Cauchy and Nonlocal Multi-Point Problems for Distributed Order Pseudo-Differential Equations, Part One

We treat the question of existence, uniqueness and construction of a solution to the Cauchy and multi-point problems for a general linear evolution equation with (in general) temporal fractional derivatives with distributed orders. Such equations have met great interest in recent years among researchers in viscoelasticity and in anomalous diffusion processes, and there are numerical analysts who consider them as a challenge. So, we find it desirable to put their theory on a strong and general mathematical basis. After an outline of relevant function spaces and the duality structure generated by them we treat, by Laplace-Fourier techniques, first the Cauchy problem, then the general multi-point problem (where the values of linear combinations of the unknown solution at different instants of time are prescribed). We condense our results in theorems on strong and on weak solutions.