Oscillatory properties of the second order nonlinear equation \[ (r(t)x^{\prime })^{\prime }+q(t)f(x)=0 \] are investigated. In particular, criteria for the existence of at least one oscillatory solution and for the global monotonicity properties of nonoscillatory solutions are established. The possible coexistence of oscillatory and nonoscillatory solutions is studied too.
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.
We study solutions tending to nonzero constants for the third order differential equation with the damping term (a1(t)(a2(t)x ′ (t))′ ) ′ + q(t)x ′ (t) + r(t)f(x(ϕ(t))) = 0 in the case when the corresponding second order differential equation is oscillatory.