Oscillation criteria for second order nonlinear differential equations with integrable coefficients.(English)Zbl 0760.34032

The paper deals with oscillation properties of the nonlinear second order differential equation $$y''+a(t)f(y)=0$$. In addition to the usual assumptions on the nonlinearity $$f(f\in C^1,\;f'\geq 0,\;yf(y)>0$$ for $$y\ne 0)$$ it is assumed that the function $$a(t)$$ is integrable, i.e., the improper integral $$\int^\infty_ta(s)\,ds=:A(t)$$ exists and is finite for each $$t\ge 0$$. The oscillation criteria of W. J. Coles [Ann. Mat. Pura Appl., IV.Ser. 82, 123–133 (1969; Zbl 0188.15304)] and G. J. Butler [Indian J. Math. 24, 1–7 (1982; Zbl 0532.34024)] are extended to a more general equation without the sign restriction on the function $$A(t)$$.

MSC:

 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems

Citations:

Zbl 0188.15304; Zbl 0532.34024
Full Text:

References:

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