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Lipschitz Stability for Non-Instantaneous Impulsive Caputo Fractional Differential Equations with State Dependent Delays

1
Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363, USA
2
Distinguished University Professor of Mathematics, Florida Institute of Technology, Melbourne, FL 32901, USA
3
Department of Applied Mathematics and Modeling, University of Plovdiv “Paisii Hilendarski”, 4000 Plovdiv, Bulgaria
4
School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, H91 CF50 Galway, Ireland
*
Author to whom correspondence should be addressed.
Received: 21 November 2018 / Revised: 24 December 2018 / Accepted: 25 December 2018 / Published: 29 December 2018
(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)
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Abstract

In this paper, we study Lipschitz stability of Caputo fractional differential equations with non-instantaneous impulses and state dependent delays. The study is based on Lyapunov functions and the Razumikhin technique. Our equations in particular include constant delays, time variable delay, distributed delay, etc. We consider the case of impulses that start abruptly at some points and their actions continue on given finite intervals. The study of Lipschitz stability by Lyapunov functions requires appropriate derivatives among fractional differential equations. A brief overview of different types of derivative known in the literature is given. Some sufficient conditions for uniform Lipschitz stability and uniform global Lipschitz stability are obtained by an application of several types of derivatives of Lyapunov functions. Examples are given to illustrate the results. View Full-Text
Keywords: non-instantaneous impulses; Caputo fractional derivative; differential equations; state dependent delays; lipschitz stability non-instantaneous impulses; Caputo fractional derivative; differential equations; state dependent delays; lipschitz stability
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Agarwal, R.; Hristova, S.; O’Regan, D. Lipschitz Stability for Non-Instantaneous Impulsive Caputo Fractional Differential Equations with State Dependent Delays. Axioms 2019, 8, 4.

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