A Liapunov functional for a singular integral equation (Q607103)

The integral equation \[ x(t) = a(t)-\int^t_0 C(t,s)g(s,x(s))\,ds \] is considered. Here \(a\in L^{2}[0,+\infty)\), while \(C(t,s)\) has a significant singularity, but is convex when \(t>s\). A Lyapunov functional is constructed and it is shown that \(g(t,x(t)-a(t))\in L^{2}[0,+\infty)\) and that \(x(t)-a(t)\to 0\) pointwise as \(t\to +\infty\). The infinite and finite delay problems are also considered.