Total stability properties for an almost periodic equation by means of limiting equations. (English) Zbl 0556.34054

D’Anna, Armando

Funkc. Ekvacioj, Ser. Int. 27, 201-209 (1984).

The behavior of a bounded solution \(\phi\) (t) of an almost periodic equation in t (\(\circ)\dot x=f(t,x)\) in \(R^ n\) is considered. Substantially the following results are proved using arguments of topological dynamics. If \(\phi\) (t) is the unique solution of (\(\circ)\) through (0,\(\phi\) (0)) and if there exists a limiting equation of (\(\circ)\) whose solution \(\psi\) (t) corresponding to \(\phi\) (t) (in a suitable sense) is totally stable, then also \(\phi\) (t) is totally stable. If in addition f(t,x) is supposed to be regular (in the sense of Sell), then the uniform asymptotic stability of \(\psi\) (t) implies the total uniform asymptotic stability of \(\phi\) (t).

MSC:

34D30	Structural stability and analogous concepts of solutions to ordinary differential equations
34C25	Periodic solutions to ordinary differential equations

Keywords:

limiting equation; bounded solution; topological dynamics; uniform asymptotic stability; total uniform asymptotic stability