Oscillatory solutions of singular equations arising in hydrodynamics.

Rachůnková, Irena; Tomeček, Jan; Stryja, Jakub

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Asymptotic dichotomy in a class of fourth-order nonlinear delay differential equations with damping.

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Singular problems on the half-line

Irena Rachůnková, Jan Tomeček (2009)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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The paper investigates singular nonlinear problems arising in hydrodynamics. In particular, it deals with the problem on the half-line of the form ${(p (t) u^{'} (t))}^{'} = p (t) f (u (t)),$ $u^{'} (0) = 0, u (\infty) = L .$ The existence of a strictly increasing solution (a homoclinic solution) of this problem is proved by the dynamical systems approach and the lower and upper functions method.

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Disconjugacy and disfocality criteria for second order singular half-linear differential equations

Ondřej Došlý, Alexander Lomtatidze (1999)

Annales Polonici Mathematici

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We establish Vallée Poussin type disconjugacy and disfocality criteria for the half-linear second order differential equation $u^{''} = {p (t) | u |}^{α} {| u^{'} |}^{1 - α} s g n u + g (t) u^{'}$ , where α ∈ (0,1] and the functions $p, g \in L_{l o c} (a, b)$ are allowed to have singularities at the end points t = a, t = b of the interval under consideration.