Periodic solutions for a second order nonlinear functional differential equation. (English) Zbl 1151.34056

\[ L(x(t)):=x''(t)+p(t)x'(t)+q(t)x(t), \]
where \(p,q:\mathbb R\to\mathbb R^+\) are continuous \(T\)-periodic functions with positive average, and \(T>0\).
The main result of this paper establishes sufficient conditions to ensure the existence of at least one \(T\)-periodic solution for the second order delay-differential equation \[ L(x(t))=r(t)x'(t-\tau(t))+f(t,x(t),x(t-\tau(t))).\tag{1} \] Here, \(r,\tau:\mathbb R\to\mathbb R\) are continuous and \(T\)-periodic, and the continuous function \(f(t,x,y)\) is \(T\)-periodic in \(t\) for all \((x,y)\in\mathbb R^2\).
The Green function for the periodic problem associated to the ordinary differential equation \(L(x(t))=\phi(t)\) is used to define a suitable abstract operator whose fixed points are the periodic solutions of (1). Then, a fixed point theorem due to Krasnosel’skii is applied to get the desired existence result. Under an additional condition, this operator is shown to be a contraction, and therefore the \(T\)-periodic solution is unique.
Reviewer: Eduardo Liz (Vigo)


34K13 Periodic solutions to functional-differential equations
34B27 Green’s functions for ordinary differential equations
Full Text: DOI


[1] Freedman, H.I.; Wu, J., Periodic solutions of single-species models with periodic delay, SIAM J. math. anal., 23, 689-701, (1992) · Zbl 0764.92016
[2] Kuang, Y., Delay differential equations with application in population dynamics, (1993), Academic Press New York
[3] Tang, B.; Kuang, Y., Existence, uniqueness and asymptotic stability of periodic solutions of periodic functional-differential systems, Tohoku math. J., 49, 217-239, (1997) · Zbl 0883.34074
[4] Wan, A.; Jiang, D., Existence of positive periodic solutions for functional differential equations, Kyushu J. math., 56, 193-202, (2002) · Zbl 1012.34068
[5] Wan, A.; Jiang, D.; Xu, X., A new existence theory for positive periodic solutions to functional differential equations, Comput. math. appl., 47, 1257-1262, (2004) · Zbl 1073.34082
[6] Cheng, S.; Zhang, G., Existence of positive periodic solutions for non-autonomous functional differential equations, Electron. J. differential equations, 59, 1-8, (2001)
[7] Zhang, G.; Cheng, S., Positive periodic solutions of nonautonomous functional differential equations depending on a parameter, Abstr. appl. anal., 7, 279-286, (2002) · Zbl 1007.34066
[8] Raffoul, Y.N., Periodic solutions for neutral nonlinear differential equations with functional delay, Electron. J. differential equations, 102, 1-7, (2003) · Zbl 1054.34115
[9] Liu, Y.; Ge, W., Positive periodic solutions of nonlinear Duffing equations with delay and variable coefficients, Tamsui oxf. J. math. sci., 20, 235-255, (2004) · Zbl 1087.34047
[10] Smart, D.R., Fixed points theorems, (1980), Cambridge University Press Cambridge · Zbl 0427.47036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.