DISSIPATIVE PERIODIC PROCESSES

The objective of this paper is to develop a general and meaningful theory of dissipative periodic systems. For ordinary periodic differential equations one studies the iterates of a map T of a state space into itself where the map T is topological and the space is locally compact (n-dimensional Euclidean space). However, for the applications the authors have in mind, the solutions will be unique only in the forward direction of time and the state spaces are not locally compact. Because of this generalization of the results for ordinary differential equations is by no means trivial. The basic theory of dissipative periodic processes on Banach spaces are developed in Sections 2 and 3 of the paper. How this applies to retarded functional differential equations of retarded type is discussed in the fourth section. Two sufficient conditions for dissipativeness are given in terms of Liapunov functions. They formalize the intuitive notion that many systems for large displacements dissipate energy. Application of these results is illustrated. (Author)
Prepared in cooperation with LeMoyne Coll., Syracuse, N. Y. Dept. of Mathematics. Sponsored in part by Grant NGL-40-002-015.