Asymptotic behavior of solutions of functional differential equations by fixed point theorems. (English) Zbl 1044.34033

In a series of papers, the authors studied stability properties of functional-differential equations by means of fixed-point theory. They enlarge that study now by also considering delay equations which may be unstable when the delay is zero. They continue to focus on challenging examples to illustrate the work, as opposed to attempting to state general theorems. In Part I, they obtain asymptotic stability using Schauder’s and Banach’s fixed-point theorems; it advances results used by Lyapunov techniques in several ways, but particularly by placing no smoothness on the delay. It also advances some of their earlier work with fixed-point theory in that it places no smoothness conditions on the nonlinear perturbation. In Part II, the authors prove boundedness and asymptotic stability using Krasnoselskii’s fixed point theorem. Schaefer’s fixed-point theorem is used to prove that there is a periodic solution when a periodic forcing is added to that equation.


34K25 Asymptotic theory of functional-differential equations
34K20 Stability theory of functional-differential equations
47H10 Fixed-point theorems
34K13 Periodic solutions to functional-differential equations