On the limit cycles for a class of eighth-order differential equations

In this article, we provide sufficient conditions for the existence of periodic solutions of the eighth-order differential equation x ( 8 ) - ( 1 + p 2 + λ 2 + μ 2 ) x ( 6 ) + A x ⃜ + B x ¨ + p 2 λ 2 μ 2 x = ɛ F ( t , x , x ˙ , x ¨ , x ⃛ , x ⃜ , x ( 5 ) , x ( 6 ) x ( 7 ) ) , {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 = p 2 λ 2 + p 2 µ 2 + λ 2 µ 2 + p 2 + λ 2 + µ 2 , B = p 2 λ 2 + p 2 µ 2 + λ 2 µ 2 + p 2 λ 2 µ 2 , 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.