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On general fractional abstract Cauchy problem. (English) Zbl 1273.34012
Summary: This paper is concerned with general fractional Cauchy problems of order $$0 < \alpha < 1$$ and type $$0 \leq \beta \leq 1$$ in infinite-dimensional Banach spaces. A new notion, named general fractional resolvent of order $$0 < \alpha < 1$$ and type $$0 \leq \beta \leq 1$$, is developed. Some of its properties are obtained. Moreover, some sufficient conditions are presented to guarantee that the mild solutions and strong solutions of homogeneous and inhomogeneous general fractional Cauchy problem exist. An illustrative example is presented.

##### MSC:
 34A08 Fractional ordinary differential equations and fractional differential inclusions 47D06 One-parameter semigroups and linear evolution equations 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 34G20 Nonlinear differential equations in abstract spaces
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