Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Oscillation and nonoscillation of even-order nonlinear delay-differential equations


Author: Bhagat Singh
Journal: Quart. Appl. Math. 31 (1973), 343-349
MSC: Primary 34K15
DOI: https://doi.org/10.1090/qam/435557
MathSciNet review: 435557
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Abstract: The purpose of this paper is to compare the equations

$\displaystyle {y^{\left( {2n} \right)}}\left( x \right) + p\left( x \right)g\left( {y\left( x \right),{y_\tau }\left( x \right)} \right) = 0$

        (1)

$\displaystyle {y^{\left( {2n} \right)}}\left( x \right) + p\left( x \right)g\left( {y\left( x \right),{y_\tau }\left( x \right)} \right) = f\left( x \right)$

        (2) for their oscillatory and nonoscillatory nature. In Eqs. (1) and (2) $ {y^{(i)}}(x) \equiv \\ \left( {{d^i}/d{x^i}} \right)y(x),i = 1,2,...,2n$; $ {t_2} > {t_1}$; $ {y_\tau }(x) \equiv y\left( {x - \tau \left( x \right)} \right)$; $ dy/dx$ and $ {d^2}y/d{x^2}$ will also be denoted by $ y'$ and $ y''$ respectively. Throughout this paper it will be assumed that $ p\left( x \right)$, $ f\left( x \right)$, $ \tau \left( x \right)$ are continuous real-valued functions on the real line $ \left( { - \infty ,\infty } \right)$; $ f\left( x \right)$, $ p\left( x \right)$ and $ \tau \left( x \right)$, in addition, are nonnegative, $ \tau \left( x \right)$ is bounded and $ f\left( x \right)$, $ p\left( x \right)$ eventually become positive to the right of the origin. In regard to the function $ g$ we assume the following: (i) $ g:{R^2} \to R$ is continuous, $ R$ being the real line, (ii) $ g\left( {\lambda x,\lambda y} \right) = {\lambda ^{2q + 1}}g\left( {x,y} \right)$ for all real $ \lambda \ne 0$ and some integer $ q \ge 1$, (iii) $ {\mathop{\rm sgn}} g\left( {x,y} \right) = {\mathop{\rm sgn}} x$, (iv) $ g\left( {x,y} \right) \to \infty $ as $ x,y \to \infty $ ; $ g$ is increasing in both arguments monotonically. Eq. (1) is called oscillatory if every nontrivial solution $ y\left( t \right) \in \left[ {{t_0},\infty } \right)$ has arbitrarily large zeros; i.e., for every such solution $ y\left( t \right)$, if $ y\left( {{t_1}} \right) = 0$ then there exists $ {t_2} > {t_1}$ such that $ y\left( {{t_2}} \right) = 0$. Eq. (1) is called nonoscillatory if it has a solution with a last zero or no zero in $ \left[ {{t_0},\infty } \right), {t_0} \ge a > 0$. A similar definition holds for eq. (2). All solutions of (1) and (2) considered henceforth are continuous and nontrivial, existing on some halfline $ \left[ {{t_0},\infty } \right)$.

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DOI: https://doi.org/10.1090/qam/435557
Article copyright: © Copyright 1973 American Mathematical Society


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