| Title:
|
Periodic singular problem with quasilinear differential operator (English) |
| Author:
|
Rachůnková, Irena |
| Author:
|
Tvrdý, Milan |
| Language:
|
English |
| Journal:
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Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
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2464-7136 (online) |
| Volume:
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131 |
| Issue:
|
3 |
| Year:
|
2006 |
| Pages:
|
321-336 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study the singular periodic boundary value problem of the form \[ \left(\phi (u^{\prime })\right)^{\prime }+h(u)u^{\prime }=g(u)+e(t),\quad u(0)=u(T),\quad u^{\prime }(0)=u^{\prime }(T), \] where $\phi \:\mathbb{R}\rightarrow \mathbb{R}$ is an increasing and odd homeomorphism such that $\phi (\mathbb{R})=\mathbb{R},$ $h\in C[0,\infty ),$ $e\in L_1J$ and $g\in C(0,\infty )$ can have a space singularity at $x=0,$ i.e. $\limsup _{x\rightarrow 0+}|g(x)|=\infty $ may hold. We prove new existence results both for the case of an attractive singularity, when $\liminf _{x\rightarrow 0+}g(x)=-\infty ,$ and for the case of a strong repulsive singularity, when $\lim _{x\rightarrow 0+}\int _x^1g(\xi )\hspace{0.56905pt}\text{d}\xi =\infty .$ In the latter case we assume that $\phi (y)=\phi _p(y)=|y|^{p-2}y,$ $p>1,$ is the well-known $p$-Laplacian. Our results extend and complete those obtained recently by Jebelean and Mawhin and by Liu Bing. (English) |
| Keyword:
|
singular periodic boundary value problem |
| Keyword:
|
positive solution |
| Keyword:
|
$\phi $-Laplacian |
| Keyword:
|
$p$-Laplacian |
| Keyword:
|
attractive singularity |
| Keyword:
|
repulsive singularity |
| Keyword:
|
strong singularity |
| Keyword:
|
lower function |
| Keyword:
|
upper function |
| MSC:
|
34B15 |
| MSC:
|
34B16 |
| MSC:
|
34B18 |
| MSC:
|
34C25 |
| idZBL:
|
Zbl 1114.34018 |
| idMR:
|
MR2248598 |
| DOI:
|
10.21136/MB.2006.134137 |
| . |
| Date available:
|
2009-09-24T22:26:55Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134137 |
| . |
| Reference:
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