scholarly journals The Generalized Gronwall Inequality and Its Application to Periodic Solutions of Integrodifferential Impulsive Periodic System on Banach Space

2008 ◽  
Vol 2008 (1) ◽  
pp. 430521 ◽  
Author(s):  
JinRong Wang ◽  
X Xiang ◽  
W Wei ◽  
Qian Chen
Keyword(s):  
Banach Space ◽  
Periodic Solutions ◽  
Periodic System ◽  
Gronwall Inequality ◽  
Generalized Gronwall Inequality ◽  
Grönwall Inequality

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 Existence of Periodic Solutions for Integrodifferential Impulsive Periodic System on Banach Space

2008 ◽  
Vol 2008 ◽  
pp. 1-19 ◽  
Author(s):  
JinRong Wang ◽  
X. Xiang ◽  
W. Wei
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Solutions ◽  
Periodic System ◽  
Evolution Operator ◽  
Mild Solutions ◽  
Periodic Systems ◽  
Gronwall Lemma ◽  
Existence Of Periodic Solutions

This paper deals with a class of integrodifferential impulsive periodic systems on Banach space. Using impulsive periodic evolution operator given by us, theT0-periodicPC-mild solution is introduced and suitablePoincaréoperator is constructed. By virtue of the generalized new Gronwall lemma with impulse andB-norm, the estimate on thePC-mild solutions is derived. Showing the continuity and compactness of thePoincaréoperator, we utilize Horn's fixed point theorem to prove the existence ofT0-periodicPC-mild solutions when thePC-mild solutions are bounded and ultimate bounded. This extends the study of periodic solutions of integrodifferential periodic system without impulse to integrodifferential periodic system with impulse on general Banach spaces. At last, an example is given for demonstration.

Periodic solutions of semilinear impulsive periodic system on Banach space

2009 ◽  
Vol 71 (12) ◽  
pp. e1344-e1353 ◽  
Author(s):  
JinRong Wang ◽  
X. Xiang ◽  
Y. Peng
Keyword(s):  
Banach Space ◽  
Periodic Solutions ◽  
Periodic System

scholarly journals Finite-Time Stability of Neutral Fractional Time-Delay Systems via Generalized Gronwalls Inequality

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Pang Denghao ◽  
Jiang Wei
Keyword(s):  
Time Delay ◽  
Finite Time ◽  
Sufficient Conditions ◽  
Gronwall Inequality ◽  
Delay Systems ◽  
Time Delay Systems ◽  
Time Stability ◽  
Finite Time Stability ◽  
Generalized Gronwall Inequality ◽  
Grönwall Inequality

This paper studies the finite-time stability of neutral fractional time-delay systems. With the generalized Gronwall inequality, sufficient conditions of the finite-time stability are obtained for the particular class of neutral fractional time-delay systems.

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
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