Oscillation criteria for second order nonlinear differential equations with integrable coefficients

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 $f\left ( y \right ) = |y{|^\gamma }\operatorname {sgn} y$ 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 $A\left ( t \right ) \geq 0$.