Existence of positive periodic solutions for nonlinear neutral dynamic equations with variable delay on a time scale
Downloads
DOI:
https://doi.org/10.26637/mjm102/008Abstract
Let \(\mathbb{T}\) be a periodic time scale. The purpose of this paper is to use Krasnoselskii's fixed point theorem to prove the existence of positive periodic solutions on time scale of the nonlinear neutral dynamic equation with variable delay
$$
(x(t)-g(t, x(t-\tau(t))))^{\triangle}=r(t) x(t)-f(t, x(t-\tau(t))) .
$$
We invert this equation to construct a sum of a contraction and a compact map which is suitable for applying the Krasnoselskii's theorem. The results obtained here extend the works of Raffoul [17] and Ardjouni and Djoudi [3].
Keywords:
Positive periodic solutions, nonlinear neutral dynamic equations, fixed point theorem, time scalesMathematics Subject Classification:
34K13, 34A34, 34K30, 34L30- Pages: 60-67
- Date Published: 01-04-2013
- Vol. 1 No. 02 (2013): Malaya Journal of Matematik (MJM)
M. Adıvar and Y. N. Raffoul, Existence of periodic solutions in totally nonlinear delay dynamic equations, Electronic Journal of Qualitative Theory of Differential Equations, 2009, No. 1, 1-20. DOI: https://doi.org/10.14232/ejqtde.2009.4.1
A. Ardjouni and A. Djoudi, Existence of periodic solutions for nonlinear neutral dynamic equations with variable delay on a time scale. Commun. Nonlinear Sci. Numer. Simulat., 17(2012), 3061-3069. DOI: https://doi.org/10.1016/j.cnsns.2011.11.026
A. Ardjouni and A. Djoudi, Existence of positive periodic solutions for a nonlinear neutral differential equation with variable delay, Applied Mathematics E-Notes, 2012(2011), 94-101.
A. Ardjouni and A. Djoudi, Periodic solutions in totally nonlinear dynamic equations with functional delay on a time scale, Rend. Sem. Mat. Univ. Politec. Torino, 68(4)(2010), 349-359.
F. M. Atici, G. Sh. Guseinov, and B. Kaymakcalan, Stability criteria for dynamic equations on time scales with periodic coefficients, Proceedings of the International Conference on Dynamic Systems and Applications, 3(1999), 43-48.
L. Bi, M. Bohner and M. Fan, Periodic solutions of functional dynamic equations with infinite delay, Nonlinear Anal., 68(2008), 1226-1245. DOI: https://doi.org/10.1016/j.na.2006.12.017
M. Bohner and A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkhäuser, Boston, 2001 DOI: https://doi.org/10.1007/978-1-4612-0201-1
M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003. DOI: https://doi.org/10.1007/978-0-8176-8230-9
T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, 2006.
F. D. Chen, Positive periodic solutions of neutral Lotka-Volterra system with feedback control, Appl. Math. Comput., 162(3)(2005), 1279-1302. DOI: https://doi.org/10.1016/j.amc.2004.03.009
F. D. Chen and J. L. Shi, Periodicity in a nonlinear predator-prey system with state dependent delays, Acta Math. Appl. Sin. Engl. Ser., 21(1)(2005), 49-60. DOI: https://doi.org/10.1007/s10255-005-0214-2
Y. M. Dib, M.R. Maroun and Y.N. Raffoul, Periodicity and stability in neutral nonlinear differential equations with functional delay, Electronic Journal of Differential Equations, Vol. 2005(2005), No. 142, 1-11.
M. Fan and K. Wang, P. J. Y. Wong and R. P. Agarwal, Periodicity and stability in periodic n-species Lotka-Volterra competition system with feedback controls and deviating arguments, Acta Math. Sin. Engl. Ser., 19(4)(2003), 801-822. DOI: https://doi.org/10.1007/s10114-003-0311-1
E. R. Kaufmann and Y. N. Raffoul, Periodic solutions for a neutral nonlinear dynamical equation on a time scale, J. Math. Anal. Appl., 319(1)(2006), 315–325. DOI: https://doi.org/10.1016/j.jmaa.2006.01.063
E. R. Kaufmann and Y. N. Raffoul, Periodicity and stability in neutral nonlinear dynamic equations with functional delay on a time scale, Electron. J. Differential Equations, Vol. 2007(2007), No. 27, 1–12.
E. R. Kaufmann, A nonlinear neutral periodic differential equation, Electron. J. Differential Equations, Vol. 2010(2010), No. 88, 1–8.
Y. N. Raffoul, Positive periodic solutions in neutral nonlinear differential equations, Electronic Journal of Qualitative Theory of Differential Equations, Vol. 2007(2007), No. 16, 1–10. DOI: https://doi.org/10.14232/ejqtde.2007.1.16
D. S. Smart, Fixed point theorems; Cambridge Tracts in Mathematics, No. 66. Cambridge University Press, London-New York, 1974.
- NA
Similar Articles
- M.C. Ranjini , A. Anguraj, Nonlocal impulsive fractional semilinear differential equations with almost sectorial operators , Malaya Journal of Matematik: Vol. 1 No. 02 (2013): Malaya Journal of Matematik (MJM)
You may also start an advanced similarity search for this article.
Metrics
Published
How to Cite
Issue
Section
License
Copyright (c) 2013 MJM
This work is licensed under a Creative Commons Attribution 4.0 International License.
