Abstract
This paper is concerned with general fractional Cauchy problems oforder $0 < \alpha < 1$ and type $0 \leq \beta \leq 1$ ininfinite-dimensional Banach spaces. A new notion, named generalfractional resolvent of order $0 < \alpha < 1$ and type $0 \leq \beta \leq1$ is developed. Some of its properties are obtained.Moreover, some sufficient conditions are presented to guarantee thatthe mild solutions and strong solutions of homogeneous and inhomogeneous general fractional Cauchy problem exist.An illustrative example is presented.
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