A Gronwall Inequality of Fractional Variable Order

2021 ◽  
Vol 7 (2) ◽  
pp. 83-86
Keyword(s):  
Gronwall Inequality ◽  
Variable Order ◽  
Grönwall Inequality

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 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 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.

scholarly journals A generalized discrete fractional Gronwall inequality and its application on the uniqueness of solutions for nonlinear delay fractional difference system

2018 ◽  
Vol 12 (1) ◽  
pp. 36-48 ◽  
Author(s):  
Jehad Alzabut ◽  
Thabet Abdeljawad
Keyword(s):  
Gronwall Inequality ◽  
Difference System ◽  
Uniqueness Of Solutions ◽  
Fractional Difference ◽  
Numerical Example ◽  
Generalized Gronwall Inequality ◽  
Grönwall Inequality ◽  
Nonlinear Delay

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.

scholarly journals Oscillation in deviating differential equations using an iterative method

2019 ◽  
Vol 27 (2) ◽  
pp. 143-169 ◽  
Author(s):  
George E. Chatzarakis ◽  
Irena Jadlovská
Keyword(s):  
Differential Equations ◽  
Iterative Method ◽  
Gronwall Inequality ◽  
Deviating Arguments ◽  
First Order ◽  
Grönwall Inequality ◽  
Nonnegative Coefficients ◽  
First Order Differential Equations

AbstractSufficient oscillation conditions involving lim sup and lim inf for first-order differential equations with non-monotone deviating arguments and nonnegative coefficients are obtained. The results are based on the iterative application of the Grönwall inequality. Examples, numerically solved in MATLAB, are also given to illustrate the applicability and strength of the obtained conditions over known ones.

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