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On the oscillatory behaviour of a second order nonlinear differential equation

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Summary

The equation y″(t)+p(t)f(y(t))=0 is considered, in which f(y) has the same sign as y, is monotone increasing, and is « strongly nonlinear », and p(t) is locally integrable suchthat \(\mathop {\lim }\limits_{T \to \infty } \mathop \smallint \limits_t^T p(s)ds\) exists. No sign condition is assumed, either on p, or its integral. Under conditions which guarantee that, in a certain sense, p is not too negative, an integrability condition is obtained which is necessary and sufficient for all extendable solutions to oscillate. In the second half of the paper, the monotonicity and strong nonlinearity assumptions on f are shown to be essential ingredients for the existence of such osoillatory criteria. Some examples illustrating the theorems are given.

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Additional information

Entrata in Redazione l’8 agosto 1973.

Research supported in part by NRC grant no. A-8130.

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Butler, G.J. On the oscillatory behaviour of a second order nonlinear differential equation. Annali di Matematica 105, 73–92 (1975). https://doi.org/10.1007/BF02414924

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Keywords

  • Differential Equation
  • Integrability Condition
  • Oscillatory Behaviour
  • Nonlinear Differential Equation
  • Essential Ingredient