Stability by Krasnoselskii's theorem in totally nonlinear neutral differential equations

In this paper we use fixed point methods to prove asymptotic stability results of the zero solution of a class of totally nonlinear neutral differential equations with functional delay. The study concerns x'(t)= -a(t)x3(t) + c(t)x'(t-r(t)) + b(t)x3(t-r(t)). The equation has proved very challenging in the theory of Liapunov’s direct method. The stability results are obtained by means of Krasnoselskii-Burton’s theorem and they improve on the work of T.A. Burton (see Theorem 4 in [Liapunov functionals, fixed points, and stability by Krasnoselskii’s theorem, Nonlinear Studies 9 (2001), 181–190]) in which he takes c=0 in the above equation