A generalized q-fractional Gronwall inequality and its applications to nonlinear delay q-fractional difference systems

In this paper, we state and prove a new discrete fractional version of the generalized Gronwall inequality. Based on this, a particular version expressed by means of discrete Mittag-Leer functions is provided. As an application, we prove the uniqueness and obtain an estimate for the solutions of nonlinear delay Caputo fractional difference system. Numerical example is presented to demonstrate the applicability of the main results.

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Finite-time stability of $ q $-fractional damped difference systems with time delay

AIMS Mathematics ◽

10.3934/math.2021696 ◽

2021 ◽

Vol 6 (11) ◽

pp. 12011-12027

Author(s):

Jingfeng Wang ◽

◽

Chuanzhi Bai

Keyword(s):

Time Delay ◽

Finite Time ◽

Sufficient Conditions ◽

Gronwall Inequality ◽

The Novel ◽

Delayed Systems ◽

Finite Time Stability ◽

Difference Systems ◽

Grönwall Inequality ◽

Uniqueness Of The Solution

<abstract><p>In this paper, we investigate and obtain a new discrete $ q $-fractional version of the Gronwall inequality. As applications, we consider the existence and uniqueness of the solution of $ q $-fractional damped difference systems with time delay. Moreover, we formulate the novel sufficient conditions such that the $ q $-fractional damped difference delayed systems is finite time stable. Our result extend the main results of the paper by Abdeljawad et al. [A generalized $ q $-fractional Gronwall inequality and its applications to nonlinear delay $ q $-fractional difference systems, J.Inequal. Appl. 2016,240].</p></abstract>

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A generalized fractional (q,h)-Gronwall inequality and its applications to nonlinear fractional delay (q,h)-difference systems

10.22541/au.160407198.85210247/v1 ◽

2020 ◽

Author(s):

Feifei Du ◽

Baoguo Jia

Keyword(s):

Gronwall Inequality ◽

Difference Systems ◽

Grönwall Inequality ◽

Fractional Delay

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Finite-Time Stability of Solutions for Nonlinear q -Fractional Difference Coupled Delay Systems

Discrete Dynamics in Nature and Society ◽

10.1155/2021/3987479 ◽

2021 ◽

Vol 2021 ◽

pp. 1-13

Author(s):

Jingfeng Wang ◽

Chuanzhi Bai

Keyword(s):

Finite Time ◽

Gronwall Inequality ◽

Delay Systems ◽

Stability Of Solutions ◽

Stability Criteria ◽

Time Stability ◽

Fractional Difference ◽

Finite Time Stability ◽

Grönwall Inequality

In this paper, we investigate and prove a new discrete q -fractional version of the coupled Gronwall inequality. By applying this result, the finite-time stability criteria of solutions for a class of nonlinear q -fractional difference coupled delay systems are obtained. As an application, an example is provided to demonstrate the effectiveness of our result.

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A generalized fractional ( q , h )–Gronwall inequality and its applications to nonlinear fractional delay ( q , h )–difference systems

Mathematical Methods in the Applied Sciences ◽

10.1002/mma.7426 ◽

2021 ◽

Author(s):

Feifei Du ◽

Baoguo Jia

Keyword(s):

Gronwall Inequality ◽

Difference Systems ◽

Grönwall Inequality ◽

Fractional Delay

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Ulam stability of Caputo q-fractional delay difference equation: q-fractional Gronwall inequality approach

Journal of Inequalities and Applications ◽

10.1186/s13660-019-2257-6 ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 1

Author(s):

Rabia Ilyas Butt ◽

Thabet Abdeljawad ◽

Manar A. Alqudah ◽

Mujeeb ur Rehman

Keyword(s):

Existence And Uniqueness ◽

Gronwall Inequality ◽

Ulam Stability ◽

Delay Difference Equation ◽

Fractional Difference ◽

Grönwall Inequality ◽

Fractional Delay ◽

Delay Difference ◽

Theoretical Results ◽

Rassias Stability

AbstractIn this article, we discuss the existence and uniqueness of solution of a delay Caputo q-fractional difference system. Based on the q-fractional Gronwall inequality, we analyze the Ulam–Hyers stability and the Ulam–Hyers–Rassias stability. An example is provided to support the theoretical results.

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