Existence and stability of fractional differential equations with Hadamard derivative

JinRong Wang, Yong Zhou, Milan Medveď

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.

Keywords


Fractional differential equations; Hadamard derivative; nonlinear integral inequality; existence; blowing-up solutions; Ulam-Hyers stability

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