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Oscillation criteria for second order nonlinear differential equations with integrable coefficients


Author: James S. W. Wong
Journal: Proc. Amer. Math. Soc. 115 (1992), 389-395
MSC: Primary 34C10
DOI: https://doi.org/10.1090/S0002-9939-1992-1086346-0
MathSciNet review: 1086346
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Abstract: Consider the second order nonlinear differential equation $ y'' + a\left( t \right)f\left( y \right) = 0$ where $ a\left( t \right) \in C[0,\infty ),f\left( y \right) \in {C^1}\left( { - \infty ,\infty } \right),f'\left( y \right) \geq 0$, and $ yf\left( y \right) > 0$ for $ {\text{y}} \ne {\text{0}}$. Furthermore, $ f\left( y \right)$ also satisfies either a superlinear or a sublinear condition, which covers the prototype nonlinear function $ f\left( y \right) = \vert y{\vert^\gamma }\operatorname{sgn} y$ with $ \gamma {\text{ > 1}}$ and $ 0 < \gamma < 1$ respectively. The coefficient $ a\left( t \right)$ is allowed to be negative for arbitrarily large values of $ t$ and is integrable in the sense that the improper interval $ \int_t^\infty {a\left( s \right)ds} = A\left( t \right)$ exists for each $ t \geq 0$. Oscillation criteria involving integrals of $ A\left( t \right)$ due to Coles and Butler for the superlinear and sublinear cases are shown to remain valid without the additional hypothesis that $ A\left( t \right) \geq 0$.


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DOI: https://doi.org/10.1090/S0002-9939-1992-1086346-0
Keywords: Second order, nonlinear, ordinary differential equations, oscillation, asymptotic behavior
Article copyright: © Copyright 1992 American Mathematical Society

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