Resonance and multiplicity in periodic boundary value problems with singularity

Irena Rachůnková; Milan Tvrdý; Ivo Vrkoč

Mathematica Bohemica (2003)

  • Volume: 128, Issue: 1, page 45-70
  • ISSN: 0862-7959

Abstract

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The paper deals with the boundary value problem u ' ' + k u = g ( u ) + e ( t ) , u ( 0 ) = u ( 2 π ) , u ' ( 0 ) = u ' ( 2 π ) , where k ∈ ℝ , g I ↦ ℝ is continuous, e ∈ 𝕃 J and lim x → 0 + ∫ x 1 g ( s ) d s = ∞ . In particular, the existence and multiplicity results are obtained by using the method of lower and upper functions which are constructed as solutions of related auxiliary linear problems.

How to cite

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Rachůnková, Irena, Tvrdý, Milan, and Vrkoč, Ivo. "Resonance and multiplicity in periodic boundary value problems with singularity." Mathematica Bohemica 128.1 (2003): 45-70. <http://eudml.org/doc/249216>.

@article{Rachůnková2003,
abstract = {The paper deals with the boundary value problem \[ u^\{\prime \prime \}+k\,u=g(u)+e(t),\quad u(0)=u(2\pi ),\,\,u^\{\prime \}(0)=u^\{\prime \}(2\pi ), \] where $k\in \mathbb \{R\}$, $g\:I\mapsto \mathbb \{R\}$ is continuous, $e\in \mathbb \{L\}J$ and $\lim _\{x\rightarrow 0+\}\int _x^1g(s)\,\hspace\{0.56905pt\}\text\{d\}s=\infty .$ In particular, the existence and multiplicity results are obtained by using the method of lower and upper functions which are constructed as solutions of related auxiliary linear problems.},
author = {Rachůnková, Irena, Tvrdý, Milan, Vrkoč, Ivo},
journal = {Mathematica Bohemica},
keywords = {second order nonlinear ordinary differential equation; periodic problem; lower and upper functions; second-order nonlinear ordinary differential equation; periodic problem; lower and upper functions},
language = {eng},
number = {1},
pages = {45-70},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Resonance and multiplicity in periodic boundary value problems with singularity},
url = {http://eudml.org/doc/249216},
volume = {128},
year = {2003},
}

TY - JOUR
AU - Rachůnková, Irena
AU - Tvrdý, Milan
AU - Vrkoč, Ivo
TI - Resonance and multiplicity in periodic boundary value problems with singularity
JO - Mathematica Bohemica
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 128
IS - 1
SP - 45
EP - 70
AB - The paper deals with the boundary value problem \[ u^{\prime \prime }+k\,u=g(u)+e(t),\quad u(0)=u(2\pi ),\,\,u^{\prime }(0)=u^{\prime }(2\pi ), \] where $k\in \mathbb {R}$, $g\:I\mapsto \mathbb {R}$ is continuous, $e\in \mathbb {L}J$ and $\lim _{x\rightarrow 0+}\int _x^1g(s)\,\hspace{0.56905pt}\text{d}s=\infty .$ In particular, the existence and multiplicity results are obtained by using the method of lower and upper functions which are constructed as solutions of related auxiliary linear problems.
LA - eng
KW - second order nonlinear ordinary differential equation; periodic problem; lower and upper functions; second-order nonlinear ordinary differential equation; periodic problem; lower and upper functions
UR - http://eudml.org/doc/249216
ER -

References

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