Abstract
In this paper, we study the extremal solutions of Cauchy problems for abstract fractional differential equations. Some definitions such as L (1)-Lipschitz-like, L (1)-Carath,odory-like and L (1)-Chandrabhan-like are introduced. By virtue of the singular integral inequalities with several nonlinearities due to Medved', the properties of solutions are given. By using a hybrid fixed point theorem due to Dhage, existence results for extremal solutions are established. Finally, we present an example to illustrate our main results.