In this article, criteria of eventual stability are established for impulsive differential systems using piecewise continuous Lyapunov functions. The sufficient conditions that are obtained significantly depend on the...In this article, criteria of eventual stability are established for impulsive differential systems using piecewise continuous Lyapunov functions. The sufficient conditions that are obtained significantly depend on the moments of impulses. An example is discussed to illustrate the theorem.展开更多
The purpose of this paper is to study the existence and iterative approximation of minimax quasi-solutions for a class of initial value problems of first order impulsive differential systems by using monotone iterativ...The purpose of this paper is to study the existence and iterative approximation of minimax quasi-solutions for a class of initial value problems of first order impulsive differential systems by using monotone iterative methods.展开更多
The existence and uniqueness in mean square of solutions to certain random impulsive differential systems is discussed in this paper. Cauchy-Schwarz inequality, Lipschtiz condition and techniques in stochastic analysi...The existence and uniqueness in mean square of solutions to certain random impulsive differential systems is discussed in this paper. Cauchy-Schwarz inequality, Lipschtiz condition and techniques in stochastic analysis are employed in achieve the desired results.展开更多
In this paper, the Razumikhin approach is applied to the study of both p-th moment and almost sure stability on a general decay for a class of impulsive stochastic functional differential systems with Markovian switch...In this paper, the Razumikhin approach is applied to the study of both p-th moment and almost sure stability on a general decay for a class of impulsive stochastic functional differential systems with Markovian switching. Based on the Lyapunov-Razumikhin methods, some sufficient conditions are derived to check the stability of impulsive stochastic functional differential systems with Markovian switching. One numerical example is provided to demonstrate the effectiveness of the results.展开更多
This paper studies the existence and uniqueness conditions for the quaternion-valued non-linear impulsive system. Firstly, a space of quaternion-valued piecewise functions is constructed and completeness of the space ...This paper studies the existence and uniqueness conditions for the quaternion-valued non-linear impulsive system. Firstly, a space of quaternion-valued piecewise functions is constructed and completeness of the space is also proved. Then by taking advantage of the Bielecki norm and fixed point theorem, existence and uniqueness criteria of quaternion-valued nonlinear impulsive system are obtained. At last, an example is given to illustrate our theoretical results.展开更多
We study the existence of solutions to the second order three-point boundary value problem：{x″（t）＋f（t,x（t）,x′（t））=0,t≠ti,△x（ti）=Ii（x（ti）,x′（ti））,i=1,2,…,k,△x′（ti）=Ji（x（ti）,x′（t））,i=1...We study the existence of solutions to the second order three-point boundary value problem：{x″（t）＋f（t,x（t）,x′（t））=0,t≠ti,△x（ti）=Ii（x（ti）,x′（ti））,i=1,2,…,k,△x′（ti）=Ji（x（ti）,x′（t））,i=1,2,…,k,x（0）=0=x（1）-αx（η）,where 0〈η〈1,α∈R,and f：[0,1]×R×R→R,Ii：R×R→R,Ji：R×R→R（i=1,2,…,k）are continuous. Our results is new and different from previous results. In particular, we obtain the Green function of the problem, which makes the problem simpler.展开更多
Aim To investigate the existence of positive solutions for impulsive neutral differential equations. Methods The Banach contraction principle was used to establish our results. Results and Conclusion The results of...Aim To investigate the existence of positive solutions for impulsive neutral differential equations. Methods The Banach contraction principle was used to establish our results. Results and Conclusion The results of the existence of positive solutions for impulsive neutral differential equations are obtained.展开更多
The basic objects of investigation in this article are nonlinear impulsive dif- ferential equations. The impulsive moments coincide with the moments of meeting of the integral curve and some of the so-called barrier c...The basic objects of investigation in this article are nonlinear impulsive dif- ferential equations. The impulsive moments coincide with the moments of meeting of the integral curve and some of the so-called barrier curves. For such type of equations, suf- ficient conditions are found under which the solutions are continuously dependent on the perturbations with respect to the initial conditions and barrier curves. The results are applied to a mathematical model of population dynamics.展开更多
A scheme for the impulsive control of nonlinear systems with time-varying delays is investigated in this paper. Based on the Lyapunov-like stability theorem for impulsive functional differential equations （FDEs）, so...A scheme for the impulsive control of nonlinear systems with time-varying delays is investigated in this paper. Based on the Lyapunov-like stability theorem for impulsive functional differential equations （FDEs）, some sufficient conditions are presented to guarantee the uniform asymptotic stability of impulsively controlled nonlinear systems with time-varying delays. These conditions are more effective and less conservative than those obtained. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed method.展开更多
In this paper, with a given manifold y = H（x）, we have constructed a response system for a continuous-time chaotic system as a drive system, and used impulsive control theory to demonstrate theoretically that this r...In this paper, with a given manifold y = H（x）, we have constructed a response system for a continuous-time chaotic system as a drive system, and used impulsive control theory to demonstrate theoretically that this response system can achieve impulsive generalized synchronization （GS） with the drive system. Our theoretical result is supported by numerical examples.展开更多
In this paper, we investigate the stability of a class of impulsive functional differential equations by using Lyapunov functional and Jensen＇s inequality. Some new stability theorems are obtained. Examples are given...In this paper, we investigate the stability of a class of impulsive functional differential equations by using Lyapunov functional and Jensen＇s inequality. Some new stability theorems are obtained. Examples are given to demonstrate the advantage of the obtained results.展开更多
This paper is devoted to study the existence and uniqueness of solutions to a boundary value problem of nonlinear fractional differential equation with impulsive effects.The arguments are based upon Schauder and Banac...This paper is devoted to study the existence and uniqueness of solutions to a boundary value problem of nonlinear fractional differential equation with impulsive effects.The arguments are based upon Schauder and Banach fixed-point theorems. We improve and generalize the results presented in[B.Ahmad,S.Sivasundaram, Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations,Nonlinear Analysis：Hybrid Systems,3（2009）,251- 258].展开更多
Positive results are proved here about the ability of balanced methods to reproduce the mean square stability of the impulsive stochastic differential equations. It is shown that the balanced methods with strong conve...Positive results are proved here about the ability of balanced methods to reproduce the mean square stability of the impulsive stochastic differential equations. It is shown that the balanced methods with strong convergence can preserve the mean square stability with the sufficiently small stepsize. Weak variants and their mean square stability are also considered. Several numerical experiments are given for illustration and show that the fully implicit methods are superior to those of the explicit methods in terms of mean-square stabilities for relatively large stepsizes especially.展开更多
In this paper we investigate a class of impulsive differential equations with Dirichlet boundary conditions. Firstly, we define new inner product of <img src="Edit_890fce38-e82b-4f36-be40-9d05e8119b88.png"...In this paper we investigate a class of impulsive differential equations with Dirichlet boundary conditions. Firstly, we define new inner product of <img src="Edit_890fce38-e82b-4f36-be40-9d05e8119b88.png" width="40" height="17" alt="" /> and prove that the norm which is deduced by the inner product is equivalent to the usual norm. Secondly, we construct the lower and upper solutions of (1.1). Thirdly, we obtain the existence of a positive solution, a negative solution and a sign-changing solution by using critical point theory and variational methods. Finally, an example is presented to illustrate the application of our main result.展开更多
In this paper, by using of monotone iterative technique, the existence and iterative approximation of the minimax quasi_solutions of the initial value problems for more general first order impulsive differential equat...In this paper, by using of monotone iterative technique, the existence and iterative approximation of the minimax quasi_solutions of the initial value problems for more general first order impulsive differential equations in Banach spaces are investigated.展开更多
In this work,stability with respect to part of the variables of nonlinear impulsive Caputo fractional differential equations is investigated.Some effective sufficient conditions of stability,uniform stability,asymptot...In this work,stability with respect to part of the variables of nonlinear impulsive Caputo fractional differential equations is investigated.Some effective sufficient conditions of stability,uniform stability,asymptotic uniform stability and Mittag Leffler stability.The approach presented is based on the specially introduced piecewise continuous Lyapunov functions.Furthermore,some numerical examples are given to show the effectiveness of our obtained theoretical results.展开更多
The existence of periodic solutions for a class of impulsive differential equations of mixed type is studied by constructing periodic sequence solutions of difference equations.
In this article,a new algorithm is presented to solve the nonlinear impulsive differential equations.In the first time,this article combines the reproducing kernel method with the least squares method to solve the sec...In this article,a new algorithm is presented to solve the nonlinear impulsive differential equations.In the first time,this article combines the reproducing kernel method with the least squares method to solve the second-order nonlinear impulsive differential equations.Then,the uniform convergence of the numerical solution is proved,and the time consuming Schmidt orthogonalization process is avoided.The algorithm is employed successfully on some numerical examples.展开更多
Consider the following equations：{λx＂（t）＋f（t,x（t））=0,t≠tiΔλx（ti）=Ii（x（ti））,i=1,2,…,kΔλx（ti）=Li（x（ti））,i=1,2,…,kx＇（0）=0=x（1）-αx（η）.Where 0 〈 η〈 1,0 〈 α 〈 1, and f ： [0,1] &#...Consider the following equations：{λx＂（t）＋f（t,x（t））=0,t≠tiΔλx（ti）=Ii（x（ti））,i=1,2,…,kΔλx（ti）=Li（x（ti））,i=1,2,…,kx＇（0）=0=x（1）-αx（η）.Where 0 〈 η〈 1,0 〈 α 〈 1, and f ： [0,1] × [0, ∞） → [0, ∞）, Ii,Li ： [0, ∞） → R, （i = 1, 2,…, k） are continuous functions. We prove the existence of eigenvalues for the problem under a weaker condition, moreover we do not require the monotonicity of the impulsive functions.展开更多
基金This work is supported by the National Natural Science Foundation of China (60474008)
文摘In this article, criteria of eventual stability are established for impulsive differential systems using piecewise continuous Lyapunov functions. The sufficient conditions that are obtained significantly depend on the moments of impulses. An example is discussed to illustrate the theorem.
文摘The purpose of this paper is to study the existence and iterative approximation of minimax quasi-solutions for a class of initial value problems of first order impulsive differential systems by using monotone iterative methods.
基金Supported by the National Natural Science Foundation of China(No.10371074)ShuGuang Plan of Shanghai City(No.04SG27)
文摘The existence and uniqueness in mean square of solutions to certain random impulsive differential systems is discussed in this paper. Cauchy-Schwarz inequality, Lipschtiz condition and techniques in stochastic analysis are employed in achieve the desired results.
文摘In this paper, the Razumikhin approach is applied to the study of both p-th moment and almost sure stability on a general decay for a class of impulsive stochastic functional differential systems with Markovian switching. Based on the Lyapunov-Razumikhin methods, some sufficient conditions are derived to check the stability of impulsive stochastic functional differential systems with Markovian switching. One numerical example is provided to demonstrate the effectiveness of the results.
基金supported by the National Natural Science Foundation of China under Grant No.61673296
文摘This paper studies the existence and uniqueness conditions for the quaternion-valued non-linear impulsive system. Firstly, a space of quaternion-valued piecewise functions is constructed and completeness of the space is also proved. Then by taking advantage of the Bielecki norm and fixed point theorem, existence and uniqueness criteria of quaternion-valued nonlinear impulsive system are obtained. At last, an example is given to illustrate our theoretical results.
基金Supported by the National Natural Science Foundation of China(10371006)
文摘We study the existence of solutions to the second order three-point boundary value problem：{x″（t）＋f（t,x（t）,x′（t））=0,t≠ti,△x（ti）=Ii（x（ti）,x′（ti））,i=1,2,…,k,△x′（ti）=Ji（x（ti）,x′（t））,i=1,2,…,k,x（0）=0=x（1）-αx（η）,where 0〈η〈1,α∈R,and f：[0,1]×R×R→R,Ii：R×R→R,Ji：R×R→R（i=1,2,…,k）are continuous. Our results is new and different from previous results. In particular, we obtain the Green function of the problem, which makes the problem simpler.
文摘Aim To investigate the existence of positive solutions for impulsive neutral differential equations. Methods The Banach contraction principle was used to establish our results. Results and Conclusion The results of the existence of positive solutions for impulsive neutral differential equations are obtained.
文摘The basic objects of investigation in this article are nonlinear impulsive dif- ferential equations. The impulsive moments coincide with the moments of meeting of the integral curve and some of the so-called barrier curves. For such type of equations, suf- ficient conditions are found under which the solutions are continuously dependent on the perturbations with respect to the initial conditions and barrier curves. The results are applied to a mathematical model of population dynamics.
基金supported by the National Natural Science Foundation of China (Grant Nos 60534010,60774048,60728307,60804006 and 60521003)the National High Technology Research and Development Program of China (Grant No 2006AA04Z183)+1 种基金Liaoning Provincial Natural Science Foundation,China (Grant No 20062018)111 Project (Grant No B08015)
文摘A scheme for the impulsive control of nonlinear systems with time-varying delays is investigated in this paper. Based on the Lyapunov-like stability theorem for impulsive functional differential equations （FDEs）, some sufficient conditions are presented to guarantee the uniform asymptotic stability of impulsively controlled nonlinear systems with time-varying delays. These conditions are more effective and less conservative than those obtained. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed method.
基金Project supported by the National Natural Science Foundation of China （Grant No 10372054）.
文摘In this paper, with a given manifold y = H（x）, we have constructed a response system for a continuous-time chaotic system as a drive system, and used impulsive control theory to demonstrate theoretically that this response system can achieve impulsive generalized synchronization （GS） with the drive system. Our theoretical result is supported by numerical examples.
基金supported by the National Natural Science Foundation of China (No. 10871063)Scientific Research Fund of Hunan Provincial Education Department (No. 07A038)
文摘In this paper, we investigate the stability of a class of impulsive functional differential equations by using Lyapunov functional and Jensen＇s inequality. Some new stability theorems are obtained. Examples are given to demonstrate the advantage of the obtained results.
基金The NSF(10971221)of ChinaThe Youth Research Found(2009QS07)of China University of Mining and Technology,Beijing
文摘This paper is devoted to study the existence and uniqueness of solutions to a boundary value problem of nonlinear fractional differential equation with impulsive effects.The arguments are based upon Schauder and Banach fixed-point theorems. We improve and generalize the results presented in[B.Ahmad,S.Sivasundaram, Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations,Nonlinear Analysis：Hybrid Systems,3（2009）,251- 258].
基金National Natural Science Foundations of China(Nos.11561028,11101101,11461032,11401267)Natural Science Foundations of Jiangxi Province,China(Nos.20151BAB201013,20151BAB201010,20151BAB201015)
文摘Positive results are proved here about the ability of balanced methods to reproduce the mean square stability of the impulsive stochastic differential equations. It is shown that the balanced methods with strong convergence can preserve the mean square stability with the sufficiently small stepsize. Weak variants and their mean square stability are also considered. Several numerical experiments are given for illustration and show that the fully implicit methods are superior to those of the explicit methods in terms of mean-square stabilities for relatively large stepsizes especially.
文摘In this paper we investigate a class of impulsive differential equations with Dirichlet boundary conditions. Firstly, we define new inner product of <img src="Edit_890fce38-e82b-4f36-be40-9d05e8119b88.png" width="40" height="17" alt="" /> and prove that the norm which is deduced by the inner product is equivalent to the usual norm. Secondly, we construct the lower and upper solutions of (1.1). Thirdly, we obtain the existence of a positive solution, a negative solution and a sign-changing solution by using critical point theory and variational methods. Finally, an example is presented to illustrate the application of our main result.
文摘In this paper, by using of monotone iterative technique, the existence and iterative approximation of the minimax quasi_solutions of the initial value problems for more general first order impulsive differential equations in Banach spaces are investigated.
文摘In this work,stability with respect to part of the variables of nonlinear impulsive Caputo fractional differential equations is investigated.Some effective sufficient conditions of stability,uniform stability,asymptotic uniform stability and Mittag Leffler stability.The approach presented is based on the specially introduced piecewise continuous Lyapunov functions.Furthermore,some numerical examples are given to show the effectiveness of our obtained theoretical results.
基金theAppliedScienceFoundationofYunnan China (97A10 16Q)
文摘The existence of periodic solutions for a class of impulsive differential equations of mixed type is studied by constructing periodic sequence solutions of difference equations.
基金This work is supported by a Young Innovative Talents Program in Universities and Colleges of Guangdong Province(2018KQNCX338)two Scientific Research-Innovation Team Projects at Zhuhai Campus,Beijing Institute of Technology(XK-2018-15,XK-2019-10).
文摘In this article,a new algorithm is presented to solve the nonlinear impulsive differential equations.In the first time,this article combines the reproducing kernel method with the least squares method to solve the second-order nonlinear impulsive differential equations.Then,the uniform convergence of the numerical solution is proved,and the time consuming Schmidt orthogonalization process is avoided.The algorithm is employed successfully on some numerical examples.
基金Supported by the NNSF of China(10371006) Supported by the Youth Teacher Science Research Foundation of Central University of Nationalities(CUN08A)
文摘Consider the following equations：{λx＂（t）＋f（t,x（t））=0,t≠tiΔλx（ti）=Ii（x（ti））,i=1,2,…,kΔλx（ti）=Li（x（ti））,i=1,2,…,kx＇（0）=0=x（1）-αx（η）.Where 0 〈 η〈 1,0 〈 α 〈 1, and f ： [0,1] × [0, ∞） → [0, ∞）, Ii,Li ： [0, ∞） → R, （i = 1, 2,…, k） are continuous functions. We prove the existence of eigenvalues for the problem under a weaker condition, moreover we do not require the monotonicity of the impulsive functions.
