Monotone iterative technique for impulsive differential equations with time variables

In this paper, we established the general comparison prinples for IVP of impulsive differential equations with time variables, which strictly extend and improve the previous comparison results obtained by V.Lakes.et.al. and S.K.Kaul([3]–[7]). With the general comparison results, we constructed the monotone iterative sequences of solutions for IVP of such equations which converges the maximal and minimal solutions, repectively.     

Banach空间非线性脉冲Volterra型积分方程整体解的存在性定理及应用

利用一个新的比较结果和Ｍｏｎｅｃｈ不动点定理证明了Ｂａｎａｃｈ空间中非线性脉冲Ｖｏｌｔｅｒｒａ型积分方程整体解的存在性定理,并给出了对Ｂａｎａｃｈ空间中一阶脉冲微分方程初值问题的应用,改进了文（１－３）中的主要结果。     

脉冲强迫非线性时滞微分方程的渐近性

本文研究一类脉冲强迫非线性时滞微分方程的渐近性,所得结果不仅适用于线性方程和非线性方程,强迫方程和非强迫方程,脉冲方程和非脉冲方程,而且改进了最近文献［８］的主要结果．     

具有变号系数中立型时滞微分方程的振动性

本文讨论一类一阶具有变号系数中立型时滞微分方程的振动性，建立了此类方程一切解振动的几族充分条件。将其结论用于具有正系数中立型时滞微分方程及具有正负系数中立型时滞微分方程，改进了[1-8]的相应定理。     

Non-periodic averaging principles for measure functional differential equations and functional dynamic equations on time scales involving impulses

We present a non-periodic averaging principle for measure functional differential equations and, using the correspondence between solutions of measure functional differential equations and solutions of functional dynamic equations on time scales (see Federson et al., 2012 [8]), we obtain a non-periodic averaging result for functional dynamic equations on time scales. Moreover, using the relation between measure functional differential equations and impulsive measure functional differential equations, we get a non-periodic averaging theorem for these equations. Also, it is a known fact that we can relate impulsive measure functional differential equations and impulsive functional dynamic equations on time scales (see Federson et al., 2013 [9]). Therefore, applying this correspondence to our averaging principle, we obtain a non-periodic averaging theorem for impulsive functional dynamic equations on time scales.     

Asymptotic Behavior of Forced Nonlinear Delay Differential Equations with Impulses

The purpose of this paper is to investigate the asymptotic behavior of solutions of the forced nonlinear delay differential equations with impulses Our results, which hold for linear and nonlinear equations, forced and unforced equations, impulsive and nonimpulsive equations, improve and generalize the known results recently obtained in [8]. Received September 7, 1997, Revised May 26, 1998, Accepted July 15, 1998     

Differential equations on variable time scales

We introduce a class of differential equations on variable   time scales with a transition condition between two consecutive parts of the scale. Conditions for existence and uniqueness of solutions are obtained. Periodicity, boundedness and stability of solutions are considered. The method of investigation is by means of two successive reductions: B$B$-equivalence of the system [E. Akalín, M.U. Akhmet, The principles of B-smooth discontinuous flows, Computers and Mathematics with Applications 49 (2005) 981–995; M.U. Akhmet, Perturbations and Hopf bifurcation of the planar discontinuous dynamical system, Nonlinear Analysis 60 (2005) 163–178; M.U. Akhmet, N.A. Perestyuk, The comparison method for differential equations with impulse action, Differential Equations 26 (9) (1990) 1079–1086] on a variable time scale to a system on a time scale, a reduction to an impulsive differential equation [M.U. Akhmet, Perturbations and Hopf bifurcation of the planar discontinuous dynamical system, Nonlinear Analysis 60 (2005) 163–178; M.U. Akhmet, M. Turan, The differential equations on time scales through impulsive differential equations, Nonlinear Analysis 65 (2006) 2043–2060]. Appropriate examples are constructed to illustrate the theory.     

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.     

随机积分和微分方程在弱拓扑下解的存在性和比较结果*

在本文中,我们对非线性随机Volterra积分方程在Banach空间的弱拓扑下的随机解证明了几个存在定理.然后作为应用,我们得到了随机微分方程的弱随机解的存在定理.还得到了这些随机方程的极值随机解的存在性和随机比较定理.我们的定理改进和推广了[4,5,10,11,12]中的相应结果.     

EXISTENCE THEOREMS OF SOLUTIONSFOR NONLINEAR IMPULSIVE VOLTERRA INTEGRAL EQUATIONS IN BANACH SPACES

In this paper, we use the Monch fixed-point theorem to prove some existence theoremsof sotuti9ns for the nonlinear impulsive Volterra integral equations in Banach spaces that improveand extend the results in [2].     

Banach空间中非线性脉冲Volterra型积分方程的可解性

本文利用“强极小锥”的概念,获得了Ｂａｎａｃｈ空间中非线性脉冲Ｖｏｌｔｅｒｒａ型积分方程整体解的存在性定理,改进了现有文献中的某些结果．     

一类脉冲微分系统的有界变差解

在比文[6]更弱的条件下讨论了固定时刻脉冲微分系统与Kurzweil广义常微分方程的关系，并建立了这类脉冲微分系统有界变差解的局部存在性和唯一性定理．     

Banach空间中混合单调脉冲微分-积分方程解的存在性

本文给出了Banach空间中混合单调脉冲微分-积分方程解、耦合最小最大解的存在性定理及单调迭代方法,改进和推广了[1]-[4]的相应结果．     

Banach空间一阶非线性脉冲微分方程周期边值问题的解

本文通过建立Ｂａｎａｃｈ空间一阶非线性脉冲微分一积分方程周期边值问题新的比较定理,给出了其最大解和最小解的存在性。     

Perturbation method for nonlocal impulsive evolution equations

This paper deals with the existence of mild solutions for a class of semilinear nonlocal impulsive evolution equations in ordered Banach spaces. The existence and uniqueness theorem of mild solution for the associated linear nonlocal impulsive evolution equation is established. With the aid of the theorem, the existence of mild solutions for nonlinear nonlocal impulsive evolution equation is obtained by using perturbation method and monotone iterative technique. The theorems proved in this paper improve and extend some related results in ordinary differential equations and partial differential equations. Moreover, we present two examples to illustrate the feasibility of our abstract results.     

The differential equations on time scales through impulsive differential equations

In this paper we investigate differential equations on certain time scales with transition conditions (DETC) on the basis of reduction to the impulsive differential equations (IDE). DETC are in some sense more general than dynamic equations on time scales [M. Bohner, A. Peterson, Dynamic equations on time scales, in: An Introduction With Applications, Birkhäuser Boston, Inc., Boston, MA, 2001, p. x+358; V. Laksmikantham, S. Sivasundaram, B. Kaymakcalan, Dynamical Systems on Measure Chains, in: Math. and its Appl., vol. 370, Kluwer Academic, Dordrecht, 1996]. The basic properties of linear systems, the existence and stability of periodic solutions, and almost periodic solutions are considered. Appropriate examples are given to illustrate the theory.     

A new class of differential equations with impulses at instants dependent on preceding pulses. Applications to management of renewable resources

In this paper, a new type of mathematical model to represent certain processes with impulsive dynamic behavior is introduced. The main assumption is that the next impulse time is determined by three fundamental elements: the present impulse time, the state at this moment, and the value to which this state is impelled. We also establish the basic results of existence, uniqueness and continuation of solutions for these new impulsive differential equations. It is observed that the new equations have interesting applications in Bioeconomics, and sometimes they include, the traditional impulsive equations in variable times.     

New result for a class of impulsive differential equation with integral boundary conditions

This paper investigates a class of second-order impulsive differential equations with integral boundary values. By using Krasnoselskii’s fixed point theorem, the existence of positive solutions for the system is obtained. The results generalize the results in Refs. [11], [12], [13]. And an example is given to illustrate the effective of our result.     

Periodic boundary value problems for impulsive fuzzy differential equations

The peculiarity of the Hukuhara derivative makes it impossible to find periodic solutions for fuzzy differential equations with the exception of very restrictive situations. In this work, we consider a boundary value problem associated with an impulsive fuzzy differential equation and approximate the extremal solutions in a fuzzy functional interval using the monotone method. Fuzzy comparison results are useful in our procedure and the expression of the solution for some impulsive periodic ‘linear’ differential problems is also provided.     

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.