Some identification problems for integro - differential operator equations ∗

We consider, in a Hilbert space H, the convolution integro-differential equation u′′(t)−h∗Au(t) = f(t), 0 ≤ t ≤ T , h∗v(t) = ∫ t 0 h(t−s)v(s) ds, where A is a linear closed densely defined (possibly selfadjoint and/or positive definite) operator in H. Under suitable assumptions on the data we solve the inverse problem consisting of finding the kernel h from the extra data (measured data) of the type g(t) := (u(t), φ), where φ is some eigenvector of A∗. An inverse problem for the first-order equation u′(t) − l ∗ Au(t) = f(t), 0 ≤ t ≤ T , is also studied when A enjoys the same properties as in the previous case.