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Oscillatory behavior of solutions of certain second order nonlinear differential equations. (English) Zbl 0855.34039

The authors study oscillatory behavior of solutions of the nonlinear second order differential equation (*) \([a(t) (y')^\sigma]' + q(t) f(y) = r(t)\), where \(a\) is an eventually positive function, the nonlinearity \(f\) satisfies \(uf(u) > 0\), \(f'(u)\) for all \(u \neq 0\), and the power \(\sigma\) is a positive ratio of the type (odd/odd) or (even/odd). A typical result is the following statement:
Theorem: Let \(\sigma\) be the quotient of two odd integers and suppose that the following assumptions are satisfied: \(\int^\infty |r(s) |ds < \infty\), \(- \infty \int^\infty q(s) ds < \infty\), there exist \(0 < \mu \leq \nu\) such that \(\mu \leq f'(u) \leq \nu\) and \[ \int^\infty {ds \over a(s)^{1/ \sigma}} = \infty = \int^\infty {ds \over a(s)}. \] If \(y\) is a nonoscillatory solution of (*) such that \(\liminf_{t \to \infty} |y(t) |> 0\) and there exists \(L > 0\) so that \(|y'(t) |\leq L^{1/(\sigma - 1)}\), then \[ \int^\infty {a(s) \bigl[ y'(s) \bigr]^{\sigma + 1} f' \bigl( y(s) \bigr) \over \biggl[ f \bigl( y(s) \bigr) \biggr]^2} ds < \infty, \quad \lim_{t \to \infty} {a(t) \bigl[ y'(t) \bigr]^\sigma \over f \bigl( y(t) \bigr)} = 0 \] and \[ {a(t) \bigl[ y'(t) \bigr]^\sigma \over f \bigl( y(t) \bigr)} + \int^\infty_t {a(s) \bigl[ y'(s) \bigr]^{\sigma + 1} f' \bigl( y(s) \bigr) \over \biggl[ f \bigl( y(s) \bigr) \biggr]^2} ds + \int^\infty_t \left[ q(s) - {r(s) \over f \bigl( y(s) \bigr)} \right] ds \] for \(t\) sufficiently large.
Reviewer: O.Došlý (Brno)

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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