Identification problems for integro-differential delay equations

The inverse problem of recovering the coefficient $f\colon [0,T]\rightarrow{\Bbb R}$ of the equation $$u'(t)=Au(t)+A_1u(t-r)+\int_{-r}^{0}a(s)A_2u(t+s)ds+f(t)z$$ is considered. Here $A\colon D(A)\subset E\rightarrow E$ is the generator of an analytic semigroup in a Banach space $E$, $A_1$ and $A_2$ are linear closed operators defined on $D(A)$. The function $a\colon (-r,0)\rightarrow{\Bbb R}$ and $ z\in E $ are given as well as the initial conditions $$u(s)=\varphi_i(s)\quad \text{for}\quad s\in(-r,0);\quad u(0)=\varphi_0.$$ The inverse problem data $$\Phi[u(t)]=g(t),\quad t\in[0,T],$$ are assumed to be known, where $ \Phi $ is a linear continuous functional on $E$. In the $L^p$-setting of the problem, existence and regularity results, as well as continuous dependence upon the data, are proved under various assumptions depending on the values of $p$. An application of the abstract results to parabolic differential equations is presented.