| Title:
|
On the Existence of Oscillatory Solutions of the Second Order Nonlinear ODE (English) |
| Author:
|
Rohleder, Martin |
| Language:
|
English |
| Journal:
|
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica |
| ISSN:
|
0231-9721 |
| Volume:
|
51 |
| Issue:
|
2 |
| Year:
|
2012 |
| Pages:
|
107-127 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
singular ordinary differential equation of the second order |
| Keyword:
|
time singularities |
| Keyword:
|
unbounded domain |
| Keyword:
|
asymptotic properties |
| Keyword:
|
damped solutions |
| Keyword:
|
oscillatory solutions |
| MSC:
|
34A12 |
| MSC:
|
34C11 |
| MSC:
|
34C15 |
| MSC:
|
34D05 |
| idZBL:
|
Zbl 06204934 |
| idMR:
|
MR3058877 |
| . |
| Date available:
|
2012-11-26T10:21:29Z |
| Last updated:
|
2014-03-12 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143071 |
| . |
| Reference:
|
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