Stability problems of functional differential equations with abstract Volterra operator (Q1924078)

The functional-differential equations \[ \dot x(t)= (Vx)(t)+(Fx)(t),\;0<t_0<t<T\leq\infty, \;x(t)=\Phi(t),\;0\leq t\leq t_0,\;x(t_0)=x^0\in\mathbb{R}^n \] are studied with either an abstract Volterra operator \(V\) and \(F=0\), \(T=\infty\), or a linear continuous operator \(V\) and a nonlinear \(F\) (nonhomogeneous case) with a Nemytzki type operator \(F\) has a most useful example. Five different stability problems are treated: stability, uniform stability, asymptotic stability, uniformly asymptotic stability, exponential asymptotic stability. For a linear \(V\) and \(F=0\) the necessary and sufficient stability conditions are presented. For the second (nonhomogeneous) case ``some asymptotic behavior of the solution'' is discussed.