Asymptotic behaviour of solutions to certain nonlinear third order neutral functional differential equation

This paper presents asymptotic behaviour of solution to certain nonlinear nonautonomous neutral functional differential equation of the third order. The third order functional differential equation is cut back to system of first order and used together with a quadratic function to acquire suitable complete Lyapunov-Krasovskii functional. This functional is tested and employed to set up conditions on the nonlinear terms to guarantee stability, asymptotic stability, uniform stability, uniform asymptotic stability of the trivial solution, ultimate boundedness and uniform ultimate boundedness of the nontrivial solutions of the equations under consideration. The results of this paper are new as there are no existing results on the type of equation discussed. Examples are also given to validate the theoretical results.