Dependence of solutions and eigenvalues of measure differential equations on measures

It is well known that solutions of ordinary differential equations are continuously dependent on finite-dimensional parameters in equations. In this paper we study the dependence of solutions and eigenvalues of second-order linear measure differential equations on measures as an infinitely dimensional parameter. We will provide two fundamental results, which are the continuity and continuous Fréchet differentiability in measures when the weak^? topology and the norm topology of total variations for measures are considered respectively. In some sense the continuity result obtained in this paper is the strongest one. As an application, we will give a natural, simple explanation to extremal problems of eigenvalues of Sturm–Liouville operators with integrable potentials.     

Banach空间阻尼弹性系统L-拟mild解的存在性

该文讨论了Banach空间中具有阻尼弹性系统L-拟mild解的存在性.这些结果改进和推广了一些相关的结论在常微分方程和偏微分方程方面.在非线性项满足单调条件和非紧性测度条件下,获得了该问题极大mild解的存在性.另外,给出例子说明该结果的可行性.     

Existence and approximation of solutions for discontinuous functional differential equations

In this paper existence of solutions of initial value problems for discontinuous functional differential equations is investigated firstly. By applying the method of upper and lower solutions, which may be discontinuous, some existence results of extremal solutions are obtained. Furthermore, we also develop a monotone iterative technique for obtaining extremal solutions which are obtained as limits of monotone sequences.     

Solvability of three point boundary value problems for second order differential equations with deviating arguments

The monotone iterative technique is used to boundary problems for second order ordinary differential equations with deviating arguments. Corresponding results are formulated when the problem has extremal solutions or weakly coupled extremal quasi-solutions.     

Spectral properties of non-local differential operators

In this article we consider the spectral properties of a class of non-local operators that arise from the study of non-local reaction-diffusion equations. Such equations are used to model a variety of physical and biological systems with examples ranging from Ohmic heating to population dynamics. The operators studied here are bounded perturbations of linear (local) differential operators. The non-local perturbation is in the form of an integral term. It is shown here that the spectral properties of these non-local operators can differ considerably from those of their local counterpart. Multiplicities of eigenvalues are studied and new oscillation results for the associated eigenfunctions are presented. These results highlight problems with certain similar results and provide an alternative formulation. Finally, the stability of steady states of associated non-local reaction-diffusion equations is discussed.     

Spectrum of One-Dimensional p-Laplacian with an Indefinite Integrable Weight

Motivated by extremal problems of weighted Dirichlet or Neumann eigenvalues, we will establish two fundamental results on the dependence of weighted eigenvalues of the one-dimensional p-Laplacian on indefinite integrable weights. One is the continuous differentiability of eigenvalues in weights in the Lebesgue spaces L ^γ with the usual norms. Another is the continuity of eigenvalues in weights with respect to the weak topologies in L ^γ spaces. Here 1 ≤ γ ≤ ∞. In doing so, we will give a simpler explanation to the corresponding spectrum problems, with the help of several typical techniques in nonlinear analysis such as the Fréchet derivative and weak* convergence.     

Lower bounds for the eigenvalues of first-order nonlinear Hamiltonian systems on time scales

We study first-order nonlinear planar Hamiltonian boundary value problems on time scales. Estimates on lower bounds for the eigenvalues of the problems are established by way of the Lyapunov inequality method. Our results are interpreted to nonlinear differential and difference planar Hamiltonian boundary value problems. As a special case, an estimate on lower bounds for eigenvalues of half-linear dynamic equations is obtained which generalizes and improves the existing ones to nonlinear Hamiltonian systems. Based on the main results, we establish existence and uniqueness of solutions of a related linear boundary value problem.     

Weak second order S-ROCK methods for Stratonovich stochastic differential equations

It is well known that the numerical solution of stiff stochastic ordinary differential equations leads to a step size reduction when explicit methods are used. This has led to a plethora of implicit or semi-implicit methods with a wide variety of stability properties. However, for stiff stochastic problems in which the eigenvalues of a drift term lie near the negative real axis, such as those arising from stochastic partial differential equations, explicit methods with extended stability regions can be very effective. In the present paper our aim is to derive explicit Runge–Kutta schemes for non-commutative Stratonovich stochastic differential equations, which are of weak order two and which have large stability regions. This will be achieved by the use of a technique in Chebyshev methods for ordinary differential equations.     

Continuity in weak topology:higher order linear systems of ODE

We will introduce a type of Fredholm operators which are shown to have a certain con- tinuity in weak topologies.From this,we will prove that the fundamental matrix solutions of k-th, k≥2,order linear systems of ordinary differential equations are continuous in coefficient matrixes with weak topologies.Consequently,Floquet multipliers and Lyapunov exponents for periodic systems are continuous in weak topologies.Moreover,for the scalar Hill's equations,Sturm-Liouville eigenvalues, periodic and anti-periodic eigenvalues,and rotation numbers are all continuous in potentials with weak topologies.These results will lead to many interesting variational problems.     

A simple approach for determining the eigenvalues of the fourth-order Sturm–Liouville problem with variable coefficients

In this paper, a simple and efficient approach is presented to compute the eigenvalues of the fourth-order Sturm–Liouville equations with variable coefficients. By transforming the governing differential equation to a system of algebraic equation, we can get the corresponding polynomial characteristic equations for kinds of boundary conditions based on the polynomial expansion and integral technique. Moreover, the lower and higher-order eigenvalues can be determined simultaneously from the multi-roots. Several examples for estimating eigenvalues are given. The convergence and effectiveness of the method are confirmed by comparing numerical results with the exact and other existing numerical results.     

Periodic boundary value problems for impulsive fuzzy differential equations

The peculiarity of the Hukuhara derivative makes it impossible to find periodic solutions for fuzzy differential equations with the exception of very restrictive situations. In this work, we consider a boundary value problem associated with an impulsive fuzzy differential equation and approximate the extremal solutions in a fuzzy functional interval using the monotone method. Fuzzy comparison results are useful in our procedure and the expression of the solution for some impulsive periodic ‘linear’ differential problems is also provided.     

Banach空间中二阶非线性脉冲微分方程的Sturm-Liouville边值问题的极解

本文利用单调迭代技巧 ,锥理论和比较定理获得了Banach空间中二阶非线性脉冲微分方程的Sturm Liouville边值问题的极解     

An approximation theory for conjugate surfaces and solutions of elliptic multiple integral problems: Application to numerical solutions of generalized Laplace's equation

An approximation theory is given for a class of elliptic quadratic forms which include the study of conjugate surfaces for elliptic multiple integral problems. These ideas follow from the quadratic form theory of Hestenes, applied to multiple integral problems by Dennemeyer, and extended with applications for approximation problems by Gregory.The application of this theory to a variety of approximation problem areas in this setting is given. These include conjugate surfaces and conjugate solutions in the calculus of variations, oscillation problems for elliptic partial differential equations, eigenvalue problems for compact operators, numerical approximation problems, and, finally, the intersection of these problem areas.In the final part of this paper the ideas are specifically applied to the construction and counting of negative vectors in order to obtain new numerical methods for solving Laplace-type equations and to obtain the “Euler-Lagrange equations” for symmetric-banded tridiagonal matrices. In this new result (which will allow the reexamination of both the theory and applications of symmetricbanded matrices) one can construct, in a meaningful way, negative vectors, oscillation vectors, eigenvectors, and extremal solutions of classical problems as well as efficient algorithms for the numerical solution of partial differential equations. Numerical examples (test runs) are given.     

On the existence of an optimal feedback control for stochastic systems

We prove the existence of an optimal control for systems of stochastic differential equations without solving the Bellman dynamic programming equation. Instead, we use direct methods for solving extremal problems.     

On extending existence theory from scalar ordinary differential equations to infinite quasimonotone systems of functional equations

In this paper we use Tarski's fixed point theorem to extend in a systematic way the existence of extremal solutions from scalar initial value problems to boundary value problems for infinite quasimonotone functional systems of differential equations.

     

Numerical methods for eighth-, tenth- and twelfth-order eigenvalue problems arising in thermal instability

Second-order finite-difference methods are developed for the numerical solutions of the eighth-, tenth- and twelfth-order eigenvalue problems arising in the study of the effect of rotation on a horizontal layer of fluid heated from below. Instability setting-in as overstability may be modelled by an eighth-order ordinary differential equation. When a uniform magnetic field also acts across the fluid in the same direction as gravity, instability setting-in as ordinary convection may be modelled by a tenth-order differential equation, while instability setting-in as overstability may be modelled by a twelfth-order differential equation. The numerical methods are developed by making direct replacements of the derivatives in the differential equations and then by computing the eigenvalues, which may incorporate Rayleigh number, horizontal wave speed and a time constant, from the resulting algebraic eigenvalue problem. The eigenvalues are also computed by writing the differential equations as systems of second-order differential equations and then using second- and fourth-order methods to obtain the eigenvalues. Numerical results obtained using the two approaches are compared with estimates appearing in the literature.     

带有"上确界"的非线性脉冲微分方程无穷边值问题

应用上下解方法和单调迭代技术研究了带有上确界的一阶非线性脉冲微分方程无穷边值问题，并获得了其极值解的存在性结果．     

Extremal problems for Fredholm eigenvalues

The study of extremal problems for Fredholm eigenvalues was initiated by Schiffer in the context of the existence of conformal maps onto canonical domains. We present a different approach to solving rather general extremal problems for Fredholm eigenvalues related to appropriate univalent functions with quasiconformal extensions. It involves the complex geometry of the universal Teichmüller space.     

二阶非线性脉冲微分方程的 Sturm-Liouville边值问题(英文)

本文采用一个改进的单调迭代技巧 ,获得了二阶非线性脉冲微分方程的 Sturm-L iouville边值问题的极解     

Crossing of eigenvalues: Measure type estimates

This paper deals with the behavior of eigenvalues for some non-homogeneous elliptic operators. More precisely, we present measure type estimates evaluating neighborhoods of the so-called resonant set. Various problems like the non-homogeneous incompressible limit of the compressible Navier-Stokes equations lead to such studies. This will be the purpose of a forthcoming paper.