Perturbations for linear difference equations

In this paper we study boundary value problems for perturbed second-order linear difference equations with a small parameter. The reduced problem obtained when the parameter is equal to zero is a first-order linear difference equation. The solution is represented as a convergent series in the small parameter, whose coefficients are given by means of solutions of the reduced problem.     

First-order advanced difference equations

We investigate the existence of solutions to nonlinear first-order difference problems with advanced arguments. Sufficient conditions when such problems have solutions (extremal or unique) are given. Linear advanced difference inequalities are also discussed. According to my knowledge, it is a first paper when a monotone iterative method is applied to nonlinear boundary value problems for first-order difference equations with advanced arguments. An example illustrates the theoretical results.     

An observation on the periodic solutions to nonlinear physical models by means of the auxiliary equation with a sixth-degree nonlinear term

It is a fact that in auxiliary equation methods, the exact solutions of different types of auxiliary equations may produce new types of exact travelling wave solutions to nonlinear equations. In this manner, various auxiliary equations of first-order nonlinear ordinary differential equation with distinct-degree nonlinear terms are examined and, by means of symbolic computation, the new solutions of original auxiliary equation of first-order nonlinear ordinary differential equation with sixth-degree nonlinear term are presented. Consequently, the novel exact solutions of the generalized Klein–Gordon equation and the active-dissipative dispersive media equation are found out for illustration purposes. They are also applicable, where conventional perturbation method fails to provide any solution of the nonlinear problems under study.     

Distributed control for a class of non-Newtonian fluids

We consider control problems with a general cost functional where the state equations are the stationary, incompressible Navier-Stokes equations with shear-dependent viscosity. The equations are quasi-linear. The control function is given as the inhomogeneity of the momentum equation. In this paper, we study a general class of viscosity functions which correspond to shear-thinning or shear-thickening behavior. The basic results concerning existence, uniqueness, boundedness, and regularity of the solutions of the state equations are reviewed. The main topic of the paper is the proof of Gâteaux differentiability, which extends known results. It is shown that the derivative is the unique solution to a linearized equation. Moreover, necessary first-order optimality conditions are stated, and the existence of a solution of a class of control problems is shown.     

一类非线性椭圆型方程奇摄动问题

A class of singularly perturbed problems for the nonlinear elliptic equations is considered. Under suitable conditions, using the theory of differential inequalities the asymptotic behavior of solution for the boundary value problems are studied, which reduced equations possess two intersecting solutions.     

Oscillation of Nonlinear Delay Impulsive Differential Equations and Inequalities

The oscillatory and asymptotic behavior of first-order nonlinear delay impulsive differential equations and inequalities is studied. Some new sharp sufficient conditions for oscillation and nonoscillation of solutions of the equations and inequalities are obtained. Many known results are improved. In particular, the conditions are necessary and sufficient if the equations and inequalities are autonomous.     

A CLASS OF NONLINEAR SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS

A class of nonlinear singularly perturbed problems for reaction diffusion equa-tions are considered. Under suitable conditions, by using the theory of differential inequal-ities, the asymptotic behavior of solutions for the initial boundary value problems arestudied, reduced problems of which possess two intersecting solutions.     

Solutions of n-point boundary value problems associated with nonlinear summary difference equations

Variation of parameter methods play a fundamental rôle in understanding solutions of perturbed nonlinear differential as well as difference equations. This paper is devoted to the study of n-point boundary value problems associated with systems of nonlinear first-order summary difference equations by using the nonlinear variation of parameter methods. New variational formulae, which provide connections between the solutions of initial value problems and n-point boundary value problems, are obtained. An iterative scheme for computing approximated solutions of the boundary value problems is also provided.     

非线性反应扩散方程奇摄动问题

研究了具有非线性反应扩散方程奇摄动问题。在适当的条件下，利用微分不等式理论，讨论了当退化问题具有两个相交解时，原初始边值问题解的渐近性态。      

First-order regularization methods for accretive inclusions in a Banach space

Equations with set-valued accretive operators in a Banach space are considered. Their solutions are understood in the sense of inclusions. By applying the resolvent of the set-valued part of the equation operator, these equations are reduced to ones with single-valued operators. For the constructed problems, a regularized continuous method and a regularized first-order implicit iterative process are proposed. Sufficient conditions for their strong convergence are obtained in the case of approximately specified data.     

反应扩散方程的奇摄动

§ 1　 IntroductionThe nonlinearsingularly perturbed problem is a very attractive subjectof study in theinternational academic circles[1 ] .During the past decade many approximate methods havebeen developed and refined,including the method of average,boundary layer method,matched asymptotic expansions,and multiple scales.Recently,many scholars,for example,Bohé[2 ] ,Butuzov and Smurov[3] ,O Malley[4] ,Butuzov,Nefedov and Schneider[5] ,Kelley[6]and so on did a great deal of work.Mo consider…     

Nonclassical statements of problems for degenerate partial differential equations

We consider problems with point initial data for first-order partial differential equations in a half-strip. The equations contain degeneration with respect to one of the variables. We prove the existence and uniqueness of smooth solutions bounded in a neighborhood of the degeneration. The structure of the considered solutions is established. Such problems arise when solving the Cauchy problem for a system of ordinary differential equations with a movable regular singular point.     

一类非线性积分微分反应扩散方程的渐近解

研究了一类非线性积分———微分反应扩散方程.在适当的条件下,利用微分不等式理论,讨论了退化解具有两个交叉解的初值问题解的渐近性态.     

A probability wave description of stochastic variables whose means satisfy a system of difference equations

Probability wave theory is used to study the behavior of stochastic vectors whose means satisfy ordinary first-order difference equations. Difference-differential equations are given for the probability waves corresponding to the difference model for the means. Analogues of the Liouville and Ehrenfest theorems are proved. A first-order difference equation for the evolution of the component dispersion of the random vector is obtained. An algorithm for solving the wave equations is proposed. The results from analyzing some solutions to the probability wave equations are presented. The relationship of the finite-difference method to the manifestation of the particle-wave dualism is discussed. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 115. No. 1, pp. 56–76. April. 1998.     

一阶脉冲泛函微分方程非线性边值问题

考虑一阶脉冲泛函微分方程非线性边值问题,利用上下解方法和单调迭代技术得到了耦合解和唯一解存在的充分条件.所得结果改进和推广了文献的相关结果.     

On first-order discrete boundary value problems

This article analyzes a nonlinear system of first-order difference equations with periodic and non-periodic boundary conditions. Some sufficient conditions are presented under which: potential solutions to the equations will satisfy certain a priori bounds; and the equations will admit at least one solution. The methods involve new dynamic inequalities and use of Brouwer degree theory. The new results are compared with those featuring in the theory of solutions to boundary value problems for differential equations.     

First-order three-point boundary value problems at resonance

We consider three-point boundary value problems for a system of first-order equations in perturbed systems of ordinary differential equations at resonance. We obtain new results for the above boundary value problems with nonlinear boundary conditions. The existence of solutions is established by applying a version of Brouwer’s Fixed Point Theorem which is due to Miranda.     

两自由度非线性振动系统主参数激励下的分岔分析

本文给出了参数激励作用下两自由度非线性振动系统,在1:2内共振条件下主参数激励低阶模态的非线性响应.采用多尺度法得到其振幅和相位的调制方程,分析发现平凡解通过树枝分岔产生耦合模态解,采用Melnikov方法研究全局分岔行为,确定了产生Smale马蹄型混沌的参数值.     

Sequences of nonlinear differential equations with related solutions

Summary A second-order nonlinear differential equation which occurs (together with variants of it) in many problems of applied mathematics, physics and engineering is here reduced to a first-order equation. This equation contains a parameter which is a quadratic rational function of two parameters appearing in the original equation. By applying a certain identity for a quadratic rational function, two (finite or infinite) sequences of nonlinear differential equations are generated whose solutions are determinable whenever the solution of any equation belonging to a sequence is known. The cases amenable to exact solution by quadrature are given. Entrata in Redazione il 16 luglio 1968.     

时标上拥有积分边界条件的一阶脉冲动力方程解的存在性

考虑时标上一阶拥有积分边界条件的脉冲动力方程,通过上下解方法结合单调迭代技术得到解存在的充分条件,所得结果包括了周期边值问题、初值问题并且改进和丰富了已有文献的结论,并举例说明其应用.