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Abstract fractional Cauchy problems with almost sectorial operators. (English) Zbl 1238.34015

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 \(1+\gamma\) 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.

MSC:

34A08 Fractional ordinary differential equations
35K90 Abstract parabolic equations
47A60 Functional calculus for linear operators
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34G20 Nonlinear differential equations in abstract spaces
47D06 One-parameter semigroups and linear evolution equations
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