# 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

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topIrena 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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