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

• Autorzy: Irena Rachůnková , Svatoslav Staněk
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Singular ordinary differential equation of the second order; Time singularities; Set of all positive solutions; Uniqueness; Blow-up solutions

Kategorie tematyczne

• 34B18: Positive solutions of nonlinear boundary value problems
• 34B16: Singular nonlinear boundary value problems
• 34A12: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2013
• Tom: 11
• Numer: 1
• Strony: 112-132

Daty

wydano

2013-01-01

online

2012-10-24

Twórcy

autor

Irena Rachůnková

• Palacký University

autor

Svatoslav Staněk

• Palacký University

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Identyfikatory

DOI

10.2478/s11533-012-0047-1