Abstract fractional Cauchy problems with almost sectorial operators

Of concern are the Cauchy problems for linear and semilinear time fractional evolution equations involving in the linear part, a linear operator A whose resolvent satisfies the estimate of growth −γ (−1<γ<0) in a sector of the complex plane, which occurs when one considers, for instance, the partial differential operators in the limit domain of dumb-bell with a thin handle or in the space of Hölder continuous functions. By constructing a pair of families of operators in terms of the generalized Mittag-Leffler-type functions and the resolvent operators associated with A (for the first time), and a deep analysis on the properties for these families, we obtain the existence and uniqueness of mild solutions and classical solutions to the Cauchy problems. Moreover, we present three examples to illustrate the feasibility of our results.