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nonlinear ordinary differential equation of higher order; asymptotic behavior of solutions; oscillatory solution
For the equation $$ y^{(n)}+|y|^{k}\mathop {\rm sgn} y=0,\quad k>1,\ n=3,4, $$ existence of oscillatory solutions $$ y=(x^*-x)^{-\alpha } h(\log (x^*-x)),\quad \alpha =\frac {n}{k-1},\ x<x^*, $$ is proved, where $x^*$ is an arbitrary point and $h$ is a periodic non-constant function on $\mathbb {R}$. The result on existence of such solutions with a positive periodic non-constant function $h$ on $\mathbb {R}$ is formulated for the equation $$ y^{(n)}=|y|^{k}\mathop {\rm sgn} y, \quad k>1,\ n=12,13,14. $$
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