Recovering a leading coefficient and a memory kernel in first-order integro-differential operator equations

We are concerned with the identification of the scalar functions a and k in the convolution first-order integro-differential equation u′(t)−a(t)Au(t)−k*Bu(t)=f(t), 0tT, k*v(t)=∫0tk(t−s)v(s) ds, in a Banach space X, where A and B are linear closed operators in X, A being the generator of an analytic semigroup of linear bounded operators. Taking advantage of two pieces of additional information, we can recover, under suitable assumptions and locally in time, both the unknown functions a and k. The results so obtained are applied to an n-dimensional integro-differential identification problem in a bounded domain in Rn.