scholarly journals Nonlinear Variation of Parameters Formula for Impulsive Differential Equations with Initial Time Difference and Application

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Peiguang Wang ◽  
Xiaowei Liu
Keyword(s):  
Differential Equations ◽  
Impulsive Differential Equations ◽  
Time Difference ◽  
Initial Time ◽  
Nonlinear Variation ◽  
Stability Properties ◽  
Variation Of Parameters ◽  
Variation Of Parameters Formula

This paper establishes variation of parameters formula for impulsive differential equations with initial time difference. As an application, one of the results is used to investigate stability properties of solutions.

Strict stability with respect to initial time difference for Caputo fractional differential equations by Lyapunov functions

2017 ◽  
Vol 24 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Donal O’Regan ◽  
Snezhana Hristova
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Time Difference ◽  
Initial Time ◽  
Advantages And Disadvantages ◽  
Strict Stability ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential

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.

scholarly journals The method of upper, lower solutions and monotone iterative scheme for higher order hyperbolic partial differential equations

1989 ◽  
Vol 47 (1) ◽  
pp. 153-170 ◽  
Author(s):  
Ravi P. Agarwal
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Iterative Scheme ◽  
Initial Conditions ◽  
Upper And Lower Solutions ◽  
Hyperbolic Partial Differential Equations ◽  
Partial Differential ◽  
Monotone Iterative ◽  
Variation Of Parameters ◽  
Variation Of Parameters Formula

AbstractUniformly monotone convergent iterative methods for obtaining multiple solutions of (n + m)th order hyperbolic partial differential equations together with initial conditions are discussed. Appropriate partial differential inequalities which connect upper and lower solutions, and variation of parameters formula is developed.

Some stability properties related to initial time difference for Caputo fractional differential equations

2018 ◽  
Vol 21 (1) ◽  
pp. 72-93 ◽  
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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