Geodesic
On the Existence of Oscillatory Solutions of the Second Order Nonlinear ODE
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 51 (2012) no. 2, pp. 107-127

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The paper investigates the singular initial problem[4pt] $(p(t)u^{\prime }(t))^{\prime }+q(t)f(u(t))=0,\ u(0)=u_0,\ u^{\prime }(0)=0$[4pt] on the half-line $[0,\infty )$. Here $u_0\in [L_0,L]$, where $L_0$, $0$ and $L$ are zeros of $f$, which is locally Lipschitz continuous on $\mathbb {R}$. Function $p$ is continuous on $[0,\infty )$, has a positive continuous derivative on $(0,\infty )$ and $p(0)=0$. Function $q$ is continuous on $[0,\infty )$ and positive on $(0,\infty )$. For specific values $u_0$ we prove the existence and uniqueness of damped solutions of this problem. With additional conditions for $f$, $p$ and $q$ it is shown that the problem has for each specified $u_0$ a unique oscillatory solution with decreasing amplitudes.
The paper investigates the singular initial problem[4pt] $(p(t)u^{\prime }(t))^{\prime }+q(t)f(u(t))=0,\ u(0)=u_0,\ u^{\prime }(0)=0$[4pt] on the half-line $[0,\infty )$. Here $u_0\in [L_0,L]$, where $L_0$, $0$ and $L$ are zeros of $f$, which is locally Lipschitz continuous on $\mathbb {R}$. Function $p$ is continuous on $[0,\infty )$, has a positive continuous derivative on $(0,\infty )$ and $p(0)=0$. Function $q$ is continuous on $[0,\infty )$ and positive on $(0,\infty )$. For specific values $u_0$ we prove the existence and uniqueness of damped solutions of this problem. With additional conditions for $f$, $p$ and $q$ it is shown that the problem has for each specified $u_0$ a unique oscillatory solution with decreasing amplitudes.
Classification : 34A12, 34C11, 34C15, 34D05
Keywords: singular ordinary differential equation of the second order; time singularities; unbounded domain; asymptotic properties; damped solutions; oscillatory solutions 
Rohleder, Martin. On the Existence of Oscillatory Solutions of the Second Order Nonlinear ODE. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 51 (2012) no. 2, pp. 107-127. http://geodesic.mathdoc.fr/item/AUPO_2012_51_2_a8/
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     url = {http://geodesic.mathdoc.fr/item/AUPO_2012_51_2_a8/}
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