Study of Periodic Solutions for a Ninth-Order Non-Autonomous Differential Equation Using First-Order Averaging Theory - Berrehail, Chems Eddine; Bouslah, Zineb (Journal of Applied Nonlinear Dynamics, 2026)
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 $x^{(9)}-λ x^{8}+ψ x^{(7)}-λ ψ x^{(6)}+φ\overset{.....}{x}-λ χ \overset{....}{x}+A(\overset{...}{x}-λ \overset{..}{x})+p^{2} b^{2}( c^{2}x-c^{2})=ε \tilde{F}$ where $\tilde{F}=F(t,x,\overset{.}{x},\overset{..}{x},\overset{...}{x},\overset{....}{x},x^{(5)},x^{(6)},x^{(7)},x^{(8)}), ψ=p^{2}+b^{2}+c^{2}+1$, $φ=p^{2}c^2+p^{2} b^{2}+b^{2}c^{2}+p^{2}+b^{2}+c^{2}$, $A=p^{2}b^2+p^{2}c^{2}+b^{2} c^{2}+p^{2}b^{2}c^{2},$ with $b$, $c $ and $p$ are rational numbers different from $-1$, $0$, $1$, and $p eq ± b ,$ $p eq ± c ,$ $b eq ± c $, $ε $ is sufficiently small and $F$ is a nonlinear non-autonomous periodic function. We present some applications to illustrate our main results.