Identification problems for singular integro-differential equations of parabolic type II

We recover unknown kernels, depending on time only, in linear singular first-order integro-differential Cauchy problems in Banach spaces. Singular means here that the integro-differential equation is not in normal form nor it can be reduced to such a form. For this class of problems we prove an existence and uniqueness theorem, in the framework of general Banach spaces, under the condition that the “resolvent operator” (cf. (1.5) admits a polar singularity at λ=0 (see Section 3)). Moreover, when the Banach space under consideration is reflexive, we can prove a local in time existence and uniqueness result when the “resolvent operator” decays as (1+|λ|)−1. Finally, we give a few applications to explicit singular partial integro-differential equations of parabolic type.