On the concept and existence of solution for impulsive fractional differential equations

This paper is motivated from some recent papers treating the problem of the existence of a solution for impulsive differential equations with fractional derivative. We firstly show that the formula of solutions in cited papers are incorrect. Secondly, we reconsider a class of impulsive fractional differential equations and introduce a correct formula of solutions for a impulsive Cauchy problem with Caputo fractional derivative. Further, some sufficient conditions for existence of the solutions are established by applying fixed point methods. Some examples are given to illustrate the results.     

The Existence and Multiplicity of Solutions for Second-order Impulsive Differential Equations on the Half-line

In this paper, we consider the existence and multiplicity of solutions for a class of nonlinear impulsive problem on the half-line. By using a variational method and a variant fountain theorem, we obtain some new criteria to guarantee that the impulsive problem has at least one solution and infinitely many solutions respectively. We extend and improve some recent results.     

Variable impulsive consensus of nonlinear multi-agent systems

In this paper, the consensus problem for nonlinear multi-agent systems with variable impulsive control method is studied. In order to decrease the communication wastage, a novel distributed impulsive protocol is designed to achieve consensus. Compared with the common impulsive consensus method with fixed impulsive instants, the variable impulsive consensus method proposed in this paper is more flexible and reliable in practical application. Based on Lyapunov stability theory and some inequality techniques, several novel impulsive consensus conditions are obtained to realize the consensus of multi-agent systems. Finally, some necessary simulations are performed to validate the effectiveness of theoretical results.     

Existence of solutions for nonlinear impulsive differential equations with Dirichlet boundary conditions

In this paper, we consider the existence of solutions for a class of nonlinear impulsive problems with Dirichlet boundary conditions. We obtain some new existence theorems of solutions for the nonlinear impulsive problem by using critical point theory. We extend and improve some recent results.     

Positive solutions of impulsive boundary value problems for first-order nonlinear functional differential equations

In this paper, we study the problem on the existence of positive solutions for a class of impulsive periodic boundary value problems of first-order nonlinear functional differential equations. By using the fixed point theorem in cones and some analysis techniques, we present some sufficient conditions which guarantee the existence of one and multiple positive solutions for the impulsive periodic boundary value problems. Our results generalize and improve some previous results. Moreover, our results show that positive solutions for the impulsive periodic boundary value problems may be yielded completely by some proper impulsive conditions (see Example 4.1 and Remark 4.2 in Sect. 4), and also implies that proper impulsive conditions are of great significance to simulate processes, optimal control, population model and so on.     

Existence and multiplicity of solutions for an impulsive boundary value problem with a parameter via critical point theory

In this paper, an impulsive boundary value problem with a parameter is considered. By using critical point theory, some criteria are obtained to guarantee that the impulsive problem has at least one solution, two solutions and infinitely many solutions when the parameter lies in different intervals. The results obtained are also valid and new for a problem discussed in the literature.     

Existence of solutions for impulsive Dirichlet problems with the parameter inequality reverse

In this paper, we consider the existence and multiplicity for second‐order nonlinear impulsive differential equations with Dirichlet boundary condition and a parameter. By using critical point theory, we give some new criteria to guarantee that the impulsive problem has at least one solution or infinitely many solutions, assuming that the impulsive functions satisfy the superlinear growth condition and the parameter inequality is reverse. Our results extend and improve some recent results. Copyright © 2013 John Wiley & Sons, Ltd.     

Multiple Solutions of Nonlinear Impulsive Differential Equations with Dirichlet Boundary Conditions via Variational Method

In this paper, we consider the existence of multiple solutions for second-order nonlinear impulsive differential equations with Dirichlet boundary condition. We obtain some existence theorems of solutions for the nonlinear problem when the impulsive functions satisfies the superlinear growth conditions by critical point theory. We extend and improve some recent results.     

Existence of solutions for the integral boundary value problems of fractional order impulsive differential equations

In this paper, we study a class of integral boundary value problem for fractional order impulsive differential equations, where both the nonlinearity and the impulsive terms contain the fractional order derivatives. By using fixed‐point theorems, the existence results of solution for the boundary value problem are established. Finally, some examples are presented to illustrate the existence results. Copyright © 2015 John Wiley & Sons, Ltd.     

Mixed monotone iterative technique for semilinear impulsive fractional evolution equations

In this paper, we deals with the existence of mild $L$-quasi-solutions to the boundary value problem for a class of semilinear impulsive fractional evolution equations in an ordered Banach space $E$. Under a new concept of upper and lower solutions, a new monotone iterative technique on the initial value problem of impulsive fractional evolution equations has been established. The results improve and extend some relevant results in ordinary differential equations and partial differential equations. As some application that illustrate our results, An example is also given.     

Global impulsive exponential synchronization of time-delayed coupled chaotic systems

In this paper, the impulsive exponential synchronization problem for time-delayed coupled chaotic systems is investigated. By establishing an impulsive differential delay inequality and using the property of P-cone, some simple conditions of impulsive exponential synchronization of two coupled chaotic systems are derived. To illustrate the effectiveness of the new scheme, some numerical examples are given.     

序Banach空间不连续脉冲积分-微分方程初值问题的解

讨论了序Banach空间不连续脉冲积分-微分方程初值问题,通过建立一个新的比较定理,在比较弱的条件下推广了相关文献的主要结果.并在比较广泛的上控制条件而且只有一个上解或下解的假设下,获得了唯一解的存在性定理,而且给出了迭代序列的误差估计.从而推广并改进了最近某些文献中的相应结果.     

A new impulsive synchronization criterion for T–S fuzzy model and its applications

This paper focuses on the problem of impulsive synchronization of T–S fuzzy systems. A new synchronization criterion is derived for T–S fuzzy systems by utilizing the concept of average impulsive interval. The proposed impulsive control scheme has a simple control structure, and is theoretically and numerically proved to be less conservative than some existing results. The method is also illustrated by applying to Lorenz system, Rössler’s system as well as permanent magnet synchronous motors system.     

An application of variational method to second-order impulsive differential equation on the half-line

In this paper, we study the multiplicity of solutions for second-order impulsive differential equation with a parameter on the half-line. By using a variational method and a three critical points theorem, we give some new criteria to guarantee that the impulsive problem has at least three classical solutions. Also an example is given in this paper to illustrate the main results.     

Output tracking control of Boolean control networks with impulsive effects

This paper studies the output tracking problem of Boolean control networks (BCNs) with impulsive effects via the algebraic state‐space representation approach. The dynamics of BCNs with impulsive effects is converted to an algebraic form. Based on the algebraic form, some necessary and sufficient conditions are presented for the feedback output tracking control of BCNs with impulsive effects. These conditions contain constant reference signal case and time‐varying reference signal case. The study of an illustrative example shows that the obtained new results are effective.     

Mixed monotone iterative technique for a class of semilinear impulsive evolution equations in Banach spaces

This paper deals with the existence of mild L-quasi-solutions to the initial value problem for a class of semilinear impulsive evolution equations in an ordered Banach space E. Under a new concept of upper and lower solutions, a new monotone iterative technique on the initial value problem of impulsive evolution equations has been established. The results improve and extend some relevant results in ordinary differential equations and partial differential equations. An example is also given.     

Global exponential synchronization of nonlinear time-delay Lur’e systems via delayed impulsive control

This paper is concerned with the global exponential synchronization problem of two identical nonlinear time-delay Lur’e systems via delayed impulsive control. Some novel impulsive synchronization criteria are obtained by introducing a discontinuous Lyapunov function and by using the Lyapunov–Razumikhin technique, which are expressed in forms of linear matrix inequalities. The derived criteria reveal the effects of impulsive input delays and impulsive intervals on the stability of synchronization error systems. Then, sufficient conditions on the existence of a delayed impulsive controller are derived by employing these newly-obtained synchronization criteria. Additionally, some synchronization criteria for two identical time-delay Lur’e systems with impulsive effects are presented by using delayed continuous feedback control. The synchronization criteria via delayed continuous feedback control can deal with the case when the impulsive control strategy fails to synchronize two identical impulsive time-delay Lur’e systems. Three numerical examples are provided to illustrate the efficiency of the obtained results.     

Variational Approach to Fourth-Order Impulsive Differential Equations with Two Control Parameters

In this paper, we are concerned with the multiplicity of solutions for a fourth-order impulsive differential equation with Dirichlet boundary conditions and two control parameters. Using variational methods and a three critical points theorem, we give some new criteria to guarantee that the impulsive problem has at least three classical solutions. We also provide an example in order to illustrate the main abstract results of this paper.