Identification of unknown terms in convolution integro-differential equations in a Banach space

We consider the following dentification problems in a general Banach space $X$: find a function $u:[0,T] \to X$ and a vector $z \in X$ such that the initial-value problems $$ u'(t)-\int_0^t h(t-s) u'(s)ds -Au(t) = f(t)z + g(t), \quad u(0) = u_0\in X$$ and $$ u'(t) -Au(t) -\int_0^t k(t-s) Au(s)ds = f(t)z + g(t), \quad u(0) = u_0\in X $$ are fulfilled, along with the nonlocal additional condition $\int_{[0,T]}u(t) d\mu(t) = \varphi \in X$, for some probability Borel probability measure $\mu$ on the interval $[0,T]$. Here $A:D(A)\subset X \to X$ is a (possibly unbounded) closed linear operator, $h$, $k$ and $f$ are scalar functions and $g$ is a $X$-valued source term. We recall that the same problem with $h=k=0$ has been previously studied by Anikonov \& Lorenzi in \cite{AL}, Prilepko, Piskarev \& Shaw in \cite{PPS}, and subsequently generalized by Lorenzi \& Vrabie in \cite{LV}. Under suitable assumptions on the structural data of the problem, we prove local-in-time existence and uniqueness for the function $u$, and an explicit representation formula for $z$ depending on $u$. Also, a continuous dependence of Lipschitz type of the solution $(u,z)$ on the data is provided. Finally, two applications to parabolic integro-differential boundary value problems are considered.