Extremal solutions of Cauchy problems for abstract fractional differential equations

Abstract

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-Carathéodory-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.