Oscillation Criteria of Fourth-Order Differential Equations with Delay Terms

Abstract

The aim of this paper is to derive oscillation criteria of the following fourth-order differential equation with delay term ${\left(r\left(x\right){\left(z^{\prime \prime \prime }\left(x\right)\right)}^{\gamma }\right)}^{\prime }+{\sum }_{i=1}^n{q}_i\left(x\right)f\left(z\left({\eta }_i\left(x\right)\right)\right)=0$ , under the assumption ${\int }_{x_0}^{\infty }{r}^{-1/\gamma }\left(s\right)\text{d}s=\infty .$ The results are based on comparison with the oscillatory behaviour of second-order delay equations and the generalised Riccati transformation. Not only do the provided theorems provide an entirely new technique but also they vastly improve on a number of previously published conclusions. We give three examples to illustrate our findings.