## 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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