Existence of Positive Periodic Solutions for a Third-Order Delay Differential Equation
Main Article Content
Abstract
In this paper, the following third-order nonlinear delay differential equation with periodic coefficients
x”²”²”²(t)+p(t)x”²”²(t)+q(t)x”²(t)+r(t)x(t)=f(t,x(t),x(t-Ï„(t)))+c(t)x”²(t-Ï„(t)),
is considered. By employing Green's function and Krasnoselskii's fixed point theorem, we state and prove the existence of positive periodic solutions to the third-order delay differential equation.
x”²”²”²(t)+p(t)x”²”²(t)+q(t)x”²(t)+r(t)x(t)=f(t,x(t),x(t-Ï„(t)))+c(t)x”²(t-Ï„(t)),
is considered. By employing Green's function and Krasnoselskii's fixed point theorem, we state and prove the existence of positive periodic solutions to the third-order delay differential equation.
Article Details
References
- A. Ardjouni and A. Djoudi, Existence of periodic solutions for a second-order nonlinear neutral differential equation with variable delay, Palestine Journal of Mathematics, 3(2) (2014), 191-197.
- A. Ardjouni, A. Djoudi and A. Rezaiguia, Existence of positive periodic solutions for two types of third-order nonlinear neutral differential equations with variable delay, Applied Mathematics E-Notes, 14 (2014), 86-96.
- A. Ardjouni and A. Djoudi, Existence of positive periodic solutions for a nonlinear neutral differential equations with variable delay, Applied Mathematics E-Notes, 12 (2012), 94-101.
- A. Ardjouni and A. Djoudi, Existence of periodic solutions for a second order nonlinear neutral differential equation with functional delay, Electronic Journal of Qualitative Theory of Differential Equations, 2012 (2012), Art. ID 31, 1-9.
- A. Ardjouni and A. Djoudi, Periodic solutions for a second-order nonlinear neutral differential equation with variable delay, Electron. J. Differential Equations, 2011 (2011), Art. ID 128, 1-7.
- 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.
- T. A. Burton, Liapunov functionals, fixed points and stability by Krasnoselskii's theorem, Nonlinear Stud. 9 (2) (2002), 181-190.
- 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.
- 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.
- Z. Cheng and J. Ren, Existence of positive periodic solution for variable-coefficient third-order differential equation with singularity, Math. Meth. Appl. Sci. 37 (2014), 2281-2289.
- Z. Cheng and Y. Xin, Multiplicity Results for variable-coefficient singular third-order differential equation with a parameter, Abstract and Applied Analysis, 2014 (2014), Article ID 527162, 1-10.
- S. Cheng and G. Zhang, Existence of positive periodic solutions for non-autonomous functional differential equations, Electron. J. Differential Equations, 2001 (2001), Art. ID 59, 1-8.
- H. Deham and A. Djoudi, Periodic solutions for nonlinear differential equation with functional delay, Georgian Mathematical Journal, 15 (4) (2008), 635-642.
- H. Deham and A. Djoudi, Existence of periodic solutions for neutral nonlinear differential equations withvariable delay, Electronic Journal of Differential Equations, 2010 (2010), Art. ID 127, 1-8.
- Y. M. Dib, M. R. Maroun and Y. N. Rafoul, Periodicity and stability in neutral nonlinear differential equations with functional delay, Electronic Journal of Differential Equations, 2005 (2005), Art. ID 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.
- H. I. Freedman, J. Wu, Periodic solutions of single-species models with periodic delay, SIAM J. Math. Anal. 23 (1992), 689-701.
- Y. Kuang, Delay Differential Equations with Application in Population Dynamics, Academic Press, New York, 1993.
- W. G. Li and Z. H. Shen, An constructive proof of the existence Theorem for periodic solutions of Duffng equations, Chinese Sci. Bull. 42 (1997), 1591-1595.
- Y. Liu, W. Ge, Positive periodic solutions of nonlinear Duffing equations with delay and variable coefficients, Tamsui Oxf. J. Math. Sci. 20 (2004), 235-255.
- J. Ren, S. Siegmund and Y. Chen, Positive periodic solutions for third-order nonlinear differential equations, Electron. J. Differential Equations, 2011 (2011), Art. ID 66, 1-19.
- D. R. Smart, Fixed Points Theorems, Cambridge University Press, Cambridge, 1980.
- Q. Wang, Positive periodic solutions of neutral delay equations (in Chinese), Acta Math. Sinica (N.S.) 6(1996), 789-795.
- Y. Wang, H. Lian and W. Ge, Periodic solutions for a second order nonlinear functional differential equation, Applied Mathematics Letters, 20 (2007), 110-115.
- W. Zeng, Almost periodic solutions for nonlinear Duffing equations, Acta Math. Sinica (N.S.) 13 (1997), 373-380.
- G. Zhang, S. Cheng, Positive periodic solutions of non autonomous functional differential equations depending on a parameter, Abstr. Appl. Anal. 7 (2002) 279-286.