Stability and existence of a periodic solution

Abstract

Many authors have discussed the existence of a periodic solution of a periodic system of differential equations under the assumption that the system has a bounded solution which has some kind of stability. LaSalle [Z] has discussed a more general case by considering an asymptotic stability property of motions. Hale [2] and the author [3] have discussed the existence of a periodic solution of functional-differential equations (more generally, the existence of an almost periodic solution) under the assumption that the system has a bounded solution which is uniformly asymptotically stable in the large by applying Liapunov’s second method. Jones [4] also has discussed the existence of a periodic solution by assuming that a set has an asymptotic stability property in some sense. Recently, Sell [5] has shown the existence of a periodic solution of a periodic system of period w under the assumption that the system has a bounded solution which is uniformly asymptotically stable or uniformly asymptotically stable in the large. However, Sell’s definition of uniform asymptotic stability differs from the usual one. (See [a). His type of stability is weaker than the uniform asymptotic stability in the usual sense and it is stronger than the asymptotic stability. Actually, as is seen below, it is also stronger than the equiasymptotic stability. Sell shows that there exists a periodic solution of period KW for some integer K > 1. In this paper, we shall show that Sell’s concept is stronger than the equiasymptotic stability and that for a periodic system Sell’s concept for uniform asymptotic stability of a bounded solution is equivalent to the usual concept, and consequently we have a periodic solution of period w under the assumption that a bounded solution is uniformly asymptotically