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

2021 ◽  
Author(s):  
Feifei Du ◽  
Baoguo Jia
Keyword(s):  
Gronwall Inequality ◽  
Difference Systems ◽  
Grönwall Inequality ◽  
Fractional Delay

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

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>

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

2019 ◽  
Vol 2019 (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.

The Gronwall inequality for weighted integrals

1971 ◽  
Vol 30 (3) ◽  
pp. 504-504 ◽  
Author(s):  
F. M. Wright ◽  
M. L. Klasi ◽  
D. R. Kennebeck
Keyword(s):  
Gronwall Inequality ◽  
Grönwall Inequality ◽  
Weighted Integrals

scholarly journals On the Hyers-Ulam Stability of Differential Equations of Second Order

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Qusuay H. Alqifiary ◽  
Soon-Mo Jung
Keyword(s):  
Differential Equations ◽  
Initial Conditions ◽  
Second Order ◽  
Gronwall Inequality ◽  
Ulam Stability ◽  
Grönwall Inequality

By using of the Gronwall inequality, we prove the Hyers-Ulam stability of differential equations of second order with initial conditions.

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