Abstract
We study linear singular first-order integro-differential Cauchy problems in Banach spaces. The adjective “singular” means here that the integro-differential equation is not in normal form neither can it be reduced to such a form. We generalize some existence and uniqueness theorems proved in Citation[5] for kernels defined on the entire half-line R + to the case of kernels defined on bounded intervals removing the strict assumption that the kernel should be Laplace-transformable.
Particular attention is paid to single out the optimal regularity properties of solutions as well as to point out several explicit applications relative to singular partial integro-differential equations of parabolic and hyperbolic type.
Acknowledgements
The first two authors are members of G.N.A.M.P.A of the Italian Istituto Nazionale di Alta Matematic (I.N.d.A.M).
Work partially supported by the Project FIRB 2001 Analisi di equazioni a derivate parziali lineari e non lineari: aspetti metodologici, modellistici, applicazioni, of the Italian Ministero dell'Universitá e della Ricerca Scientifica e Tecnologica and by University of Bologna Funds for selected research topics.
Notes
However, in Section 3 less regular kernels k will be considered.