On the Existence of Oscillatory Solutions of the Second Order Nonlinear ODE

Rohleder, Martin

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 51 (2012) no. 2, pp. 107-127.

Résumé

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.

Détail

Classification : 34A12, 34C11, 34C15, 34D05
Keywords: singular ordinary differential equation of the second order; time singularities; unbounded domain; asymptotic properties; damped solutions; oscillatory solutions