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Oscillatory solutions of singular equations arising in hydrodynamics. (English) Zbl 1203.34058
Motivated by the recent papers of the first two authors [Math. Comput. Modelling 51, No. 5--6, 658--669 (2010; Zbl 1190.34029), and Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 3--4 (A), 2114--2118 (2010; Zbl 1186.34014)], the initial value problem for the second order nonlinear differential equation $$ (p(t)u')'=p(t)f(u),\quad t\in [0,\infty), \tag{$*$} $$ is investigated. The functions $p,f$ satisfy certain restrictions which imply, among others, that $t=0$ is a singular point of the investigated equation. One of the main results of the paper gives additional conditions on the nonlinearity $f$ under which the singular initial value problem associated with $(*)$ has oscillatory solutions with decreasing amplitudes.

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
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