The authors consider the Cauchy problem
with . Here is a fractional time derivative of order , generates a compact analytic semigroup on a Banach space , ( the domain of , ); , and . In the example,
The authors first give a technical definition of mild solutions of the Cauchy problem, and then – under conditions too lengthy to be included here – prove the existence of a mild solution. The proof is by Schauders fixed point theorem.