Abstract
In this paper, we investigate four different types of Ulam stability, i.e., Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability for a class of nonlinear implicit fractional differential equations with non-instantaneous integral impulses and nonlinear integral boundary condition. We also establish certain conditions for the existence and uniqueness of solutions for such a class of fractional differential equations using Caputo fractional derivative. The arguments are based on generalized Diaz-Margolis’s fixed point theorem. We provide two examples, which shows the validity of our main results.
Highlights
Fractional calculus was originated from a question of L’Hospital, in which he asked about the generalization of integral order differentiation
L’Hospital asked a question, “What should happen if the order is September replied, “It will be a paradox, from which later useful consequences will be drawn” [ ]
Fractional calculus is as old as the conventional calculus, and it is the generalization of integral order differentiation and integration to arbitrary non-integer order
Summary
Fractional calculus was originated from a question of L’Hospital, in which he asked about the generalization of integral order differentiation. In [ ] Mardanov et al considered impulsive fractional differential equations with two points integral boundary conditions of the following form: The notations Isβk– ,tk and I β,T are given to fractional integrals of order β with limits sk– to tk and to T , respectively, and
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