1967/01/01 by George R. Sell · 4 citations
Mathematics · #advanced mathematical theories #Advanced Topology and Set Theory #Mathematical Dynamics and Fractals
paper · doi:10.1090/s0002-9947-1967-0212314-4
openalex publication_date 1967/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
1. Introduction.In the preceding paper [22] we showed that the solutions of every admissible (nonautonomous) differential equation x'=/(x, t), defined on WxR, can be viewed as a local dynamical systemtt on Wx g, where Wis an open set in Rn and g is the space of translates off.By introducing various topologies on t, we were able to study the topological-dynamical characteristics of the flow "■*(/> t)=ft on ft.By introducing the concept of the "hull" off we were able to determine whether the flow/ be positively compact, compact, recurrent, almost periodic, etc.In this paper we are primarily interested in the asymptotic behavior of the solutions of x'=/(x, 0-In particular we shall investigate the relationship between the solution <p(x,f, t) of x'=/(x, 0 and the corresponding motion tt(x,/; 0 m Wx g.One concept, which will play a central role in this investigation, is that of the "set of limiting equations," which is the cu-limit set of the given differential equation( 2).This study is carried out in § §2 and 3.The concept of "asymptotically autonomous differential equations" introduced by L. Markus [8] can be described as those differential equations for which the a)-limit set consists of a single point.This immediately admits generalizations to asymptotically-periodic, asymptotically-almost periodic and asymptoticallyrecurrent differential equations.One is able to derive more detailed information about the solutions of these equations.In §4 we shall investigate the question of the existence of almost periodic solutions and periodic solutions of differential equations.This study is related to the work of L. G. Deysach and G. R. Sell [4], J. L. Massera [9], R. K. Miller [12], C. R. Putnam [17], G. Seifert [19], and G. R. Sell [21].In §5 we shall briefly discuss some extensions of our work.Since this paper relies heavily on the results of [22], we shall presume knowledge of the earlier paper.We shall use the definitions and the notation of the preceding paper.