Ultimate boundedness of impulsive fractional differential equations

This article investigates nonlinear impulsive Caputo fractional differential equations. Utilizing Lyapunov functions, Laplace transforms of fractional derivatives and boundedness of Mittag-Leffler functions, several sufficient conditions are derived to ensure the global ultimate boundedness and the exponential stability of the systems. An example is given to explain the obtained results.     

BOUNDEDNESS OF IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAYS

In this paper, we study the boundedness of a class of impulsive functional differential equations with infinite delays. We establish a uniform boundedness theorem and a uniformly ultimate boundedness theorem, which show that a certain impulsive perturbation can make an unbounded system into uniformly bounded, even uniformly ultimate bounded.     

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.     

滞后型脉冲泛函微分方程有界性的Lyapunov逆定理

本文通过建立滞后型脉冲泛函微分方程饱和解的存在唯一性定理,在广义常微分方程与滞后型脉冲泛函微分方程等价的基础上,研究了滞后型脉冲泛函微分方程关于一致有界性的Lyapunov逆定理.     

Local and global existence of mild solutions for impulsive fractional semilinear integro-differential equation

In this paper, we study the local and global existence of mild solutions for impulsive fractional semilinear integro-differential equations in an arbitrary Banach space associated with operators generating compact semigroup on the Banach space. Also, we review some applications of fractional differential equations.     

General methods for converting impulsive fractional differential equations to integral equations and applications

In this paper, we propose the concepts of Caputo fractional derivatives and Caputo type Hadamard fractional derivatives for piecewise continuous functions. We obtain general solutions of four classes of impulsive fractional differential equations (Theorem 3.1–Theorem 3.4) respectively. These results are applied to converting boundary value problems for impulsive fractional differential equations to integral equations. Some comments are made on recently published papers (see Section 4).     

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.     

On the impulsive fractional anti-periodic BVP modelling with constant coefficients

This paper is inspired by the recent papers on the BVP for impulsive fractional differential equations. We present a general framework to find the solutions of anti-periodic BVP for linear impulsive fractional differential equations with constant coefficients. Some sufficient conditions for existence of the solutions for nonlinear problem are established. Finally, numerical examples are given to illustrate the results.     

Global stability results for impulsive differential equations

Global results concerning stability properties of impulsive differential equations are established, employing piecewise continuous Lyapunov functions which are then applied for proving stability and boundedness properties     

一类含高阶Riemann-Liouville型分数阶导数脉冲微分方程的通解

本文首先运用迭代法获得一类含多项Riemann-Liouville型分数阶导数的微分方程的连续通解,然后应用数学归纳法得到这类脉冲微分方程的分片连续通解. 所得结果归结于脉冲分数阶微分方程领域,对分数阶微分方程研究者有参考意义.     

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.     

随机脉冲泛函微分方程的{\boldmath p}阶矩有界性

随机脉冲泛函微分方程是一个具有广泛应用前景的数学模型. 该文利用带Razumikhin条件的Liapunov直接法和比较原理, 得到了随机脉冲泛函微分方程的解的一致(一致且最终、一致且一致最终) p阶矩有界的充分条件, 其中在获得一致有界性和一致最终有界性时, 对dV(t, x(t))/dt 的限制条件也较少, 因此研究结果非常便于应用.     

Existence of solutions for nonlinear fractional stochastic differential equations

The fractional stochastic differential equations have wide applications in various fields of science and engineering. This paper addresses the issue of existence of mild solutions for a class of fractional stochastic differential equations with impulses in Hilbert spaces. Using fractional calculations, fixed point technique, stochastic analysis theory and methods adopted directly from deterministic fractional equations, new set of sufficient conditions are formulated and proved for the existence of mild solutions for the fractional impulsive stochastic differential equation with infinite delay. Further, we study the existence of solutions for fractional stochastic semilinear differential equations with nonlocal conditions. Examples are provided to illustrate the obtained theory.     

积分方程的周期解与一致最终有界性

讨论了具有无限时滞的Ｖｏｌｔｅｒｒａ积分方程的周期解和一致最终有界性．把泛函微分方程中的一个著名定理,即一致有界性和一致最终有界性保证周期解的存在性,推广到积分方程．     

Boundedness of solutions for impulsive differential equations with integral jump conditions

The boundedness of solutions for certain nonlinear impulsive differential equations are obtained, the jumping conditions at discontinuous points are related to the integral of the past states, rather than a left hand limit at the discontinuous points. These results are obtained by new built impulsive integral inequalities with integral jumping conditions using the method of successive iteration.     

Boundedness of Nonlinear Differential Systems with Impulsive Effect on Random Moments

The model of nonlinear differential systems with impulsive effect on random moments is broughtforward in this paper.Then,sufficient conditions for(uniform,uniform and ultimate,and uniform and uniformlyultimate) p-moment boundedness of the systems are presented.Finally,an example is discussed to showapplications of two obtained results.     

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.     

Existence,uniqueness and boundedness results for impulsive delay differential equations

This paper considers general impulsive delay differential equations. By utilizing a non-classical approach, the theory of existence and uniqueness of solutions are developed. Criteria on boundedness of solutions are also established through the use of Lyapunov functionals.     

一类二阶脉冲微分方程解的渐近性态

研究得到了一类二阶脉冲微分方程,的正一切解有界或趋于零的充分条件.     

带有边界值问题的脉冲分数阶微分方程正解的存在性

本文研究了具有边界值条件的脉冲分数阶微分方程.利用Kuratowski非紧性测度理论和Sadovskii不动点定理,得到了脉冲分数阶微分方程正解的存在性的结果,推广了已有文献的结论.