Periodic singular problem with quasilinear differential operator

Irena Rachůnková; Milan Tvrdý

Mathematica Bohemica (2006)

  • Volume: 131, Issue: 3, page 321-336
  • ISSN: 0862-7959

Abstract

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We study the singular periodic boundary value problem of the form φ ( u ' ) ' + h ( u ) u ' = g ( u ) + e ( t ) , u ( 0 ) = u ( T ) , u ' ( 0 ) = u ' ( T ) , where φ ℝ → ℝ is an increasing and odd homeomorphism such that φ ( ℝ ) = ℝ , h ∈ C [ 0 , ∞ ) , e ∈ L 1 J and g ∈ C ( 0 , ∞ ) can have a space singularity at x = 0 , i.e.  lim sup x → 0 + | g ( x ) | = ∞ may hold. We prove new existence results both for the case of an attractive singularity, when lim inf x → 0 + g ( x ) = - ∞ , and for the case of a strong repulsive singularity, when lim x → 0 + ∫ x 1 g ( ξ ) d ξ = ∞ . In the latter case we assume that φ ( y ) = φ 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.

How to cite

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Rachůnková, Irena, and Tvrdý, Milan. "Periodic singular problem with quasilinear differential operator." Mathematica Bohemica 131.3 (2006): 321-336. <http://eudml.org/doc/249905>.

@article{Rachůnková2006,
abstract = {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.},
author = {Rachůnková, Irena, Tvrdý, Milan},
journal = {Mathematica Bohemica},
keywords = {singular periodic boundary value problem; positive solution; $\phi $-Laplacian; $p$-Laplacian; attractive singularity; repulsive singularity; strong singularity; lower function; upper function; singular periodic boundary value problem; positive solution; -Laplacian},
language = {eng},
number = {3},
pages = {321-336},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Periodic singular problem with quasilinear differential operator},
url = {http://eudml.org/doc/249905},
volume = {131},
year = {2006},
}

TY - JOUR
AU - Rachůnková, Irena
AU - Tvrdý, Milan
TI - Periodic singular problem with quasilinear differential operator
JO - Mathematica Bohemica
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 131
IS - 3
SP - 321
EP - 336
AB - 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.
LA - eng
KW - singular periodic boundary value problem; positive solution; $\phi $-Laplacian; $p$-Laplacian; attractive singularity; repulsive singularity; strong singularity; lower function; upper function; singular periodic boundary value problem; positive solution; -Laplacian
UR - http://eudml.org/doc/249905
ER -

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