Remarks on nonoscillation theorems for a second order nonlinear differential equation

Abstract:

This paper proves two results concerning nonoscillation of solutions of the second order nonlinear differential equation (†) \[ y + a(t){\left | y \right |^\gamma }\;\operatorname {sgn} y = 0,\quad \gamma > 0,\] 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.