Operator Hamiltonians and anticanonic equations with periodic coefficients

In a Hilbert space we study Hamiltonians and anticanonic equations with periodic coefficients. We prove existence theorems for the solutions of ill-posed Cauchy problems for the given equations. Following Krein we define the notion of the genus of the spectrum points of the monodromy operator of an equation of the class being studied. We formulate existence and uniqueness theorems for the solutions when determining the reflected and the transmitted waves for a specified incident wave. The theory developed is applied to the study of cylindrical waveguides with a periodic filling.Translated from Problemy Matematicheskogo Analiza. No. 4: Integralnye i Differentsial'nye Operatory. Differentsial'nye Uraveniya, pp. 9–36, 1973.     

A global solution curve for a class of periodic problems, including the pendulum equation

Using continuation methods and bifurcation theory, we study the exact multiplicity of periodic solutions, and the global solution structure, for a class of periodically forced pendulum-like equations. Our results apply also to the first order equations. We also show that by choosing a forcing term, one can produce periodic solutions with any number of Fourier coefficients arbitrarily prescribed.     

具阶段结构害虫防治模型的脉冲效应

对于用微分方程描述的种群生态动力系统，其研究结果已十分丰富，但自然界中的许多变化规律都呈现出脉冲效应，因此用脉冲微分方程描述某些运动状态在固定或不固定时刻的快速变化或跳跃更切合实际，尤其在刻画种群生长和流行病动力学行为方面，脉冲微分方程的描述显得更科学更真实，具有脉冲效应的种群动力学模型的研究目前还处于刚刚起步阶段，本对符合实际的有脉冲效应的具阶段结构的常系数害早防治模型进行了研究，得到了系统存在周期解的充分条件，系统存在唯一周期解的充分条件，系统周期解轨道渐近稳定的充分条件。     

Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids

Many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite. Using classical methods, the authors develop a general theory of such problems. This theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto-fluid dynamics including new constructive local existence theorems for the time-singular limit equations. In particular, the authors give an entirely self-contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the Mach number becomes small. The theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures. Similar results are developed also for the compressible and incompressible Navier-Stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero.     

AN OSCILLATION THEOREM FOR A TWO-PARAMETER SYSTEM OF DIFFERENTIAL EQUATIONS WITH PERIODIC BOUNDARY CONDITIONS

Abstract

The oscillation theory for two simultaneous systems of second order linear differential equations in two parameters with periodic boundary conditions is well known when the coefficients of the differential equations are subjected to the usual definiteness condition. However, in practical applications the usual definiteness condition may fail to hold, and hence in this paper we consider the oscillation theory under another important definiteness condition.     

关于高阶常系数线性中立型方程周期解的讨论

本文讨论高阶常系数线性中立型方程的周期解问题,作者利用Fourier级数理论给出周期解存在,唯一的充分必要条件,所得结果包含和推广了文献[1]中的结果。     

时标上具有阶段结构的三种群捕食系统的周期解

研究了时标上具有阶段结构的三种群捕食系统.运用时标上连续拓扑度定理,得到了系统存在周期解的充分条件.其研究方法使系统的连续时间情形和离散时间情形的周期解问题得到了统一,被广泛地应用来研究微分方程和差分方程的周期解的存在问题.     

Oscillation and non‐oscillation results for solutions of perturbed half‐linear equations

The purpose of this paper is to describe the oscillatory properties of second‐order Euler‐type half‐linear differential equations with perturbations in both terms. All but one perturbations in each term are considered to be given by finite sums of periodic continuous functions, while coefficients in the last perturbations are considered to be general continuous functions. Since the periodic behavior of the coefficients enables us to solve the oscillation and non‐oscillation of the considered equations, including the so‐called critical case, we determine the oscillatory properties of the equations with the last general perturbations. As the main result, we prove that the studied equations are conditionally oscillatory in the considered very general setting. The novelty of our results is illustrated by many examples, and we give concrete new corollaries as well. Note that the obtained results are new even in the case of linear equations.     

Some results in Floquet theory,with application to periodic epidemic models

In this paper, we present several new results to the classical Floquet theory on the study of differential equations with periodic coefficients. For linear periodic systems, the Floquet exponents can be directly calculated when the coefficient matrices are triangular. Meanwhile, the Floquet exponents are eigenvalues of the integral average of the coefficient matrices when they commute with their antiderivative matrices. For the stability analysis of constant and nontrivial periodic solutions of nonlinear differential equations, we derive a few results based on linearization. We also briefly discuss the properties of Floquet exponents for delay linear periodic systems. To demonstrate the application of these analytical results, we consider a new cholera epidemic model with phage dynamics and seasonality incorporated. We conduct mathematical analysis and numerical simulation to the model with several periodic parameters.     

On the oscillation of some linear difference equations with periodic coefficients

Some linear difference equations with periodic coefficients (not necessarily nonnegative) are considered. Necessary conditions and sufficient conditions for the oscillation of the solutions are established. Conditions under which all nonoscillatory solutions tend to zero at ∞ are also presented. The results obtained are the discrete analogues of the oscillation results for some linear delay differential equations with periodic coefficients, which were given earlier by the second author [Oscillations of some delay differential equations with periodic coefficients, J. Math. Anal. Appl. 162 (1991) 452–475].     

A finite element method for elliptic problems with rapidly oscillating coefficients

In this paper, we consider solving second-order elliptic problems with rapidly oscillating coefficients. Under the assumption that the oscillating coefficients are periodic, on the basis of classical homogenization theory, we present a finite element method whose key is to combine a numerical approximation of the 1-order approximate solution of those equations and a numerical approximation of the classical boundary corrector of those equations from different meshes exploiting the need for different levels of resolution. Numerical experiments are included to illustrate the competitive behavior of the proposed finite element method.     

Pseudo almost periodic in distribution solutions and optimal solutions to impulsive partial stochastic differential equations with infinite delay

In this paper, we study a general class of impulsive partial stochastic differential equations with infinite delay and pseudo almost periodic coefficients in Hilbert spaces. Firstly, a more appropriate concept of pseudo almost periodic in distribution for stochastic processes of infinite class is introduced. Secondly, the existence of pseudo almost periodic in distribution mild solutions is investigated by utilizing the interpolation theory, the stochastic analysis techniques and fixed point theorem. The existence of optimal mild solutions of the systems is also proved. Finally, an example is provided to show the effectiveness of the theoretical results.     

具多偏差变元的n-维p-Laplacian方程周期解的存在性

研究了一类具多偏差变元的n-维p-Laplacian方程周期解的存在性,利用迭合度理论得到了存在周期解的新条件.     

Unified Orbital Description of the Envelope Dynamics in Two‐Dimensional Simple Periodic Lattices

The propagation of wave envelopes in two‐dimensional (2‐D) simple periodic lattices is studied. A discrete approximation, known as the tight‐binding (TB) approximation, is employed to find the equations governing a class of nonlinear discrete envelopes in simple 2‐D periodic lattices. Instead of using Wannier function analysis, the orbital approximation of Bloch modes that has been widely used in the physical literature, is employed. With this approximation the Bloch envelope dynamics associated with both simple and degenerate bands are readily studied. The governing equations are found to be discrete nonlinear Schrödinger (NLS)‐type equations or coupled NLS‐type systems. The coefficients of the linear part of the equations are related to the linear dispersion relation. When the envelopes vary slowly, the continuous limit of the general discrete NLS equations are effective NLS equations in moving frames. These continuous NLS equations (from discrete to continuous) also agree with those derived via a direct multiscale expansion. Rectangular and triangular lattices are examples.     

关于几类Riccati方程和二阶常微分方程的周期解

本文讨论了具有周期系数的几类Riccati方程的周期解问题,并给出几类二阶常微分方程有周期解的条件.     

Fourier expansion based recursive algorithms for periodic Riccati and Lyapunov matrix differential equations

Combining Fourier series expansion with recursive matrix formulas, new reliable algorithms to compute the periodic, non-negative, definite stabilizing solutions of the periodic Riccati and Lyapunov matrix differential equations are proposed in this paper. First, periodic coefficients are expanded in terms of Fourier series to solve the time-varying periodic Riccati differential equation, and the state transition matrix of the associated Hamiltonian system is evaluated precisely with sine and cosine series. By introducing the Riccati transformation method, recursive matrix formulas are derived to solve the periodic Riccati differential equation, which is composed of four blocks of the state transition matrix. Second, two numerical sub-methods for solving Lyapunov differential equations with time-varying periodic coefficients are proposed, both based on Fourier series expansion and the recursive matrix formulas. The former algorithm is a dimension expanding method, and the latter one uses the solutions of the homogeneous periodic Riccati differential equations. Finally, the efficiency and reliability of the proposed algorithms are demonstrated by four numerical examples.     

Semiconjugate factorizations of higher order linear difference equations in rings

We use a new nonlinear method to study linear difference equations with variable coefficients in a non-trivial ring R. If the homogeneous part of the linear equation has a solution in the unit group of a ring with identity (a unitary solution), then we show that the equation decomposes into two linear equations of lower orders. This decomposition, known as a semiconjugate factorization in the nonlinear theory, is based on sequences of ratios of consecutive terms of a unitary solution. Such sequences, which may be called eigensequences, are well suited to variable coefficients; for instance, they provide a natural context for the expression of the Poincaré–Perron theorem. As applications, we obtain new results for linear difference equations with periodic coefficients and for linear recurrences in rings of functions (e.g. the recurrence for the modified Bessel functions).     

余模范畴中的余倾斜预包络和余倾斜挠类

众所周知, Assem-Smal定理在倾斜理论中有重要的作用.本文的目的是建立一个在余模范畴中的Assem-Smal定理的版本,并通过利用预包络理论来刻画余模范畴中的余倾斜挠类.     

Generalized G-convergence for quasilinear elliptic differential operators

Generalized G-convergence for a quasilinear elliptic differential equation is defined and studied. The equation describes heat conduction in the cores of large electric transformers. The coefficients of the equation depend on temperature and the corresponding differential operator is neither potential nor monotone. A theory which generalizes the classical G-convergence is proposed. The theory is applied to the homogenization of the quasilinear elliptic differential equation with periodic coefficients.     

Differential games on partially overlapping sets

We suggest four new notions of optimality (equilibrium) and use them to construct a theory providing the existence and (almost always) the uniqueness of solutions of game problems (both static problems and problems described by differential equations) with partially overlapping game sets of the players; such problems, which are used when modeling conflict problems with incidental interests (or incidental profits) of the players, have not been studied in the classical theory.