Linear Volterra equations and integrated solution families
Consider the linear Volterra equation \[ u(t)=f(t)+A\int^t_0 a(t-s)u(s)ds\quad\text{for }t\geq 0.\tag{\(*\)} \] Here \(A\) is an unbounded closed operator in a Banach space \(X\), \(u\) and \(f\) take their values in \(X\), and \(a\) is a scalar function. Often the operator \(A\) is alternatively moved inside the integral. There is a variation of constants formula \[ u(t)= {d\over dt}\int^t_0 S(t-s)f(s)ds, \] where \(S\) is the solution family (the resolvent operator), characterized by the fact that \(AS(t)=S(t)A\) on the domain of \(A\) for all \(t\geq 0\), and \(S(t)x=x+A\int^t_0 a(t-s)S(s)x ds\) for all \(x\in X\) and \(t\geq 0\). To get a more general variation of constants formula one can use an \(n\)-times integrated solution family \(S_n\) instead (which can be thought of as the \(n\)th integral of a solution family), and write the variation of constants formula in the form \[ u(t)={d^{n+1}\over dt^{n+1}}\int^t_0 S(t-s)f(s)ds, \] which can then be interpreted in a weaker or stronger sense. This paper studies the existence and uniqueness of \(n\)-times integrated solution families and their relations to the existence of more or less smooth solutions of \((*)\). As two special cases one gets \(n\)-times integrated semigroups (take \(a(t)=1\)) or cosine families (take \(a(t)=t\)). A number of perturbation results are given where \(A\) is replaced by either \(A+B\) or \(CA\). The adjoint problem is studied as well.
scientific article; zbMATH DE number 943669
scientific article