Properties of the set of positive solutions to Dirichlet boundary value problems with time singularities

Irena Rachůnková; Svatoslav Staněk

Open Mathematics (2013)

  • Volume: 11, Issue: 1, page 112-132
  • ISSN: 2391-5455

Abstract

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The paper investigates the structure and properties of the set S of all positive solutions to the singular Dirichlet boundary value problem u″(t) + au′(t)/t − au(t)/t 2 = f(t, u(t),u′(t)), u(0) = 0, u(T) = 0. Here a ∈ (−∞,−1) and f satisfies the local Carathéodory conditions on [0,T]×D, where D = [0,∞)×ℝ. It is shown that S c = {u ∈ S: u′(T) = −c} is nonempty and compact for each c ≥ 0 and S = ∪c≥0 S c. The uniqueness of the problem is discussed. Having a special case of the problem, we introduce an ordering in S showing that the difference of any two solutions in S c,c≥ 0, keeps its sign on [0,T]. An application to the equation v″(t) + kv′(t)/t = ψ(t)+g(t, v(t)), k ∈ (1,∞), is given.

How to cite

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Irena Rachůnková, and Svatoslav Staněk. "Properties of the set of positive solutions to Dirichlet boundary value problems with time singularities." Open Mathematics 11.1 (2013): 112-132. <http://eudml.org/doc/269369>.

@article{IrenaRachůnková2013,
abstract = {The paper investigates the structure and properties of the set S of all positive solutions to the singular Dirichlet boundary value problem u″(t) + au′(t)/t − au(t)/t 2 = f(t, u(t),u′(t)), u(0) = 0, u(T) = 0. Here a ∈ (−∞,−1) and f satisfies the local Carathéodory conditions on [0,T]×D, where D = [0,∞)×ℝ. It is shown that S c = \{u ∈ S: u′(T) = −c\} is nonempty and compact for each c ≥ 0 and S = ∪c≥0 S c. The uniqueness of the problem is discussed. Having a special case of the problem, we introduce an ordering in S showing that the difference of any two solutions in S c,c≥ 0, keeps its sign on [0,T]. An application to the equation v″(t) + kv′(t)/t = ψ(t)+g(t, v(t)), k ∈ (1,∞), is given.},
author = {Irena Rachůnková, Svatoslav Staněk},
journal = {Open Mathematics},
keywords = {Singular ordinary differential equation of the second order; Time singularities; Set of all positive solutions; Uniqueness; Blow-up solutions; singular ordinary differential equation; positive solutions; blow up solutions; uniqueness},
language = {eng},
number = {1},
pages = {112-132},
title = {Properties of the set of positive solutions to Dirichlet boundary value problems with time singularities},
url = {http://eudml.org/doc/269369},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Irena Rachůnková
AU - Svatoslav Staněk
TI - Properties of the set of positive solutions to Dirichlet boundary value problems with time singularities
JO - Open Mathematics
PY - 2013
VL - 11
IS - 1
SP - 112
EP - 132
AB - The paper investigates the structure and properties of the set S of all positive solutions to the singular Dirichlet boundary value problem u″(t) + au′(t)/t − au(t)/t 2 = f(t, u(t),u′(t)), u(0) = 0, u(T) = 0. Here a ∈ (−∞,−1) and f satisfies the local Carathéodory conditions on [0,T]×D, where D = [0,∞)×ℝ. It is shown that S c = {u ∈ S: u′(T) = −c} is nonempty and compact for each c ≥ 0 and S = ∪c≥0 S c. The uniqueness of the problem is discussed. Having a special case of the problem, we introduce an ordering in S showing that the difference of any two solutions in S c,c≥ 0, keeps its sign on [0,T]. An application to the equation v″(t) + kv′(t)/t = ψ(t)+g(t, v(t)), k ∈ (1,∞), is given.
LA - eng
KW - Singular ordinary differential equation of the second order; Time singularities; Set of all positive solutions; Uniqueness; Blow-up solutions; singular ordinary differential equation; positive solutions; blow up solutions; uniqueness
UR - http://eudml.org/doc/269369
ER -

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