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Existence of oscillatory solutions of singular nonlinear differential equations. (English) Zbl 1222.34035

Summary: Asymptotic properties of solutions of the singular differential equation
\[ (p(t)u'(t))' = p(t)f(u(t)) \]
are described. Here, \(f\) is Lipschitz continuous on \(\mathbb R\) and has at least two zeros 0 and \(L > 0\). The function \(p\) is continuous on \([0, \infty)\) and has a positive continuous derivative on \((0, \infty)\) and \(p(0) = 0\). Further conditions on \(f\) and \(p\) under which the equation has oscillatory solutions converging to \(0\) are given.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
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