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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.

In this manuscript, we give some sufficient conditions for existence, uniqueness and various kinds of Ulam stability for a toppled system of fractional order boundary value problems involving the Riemann-Liouville fractional derivative. Applying the Banach contraction principle and the Leray-Schauder result of cone type, uniqueness and existence results are proved for the proposed toppled system. Stability is investigated by using the classical technique of nonlinear functional analysis. The results obtained are well illustrated with the aid of an example.

This paper deals with a new class of non-linear impulsive sequential fractional differential equations with multi-point boundary conditions using Caputo fractional derivative, where impulses are non instantaneous. We develop some sufficient conditions for existence, uniqueness and different types of Ulam stability, namely Hyers–Ulam stability, generalized Hyers–Ulam stability, Hyers–Ulam–Rassias stability and generalized Hyers–Ulam–Rassias stability for the given problem. The required conditions are obtained using fixed point approach. The validity of our main results is shown with the aid of few examples.

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Communicated by M. KiraneThis paper is devoted to establish Bielecki-Ulam-Hyers-Rassias stability, generalized Bielecki-Ulam-Hyers-Rassias stability, and Bielecki-Ulam-Hyers stability on a compact interval OE0, T, for a class of higher-order nonlinear differential equations with fractional integrable impulses. The phrase 'fractional integrable' brings one to fractional calculus. Hence, applying usual methods for analysis offers many difficulties in proving the results of existence and uniqueness of solution and stability theorems. Picard operator is applied in showing existence and uniqueness of solution. Stability results are obtained by using the tools of fractional calculus and Hölder's inequality of integration. Along with tools of fractional calculus, Bielecki's normed Banach spaces are considered, which made the results more interesting.

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In this paper, existence and uniqueness of solution for a coupled impulsive Hilfer–Hadamard type fractional differential system are obtained by using Kransnoselskii’s fixed point theorem. Different types of Hyers–Ulam stability are also discussed.We provide an example demonstrating consistency to the theoretical findings.