Some nonoscillation theorems for the higher order nonlinear functional differential equations

Abstract

We investigate the equation $$\left( {r_{n - 1} \left( t \right)\left( {r_{n - 2} \left( t \right)\left( {...} \right)\left( {r_2 \left( t \right)\left( {r_1 \left( t \right)x'\left( t \right)} \right)'...} \right)'} \right)'} \right)' + a\left( t \right)h\left( {x\left( t \right)} \right)p\left( {x'\left( t \right)} \right) + b\left( t \right)f\left( {x\left( {g\left( t \right)} \right)} \right) = c\left( t \right)$$ and give sufficient conditions for the approach to zero of all nonoscillatory solutions or all bounded nonoscillatory solutions as t → ∞. Let us consider the following two cases. Case1. b(t) is oscillatory on [τ, ∞). Case2. b(t) is nonnegative on [τ, ∞).