Properties of Third-Order Nonlinear Functional Differential Equations with MixedArguments

Abstract

The aim of this paper is to offer sufficient conditions for property (B) and/or the oscillation of the third-order nonlinear functional differential equation with mixed arguments ${\left[a\left(t\right){\left[x^{\prime \prime }\left(t\right)\right]}^{\gamma }\right]}^{\prime }=q\left(t\right)f\left(x\left[\tau \right(t\left)\right]\right)+p\left(t\right)h\left(x\left[\sigma \right(t\left)\right]\right)$ . Both cases ${{\displaystyle \int }}^{\infty }{a}^{-1/\gamma }\left(s\right)\text{d}s=\infty$ and ${{\displaystyle \int }}^{\infty }{a}^{-1/\gamma }\left(s\right)\text{d}s<\infty$ are considered. We deduce properties of the studied equations via new comparison theorems. The results obtained essentially improve and complement earlier ones.