Existence and stability of fractional differential equations with Hadamard derivative

Abstract

In this paper, we study nonlinear fractional differential equations with Hadamard derivative and Ulam stability in the weighted space of continuous functions. Firstly, some new nonlinear integral inequalities with Hadamard type singular kernel are established, which can be used in the theory of certain classes of fractional differential equations. Secondly, some sufficient conditions for existence of solutions are given by using fixed point theorems via a prior estimation in the weighted space of the continuous functions. Meanwhile, a sufficient condition for nonexistence of blowing-up solutions is derived. Thirdly, four types of Ulam-Hyers stability definitions for fractional differential equations with Hadamard derivative are introduced and Ulam-Hyers stability and generalized Ulam-Hyers-Rassias stability results are presented. Finally, some examples and counterexamples on Ulam-Hyers stability are given.