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Keywords:
third order nonlinear differential equation; vanishing at infinity solution; Kneser solution; oscillatory solution
Summary:
We study asymptotic and oscillatory properties of solutions to the third order differential equation with a damping term $$x'''(t)+q(t)x'(t)+r(t)|x|^{\lambda }(t)\mathop {\rm sgn} x(t)=0 ,\quad t\geq 0.$$ We give conditions under which every solution of the equation above is either oscillatory or tends to zero. In case $\lambda \leq 1$ and if the corresponding second order differential equation $h''+q(t)h=0$ is oscillatory, we also study Kneser solutions vanishing at infinity and the existence of oscillatory solutions.
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