On a class of time-fractional differential equations

In this paper we investigate Cauchy problem for a class of time-fractional differential equation D(t)(alpha)u(t) + c(1)D(t)(beta 1)u(t) +...+ c(d)D(t)(beta d)u(t) = Au(t), t > 0, u((j))(0) = x(j), j = 0, ..., m-1, where A is a closed densely defined linear operator in a Banach space X, alpha > beta (1) > ... > beta (d) > 0, c (j) are constants and m = aOEI +/- aOES. 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.