# New technique in asymptotic stability for third-order nonlinear delay differential equations

### Abstract

The stability of the zero solution for the third order nonlinear delay differential equation

x+f(t,x,x,x)x+h(t,x,x)x+c(t)g(x(t-τ(t)))=0,

with variable delay τ(t)≥0, is investigated. Omitting assumptions as differentiability on τ or inversibility of functions t-τ(t), makes the variation of parameters method difficult to apply to the equation. To overcome the difficulties we choose conditions for f,h,g and a space of functions carefully amplified so that the equation takes a suitable form that facilitates the inversion of the equation into an equivalent one from which we derive a fixed point mapping. The end result is not only existence and uniqueness of solutions of the equation, but also for boundedness, stability and convergence to zero of the zero solution of that equation. The technique we use here avoids many difficulties which we often encounter in studying classes of higher order nonlinear equations and offers, what we hope, a new way to investigate the stability by fixed point theory for delay equations.