Total stability property in limiting equations of integrodifferential equations. (English) Zbl 0709.45012

It is known that the total stability of a bounded solution of an ordinary differential equation or of a functional differential equation with delay can be deduced from the total stability in a certain limiting equation, which is obtained by employing the Bohr topology. This is not true when one uses the compact open topology.
This paper is concerned with the relationships between the stability properties of solutions of the following nonlinear integrodifferential equation: \(\dot x(t)=f(t,x(t))+\int^{0}_{- \infty}F(t,s,x(t+s),x(t))ds+h(t,x_ t),\) where \(x_ t(s)=x(t+s),\) \(- \infty <s\leq 0\), and those of its limiting equations. The author also obtains some results on the existence of an almost periodic solution of an almost periodic integrodifferential equation.
Reviewer: S.Anitą


45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations
45M10 Stability theory for integral equations
45M15 Periodic solutions of integral equations