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

The purpose of this paper is to compare the equations \[ {y^{\left ( {2n} \right )}}\left ( x \right ) + p\left ( x \right )g\left ( {y\left ( x \right ),{y_\tau }\left ( x \right )} \right ) = 0\] (1) \[ {y^{\left ( {2n} \right )}}\left ( x \right ) + p\left ( x \right )g\left ( {y\left ( x \right ),{y_\tau }\left ( x \right )} \right ) = f\left ( x \right )\] (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) $\operatorname {sgn} g\left ( {x,y} \right ) = \operatorname {sgn} x$, (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 $\left [ {{t_0},\infty } \right )$.