Stability Theorem for a Class of Nonautonomous Neutral Differential Equations with Unbounded Delay

Consider the one-dimensional nonautonomous neutral differential equation d dt xŽt. f Žt, xŽ pŽt... gŽt, xŽqŽt... 0, t t0 , where p, q : t0 , . R are continuous and strictly increasing, pŽt. t, qŽt. t for all t t0 , and xgŽt, x. 0 for t t0 , x R. In this paper, the authors obtain sufficient conditions for the zero solution of this equation with unbounded delay to be uniformly stable as well as asymptotically stable.