Oscillation theorems concerning non-linear differential equations of the second order

Abstract

This paper concerns the oscillation of solutions of the differential eq. $\left[ r\left( t\right) \psi \left(x\left( t\right) \right) f\text{ }( \overset{\cdot }{x}(t))\right]^{\cdot }+q\left( t\right) \varphi (g\left( x\left( t\right) \right), r\left( t\right) \psi \left( x\left( t\right) \right) f(\overset{\cdot }{x}(t)))=0,$ where $u\varphi \left( u,v\right) \gt 0$ for all $u\neq 0$, $xg\left( x\right) \gt 0$, $xf\left( x\right)\gt 0$ for all $x\neq 0$, $\psi \left( x\right) \gt 0$ for all $x\in \mathbb{R}$, $r\left( t\right) \gt 0$ for $t\geq t_{0}\gt 0$ and $q$ is of arbitrary sign. Our results complement the results in [A.G. Kartsatos, On oscillation of nonlinear equations of second order, J. Math. Anal. Appl. 24 (1968), 665–668], and improve a number of existing oscillation criteria. Our main results are illustrated with examples.