Strong singularities in mixed boundary value problems

Irena Rachůnková

Mathematica Bohemica (2006)

  • Volume: 131, Issue: 4, page 393-409
  • ISSN: 0862-7959

Abstract

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We study singular boundary value problems with mixed boundary conditions of the form ( p ( t ) u ' ) ' + p ( t ) f ( t , u , p ( t ) u ' ) = 0 , lim t → 0 + p ( t ) u ' ( t ) = 0 , u ( T ) = 0 , where [ 0 , T ] ⊂ ℝ . We assume that ℝ 2 , f satisfies the Carathéodory conditions on ( 0 , T ) × p ∈ C [ 0 , T ] and 1 / p need not be integrable on [ 0 , T ] . Here f can have time singularities at t = 0 and/or t = T and a space singularity at x = 0 . Moreover, f can change its sign. Provided f is nonnegative it can have even a space singularity at y = 0 . We present conditions for the existence of solutions positive on [ 0 , T ) .

How to cite

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Rachůnková, Irena. "Strong singularities in mixed boundary value problems." Mathematica Bohemica 131.4 (2006): 393-409. <http://eudml.org/doc/249898>.

@article{Rachůnková2006,
abstract = {We study singular boundary value problems with mixed boundary conditions of the form \[ (p(t)u^\{\prime \})^\{\prime \}+ p(t)f(t,u,p(t)u^\{\prime \})=0, \quad \lim \_\{t\rightarrow 0+\}p(t)u^\{\prime \}(t)=0, \quad u(T)=0, \] where $[0,T]\subset \{\mathbb \{R\}\}.$ We assume that $\{\mathbb \{R\}\}^2,$$f$ satisfies the Carathéodory conditions on $(0,T)\times $$p\in C[0,T]$ and $\{1/p\}$ need not be integrable on $[0,T].$ Here $f$ can have time singularities at $t=0$ and/or $t=T$ and a space singularity at $x=0$. Moreover, $f$ can change its sign. Provided $f$ is nonnegative it can have even a space singularity at $y=0.$ We present conditions for the existence of solutions positive on $[0,T).$},
author = {Rachůnková, Irena},
journal = {Mathematica Bohemica},
keywords = {singular mixed boundary value problem; positive solution; lower function; upper function; convergence of approximate regular problems; positive solution; lower function; upper function},
language = {eng},
number = {4},
pages = {393-409},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Strong singularities in mixed boundary value problems},
url = {http://eudml.org/doc/249898},
volume = {131},
year = {2006},
}

TY - JOUR
AU - Rachůnková, Irena
TI - Strong singularities in mixed boundary value problems
JO - Mathematica Bohemica
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 131
IS - 4
SP - 393
EP - 409
AB - We study singular boundary value problems with mixed boundary conditions of the form \[ (p(t)u^{\prime })^{\prime }+ p(t)f(t,u,p(t)u^{\prime })=0, \quad \lim _{t\rightarrow 0+}p(t)u^{\prime }(t)=0, \quad u(T)=0, \] where $[0,T]\subset {\mathbb {R}}.$ We assume that ${\mathbb {R}}^2,$$f$ satisfies the Carathéodory conditions on $(0,T)\times $$p\in C[0,T]$ and ${1/p}$ need not be integrable on $[0,T].$ Here $f$ can have time singularities at $t=0$ and/or $t=T$ and a space singularity at $x=0$. Moreover, $f$ can change its sign. Provided $f$ is nonnegative it can have even a space singularity at $y=0.$ We present conditions for the existence of solutions positive on $[0,T).$
LA - eng
KW - singular mixed boundary value problem; positive solution; lower function; upper function; convergence of approximate regular problems; positive solution; lower function; upper function
UR - http://eudml.org/doc/249898
ER -

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