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

Rohleder, Martin. "On the Existence of Oscillatory Solutions of the Second Order Nonlinear ODE." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 51.2 (2012): 107-127. <http://eudml.org/doc/247195>.

@article{Rohleder2012,
abstract = {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.},
author = {Rohleder, Martin},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {singular ordinary differential equation of the second order; time singularities; unbounded domain; asymptotic properties; damped solutions; oscillatory solutions; singular ordinary differential equation; second order; time singularity; unbounded domain; asymptotic properties; oscillatory solutions},
language = {eng},
number = {2},
pages = {107-127},
publisher = {Palacký University Olomouc},
title = {On the Existence of Oscillatory Solutions of the Second Order Nonlinear ODE},
url = {http://eudml.org/doc/247195},
volume = {51},
year = {2012},
}

TY - JOUR
AU - Rohleder, Martin
TI - On the Existence of Oscillatory Solutions of the Second Order Nonlinear ODE
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2012
PB - Palacký University Olomouc
VL - 51
IS - 2
SP - 107
EP - 127
AB - 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.
LA - eng
KW - singular ordinary differential equation of the second order; time singularities; unbounded domain; asymptotic properties; damped solutions; oscillatory solutions; singular ordinary differential equation; second order; time singularity; unbounded domain; asymptotic properties; oscillatory solutions
UR - http://eudml.org/doc/247195
ER -