非线性泛函差分方程解的性质

In this paper, some sufficient and necessary conditions are established for the oscillatory of solutions for nonlinear functional difference equations, which extend and improve some corresponding results obtained and are discrete analogues of the corresponding results for the continuous version.     

Soliton solutions to generalized discrete Korteweg-de Vries equations

New discrete equations of the simplest three-point form are considered that generalize the discrete Korteweg-de Vries equation. The properties of solitons, kinks, and oscillatory waves are numerically examined for three types of interactions between neighboring chain elements. An analogy with solutions to limiting continual equations is drawn.     

Time averages and Boltzmann extremals for Markov chains,discrete Liouville equations,and the Kac circular model

Time averages are proved to coincide with Boltzmann extremals for Markov chains, discrete Liouville equations, and their generalizations. A variational principle is proposed for finding stationary solutions in these cases.     

Numerical treatment of oscillatory functional differential equations

In this paper we are concerned with oscillatory functional differential equations (that is, those equations where all the solutions oscillate) under a numerical approximation. Our interest is in the preservation of qualitative properties of solutions under a numerical discretisation. We give conditions under which an equation is oscillatory, and consider whether the discrete schemes derived using linear ?-methods will also be oscillatory. We conclude with some general theory.     

Finite-difference scheme for two-scale homogenized equations of one-dimensional motion of a thermoviscoelastic Voigt-type body

The Bakhvalov-Eglit two-scale homogenized equations are used to describe the motion of layered periodic compressible media with rapidly oscillating data. A new finite-difference scheme for a system of such equations is proposed and analyzed in the case of a thermoviscoelastic Voigt-type body. A priori estimates of solutions are derived for nonsmooth data. The existence and uniqueness of discrete solutions are established. A theorem is proved on the convergence of a subsequence of discrete solutions to a weak solution of the problem under study. Simultaneously, a new theorem on the existence of global weak solutions is deduced.     

Oscillatory theorems of a class of even-order neutral equations

A class of even-order nonlinear neutral differential equations with distributed deviating arguments is studied, and oscillatory criteria for solutions of such equations are established.     

Concerning solutions of third order self-adjoint differential equations

Summary This work is concerned with the behavior of solutions of third order self-adjoint differential equations. It is well known that products of solutions of a second order self-adjoint equation are solutions of the third order equation. A converse of this fact is established. Existence of oscillatory solutions and asymptotic behavior of nonoscillatory solutions is established. Separation and comparison theorems are obtained for such equations. Entrata in Redazione il 13 aprile 1972.     

Oscillatory and Asymptotic Behavior of Solutions for Nonlinear Impulsive Delay Differential Equations

The oscillatory and asymptotic behavior of the solutions for third order nonlinear impulsive delay differential equations are investigated. Some novel criteria for all solutions to be oscillatory or be asymptotic are established, Three illustrative examples are proposed to demonstrate the effectiveness of the conditions.     

Oscillatory solutions of fractional integro-differential equations

We give necessary conditions to get oscillatory solutions of a class of fractional order neutral differential equations with continuously distributed delay by means of the fractional derivative with respect to a given function. In particular, oscillatory solutions of the considered fractional equations with Caputo and Hadamard type of fractional derivatives are established. Some explicit examples are given to illustrate the main results.     

非线性泛函微分方程解的性态

本文给出了一类非线性泛函微分方程解的性态的若干充要条件,推广了[1]的某些结论.     

OSCILLATIONS OF FIRST ORDER NONLINEAR RETARDED DIFFERENTIAL EQUATIONS

ＯＳＣＩＬＬＡＴＩＯＮＳＯＦＦＩＲＳＴＯＲＤＥＲＮＯＮＬＩＮＥＡＲＲＥＴＡＲＤＥＤＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳ￥ＷａｎｇＱｉｒｕ（王其如）（ＬｕｏｙａｎｇＴｅａｃｈｅｒｓ＇Ｃｏｌｌｅｇｅ，洛阳师范高等专科学校，邮编：４７１０２２）Ａｂｓ...     

A new consistent discrete‐velocity model for the Boltzmann equation

This paper discusses the convergence of a new discrete‐velocity model to the Boltzmann equation. First the consistency of the collision integral approximation is proved. Based on this we prove the convergence of solutions for a modified model to renormalized solutions of the Boltzmann equation. In a numerical example, the solutions to the discrete problems are compared with the exact solution of the Boltzmann equation in the space‐homogeneous case. Copyright © 2002 John Wiley & Sons, Ltd.     

Oscillation criteria for first-order neutral differential equations

This paper discusses a class of first-order neutral differential equations with variable coefficients and variable deviations. A series of sufficient conditions are established for all solutions of the equations to be oscillatory, and some of the conditions are sharp.     

Dispersive Properties for Discrete Schr?dinger Equations

In this paper we prove dispersive estimates for the system formed by two coupled discrete Schr?dinger equations. We obtain estimates for the resolvent of the discrete operator and prove that it satisfies the limiting absorption principle. The decay of the solutions is proved by using classical and some new results on oscillatory integrals.     

Uncertainty principle and kinetic equations

A large number of mathematical studies on the Boltzmann equation are based on the Grad's angular cutoff assumption. However, for particle interaction with inverse power law potentials, the associated cross-sections have a non-integrable singularity corresponding to the grazing collisions. Smoothing properties of solutions are then expected. On the other hand, the uncertainty principle, established by Heisenberg in 1927, has been developed so far in various situations, and it has been applied to the study of the existence and smoothness of solutions to partial differential equations. This paper is the first one to apply this celebrated principle to the study of the singularity in the cross-sections for kinetic equations. Precisely, we will first prove a generalized version of the uncertainty principle and then apply it to justify rigorously the smoothing properties of solutions to some kinetic equations. In particular, we give some estimates on the regularity of solutions in Sobolev spaces w.r.t. all variables for both linearized and nonlinear space inhomogeneous Boltzmann equations without angular cutoff, as well as the linearized space inhomogeneous Landau equation.     

Asymptotische Formeln für Eigenwerte des TRICOMIschen Differentialoperators erster Art mit Restgliedabschätzung

A class of n-th order (n > 1) differential equations with deviating arguments is considered and sufficient conditions are established in order that all bounded quickly oscillatory solutions (i.e. oscillatory solutions whose consecutive zeros have distance which approaches zero) tend to zero at ∞.     

Oscillations of the Solutions of Nonlinear Delay Hyperbolic Partial Differential Equations

In this paper, oscillatory properties of solutions of certain hyperbolic partial differential equations with multi-delays are investigated and a series of sufficient conditions for oscillations of the equations are established. The results fully indicate that the oscillations are caused by delays.     

Solutions of the Boltzmann equation with infinite energy: Existence, stability and long time behavior

This paper is devoted to studying a class of solutions to the nonlinear Boltzmann equation having infinite kinetic energy, these solutions have an upper Maxwellian bound with infinite kinetic energy. Firstly, the existence and stability of this kind of solutions are established near vacuum. Secondly, it is proved that this kind of solutions are stable for any initial data, as a consequence, the Boltzmann equation has at most one solution with infinite kinetic energy. Finally, the long time behavior of the solutions is also established.     

非线性脉冲中立型双曲方程的振动性

讨论非线性时滞脉冲中立型双曲方程的振动性质,利用平均值方法以及泛函微分不等式获得了该类方程在一类非线性边值条件下所有解振动的充分判据.结果表明,振动是由滞量和脉冲引起的.     

Dominant and recessive solutions for second order self-adjoint linear difference equations

This paper establishes the existence of recessive and dominant solutions for oscillatory second order homogeneous linear difference equations. Growth properties and comparison results concerning these solutions are established. Such information is then used to generalize the limit point results in the literature.