Opuscula Math. 33, no. 2 (2013), 255-272
Stability by Krasnoselskii's theorem in totally nonlinear neutral differential equations
Abstract. 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 \[x'(t)=a(t)x^3(t)+c(t)x'(t-r(t))+b(t)x^3(t-r(t)).\] 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.
Keywords: fixed point, stability, nonlinear neutral equation, Krasnoselskii-Burton theorem.
Mathematics Subject Classification: 47H10, 34K20, 34K30, 34K40.