Initial time difference quasilinearization for Caputo Fractional Differential Equations

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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Practical stability with respect to initial time difference for Caputo fractional differential equations

Communications in Nonlinear Science and Numerical Simulation ◽

10.1016/j.cnsns.2016.05.005 ◽

2017 ◽

Vol 42 ◽

pp. 106-120 ◽

Cited By ~ 15

Author(s):

Ravi Agarwal ◽

D. O’Regan ◽

S. Hristova ◽

M. Cicek

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Time Difference ◽

Practical Stability ◽

Initial Time ◽

Caputo Fractional Differential Equations ◽

Fractional Differential

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Qualitative results for finite system of R-L fractional differential equations with initial time difference

Journal of Mathematical and Computational Science ◽

10.28919/jmcs/5901 ◽

2021 ◽

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Finite System ◽

Time Difference ◽

Initial Time ◽

Fractional Differential

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Method of lower - upper solutions of fractional differential equations with initial time difference and applications

Malaya Journal of Matematik ◽

10.26637/mjm0803/0080 ◽

2020 ◽

Vol 8 (3) ◽

pp. 1196-1199

Author(s):

Nanware J.A. ◽

Dawkar B.D.

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Time Difference ◽

Initial Time ◽

Lower Upper ◽

Fractional Differential

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Initial time difference quasilinearization method for fractional differential equations involving generalized Hilfer fractional derivative

Computational and Applied Mathematics ◽

10.1007/s40314-019-1004-4 ◽

2019 ◽

Vol 39 (1) ◽

Cited By ~ 3

Author(s):

Kishor D. Kucche ◽

Ashwini D. Mali

Keyword(s):

Differential Equations ◽

Fractional Derivative ◽

Fractional Differential Equations ◽

Time Difference ◽

Initial Time ◽

Quasilinearization Method ◽

Hilfer Fractional Derivative ◽

Fractional Differential

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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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Implicit nonlinear fractional differential equations of variable order

Boundary Value Problems ◽

10.1186/s13661-021-01540-7 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Amar Benkerrouche ◽

Mohammed Said Souid ◽

Kanokwan Sitthithakerngkiet ◽

Ali Hakem

Keyword(s):

Differential Equations ◽

Boundary Value Problem ◽

Fractional Differential Equations ◽

Boundary Value ◽

Variable Order ◽

Stability Of Solutions ◽

Nonlinear Fractional Differential Equations ◽

Caputo Fractional Differential Equations ◽

Fractional Differential ◽

The Stability

AbstractIn this manuscript, we examine both the existence and the stability of solutions to the implicit boundary value problem of Caputo fractional differential equations of variable order. We construct an example to illustrate the validity of the observed results.

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