Existence and compactness of solutions to impulsive differential equations with nonlocal conditions

The new type of nonlinear integral inequalities of Volterra–Fredholm type for discontinuous functions is investigated. Then, by using these inequalities and Schaefer fixed‐point theorem, we present new existence results for impulsive semilinear differential equations with nonlocal conditions. Moreover, the compactness of solution sets can be shown in some certain conditions. Copyright © 2015 John Wiley & Sons, Ltd.     

Picard and weakly Picard operators technique for nonlinear differential equations in Banach spaces

In this paper we study nonlocal Cauchy problems for differential equations in Banach spaces. Using Picard and weakly Picard operators technique and suitable Bielecki norms, some existence, uniqueness and data dependence results are obtained under some mild conditions. As an application, we also discuss a class of impulsive Cauchy problems by adapting the same methods.     

On the natural solution of an impulsive fractional differential equation of order q ∈ (1, 2)

This paper is motivated from some recent papers treating the impulsive Cauchy problems for some differential equations with fractional order q ∈ (1, 2). A better definition of solution for impulsive fractional differential equation is given. We build up an effective way to find natural solution for such problems. Then sufficient conditions for existence of the solutions are established by applying fixed point methods. Four examples are given to illustrate the results.     

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.     

Stability of impulsive delay differential equations

In this paper, we introduce impulsive delayed matrix function and give its norm estimation. With the help of impulsive delayed Cauchy matrix and the variation of constants method, we obtain representation of solutions to linear impulsive delay differential equations. Moreover, we derive some sufficient conditions to guarantee the trivial solution is locally asymptotically stable. Finally, two examples are given to illustrate our theoretical results.     

Existence of Solutions for Fractional Integro-differential Equations with Impulsive and Integral Conditions

This paper presents the existence of solutions for a class of Cauchy problems with integral condition for impulsive fractional integro-differential equations. Based on definition of solution for impulsive fractional integro- differential equations, the existence theorems of solutions of fractional differ- ential equation are obtained by applying fixed point methods. Finally, three examples are given to demonstrate the feasibility of the obtained results.     

Generalized Cauchy problems involving nonlocal and impulsive conditions

We study the abstract Cauchy problem involving a class of nonlinear differential inclusions, with impulsive and nonlocal conditions. By using MNC estimates, the existence result and continuous dependence on initial data of the solution set are proved.     

Existence of Global Solutions for a Class of Abstract Second-Order Nonlocal Cauchy Problem with Impulsive Conditions in Banach Spaces

In this paper, we are concerned with a class of abstract second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces. First, we study the existence of mild solutions for a class of second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces on an interval [0,a]. Later, we study a couple of cases where we can establish the existence of global solutions for a class of abstract second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces. We apply our theory to study the existence of solutions for impulsive partial differential equations.     

A new continuous dependence result for impulsive retarded functional differential equations

We consider a large class of impulsive retarded functional differential equations (IRFDEs) and prove a result concerning uniqueness of solutions of impulsive FDEs. Also, we present a new result on continuous dependence of solutions on parameters for this class of equations. More precisely, we consider a sequence of initial value problems for impulsive RFDEs in the above setting, with convergent right-hand sides, convergent impulse operators and uniformly convergent initial data. We assume that the limiting equation is an impulsive RFDE whose initial condition is the uniform limit of the sequence of the initial data and whose solution exists and is unique. Then, for sufficient large indexes, the elements of the sequence of impulsive retarded initial value problem admit a unique solution and such a sequence of solutions converges to the solution of the limiting Cauchy problem.     

A Robin boundary value problem in the upper half plane for the Bitsadze equation

In this article we give the solvability conditions and an integral representation of the solution of a Robin problem for the Bitsadze equation in the upper half plane. In order to do that, we use classical results of complex analysis and carry out the composition of two Robin problems for the Cauchy Riemann operator.     

Periodic nonautonomous differential equations with noninstantaneous impulsive effects

In this paper, we study a new class of periodic nonautonomous differential equations with periodic noninstantaneous impulsive effects. A concept of noninstantaneous impulsive Cauchy matrix is introduced, and some basic properties are considered. We give the representation of solutions to the homogeneous problem and nonhomogeneous problem by using noninstantaneous impulsive Cauchy matrix, and the variation of constants method, adjoint systems, and periodicity of solutions is verified under standard periodicity conditions. Further, we show the existence and uniqueness of solutions of semilinear problem and establish existence result for periodic solutions via Brouwer fixed point theorem and uniqueness and global asymptotic stability via Banach fixed point theorem.     

脉冲平方Logistic型时滞微分方程的全局吸引性

本文研究平方 Logistic型时滞微分方程在脉冲条件下的全局吸引性 ,获得了保证方程每一解趋于 0的充分条件 .     

非线性Schr\"{o}dinger 方程解的整体存在和爆破的门槛结果

研究了非线性 Schr\"{o}dinger 方程的柯西问题. 通过引进位势井及其外部集合, 得到了解的整体存在性和爆破的门槛结果.     

Almost periodic solutions for a class of non-instantaneous impulsive differential equations

Abstract

This paper is concerned with almost periodic solutions for nonlinear non-instantaneous impulsive differential equations with variable structure. With the help of the notation of non-instantaneous impulsive Cauchy matrix, mild sufficient conditions are derived to guarantee the existence, uniqueness of asymptotically stable almost periodic solutions. Both example and numerical simulation are given to illustrate our effectiveness of the above results. As one expects, the results presented here have extended and improved some previous results for instantaneous impulsive differential equations.     

Cauchy and Fredholm methods for Euler equations

The mathematical treatment of many problems in mathematical physics requires the minimization of a quadratic functional. It is shown that the optimizing function can be viewed as the solution of the familiar Euler equation, subject to boundary conditions, or as the solution of a certain Fredholm integral equation, or as the solution of an initial-value (Cauchy) problem. Each formulation has certain analytic and computational advantages and disadvantages.     

Local and global existence of mild solution to an impulsive fractional functional integro-differential equation with nonlocal condition

This paper is concerned with the local and global existence of mild solution for an impulsive fractional functional integro differential equations with nonlocal condition. We establish a general framework to find the mild solutions for impulsive fractional integro-differential equations, which will provide an effective way to deal with such problems. The results are obtained by using the fixed point technique and solution operator on a complex Banach space.     

Impulsive surfaces on dynamical systems

This work is devoted to the construction of impulsive sets in \({\mathbb{R}^n}\). In the literature, there are many examples of impulsive dynamical systems whose impulsive sets are chosen in an abstract way, and in this paper we present sufficient conditions to characterize impulsive sets in \({\mathbb{R}^n}\) which satisfy some “tube conditions” and ensure a good behavior of the flow. Moreover, we present some examples to illustrate the theoretical results.     

On improperly posed Caucfay problems and their approximate solution

It is well known that, in general, the Cauchy problem for theLaplace equation does not allow a solution and therefore isill-posed in both the Hadamard and the Tikhonov senses. Thepresent work focuses on the question whether the problem hasany meaningful approximate solution for arbitrary boundary conditions.Firstly, it is shown that it is possible to construct an analyticfunction which assumes some prescribed value on part of theboundary of a simply-connected domain. This problem is thenshown to be equivalent to the Cauchy problem under consideration,the solution to which can thus be invariably approximated toany degree of accuracy on the unit circle centred at the originwhen both the potential and the flux are specified as square-integrablefunctions over half the unit circle boundary. The uniquenessof the exact solution to the problem is also established. Theseresults are actually true for any simply-connected domain whichcan be conformally mapped onto the unit circle so that the partof its boundary with prescribed potential and flux correspondsto one-half of the unit circle boundary. Finally, the feasibilityof a boundary element formulation for a generic type of ill-posedboundary value problems is briefly discussed.     

Permanence and stability of an Ivlev-type predator–prey system with impulsive control strategies

In this paper, we study a predator–prey system with an Ivlev-type functional response and impulsive control strategies containing a biological control (periodic impulsive immigration of the predator) and a chemical control (periodic pesticide spraying) with the same period, but not simultaneously. We find conditions for the local stability of the prey-free periodic solution by applying the Floquet theory of an impulsive differential equation and small amplitude perturbation techniques to the system. In addition, it is shown that the system is permanent under some conditions by using comparison results of impulsive differential inequalities. Moreover, we add a forcing term into the prey population’s intrinsic growth rate and find the conditions for the stability and for the permanence of this system.     

Global existence,nonexistence and asymptotic behavior of solutions for the Cauchy problem of semilinear heat equations

In this paper, we study the Cauchy problem of semilinear heat equations. By introducing a family of potential wells, we first prove the invariance of some sets and isolating solutions. Then we obtain a threshold result for the global existence and nonexistence of solutions. Finally we discuss the asymptotic behavior of the solution.