Solutions of mixed-type functional differential equations with state-dependence

Abstract

In this paper we study existence and asymptotic behaviors of solutions for mixed-type functional differential equation with state-dependence x˙(t)=f(t,x(T1(t,x(t))),...,x(Tn(t,x(t)))). Because of complexity of state-dependent terms, it is so difficult to discuss its stability by Lyapunov direct method. Through an order-preserving transformation, we generalize Gronwall inequality to obtain boundedness and asymptotics of solutions. It can be regarded as an extension to Lyapunov indirect method. Moreover, we indicate that Krasnoselskii fixed point theorem also can be used to study existence of sign-changing solutions. Finally, we apply our method to iterative functional differential equations and provide two concrete examples.