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

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.

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

Axioms ◽  
2018 ◽  
Vol 8 (1) ◽  
pp. 4 ◽  
Author(s):  
◽  
◽  
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Distributed Delay ◽  
Lipschitz Stability ◽  
Razumikhin Technique ◽  
State Dependent ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential

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.

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

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5217-5239 ◽  
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.

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

2017 ◽  
Vol 24 (1) ◽  
pp. 1-13 ◽  
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.

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

2018 ◽  
Vol 21 (1) ◽  
pp. 72-93 ◽  
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.

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

2016 ◽  
Vol 19 (2) ◽  
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.

scholarly journals Applications of Lyapunov Functions to Caputo Fractional Differential Equations

Mathematics ◽  
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.

scholarly journals A Mixed Monotone Operator Method for the Existence and Uniqueness of Positive Solutions to Impulsive Caputo Fractional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Jieming Zhang ◽  
Chen Yang ◽  
Chengbo Zhai
Keyword(s):  
Differential Equations ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Operator Method ◽  
Monotone Operators ◽  
Mixed Monotone Operator ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential

We establish some sufficient conditions for the existence and uniqueness of positive solutions to a class of initial value problem for impulsive fractional differential equations involving the Caputo fractional derivative. Our analysis relies on a fixed point theorem for mixed monotone operators. Our result can not only guarantee the existence of a unique positive solution but also be applied to construct an iterative scheme for approximating it. An example is given to illustrate our main result.

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