On a class of time-fractional differential equations

In this paper we investigate Cauchy problem for a class of time-fractional differential equation (0.1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{gathered} D_t^\alpha u(t) + c_1 D_t^{\beta _1 } u(t) + \cdots + c_d D_t^{\beta _d } u(t) = Au(t), t > 0, \hfill \\ u^{(j)} (0) = x_j , j = 0, \cdots ,m - 1, \hfill \\ \end{gathered}$$\end{document} where A is a closed densely defined linear operator in a Banach space X, α > β1 > ... > βd > 0, cj are constants and m = ⌈α⌊. A new type of resolvent family corresponding to well-posedness of (0.1) is introduced. We derive the generation theorems, algebraic equations and approximation theorems for such resolvent families. Moreover, we give the exact solution for a kind of generalized fractional telegraph equations. Some examples are given as illustrations.