On the behaviour of solutions to a kind of third order neutral stochastic differential equation with delay

Abstract

This article demonstrates the behaviour of solutions to a kind of nonlinear third order neutral stochastic differential equations. Setting x^{prime }(t)=y(t), y^{prime }(t) =z(t) the third order differential equation is ablated to a system of first order differential equations together with its equivalent quadratic function to derive a suitable downright Lyapunov functional. This functional is utilised to obtain criteria which guarantee stochastic stability of the trivial solution and stochastic boundedness of the nontrivial solutions of the discussed equations. Furthermore, special cases are provided to verify the effectiveness and reliability of our hypotheses. The results of this paper complement the existing decisions on system of nonlinear neutral stochastic differential equations with delay and extend many results on third order neutral and stochastic differential equations with and without delay in the literature.