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Remarks on nonoscillation theorems for a second order nonlinear differential equation


Author: James S. W. Wong
Journal: Proc. Amer. Math. Soc. 83 (1981), 541-546
MSC: Primary 34C10
DOI: https://doi.org/10.1090/S0002-9939-1981-0627687-0
MathSciNet review: 627687
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Abstract: This paper proves two results concerning nonoscillation of solutions of the second order nonlinear differential equation ()

$\displaystyle y + a(t){\left\vert y \right\vert^\gamma }\;\operatorname{sgn} y = 0,\quad \gamma > 0,$

where $ a(t)$ is positive, continuous and locally of bounded variation, and sgn $ y$ denotes the sign of the function $ y(t)$. Assume also that $ a(t)$ satisfies $ \smallint _0^\infty {a^{ - 1}}(s)\;d{a_ + }(s) < \infty $. The main results are Theorem A. Let $ 0 < \gamma < 1$. If $ {\lim _{t \to \infty }}{t^2}a(t) = 0$, then () is nonoscillatory. Theorem B. Let $ \gamma > 1$. If $ {\lim _{t \to \infty }}{t^{\gamma + 1}}a(t) = 0$, then () is nonoscillatory.

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DOI: https://doi.org/10.1090/S0002-9939-1981-0627687-0
Keywords: Second order, nonlinear, differential equation, oscillation
Article copyright: © Copyright 1981 American Mathematical Society

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