scholarly journals Linearization of Impulsive Differential Equations with Ordinary Dichotomy

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Yongfei Gao ◽  
Xiaoqing Yuan ◽  
Yonghui Xia ◽  
P. J. Y. Wong
Keyword(s):  
Nonlinear System ◽  
Linear System ◽  
Differential Equations ◽  
Impulsive Differential Equations ◽  
Impulsive System ◽  
Linearization Theorem

This paper presents a linearization theorem for the impulsive differential equations when the linear system has ordinary dichotomy. We prove that when the linear impulsive system has ordinary dichotomy, the nonlinear systemx˙(t)=A(t)x(t)+f(t,x),t≠tk,Δx(tk)=A~(tk)x(tk)+f~(tk,x),k∈ℤ, is topologically conjugated tox˙(t)=A(t)x(t),t≠tk,Δx(tk)=A~(tk)x(tk),k∈ℤ, whereΔx(tk)=x(tk+)-x(tk-),x(tk-)=x(tk), represents the jump of the solutionx(t)att=tk. Finally, two examples are given to show the feasibility of our results.

scholarly journals Linearization of Nonautonomous Impulsive System with Nonuniform Exponential Dichotomy

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Yongfei Gao ◽  
Yonghui Xia ◽  
Xiaoqing Yuan ◽  
P. J. Y. Wong
Keyword(s):  
Linear System ◽  
Differential Equations ◽  
Exponential Dichotomy ◽  
Impulsive Differential Equations ◽  
Impulsive System ◽  
Topological Conjugacy ◽  
Equivalent Function ◽  
Nonlinear Impulsive System ◽  
Nonuniform Exponential Dichotomy

This paper gives a version of Hartman-Grobman theorem for the impulsive differential equations. We assume that the linear impulsive system has a nonuniform exponential dichotomy. Under some suitable conditions, we proved that the nonlinear impulsive system is topologically conjugated to its linear system. Indeed, we do construct the topologically equivalent function (the transformation). Moreover, the method to prove the topological conjugacy is quite different from those in previous works (e.g., see Barreira and Valls, 2006).

A MONOTONE-ITERATIVE METHOD FOR FINDING PERIODIC SOLUTION OF AN IMPULSIVE DIFFERENTIAL EQUATIONS AND APPLICATION

2008 ◽  
Vol 01 (02) ◽  
pp. 247-256
Author(s):  
JIAWEI DOU
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Upper And Lower Solutions ◽  
Impulsive Differential Equations ◽  
Iteration Process ◽  
Impulsive System ◽  
Monotone Iterative Method ◽  
Initial Value ◽  
Monotone Iterative ◽  
Monotone Iterations

In this paper, using the method of upper and lower solutions and its associated monotone iterations, we establish a new monotone-iterative scheme for finding periodic solutions of an impulsive differential equations. This method leads to the existence of maximal and minimal periodic solutions which can be computed from a linear iteration process in the same fashion as for impulsive differential equations initial value problem. This method is constructive and can be used to develop a computational algorithm for numerical solution of the periodic impulsive system. Our existence result improves a result established in [1]. The result is applied to a model of mutualism of Lotka–Volterra type which involves interactions among a mutualist-competitor, a competitor and a mutualist, the existence of positive periodic solutions of the model is obtained.

scholarly journals Invariant Sets of Impulsive Differential Equations with Particularities inω-Limit Set

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Mykola Perestyuk ◽  
Petro Feketa
Keyword(s):  
Asymptotic Stability ◽  
Euclidean Space ◽  
Differential Equations ◽  
Direct Product ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Impulsive System ◽  
Invariant Sets ◽  
Limit Set ◽  
System Of Differential Equations

Sufficient conditions for the existence and asymptotic stability of the invariant sets of an impulsive system of differential equations defined in the direct product of a torus and an Euclidean space are obtained.

scholarly journals The general solution for impulsive differential equations with Riemann-Liouville fractional-order q ∈ (1,2)

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Xianmin Zhang ◽  
Praveen Agarwal ◽  
Zuohua Liu ◽  
Hui Peng
Keyword(s):  
Approximate Solution ◽  
General Solution ◽  
Differential Equations ◽  
Fractional Order ◽  
Impulsive Differential Equations ◽  
Impulsive System ◽  
Limit Case ◽  
Approach Zero

AbstractIn this paper we consider the generalized impulsive system with Riemann-Liouville fractional-order q ∈ (1,2) and obtained the error of the approximate solution for this impulsive system by analyzing of the limit case (as impulses approach zero), as well as find the formula for a general solution. Furthermore, an example is given to illustrate the importance of our results.

scholarly journals On Nonlinear Forced Impulsive Differential Equations under Canonical and Non-Canonical Conditions

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2066
Author(s):  
Shyam Sundar Santra ◽  
Hammad Alotaibi ◽  
Samad Noeiaghdam ◽  
Denis Sidorov
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Impulsive System ◽  
Bounded Solutions ◽  
Oscillatory Behaviour ◽  
Impulsive Systems ◽  
Forcing Term

This study is connected with the nonoscillatory and oscillatory behaviour to the solutions of nonlinear neutral impulsive systems with forcing term which is studied for various ranges of of the neutral coefficient. Furthermore, sufficient conditions are obtained for the existence of positive bounded solutions of the impulsive system. The mentioned example shows the feasibility and efficiency of the main results.

scholarly journals The General Solution of Impulsive Systems with Caputo-Hadamard Fractional Derivative of Orderq∈C  (R(q)∈(1,2))

2016 ◽  
Vol 2016 ◽  
pp. 1-20 ◽  
Author(s):  
Xianmin Zhang ◽  
Xianzhen Zhang ◽  
Zuohua Liu ◽  
Hui Peng ◽  
Tong Shu ◽  
...  
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Impulsive Differential Equations ◽  
Impulsive System ◽  
Limit Case ◽  
Impulsive Systems ◽  
Hadamard Fractional Derivative ◽  
Approach Zero

Motivated by some preliminary works about general solution of impulsive system with fractional derivative, the generalized impulsive differential equations with Caputo-Hadamard fractional derivative ofq∈C  (R(q)∈(1,2)) are further studied by analyzing the limit case (as impulses approach zero) in this paper. The formulas of general solution are found for the impulsive systems.

Normalization Bernstein Basis For Solving Fractional Fredholm-Integro Differential Equation

2018 ◽  
pp. 491
Author(s):  
Abdul Khaleq O. Al-Jubory ◽  
Shaymaa Hussain Salih
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Linear System ◽  
Differential Equations ◽  
Bernstein Basis ◽  
Traditional Methods ◽  
Integro Differential Equation

In this work, we employ a new normalization Bernstein basis for solving linear Freadholm of fractional integro-differential equations  nonhomogeneous  of the second type (LFFIDEs). We adopt Petrov-Galerkian method (PGM) to approximate solution of the (LFFIDEs) via normalization Bernstein basis that yields linear system. Some examples are given and their results are shown in tables and figures, the Petrov-Galerkian method (PGM) is very effective and convenient and overcome the difficulty of traditional methods. We solve this problem (LFFIDEs) by the assistance of Matlab10.   

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