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.
@article{bwmeta1.element.doi-10_2478_s11533-012-0047-1,
author = {Irena Rach\r unkov\'a and Svatoslav Stan\v ek},
title = {Properties of the set of positive solutions to Dirichlet boundary value problems with time singularities},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {112-132},
zbl = {1271.34027},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0047-1}
}
Irena Rachůnková; Svatoslav Staněk. Properties of the set of positive solutions to Dirichlet boundary value problems with time singularities. Open Mathematics, Tome 11 (2013) pp. 112-132. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0047-1/
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