Application of self-adjoint boundary value problems to investigation of stability of periodic delay systems

The problem on determining conditions for the asymptotic stability of linear periodic delay systems is considered. Solving this problem, we use the function space of states. Conditions for the asymptotic stability are determined in terms of the spectrum of the monodromy operator. To find the spectrum, we construct a special boundary value problem for ordinary differential equations. The motion of eigenvalues of this problem is studied as the parameter changes. Conditions of the stability of the linear periodic delay system change when an eigenvalue of the boundary value problem intersects the circumference of the unit disk. We assume that, at this moment, the boundary value problem is self-adjoint. Sufficient coefficient conditions for the asymptotic stability of linear periodic delay systems are given.     

Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary conditions

By taking as a “prototype problem” a one-delay linear autonomous system of delay differential equations we present the problem of computing the characteristic roots of a retarded functional differential equation as an eigenvalue problem for a derivative operator with non-local boundary conditions given by the particular system considered. This theory can be enlarged to more general classes of functional equations such as neutral delay equations, age-structured population models and mixed-type functional differential equations.It is thus relevant to have a numerical technique to approximate the eigenvalues of derivative operators under non-local boundary conditions. In this paper we propose to discretize such operators by pseudospectral techniques and turn the original eigenvalue problem into a matrix eigenvalue problem. This approach is shown to be particularly efficient due to the well-known “spectral accuracy” convergence of pseudospectral methods. Numerical examples are given.     

One-dimensional quasilinear eigenvalue and eigenfunction problem

We consider a one-dimensional quasilinear eigenvalue and eigenfunction problem. The simplification of the problem by replacing a second-order elliptic operator by a one-dimensional one permits carrying out an efficient and unified study of the structure of the spectrum and the multiplicity of eigenvalues for a wide range of functions specifying the nonlinearity. This opens up the possibility of preliminarily analyzing specific properties of solutions of such problems for partial differential equations, in particular, the equilibrium of a toroidal plasma, since the spectrum of the problem and its multiplicity depend only weakly on the space dimension.     

Eigenvalue Problem of Doubly Stochastic Hamiltonian Systems with Boundary Conditions

In this paper, we investigate the eigenvalue problem of forward-backward doubly stochastic dii~erential equations with boundary value conditions. We show that this problem can be represented as an eigenvalue problem of a bounded continuous compact operator. Hence using the famous Hilbert-Schmidt spectrum theory, we can characterize the eigenvalues exactly.     

Singular numbers of the monodromy operator and sufficient conditions of the asymptotic stability of periodic system of differential equations with fixed delay

To obtain sufficient conditions for the asymptotic stability of linear periodic systems with fixed delay commensurable with the period of coefficients, singular numbers of the monodromy operator are used. To find these numbers, a self-adjoint boundary value problem for ordinary differential equations is applied. We study the motion of eigenvalues of this boundary value problem under a variation of a parameter. Obtaining sufficient conditions for the asymptotic stability is reduced to finding the bifurcation value of the parameter for the boundary value problem.     

Die Konvergenz des Rand- und Eigenwertproblems linearer gewöhnlicher Differenzengleichungen

Summary In this paper the convergence of finite difference approximations for the general eigenvalue and boundary value problem of ordinary differential equations is proved under the condition of consistency and stability. The eigenvalues are shown to converge preserving multiplicity. Estimates are given for the rate of convergence of difference quotients and eigenvalues.     

Positive solutions of m-point boundary value problems for a class of nonlinear fractional differential equations

This paper deals with the existence and multiplicity of positive solutions for a class of nonlinear fractional differential equations with m-point boundary value problems. We obtain some existence results of positive solution by using the properties of the Green’s function, u ₀-bounded function and the fixed point index theory under some conditions concerning the first eigenvalue with respect to the relevant linear operator.     

Spectrum of the 1‐Laplacian and Cheeger's Constant on Graphs

We develop a nonlinear spectral graph theory, in which the Laplace operator is replaced by the 1 ? Laplacian Δ₁. The eigenvalue problem is to solve a nonlinear system involving a set valued function. In the study, we investigate the structure of the solutions, the minimax characterization of eigenvalues, the multiplicity theorem, etc. The eigenvalues as well as the eigenvectors are computed for several elementary graphs. The graphic feature of eigenvalues are also studied. In particular, Cheeger's constant, which has only some upper and lower bounds in linear spectral theory, equals to the first nonzero Δ₁ eigenvalue for connected graphs.     

Spectral Asymptotics of Self-Adjoint Fourth Order Differential Operators with Eigenvalue Parameter Dependent Boundary Conditions

A fourth-order regular ordinary differential operator with eigenvalue dependent boundary conditions is considered. This problem is realized by a quadratic operator pencil with self-adjoint operators. The location of the eigenvalues is discussed and the first four terms of the eigenvalue asymptotics are evaluated explicitly.     

One‐dimensional approximations of the eigenvalue problem of curved rods

In this work we analyse the asymptotic behaviour of eigenvalues and eigenfunctions of the linearized elasticity eigenvalue problem of curved rod‐like bodies with respect to the small thickness ? of the rod. We show that the eigenfunctions and scaled eigenvalues converge, as ? tends to zero, toward eigenpairs of the eigenvalue problem associated to the one‐dimensional curved rod model which is posed on the middle curve of the rod. Because of the auxiliary function appearing in the model, describing the rotation angle of the cross‐sections, the limit eigenvalue problem is non‐classical. This problem is transformed into a classical eigenvalue problem with eigenfunctions being inextensible displacements, but the corresponding linear operator is not a differential operator. Copyright © 2001 John Wiley & Sons, Ltd.     

A Note on Random Perturbations of a Multiple Eigenvalue of a Hermitian Operator

The problem of a random Hermitian perturbation of a multiple isolated eigenvalue of a Hermitian operator is considered. It is shown that the combined multiplicities of the perturbed eigenvalues converge in probability to the multiplicity of the eigenvalue of the target operator. Also the asymptotic distribution of a certain average of these eigenvalues, centered at the target, is obtained. As a tool differentiation of analytic functions of operators is employed in conjunction with an ensuing “delta-method”. The result is of a probabilistic rather than statistical nature.     

The Eigenvalue Problem for a One-Dimensional Differential Operator with a Variable Coefficient and Nonlocal Integral Conditions*

We consider conditions for the existence of the eigenvalue λ = 0 in the eigenvalue problem for a differential operator with a variable coefficient and integral conditions. We analyze how these conditions depend on such properties of the coefficient p(x) as monotonicity and symmetry and observe some other properties of the spectrum of the eigenvalue problem. Particularly, we show by a numerical experiment that the fundamental theorem on the increase of the eigenvalues in the case of increasing coefficient p(x) is not valid for the eigenvalue problem with nonlocal conditions.     

二阶非线性微分方程组三点边值问题的多个正解

研究的是二阶非线性微分方程组的边值问题,在适合的条件下,应用抽象不动点理论以及线性算子的第一特征值的条件,得出了方程组的多个正解的存在性．     

Mellin-type differential equations and associated sampling expansions

This paper is devoted to a self-contained approach to Mellin-type differential equations and associated ssampling expansions. Here the first order differential operator is not the normal d/dx but D_M,c=xd/dx+c,c E R being connected with the definition of the Mellin transform. Existence and uniqueness theorems are established for a system of first order Mellin equations and the properties of nth order linear equations are investigated. Then self adjoint Mellin-type second order Sturm-Liouville eigenvalue problems are considered and properties of the eigenvalues, eigenfunctions and Green's functions are derived. As applications. sampling representations for two classes of integral transforms arising from the eigenvalue problem are introduced. In the first class the kernesl are solutions of the problem and in the second they are expressed in terms of green's function.     

Computable bounds for eigenvalues and eigenfunctions of elliptic differential operators

Summary We are concerned with bounds for the error between given approximations and the exact eigenvalues and eigenfunctions of self-adjoint operators in Hilbert spaces. The case is included where the approximations of the eigenfunctions don't belong to the domain of definition of the operator. For the eigenvalue problem with symmetric elliptic differential operators these bounds cover the case where the trial functions don't satisfy the boundary conditions of the problem. The error bounds suggest a certain defectminization method for solving the eigenvalue problems. The method is applied to the membrane problem.     

On the eigenvalue problem of a class of linear partial difference equations

In this paper, the eigenvalue problem of a class of linear partial difference equations is studied. The results concern the existence of eigenvalues, their character (real, positive), as well as the behavior of its eigenfunctions (positivity, oscillation). Moreover a theorem is given concerning the existence of a unique solution of an associated non-homogeneous partial difference equation. The results generalize previously known results for ordinary linear difference equations. The method used is a functional-analytic one, which transforms the eigenvalue problem for the difference equation into the equivalent problem of the eigenvalues of an operator defined on an abstract separable Hilbert space.     

Asymptotics of Eigenvalues of Differential Operator With Alternating Weight Function

We study a differential operator of the sixth order with an alternating weight function. The potential of the operator has a first-order discontinuity at some point of the segment, where the operator is being considered. The boundary conditions are separated. We study the asymptotics of solutions to the corresponding differential equations and the asymptotics of eigenvalues of the considered differential operator.     

Calculation of eigenvalues of a discrete self-adjoint operator perturbed by a bounded operator

We generalize the method of regularized traces which calculates eigenvalues of a perturbed discrete operator for the case of an arbitrary multiplicity of eigenvalues of the unperturbed operator. We obtain a system of equations, enabling one to calculate eigenvalues of the perturbed operator with large ordinal numbers. As an example, we calculate eigenvalues of a perturbed Laplace operator in a rectangle.     

Spectral Analysis of the Differential Operator in Wavelet Bases

We analyze the spectral properties of the differential operator in wavelet bases. The problem is studied on a periodic domain, with periodized wavelets. An algorithm for finding the eigenvalue function of the differential operator is presented, and general conditions that ensure a "nicely behaving" eigenvalue function are derived.