Direct and Inverse Fractional Abstract Cauchy Problems

We are concerned with a fractional abstract Cauchy problem for possibly degenerate equations in Banach spaces. This form of degeneration may be strong and some convenient assumptions about the involved operators are required to handle the direct problem. Moreover, we succeeded in handling related inverse problems, extending the treatment given by Alfredo Lorenzi. Some basic assumptions on the involved operators are also introduced allowing application of the real interpolation theory of Lions and Peetre. Our abstract approach improves previous results given by Favini-Yagi by using more general real interpolation spaces with indices theta, p, p is an element of(0, infinity] instead of the indices theta, infinity. As a possible application of the abstract theorems, some examples of partial differential equations are given.