Direct and inverse problems for evolution integro-differential equations of the first-order in time

Il contributo è un articolo scientifico originale, non pubblicato altrove in nessuna forma, e sottoposto a referee. This paper deals with an identification problem for integro-differential equations. More precisely the authors study an abstract evolution equation of the type $$u'(t)=A_1u(t)+\int_0^th(t-s)A_2u(s)ds+f(t)$$ in a bounded time interval $[0,T]$ in a Hilbert space framework. One of the main assumptions on the operators $A_1$ and $A_2$ is that $A_2$ dominates $A_1$. The authors consider both the direct and the inverse problems. The first problem assumes that the kernel $h$ is a known datum and that the initial condition on $u$ is given, while the inverse problem consists in determining the functions $u$ and $h$ simultaneously given additional conditions on $u$. The main results are existence and uniqueness for the solutions of both problems.