CONVERGENCE OF SOLUTIONS OF CERTAIN SYSTEM OF THIRD ORDER NON-LINEAR ORDINARY DIFFERENTIAL EQUATIONS

The object of this paper is to give an n-dimensional analogue of a convergence result of Tejumola [8] for some third-order differential equations. The result extends earlier results of Afuwape [1] and Ezeilo [4] to some system of third-order differential equations.     

高阶非线性中立型方程正解的渐近性及存在性

本文利用非线性分析中半序方法和增算子不动点定理，对高阶非线性中立型方程正解的渐近性进行了详细分类，并且给出了各种类型正解存在的充要条件和具体例子．     

ON THE EXISTENCE AND UNIQUENESS OF THE SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH RESPECT TO SEMIMARTINGALES

Papers [1, 2] deal with the problem of the existence and uniqueness of the solutions of two different stochastic differential equations with respect to semimarting?ales. The equation discussed in this paper unifies the equations in papers [1, 2]. This paper gives us the sufficient conditions for the existence and uniqueness of its solutions, so that the results in papers [1,2] can be obtained by theorem 1 and theorem 1; of this paper respectively.     

MULTIPLE POSITIVE SOLUTIONS OF NONRESONANT SINGULAR BOUNDARY VALUE PROBLEM OF SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

This paper studies the existence of multiple positive solutions of nonresonant singular boundary value problem of second order ordinary differential equations. A sufficient condition for the existence of C[0,1] multiple positive solutions as well as C1[0, 1] multiple positive solutions is given by means of the fixed point theorems on cones.     

THE EXISTENCE OF POSITIVE SOLUTIONS TO BOUNDARY VALUE PROBLEM OF SECOND ORDER DIFFERENTIAL EQUATIONS

In this paper, a fixed point theorem in C1[0,1] space is established using the properties of the fixed point index. This theorem is applied to prove the existence of positive solution to the boundary value problems of second order differential equations.     

中立型泛函微分方程的多步法

在探求泛函微分方程的数值解法时,常常设法把常微分方程的许多古典的有效方法经过改造移殖到泛函微分方程。常微分方程的各种数值方法本质上都是力图使变量离散化。所以,Henrici干脆称之为离散变量法。与常微分方程不同,计算泛函微分方程的解在第n点上的值,不仅与前面某k个点上的值有关,而且往往与这些分点之间的值有关。这是推广过程中遇到的一大困难。1964年Feldstein对时滞微分方程提出了所谓连续Euler法,引进了插值思想,使离散方法连续化,克服了上述困难。1973年Castleton和Crimin把这一方法推广到中立型泛函微分方程。特别是Cryer和Tavernini及     

Constructive methods for certain Bergman operators

This paper is concerned with the class of linear partial differential equations of second order such that there exist Bergman operators with polynomial kernels (cf, [12]). In an earlier paper [ll] the authors have shown that these equations also admit differential operators as introduced by K. W. Bauer [I]. In the present paper, relations between different types of representations of solutions are investigated. These representations are of interest in developing a function theory of solutions; cf., for instance, K. W. Bauer [I] and S. Ruscheweyh [19]. They are also essential to global extensions of local results obtained by means of Bergman operators of the first kind. The inversion problem for those operators is solved, and it is shown that all solutions of equations of that class which are holomorphic in a domain of C² can be represented by operators with polynomial kernels. Furthermore, a construction principle for deriving the equations investigated by K. W. Bauer [2] is obtained; this yields corresponding representations of solutions by differential and integral operators in a systematic fashion     

一类四阶奇异Sturm-Liouville边值问题正解存在的充分必要条件

研究了一类非线性四阶微分方程奇异Sturm-Liouville边值问题,利用锥上的不动点定理得到了这类方程的C~3[0,1]正解和C~2[0,1]正解存在的充分必要条件.     

Reducing systems of linear differential equations to a passive form

In this paper, an application of the Riquer-Thomas-Janet theory is described for the problem of transforming a system of partial differential equations into a passive form, i.e., to a special form which contains explicitly both the equations of the initial system and all their nontrivial differential consequences. This special representation of a system markedly facilitates the subsequent integration of the corresponding differential equations. In this paper, the modern approach to the indicated problem is presented. This is the approach adopted in the Knuth-Bendix procedure [13] for critical-pair/completion and then Buchberger's algorithm for completion of polynomial ideal bases [13] (or, alternatively, for the construction of Groebner bases for ideals in a differential operator ring [14]). The algorithm of reduction to the passive form for linear system of partial differential equations and its implementation in the algorithmic language REFAL, as well as the capabilities of the corresponding program, are outlined. Examples illustrating the power and efficiency of the system are presented.     

OPTIMAL CONTROL PROBLEM OF PARABOLIC DIFFERENTIAL EQUATION WITH TWO POINT BOUNDARY CONDITION

This paper deals with optimal control problem of parabolic differential equation with two point boundary conditions (in the time variable). The results here extend those in [3] on optimal control of the heat equations. Moreover, the technique used in this paper is based on some smooth approximations of “tangent cones” in the sense of Clarke and some maximal monotone operators.     

反射倒向随机微分方程的逆比较问题(Ⅰ)

在Briand,Coquet,Hu,Memin,Peng[1],Coquet,Hu,Memin,Peng[2],Chen[3],Jiang [8]等中,研究了倒向随机微分方程的逆比较定理,就是通过比较倒向随机微分方程的解来比较倒向随机微分方程的生成元问题．在文[9]中Li和Tang首次研究了反射倒向随机微分方程的逆比较问题．本文考虑在更一般的条件下,反射倒向随机微分方程的生成元的逆比较问题．     

Two comparison theorems of bsdes

In this paper, by the equations of Mao [9] and Peng [5], we use the martingale method to establish the comparison theorems of backward stochastic differential equations (BSDEs). We generalize the results of Cao-Yan [1].     

Local first integrals of differential systems and diffeomorphisms

In this paper using theory of linear operators and normal forms we generalize a result of Poincaré [11] about the non-existence of local first integrals for systems of differential equations in a neighbourhood of a singular point. As an application of the generalized result, and under more weak conditions we obtain a result of Furta [8] about local first integrals of semi-quasi-homogeneous systems. Moreover, for diffeomorphisms and periodic differential systems we give definitions of their first integrals, and generalize the previous results about systems of differential equations to diffeomorphisms in a neighbourhood of a fixed point and to periodic differential systems in a neighbourhood of a constant solution.     

STRUCTURE OF POSITIVE SOLUTIONS OF A CLASS OF QUASILINEAR ORDINARY DIFFERENTIAL EQUATIONS

The existence and uniqueness of positive solutions for a class of quasilinearordinary differential equations with a large parameter are obtained. It is shownthat the flat core of positive solutions can exist. So some results of [1-3] aresharpened.     

模糊微分方程的最小与最大解

文献「１，２」利用Ｌｉｐｓｃｈｉｔｚ条件研究了模糊微分方程初值问题解的存在与唯一性。本文利用单调迭代方法研究了上述问题在序区间中的最小与最大解存在性问题。     

Exponential stability for stochastic neutral partial functional differential equations

In this paper, we consider a class of stochastic neutral partial functional differential equations in a real separable Hilbert space. Some conditions on the existence and uniqueness of a mild solution of this class of equations and also the exponential stability of the moments of a mild solution as well as its sample paths are obtained. The known results in Govindan [T.E. Govindan, Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics 77 (2005) 139-154], Liu and Truman [K. Liu, A. Truman, A note on almost sure exponential stability for stochastic partial functional differential equations, Statist. Probab. Lett. 50 (2000) 273-278] and Taniguchi [T. Taniguchi, Almost sure exponential stability for stochastic partial functional differential equations, Stoch. Anal. Appl. 16 (1998) 965-975; T. Taniguchi, Asymptotic stability theorems of semilinear stochastic evolution equations in Hilbert spaces, Stochastics 53 (1995) 41-52] are generalized and improved.     

THE STABILITY AND CONVERGENCE OF COMPUTINGLONG-TIME BEHAVIOUR

1.IntroductionIn1978,Hcff[3]consideredthelong--timebehaviorcomputationofnonlinearreactiondiffusionequations,whichissupposedtohaveaninvariantregionS,i.e.anylocalsolutionarisingfromapointinSisconstrainedtoliein.Hcffconstructedafamilyoffinitdifferencesc...     

On a class of backward doubly stochastic differential equations

In this paper, a new class of backward doubly stochastic differential equations is studied. This type of equations has a more general form of the forward Itô integrals compared to the ones which have been studied until now. We conclude that unique solutions of these equations can be represented with the help of solutions of the corresponding backward doubly stochastic differential equations, considered earlier in paper [5] by Pardoux and Peng. Some comparison theorems are also given, as well as a probabilistic interpretation for solutions of the corresponding quasilinear stochastic partial differential equations.     

On the convergence of multi-grid methods with transforming smoothers

Summary In the present paper we give a convergence theory for multi-grid methods with transforming smoothers as introduced in [31] applied to a general system of partial differential equations. The theory follows Hackbusch's approach for scalar pde and allows a convergence proof for some well-known multi-grid methods for Stokes- and Navier-Stokes equations as DGS by Brandt-Dinar, [5], TILU from [31] and the SIMPLE-methods by Patankar-Spalding, [23].This work was supported in part by Deutsche Forschungsgemeinschaft     

Weak self-adjointness and conservation laws for a porous medium equation

The concepts of self-adjoint and quasi self-adjoint equations were introduced by Ibragimov (2006, 2007) [4], [7]. In Ibragimov (2007) [6] a general theorem on conservation laws was proved. In Gandarias (2011) [3] we generalized the concept of self-adjoint and quasi self-adjoint equations by introducing the definition of weak self-adjoint equations. In this paper we find the subclasses of weak self-adjoint porous medium equations. By using the property of weak self-adjointness we construct some conservation laws associated with symmetries of the differential equation.