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The objective of this work is to study the periodic solutions for a class of sixth-order autonomous ordinary differential equations x(6)+(1+p2+q2)x… .+(p2+q2+p2q2)x¨+p2q2x=εF(x,ẋ,x¨,x…,x… .,x(5)), where p and q are rational numbers different from 1, 0, −1 and p ≠ q, ε is a small enough parameter and F ∈ C² is a nonlinear autonomous function.

The authors shall use the averaging theory to study the periodic solutions for a class of perturbed sixth-order autonomous differential equations (DEs). The averaging theory is a classical tool for the study of the dynamics of nonlinear differential systems with periodic forcing. The averaging theory has a long history that begins with the classical work of Lagrange and Laplace. The averaging theory is used to the study of periodic solutions for second and higher order DEs.

All the main results for the periodic solutions for a class of perturbed sixth-order autonomous DEs are presenting in the Theorem 1. The authors present some applications to illustrate the main results.

The authors studied Equation 1 which depends explicitly on the independent variable t. Here, the authors studied the autonomous case using a different approach.

This study aims to provide sufficient conditions for the existence of periodic solutions of the fifth-order differential equation.

The authors shall use the averaging theory, more precisely Theorem $6$.

The main results on the periodic solutions of the fifth-order differential equation (equation (1)) are given in the statement of Theorem 1 and 2.

In this article, the authors provide sufficient conditions for the existence of periodic solutions of the fifth-order differential equation.