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Journal of Applied Nonlinear Dynamics 15(3) (2026) 723--738 | DOI:10.5890/JAND.2026.09.014

Chems Eddine Berrehail, Zineb Bouslah

Department of Mathematics, Applied Mathematics Laboratory, Badji Mokhtar University, Annaba, Algeria

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Abstract

In this paper, we use the averaging theory of first order to study the periodic solutions of the perturbed ninth-order non-autonomous differential equation% \begin{equation*} x^{(9)}-\lambda x^{8}+\psi x^{(7)}-\lambda \psi x^{(6)}+\phi\overset{.....}{x}-\lambda \chi \overset{....}{x}+A(\overset{...}{x}-\lambda \overset{..}{x})+p^{2} b^{2}( c^{2}x-c^{2})=\varepsilon \tilde{F} \end{equation*} where $\tilde{F}=F(t,x,\overset{.}{x},\overset{% ..}{x},\overset{...}{x},\overset{....}{x},x^{(5)},x^{(6)},x^{(7)},x^{(8)}), \psi=p^{2}+b^{2}+c^{2}+1$, $\phi=p^{2}c^2+p^{2} b^{2}+b^{2}c^{2}+p^{2}+b^{2}+c^{2}$, $A=p^{2}b^2+p^{2}c^{2}+b^{2} c^{2}+p^{2}b^{2}c^{2},$ with $b$, $c $ and $p$ are rational numbers different from $-1$, $0$, $1$, and $p\neq \pm b ,$ $p\neq \pm c ,$ $b \neq \pm c $, $\varepsilon $ is sufficiently small and $F$ is a nonlinear non-autonomous periodic function. We present some applications to illustrate our main results.

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