AbstractIn this article, we provide sufficient conditions for the existence of periodic solutions of the eighth-order differential equation {x^{\left( 8 \right)}} - \left( {1 + {p^2} + {\lambda ^2} + {\mu ^2}} \right){x^{\left( 6 \right)}} + A\ddddot x + B\ddot x + {p^2}{\lambda ^2}{\mu ^2}x = \varepsilon F\left( {t,x,\dot x,\ddot x,\dddot x,\ddddot x,{x^{\left( 5 \right)}},{x^{\left( 6 \right)}}{x^{\left( 7 \right)}}} \right), where A = p2λ2 + p2µ2 + λ2µ2 + p2 + λ2 + µ2, B = p2 λ2 + p2µ2 + λ2µ2 + p2λ2µ2, with λ, µ and p are rational numbers different from −1, 0, 1, and p ≠ ±λ, p ≠±µ, λ ≠±µ, ɛ is sufficiently small and F is a nonlinear non-autonomous periodic function. Moreover we provide some applications.

PurposeThis study aims to provide sufficient conditions for the existence of periodic solutions of the fifth-order differential equation.Design/methodology/approachThe authors shall use the averaging theory, more precisely Theorem $6$.FindingsThe main results on the periodic solutions of the fifth-order differential equation (equation (1)) are given in the statement of Theorem 1 and 2.Originality/valueIn this article, the authors provide sufficient conditions for the existence of periodic solutions of the fifth-order differential equation.