临界状态下一阶时滞微分方程的线性化振动性

本文首先在临界状态下建立了一阶非线性非自治时滞微分方程ｘ＇（ｔ）＋＝１ｐｉｘ（ｔ－Ｔｉ）＋ｆ（ｔ,ｘ（ｔ－σ１（ｔ））…,ｘ（ｔ－σｎ（ｔ））＝０与一个相关的二阶常微分方程振动性等价定理,进而给出了一阶非线性自治微分方程与相应的线性方程振动性等价的充分条件,从而较好地回答了张炳根在文［２］中提出的一个公开问题．     

INSTABILITY TO DIFFERENTIAL EQUATIONS OF FIFTH AND SIXTH ORDER WITH VARIABLE DEVIATING ARGUMENTS

In this paper, we discuss the instability of the zero solutions to certain nonlinear differential equations of fifth and sixth order with multiple retarded arguments. By the Lyapunov's direct method, sufficient conditions for instability of the zero solutions are obtained.     

Global solutions of a singular initial value problem to second order nonlinear delay differential equations

The paper is concerned with an initial value problem to second order nonlinear singular delay differential equations. By the use of the Schauder fixed point theorem, a result for the existence of global solutions is derived. Also, via the Banach contraction principle, another result concerning the existence and uniqueness of global solutions is established. Moreover, applications of these results to a particular case of second order nonlinear singular delay differential equations as well as to the special case of second order nonlinear singular ordinary differential equations are presented. Finally, some specific applications to certain equations and two examples are given to demonstrate the applicability of the results of the paper.     

On the behavior of solutions of a first order nonlinear neutral delay differential equation

The authors obtain results on the asymptotic behavior of the non-oscillatory solutions of a first order nonlinear neutral delay differential equation. A theorem giving sufficient conditions for all bounded solutions to be oscillatory is also proved.     

Existence of positive solutions to a boundary value problem for a delayed singular high order fractional differential equation with sign-changing nonlinearity

In this paper, we discuss the existence of positive solutions to the boundary value problem for a high order fractional differential equation with delay and singularities including changing sign nonlinearity. By using the properties of the Green function, Guo-krasnosel"skii fixed point theorem, Leray-Schauder"s nonlinear alternative theorem, some existence results of positive solutions are obtained, respectively.     

On liapunov functional in theory of stability of systems with time-lag

It is well known, that in the theory of stability in differential equations, Liapunov's second method may be the most important. The center problem of Liapunov's second method is construction of Liapunov function for concrete problems. Beyond any doubt, construction of Liapunov functions is an art. In the case of functional differential equations, there were also many attempts to establish various kinds of Liapunov type theorems. Recently Burton [2] presented an excellent theorem using the Liapunov functional to solve the asymptotic stability of functional differential equation with bounded delay. However, the construction of such a Liapunov functional is still very hard for concrete problems. In this paper, by utilizing this theorem due to Burton, we construct concrete Liapunov functional for certain and nonlinear delay differential equations and derive new sufficient conditions for asymptotic stability. Those criteria improve the result of literature [1] and they are with simple forms, easily checked and applicable.This project is supported by the National Natural Science Foundation of China.     

ON THE INSTABILITY OF SOLUTIONS TO A NONLINEAR VECTOR DIFFERENTIAL EQUATION OF FOURTH ORDER

This paper presents a new result related to the instability of the zero solution to a nonlinear vector differential equation of fourth order.Our result includes and improves an instability result in the previous literature,which is related to the instability of the zero solution to a nonlinear scalar differential equation of fourth order.     

LINEARIZED OSCILLATIONS FOR ODD-ORDERNEUTRAL DELAY DIFFERENTIAL EQUATIONSWITH ASYMPTOTIC PERIODIC COEFFICIENTS

1IntroductionSince1987,thelinearizedoscillationtheoryofnonlineardelaydifferentialequations(DDEs)hasabsorbedmanymathematicians(see[1,2]),itisparalleltotheso-calledlinearizedstabilitytheory,andhasbecomeamainstudyofoscillationtheoryofDDEs.ByusingthenecessaryandsufficientconditionsfortheoscillationoflinearDDEswithconstantcoefficients,theliteraturesconcentratedtoprovethatcertainnonlinearDDEhasthesameoscillatorycharacterasanassociatedlinearizedDDEwithconstantcoefficients.However)whenthelineari…     

EXISTENCE AND UNIQUENESS OF SOLUTIONS TO A CLASS OF NONLINEAR FRACTIONAL ORDER DIFFERENTIAL EQUATIONS WITH DELAY

A class of nonlinear fractional order differential equations with delay is investigated in this paper.Using Leray-Schauder fixed point theorem and the contraction mapping theorem,we obtain some sufficient conditions for the existence and uniqueness of solutions to the fractional order differential equations.     

Oscillation Criteria for Impulsive Parabolic Differential Equations with Delay

In this paper, we investigate a class of nonlinear impulsive parabolic systems with delay. Several oscillation criteria are established for such systems subject to two different boundary conditions by employing Gauss' divergence theorem and certain impulsive delay differential inequalities.     

Almost periodic solutions for undamped nonhomogeneous delay-differential equations

We first establish a result giving conditions that certain undamped delay differential equations with almost periodic time dependence have unique almost periodic solutions. Using this result we obtain conditions that a second order scalar nonlinear delay differential equation with almost periodic forcing will have a unique almost periodic solution having saddle-type stability properties. These results use the method of averaging.

     

一类三阶非线性微分方程解的不稳定性*

文献[1]讨论了非线性缓变系统的渐近稳定性,文献[2]讨论了三阶变系数线性微分方程解的不稳定性。本文应用文献[1]、[2]的方法讨论一类三阶非线性微分方程解的不稳定性。     

Note on the Ladas' paper on oscillation and asymptotic behavior of solutions of differential equations with retarded argument

This paper consists of three theorems. The first theorem gives the asymptotic behavior of solutions of the nonlinear delay differential equation. The other theorems give sufficient conditions for oscillatory solutions of variants of that equation.     

On the Nature of Buffering

We state a theorem on the instability of self-similar cycles and tori of a certain type in a system which is a quasinormal form of a boundary-value problem for a nonlinear wave equation in the square.     

Existence of solutions of a boundary value problem on the half-line to second order nonlinear delay differential equations

By using the Schauder fixed point theorem, we establish a result for the existence of solutions of a boundary value problem on the half-line to second order nonlinear delay differential equations. We also present the application of our result to the special case of second order nonlinear ordinary differential equations as well as to a specific class of second order nonlinear delay differential equations. Moreover, we give a general example which demonstrates the applicability of our result. Received: 10 May 2004     

Some oscillatory properties of second order non linear differential equations

Summary The aim of this note is to show that, under certain conditions, the comparìson theorems between a nonlinear second order differential equation of certain type and a linear second order differential equation hold. From this we shall be able to guarentee the oscillatory or nonoscillatory character of certain or all solutions of the considered nonlinear differential equation. Work done at the Instituto Matematico ? Ulisse Dini ? in Firenze.     

Controllability of second‐order impulsive neutral functional differential inclusions in Banach spaces

In this paper, we shall prove the controllability of second‐order impulsive neutral functional differential inclusions with infinite delay in Banach spaces. The results are obtained by using the theory of strongly continuous cosine families and a fixed point theorem for multi‐valued mappings. As an example, we also consider an second‐order impulsive neutral functional differential equation with infinite delay. Copyright © 2009 John Wiley & Sons, Ltd.     

非线性分数阶微分方程三点边值问题解的存在性

讨论了一类非线性分数阶微分方程三点边值问题解的存在性.微分算子是Riemann-Liouville导算子并且非线性项依赖于低阶分数阶导数.通过将所考虑的问题转化为等价的Fredholm型积分方程,利用Schauder不动点定理获得该三点边值问题至少存在一个解.     

Analysis of the time fractional nonlinear diffusion equation from diffusion process

Under investigation in this paper is a time fractional nonlinear diffusion equation which can be utilized to express various diffusion processes. The symmetry of this considered equation has been obtained via fractional Lie group approach with the sense of Riemann-Liouville (R-L) fractional derivative. Based on the symmetry, this equation can be changed into an ordinary differential equation of fractional order. Moreover, some new invariant solutions of this considered equation are found. Lastly, utilising the Noether theorem and the general form of Noether type theorem, the conservation laws are yielded to the time fractional nonlinear diffusion equation, respectively. Our discovery that there are no conservation laws under the general form of Noether type theorem case. This result tells us the symmetry of this equation is not variational symmetry of the considered functional. These rich results can give us more information to interpret this equation.     

一类非线性分数阶泛函微分方程的正解的存在性

本文研究了一类非线性分数阶泛函微分方程.利用锥拉伸与锥压缩不动点定理,得到这类分数阶泛函微分方程的正解的存在性,推广了马如云的结果.