# Periodic Solutions for Third-Order Nonlinear Delay Differential Equations with Variable Coefficients

Document Type: Original Article

Authors

1 Department of Mathematics and Informatics, University of Souk Ahras, P.O. Box 1553, Souk Ahras, 41000, Algeria.

2 Department of Mathematics, University of Annaba, P.O. Box 12, Annaba, 23000, Algeria.

Abstract

In this paper, the following third-order nonlinear delay differential equation
with periodic coefficients%
\begin{align*}
& x^{\prime\prime\prime}(t)+p(t)x^{\prime\prime}(t)+q(t)x^{\prime
}(t)+r(t)x(t)\\
& =f\left( t,x\left( t\right) ,x(t-\tau(t))\right) +\frac{d}{dt}g\left(
t,x\left( t-\tau\left( t\right) \right) \right) ,
\end{align*}
is considered. By employing Green's function, Krasnoselskii's fixed point
theorem and the contraction mapping principle, we state and prove the
existence and uniqueness of periodic solutions to the third-order nonlinear
delay differential equation.

Keywords

Main Subjects