On Existence and Asymptotic Properties of Kneser Solutions to Singular Second Order ODE.

Jana Vampolová

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2013)

  • Volume: 52, Issue: 1, page 135-152
  • ISSN: 0231-9721

Abstract

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We investigate an asymptotic behaviour of damped non-oscillatory solutions of the initial value problem with a time singularity p ( t ) u ' ( t ) ' + p ( t ) f ( u ( t ) ) = 0 , u ( 0 ) = u 0 , u ' ( 0 ) = 0 on the unbounded domain [ 0 , ∞ ) . Function f is locally Lipschitz continuous on ℝ and has at least three zeros L 0 < 0 , 0 and L > 0 . The initial value u 0 ∈ ( L 0 , L ) ∖ { 0 } . Function p is continuous on [ 0 , ∞ ) , has a positive continuous derivative on ( 0 , ∞ ) and p ( 0 ) = 0 . Asymptotic formulas for damped non-oscillatory solutions and their first derivatives are derived under some additional assumptions. Further, we provide conditions for functions p and f , which guarantee the existence of Kneser solutions.

How to cite

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Vampolová, Jana. "On Existence and Asymptotic Properties of Kneser Solutions to Singular Second Order ODE.." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 52.1 (2013): 135-152. <http://eudml.org/doc/260693>.

@article{Vampolová2013,
abstract = {We investigate an asymptotic behaviour of damped non-oscillatory solutions of the initial value problem with a time singularity $\left( p(t)u^\{\prime \}(t) \right)^\{\prime \} + p(t)f ( u(t) )=0$, $u(0)=u_0$, $u^\{\prime \}(0)=0$ on the unbounded domain $[0,\infty )$. Function $f$ is locally Lipschitz continuous on $\mathbb \{R\}$ and has at least three zeros $L_0 <0$, $0$ and $L>0$. The initial value $u_0\in (L_0, L)\setminus \lbrace 0\rbrace $. Function $p$ is continuous on $[0,\infty ),$ has a positive continuous derivative on $(0,\infty )$ and $p(0)=0$. Asymptotic formulas for damped non-oscillatory solutions and their first derivatives are derived under some additional assumptions. Further, we provide conditions for functions $p$ and $f$, which guarantee the existence of Kneser solutions.},
author = {Vampolová, Jana},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {singular ordinary differential equation of the second order; time singularities; unbounded domain; asymptotic properties; Kneser solutions; damped solutions; non-oscillatory solutions; singular differential equation; Kneser solution; nonoscillatory solution},
language = {eng},
number = {1},
pages = {135-152},
publisher = {Palacký University Olomouc},
title = {On Existence and Asymptotic Properties of Kneser Solutions to Singular Second Order ODE.},
url = {http://eudml.org/doc/260693},
volume = {52},
year = {2013},
}

TY - JOUR
AU - Vampolová, Jana
TI - On Existence and Asymptotic Properties of Kneser Solutions to Singular Second Order ODE.
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2013
PB - Palacký University Olomouc
VL - 52
IS - 1
SP - 135
EP - 152
AB - We investigate an asymptotic behaviour of damped non-oscillatory solutions of the initial value problem with a time singularity $\left( p(t)u^{\prime }(t) \right)^{\prime } + p(t)f ( u(t) )=0$, $u(0)=u_0$, $u^{\prime }(0)=0$ on the unbounded domain $[0,\infty )$. Function $f$ is locally Lipschitz continuous on $\mathbb {R}$ and has at least three zeros $L_0 <0$, $0$ and $L>0$. The initial value $u_0\in (L_0, L)\setminus \lbrace 0\rbrace $. Function $p$ is continuous on $[0,\infty ),$ has a positive continuous derivative on $(0,\infty )$ and $p(0)=0$. Asymptotic formulas for damped non-oscillatory solutions and their first derivatives are derived under some additional assumptions. Further, we provide conditions for functions $p$ and $f$, which guarantee the existence of Kneser solutions.
LA - eng
KW - singular ordinary differential equation of the second order; time singularities; unbounded domain; asymptotic properties; Kneser solutions; damped solutions; non-oscillatory solutions; singular differential equation; Kneser solution; nonoscillatory solution
UR - http://eudml.org/doc/260693
ER -

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