Global solutions of nonlinear second-order impulsive integro-differential equations of mixed type in Banach spaces

In this paper, by using the generalization of Darbo’s fixed point theorem, we establish the existence of global solutions of an initial value problem for a class of second-order impulsive integro-differential equations of mixed type in a real Banach space. Our results generalize and improve on the results of Guo et al. [F. Guo, L.S. Liu, Y.H. Wu, P. Siew, Global solutions of initial value problems for nonlinear second-order impulsive integro-differential equations of mixed type in Banach spaces, Nonlinear Anal. 61 (2005) 1363–1382] in the sense that the conditions for existence of global solution in our theorem is simpler and less strict. To demonstrate the application of the theorem, we give the global solutions of two mixed boundary value problems for two classes of fourth order impulsive integro-differential equations.     

Initial value problems for nonlinear second order impulsive integro-differential equations of mixed type in Banach spaces

In this paper, through solving equations step by step, without any assumption of compactness-type conditions, we obtain unique solution of initial value problems of nonlinear second order impulsive integral-differential in Banach spaces. The results obtained generalize and improve the corresponding results of Guo and Chai in papers [D.J. Guo, Initial value problems for nonlinear second-order impulsive integro-differential equations in Banach spaces, J. Math. Anal. Appl. 200 (1996) 1–13; G.Q. Chai, Initial value problems for nonlinear second order impulsive integro-differential equations in Banach space, Acta Math. Sinica 20 (3) (2000) 351–359 (in Chinese)].     

Periodic boundary value problems for second-order impulsive integro-differential equations

By developing a new comparison result and using the monotone iterative technique, we are able to obtain existence of minimal and maximal solutions of periodic boundary value problems for second-order nonlinear impulsive integro-differential equations of mixed type.     

Existence of solutions of boundary value problems with integral boundary conditions for second-order impulsive integro-differential equations in Banach spaces

This paper investigates the existence of solutions for a class of second-order boundary-value problems with integral boundary conditions of nonlinear impulsive integro-differential equations in Banach spaces. The arguments are based upon the fixed point theorem of strict set contraction operators. Meanwhile, an example is worked out to demonstrate the main results.     

Existence of positive solution for second-order nonlinear impulsive singular differential equations of mixed type in Banach spaces

In this paper, by constructing a closed convex set and using the fixed point theory of completely continuous operators, we investigate the existence of positive solutions for an initial value problem of second-order nonlinear impulsive singular integro-differential equations in a Banach space. The method used in this paper is different from that in the literature.     

二阶非线性脉冲积分-微分方程初值问题的解

该文利用单调迭代技术讨论了Banach空间中含有非线性一阶微分项x'的二阶脉冲积分-微分方程初值问题的最大最小解的存在性问题. 作为该文主要结论的应用,作者给出了一个例子.     

二阶脉冲积微分方程初值问题

用不动点理论研究 Banach空间中定义在无穷区间上的二阶脉冲积微分方程初值问题 ,建立了方程(1 )的解的存在唯一性定理 .     

二阶脉冲积分微分方程周期边值问题（英文）

In this paper, the properties of solution of periodic boundary value problem for second-order impulsive integro-differential equation are discussed. Using the iterative analysis method, the existence and uniqueness of periodic solution and the sufficient condition for uniform stability of trivial solution are obtained, which extend the previous results on integro-differential equation in periodic boundary value problem.     

Banach空间中N阶脉冲积分-微分方程边值问题的解

运用Monch不动点定理，获得了Banach空间中一类N阶非线性混合型脉冲积分-微分方程边值问题解的存在性．最后给出一个三阶无穷脉冲积分-微分方程边值问题的例子来说明文中所给的条件是合理的．     

Banach空间中脉冲积分-微分方程的初值问题

给出了 Banach空间中一阶线性脉冲积分 -微分方程初值问题解的存在唯一性的一个新证法 ,改进了已有结果 .利用它讨论了一阶非线性脉冲积分 -微分方程初值问题的解 ,所得结果大大推广了已有的相关结果 .     

Banach空间二阶脉冲积微分方程终值问题

利用上下解方法讨论了抽象空间中定义在无穷区间上且有无穷个脉冲点的一类二阶脉冲积微分方程终值问题,获得了其最小最大解的存在性.     

New comparison results for impulsive integro-differential equations and applications

We prove some new maximum principles for ordinary integro-differential equations. This allows us to introduce a new definition of lower and upper solutions which leads to the development of the monotone iterative technique for a periodic boundary value problem related to a nonlinear first-order impulsive integro-differential equation.     

Second-order nonlinear impulsive evolution equations with time-varying generating operators and optimal controls

In this article, the second-order nonlinear impulsive evolution differential equations with time-varying generating operators is considered. Constructing evolution systems generated by time-varying operator matrix, we introduce suitable mild solution of the second-order nonlinear impulsive evolution differential equations. The existence and uniqueness of the mild solutions and the continuous dependence on initial value are proved. The existence of the optimal controls for a Lagrange problem of the systems governed by the second-order nonlinear impulsive evolution equations is also presented. An example is given for demonstration.     

Global solutions of initial value problems for nonlinear second-order impulsive integro-differential equations of mixed type in Banach spaces

In this paper, we develop a theorem for the existence of global solutions of an initial value problems for a class of second-order impulsive integro-differential equations of mixed type in a real Banach space. The results obtained are the generalizations and improvements of various recent results. As an application of the theorem, the existence of global solutions of two mixed boundary value problems for two classes of fourth-order impulsive differential equations are established.     

Banach空间非线性二阶脉冲奇异积分-微分方程积分边值问题的正解

The existence of positive solutions to a boundary value problem of second-order impulsive singular integro-differential equation with integral boundary conditions in a Banach space is obtained by means of fixed point theory.Moreover,an application is also given to illustrate the main result.     

Banach空间二阶非线性脉冲积分-微分方程初值问题

该文在较宽松的条件下,利用不动点理论,得到了Banach空间中二阶非线性脉冲积分-微分方程初值问题解的存在性定理.作者去掉了脉冲项的紧性和增长性限制, 因而本质上改进和推广了某些现有的结果.     

Banach空间非线性脉冲Volterra型积分方程和积分-微分方程的可解性

本文在较宽松的条件下,研究了Banach空间非线性脉冲Volterra型积分方 程和脉冲积分一微分方程解的存在性,建立了新的存在性定理,本质上改进了某些已 知的结果.     

Positive periodic solution of a nonlinear integro-differential prey-competition impulsive model with infinite delays

By using the method of coincidence degree, some conditions are obtained for the existence of positive periodic solution of a nonlinear integro-differential prey-competition impulsive model with infinite delays.     

Exponential stability of nonlinear impulsive neutral integro-differential equations

In this paper, a nonlinear impulsive neutral integro-differential equation with time-varying delays is considered. By establishing a singular impulsive delay integro-differential inequality and transforming the n$n$-dimensional impulsive neutral integro-differential equation to a 2n$2n$-dimensional singular impulsive delay integro-differential equation, some sufficient conditions ensuring the global exponential stability in PC¹$P{C}^1$ of the zero solution of an impulsive neutral integro-differential equation are obtained. The results extend and improve the earlier publications. An example is also discussed to illustrate the efficiency of the obtained results.     

Existence of solutions for anti-periodic boundary value problems of nonlinear impulsive functional integro-differential equations of mixed type

We discuss the existence of minimal and maximal solutions for a class of first order nonlinear impulsive functional integro-differential equations of mixed type with anti-periodic boundary conditions. The main tool of study is the monotone iterative technique.