In the first part of this paper, after a brief overview of the construction of functional calculus about almost sectorial operators, the authors state some results about the analytic semigroups of growth order
and summarize some properties on Caputo fractional derivative and two special functions. Next, a pair of families of operators is constructed and some properties for these families are given. This enables the authors to introduce a concept of solution, which is used to analyze the existence of mild solutions and classical solutions to the Cauchy problem. The corresponding semilinear problem is studied in the next section of this paper. First, the existence of mild solutions is investigated, and then the existence of classical solutions. Some examples are provided in the final section of this paper. They cover the cases of systems of fractional partial differential equations, fractional initial-boundary value problems, and fractional Cauchy problems.