New conditions for the exponential stability of fractionally perturbed ODEs

Medveď, Milan; Brestovanská, Eva

doi:10.14232/ejqtde.2018.1.84

Cited by 4 publications

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“…The method of successive approximations dates to the work of Picard's [34] and it has recently been applied also in the theory of fractional differential Differential Equations with Tempered Ψ −Caputo Fractional Derivative 643 equations (see e.g. [22,29,31]). We apply this method in the following second proof of Theorem 3.…”

Section: Fractional Differential Equations With Tempered ψ−Caputo Derivativementioning

confidence: 99%

“…It is well-known that the classical fractional differential equations can have asymptotically stable solutions, but not exponentially. Some sufficient conditions for the exponential stability of solutions of fractionally perturbed ODEs with tempered Riemann-Liouville fractional integrals and several constant delays are proved in the papers [28,29,30,31]. A new type of derivative, called the Hilfer fractional derivative, was defined by R. Hilfer [16].…”

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Differential Equations With Tempered Ψ-Caputo Fractional Derivative

Medveď

Brestovanská

2021

Mathematical Modelling and Analysis

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In this paper we define a new type of the fractional derivative, which we call tempered Ψ−Caputo fractional derivative. It is a generalization of the tempered Caputo fractional derivative and of the Ψ−Caputo fractional derivative. The Cauchy problem for fractional differential equations with this type of derivative is discussed and some existence and uniqueness results are proved. We present a Henry-Gronwall type inequality for an integral inequality with the tempered Ψ−fractional integral. This inequality is applied in the proof of an existence theorem. A result on a representation of solutions of linear systems of Ψ−Caputo fractional differential equations is proved and in the last section an example is presented.

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Section: Fractional Differential Equations With Tempered ψ−Caputo Derivativementioning

confidence: 99%

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confidence: 99%