Geodesic
On monotone solutions of nonlinear ordinary differential equations of order $n$.
Izvestiya. Mathematics , Tome 3 (1969) no. 6, pp. 1293-1317.

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We establish in this paper existence and uniqueness criteria and study the behavior for $t\to +\infty$ of the solution $u(t)$ of the differential equation $u^{(n)}=f(t,u,u',\dots,u^{(n-1)})$, defined in the interval $(0,+\infty)$ and satisfying the conditions $\lim\limits_{t\to +0}u(t)=u_0$ $(-1)^{k}u^{(k)}(t)\geqslant 0$, for $t>0$ $(k=0,1,\dots,n-1)$. 
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I. T. Kiguradze. On monotone solutions of nonlinear ordinary differential equations of order $n$.. Izvestiya. Mathematics , Tome 3 (1969) no. 6, pp. 1293-1317. http://geodesic.mathdoc.fr/item/IM2_1969_3_6_a8/

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