, M. N. and Raffoul, Y. N., Periodic solutions of neutral nonlinear system of differential equations with functional delay. J. Math. Anal. Appl. 331 (2007), no. 2, 1175–1186. http://dx.doi.org/10.1016/j.jmaa.2006.09.030 Google Scholar
[10] Kaufmann, E. and Raffoul, Y. N., Periodicity and stability in neutral nonlinear dynamic equations with functional delay on a time scale. Electron. J. Differential Equations 2007, no. 27, 1–12. Google Scholar
[11] Kaufmann, E. and Raffoul, Y. N., Periodic solutions for a neutral nonlinear dynamical equations on time scale. J. Math. Anal. Appl. 319 (2006), no. 1, 315–325. http://dx.doi.org/10.1016/j.jmaa.2006.01.063 Google Scholar
[12] Miller, R. K., Nonlinear Volterra integral equations. W. A. Benjamin Inc.,Menlo Park, CA, 1971.Google Scholar
[13] Miller, R. K., Nohel, J. A., and J. S.W.Wong, Perturbations of Volterra integral equations. J. Math. Anal. Appl. 25 (1969), 676–691. http://dx.doi.org/10.1016/0022-247X(69)90265-0 Google Scholar
[14] Peterson, A. and Tisdell, C., Boundedness and uniqueness of solutions to dynamic equations. J. Difference Equ. Appl. 10 (2004), no. 13–15. 1295–1306. http://dx.doi.org/10.1080/10236190410001652793 Google Scholar
[15] Raffoul, Y. N., Periodicity in nonlinear systems with infinite delay. Adv. Dyn. Syst. Appl. 3 (2008), no. 1, 185–194. Google Scholar
[16] Schaefer, H., Über die Methode der a priori-Schranken. Math. Ann. 129 (1955), 415–416. http://dx.doi.org/10.1007/BF01362380 Google Scholar
[17] Smart, D. R., Fixed point theorems. Cambridge Tracts ion Mathematics, 66, Cambridge University Press, London-New Yok, 1980.Google Scholar
[18] Strauss, A., On a perturbed Volterra integral equation. J. Math. Anal. Appl. 30 (1970), 564–575. http://dx.doi.org/10.1016/0022-247X(70)90141-1 Google Scholar
[19] Zhang, B., Asymptotic stability criteria and integrability properties of the resolvent of Volterra and functional equations. Funkcial. Ekvac. 40 (1997), no. 3, 335–351. Google Scholar