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

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.

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Caputo fractional differential equations with non-instantaneous impulses and strict stability by Lyapunov functions

Filomat ◽

10.2298/fil1716217a ◽

2017 ◽

Vol 31 (16) ◽

pp. 5217-5239 ◽

Cited By ~ 5

Author(s):

Ravi Agarwal ◽

Snehana Hristova ◽

Donal O’Regan

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Lyapunov Functions ◽

Sufficient Conditions ◽

Time Intervals ◽

Dini Derivative ◽

Comparison Results ◽

Strict Stability ◽

Caputo Fractional Differential Equations ◽

Fractional Differential

In this paper the statement of initial value problems for fractional differential equations with noninstantaneous impulses is given. These equations are adequate models for phenomena that are characterized by impulsive actions starting at arbitrary fixed points and remaining active on finite time intervals. Strict stability properties of fractional differential equations with non-instantaneous impulses by the Lyapunov approach is studied. An appropriate definition (based on the Caputo fractional Dini derivative of a function) for the derivative of Lyapunov functions among the Caputo fractional differential equations with non-instantaneous impulses is presented. Comparison results using this definition and scalar fractional differential equations with non-instantaneous impulses are presented and sufficient conditions for strict stability and uniform strict stability are given. Examples are given to illustrate the theory.

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Lyapunov Functions and Lipschitz Stability for Riemann–Liouville Non-Instantaneous Impulsive Fractional Differential Equations

Symmetry ◽

10.3390/sym13040730 ◽

2021 ◽

Vol 13 (4) ◽

pp. 730

Author(s):

Ravi Agarwal ◽

Snezhana Hristova ◽

Donal O’Regan

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Lyapunov Functions ◽

Sufficient Conditions ◽

Initial Time ◽

Lipschitz Stability ◽

Comparison Results ◽

Liouville Fractional Derivative ◽

Fractional Differential ◽

Definition Of

In this paper a system of nonlinear Riemann–Liouville fractional differential equations with non-instantaneous impulses is studied. We consider a Riemann–Liouville fractional derivative with a changeable lower limit at each stop point of the action of the impulses. In this case the solution has a singularity at the initial time and any stop time point of the impulses. This leads to an appropriate definition of both the initial condition and the non-instantaneous impulsive conditions. A generalization of the classical Lipschitz stability is defined and studied for the given system. Two types of derivatives of the applied Lyapunov functions among the Riemann–Liouville fractional differential equations with non-instantaneous impulses are applied. Several sufficient conditions for the defined stability are obtained. Some comparison results are obtained. Several examples illustrate the theoretical results.

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Mittag-Leffler stability for non-instantaneous impulsive Caputo fractional differential equations with delays

Mathematica Slovaca ◽

10.1515/ms-2017-0249 ◽

2019 ◽

Vol 69 (3) ◽

pp. 583-598 ◽

Cited By ~ 2

Author(s):

Ravi Agarwal ◽

Snezhana Hristova ◽

Donal O’Regan

Keyword(s):

Differential Equations ◽

Lyapunov Functions ◽

Sufficient Conditions ◽

Razumikhin Technique ◽

Fractional Equations ◽

Caputo Fractional Differential Equations ◽

Fractional Delay ◽

Fractional Differential ◽

Delay Differential ◽

Fractional Delay Differential Equations

Abstract Caputo fractional delay differential equations with non-instantaneous impulses are studied. Initially a brief overview of the basic two approaches in the interpretation of solutions is given. A generalization of Mittag-Leffler stability with respect to non-instantaneous impulses is given and sufficient conditions are obtained. Lyapunov functions and the Razumikhin technique will be applied and appropriate derivatives among the studied fractional equations is defined and applied. Examples are given to illustrate our results.

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Strict stability with respect to initial time difference for Caputo fractional differential equations by Lyapunov functions

Georgian Mathematical Journal ◽

10.1515/gmj-2016-0080 ◽

2017 ◽

Vol 24 (1) ◽

pp. 1-13 ◽

Cited By ~ 1

Author(s):

Ravi P. Agarwal ◽

Donal O’Regan ◽

Snezhana Hristova

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Lyapunov Functions ◽

Sufficient Conditions ◽

Time Difference ◽

Initial Time ◽

Advantages And Disadvantages ◽

Strict Stability ◽

Caputo Fractional Differential Equations ◽

Fractional Differential

AbstractThe strict stability properties are generalized to nonlinear Caputo fractional differential equations in the case when both initial points and initial times are changeable. Using Lyapunov functions, some criteria for strict stability, eventually strict stability and strict practical stability are obtained. A brief overview of different types of derivatives in the literature related to the application of Lyapunov functions to Caputo fractional equations are given, and their advantages and disadvantages are discussed with several examples. The Caputo fractional Dini derivative with respect to to initial time difference is used to obtain some sufficient conditions.

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Some stability properties related to initial time difference for Caputo fractional differential equations

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2018-0005 ◽

2018 ◽

Vol 21 (1) ◽

pp. 72-93 ◽

Cited By ~ 5

Author(s):

Ravi Agarwal ◽

Snezhana Hristova ◽

Donal O’Regan

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Fractional Differential Equations ◽

Sufficient Conditions ◽

Time Difference ◽

Initial Time ◽

Lipschitz Stability ◽

Caputo Fractional Differential Equations ◽

Fractional Differential ◽

Order Extension

Abstract Lipschitz stability and Mittag-Leffler stability with initial time difference for nonlinear nonautonomous Caputo fractional differential equation are defined and studied using Lyapunov like functions. Some sufficient conditions are obtained. The fractional order extension of comparison principles via scalar fractional differential equations with a parameter is employed. The relation between both types of stability is discussed theoretically and it is illustrated with examples.

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A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2016-0017 ◽

2016 ◽

Vol 19 (2) ◽

Cited By ~ 33

Author(s):

Ravi Agarwal ◽

Snezhana Hristova ◽

Donal O’Regan

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Lyapunov Functions ◽

Direct Method ◽

Sufficient Conditions ◽

Dini Derivative ◽

Advantages And Disadvantages ◽

Impulsive Fractional Differential Equations ◽

Caputo Fractional Differential Equations ◽

Fractional Differential

AbstractWe present an overview of the literature on solutions to impulsive Caputo fractional differential equations. Lyapunov direct method is used to obtain sufficient conditions for stability properties of the zero solution of nonlinear impulsive fractional differential equations. One of the main problems in the application of Lyapunov functions to fractional differential equations is an appropriate definition of its derivative among the differential equation of fractional order. A brief overview of those used in the literature is given, and we discuss their advantages and disadvantages. One type of derivative, the so called Caputo fractional Dini derivative, is generalized to impulsive fractional differential equations. We apply it to study stability and uniform stability. Some examples are given to illustrate the results.

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Applications of Lyapunov Functions to Caputo Fractional Differential Equations

Mathematics ◽

10.3390/math6110229 ◽

2018 ◽

Vol 6 (11) ◽

pp. 229

Author(s):

Ravi Agarwal ◽

Snezhana Hristova ◽

Donal O’Regan

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Lyapunov Functions ◽

Sufficient Conditions ◽

Basic Question ◽

Caputo Fractional Differential Equations ◽

Fractional Differential ◽

Definition Of ◽

The Given ◽

Fractional Equation

One approach to study various stability properties of solutions of nonlinear Caputo fractional differential equations is based on using Lyapunov like functions. A basic question which arises is the definition of the derivative of the Lyapunov like function along the given fractional equation. In this paper, several definitions known in the literature for the derivative of Lyapunov functions among Caputo fractional differential equations are given. Applications and properties are discussed. Several sufficient conditions for stability, uniform stability and asymptotic stability with respect to part of the variables are established. Several examples are given to illustrate the theory.

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Ulam type stability for non-instantaneous impulsive Caputo fractional differential equations with finite state dependent delay

Georgian Mathematical Journal ◽

10.1515/gmj-2020-2061 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Ravi Agarwal ◽

Snezhana Hristova ◽

Donal O’Regan

Keyword(s):

Lower Bound ◽

Differential Equations ◽

Fractional Differential Equations ◽

Sufficient Conditions ◽

State Dependent ◽

State Dependent Delay ◽

Caputo Fractional Differential Equations ◽

Finite State ◽

Fractional Differential ◽

Ulam Type Stability

AbstractFour Ulam type stability concepts for non-instantaneous impulsive fractional differential equations with state dependent delay are introduced. Two different approaches to the interpretation of solutions are investigated. We study the case of an unchangeable lower bound of the Caputo fractional derivative and the case of a lower bound coinciding with the point of jump for the solution. In both cases we obtain sufficient conditions for Ulam type stability. An example is also provided to illustrate both approaches.

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Practical stability for Riemann–Liouville delay fractional differential equations

Arabian Journal of Mathematics ◽

10.1007/s40065-021-00320-6 ◽

2021 ◽

Author(s):

Ravi Agarwal ◽

Snezhana Hristova ◽

Donal O’Regan

Keyword(s):

Differential Equations ◽

Fractional Derivative ◽

Fractional Differential Equations ◽

Lyapunov Functions ◽

Initial Conditions ◽

Sufficient Conditions ◽

Practical Stability ◽

Razumikhin Technique ◽

Fractional Differential ◽

Derivatives Of

AbstractIn this paper, we study a system of nonlinear Riemann–Liouville fractional differential equations with delays. First, we define in an appropriate way initial conditions which are deeply connected with the fractional derivative used. We introduce an appropriate generalization of practical stability which we call practical stability in time. Several sufficient conditions for practical stability in time are obtained using Lyapunov functions and the modified Razumikhin technique. Two types of derivatives of Lyapunov functions are used. Some examples are given to illustrate the introduced definitions and results.

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