Integral equations, LpLp-forcing, remarkable resolvent: Liapunov functionals

In this paper we study an integral equation of the form x(t)=a(t)−∫0tC(t,s)x(s)ds with resolvent R(t,s)R(t,s) and variation-of-parameters formula x(t)=a(t)−∫0tR(t,s)a(s)ds. We give a variety of conditions under which the mapping (Pϕ)(t)=ϕ(t)−∫0tR(t,s)ϕ(s)ds maps a vector space containing unbounded functions into an LpLp space. It is known from the ideal theory of Ritt that R(t,s)R(t,s) is arbitrarily complicated. Thus, it is widely supposed that this integral is also extremely complicated. In fact, it is not. That integral can be a very close approximation to ϕϕ even when ϕϕ is unbounded. These unbounded functions are essentially harmless perturbations.