Periodic solutions of third order differential equations

Abstract

In this paper, we study the existence of periodic solutions for the following piecewise third-order differential equation: $$ \dddot{x}+\dot{x}-\varepsilon\sum\limits_{i=1}^{2}c_i|x|^i=0, $$ with $\varepsilon$ a real parameter sufficiently small, $c_1$ and $c_2$ real numbers. By applying new results from the averaging theory for continuous differential systems, we prove the existence of at most one periodic solution for the differential equation. An example is given to illustrate the established result.

Keywords

• Averaging theory, Periodic Solutions, Differential equations.