Liapunov functionals and periodicity in integral equations (Q1335556)

A Lyapunov functional is constructed for the scalar nonconvolution integral equation \[ x(t) = a(t) - \int^ t_{-\infty} D(t,s) g \bigl( s,x (s) \bigr) ds \quad (t \geq t_ 0). \] The kernel \(D\) is of the type \(D(t,s) = B(t,s) - Q(t,s)\), where \(B_ s (t,s) \geq 0\), \(B_{st} (t,s) \leq 0\), and \(Q(t,s)\) is a ``perturbation term.'' First a basic inequality is proved, and then this inequality is used to derive a number of asymptotic results concerning boundedness and some additional asymptotic properties of \(x\), such as \(\int^ \infty_{t_ 0} x(t) g(t,x(t)) dt < \infty\) and \(x(t) \to 0\) as \(t \to \infty\). The existence of periodic solutions is also studied, and some extensions to systems of linear equations are included.