Inverse spectral problems for differential pencils with boundary conditions dependent on the spectral parameter

In this paper, we discuss two inverse problems for differential pencils with boundary conditions dependent on the spectral parameter. We will prove the Hochstadt–Lieberman type theorem of 1 – 3 except for arbitrary one eigenvalue and the Borg type theorem of 1 – 3 except for at most arbitrary two eigenvalues, respectively. The new results are generalizations of the related results. Copyright © 2016 John Wiley & Sons, Ltd.     

Uniqueness theorems for differential pencils with eigenparameter boundary conditions and transmission conditions

Inverse spectral problems are considered for differential pencils with boundary conditions depending polynomially on the spectral parameter and with a finite number of transmission conditions. We give formulations of the associated inverse problems such as Titchmarsh–Weyl theorem, Hochstadt–Lieberman theorem and Mochizuki–Trooshin theorem, and prove corresponding uniqueness theorems. The obtained results are generalizations of the similar results for the classical Sturm–Liouville operator on a finite interval.     

A modification of one method for solving nonlinear self-adjoint eigenvalue problems for hamiltonian systems of ordinary differential equations

A modification of the method proposed earlier by the author for solving nonlinear self-adjoint eigenvalue problems for linear Hamiltonian systems of ordinary differential equations is examined. The basic assumption is that the initial data (that is, the system matrix and the matrices specifying the boundary conditions) are monotone functions of the spectral parameter.     

Fredholm boundary value problems for perturbed systems of dynamic equations on time scales

This paper offers conditions ensuring the existence of solutions of linear boundary value problems for systems of dynamic equations on time scales. Utilizing a method of Moore–Penrose pseudo‐inverse matrices leads to an analytical form of a criterion for the existence of solutions in a relevant space and, moreover, to the construction of a family of linearly independent solutions of such problems in a general case with the number of boundary conditions (defined by a linear vector functional) not coinciding with the number of unknowns of a system of dynamic equations. As an example of an application of the presented results, the problem of bifurcation of solutions of boundary value problems for systems of dynamic equations on time scales with a small parameter is considered.     

Sturmian theory for linear Hamiltonian systems without controllability

In this paper we present Sturmian separation and comparison theorems for linear Hamiltonian systems when no controllability assumption is imposed. This generalizes the traditional results of W. T. Reid for controllable (or normal) linear Hamiltonian systems to the possibly abnormal case. Our new theory is based on several recent results on linear Hamiltonian systems without controllability by W. Kratz, M. Wahrheit, V. Zeidan, and the author regarding the piecewise constant kernel for conjoined bases, the oscillation and eigenvalue theorems, and the Rayleigh principle. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

On the self-adjoint nonlinear eigenvalue problem for Hamiltonian systems of ordinary differential equations with singularities

Certain properties of the nonlinear self-adjoint eigenvalue problem for Hamiltonian systems of ordinary differential equations with singularities are examined. Under certain assumptions on the way in which the matrix of the system and the matrix specifying the boundary condition at a regular point depend on the spectral parameter, a numerical method is proposed for determining the number of eigenvalues lying on a prescribed interval of the spectral parameter.     

On the number of an eigenvalue of a general nonlinear self-adjoint spectral problem for systems of ordinary differential equations

In the case of a general nonlinear self-adjoint spectral problem for systems of ordinary differential equations with boundary conditions independent of the spectral parameter, we introduce the notion of the number of an eigenvalue. Methods for the computation of the numbers of eigenvalues lying in a given range of the spectral parameter and for finding the eigenvalue with a given number, which were earlier suggested by the author for Hamiltonian systems, are generalized to the considered problem. We introduce the notion of an index of a problem for a general nontrivially solvable linear homogeneous self-adjoint boundary value problem.     

非正曲率流形及其子流形上有界区域的特征值

本文研究了完备单连通具有非正曲率黎曼流形及其子流形上有界区域的特征值问题.利用广义Hessian比较定理,获得了局部特征值的下界估计式,将McKean[2]的定理在局部上推广到了非正曲率的情形.     

Stability, resonance and Lyapunov inequalities for periodic conservative systems

This paper is devoted to the study of Lyapunov type inequalities for periodic conservative systems. The main results are derived from a previous analysis which relates the best Lyapunov constants to some especial (constrained or unconstrained) minimization problems. We provide some new results on the existence and uniqueness of solutions of nonlinear resonant and periodic systems. Finally, we present some new conditions which guarantee the stable boundedness of linear periodic conservative systems.     

Spectral Problems for Systems of Differential Equations y′ + A0y = λA1y with λ–Polynomial Boundary Conditions

This paper deals with the spectral properties of boundary eigenvalue problems for systems of first order differential equations with boundary conditions which depend on the spectral parameter polynomially. It is not assumed that is injective or surjective. The main results concern the completeness minimality and Riesz basis properties of the corresponding eigenfunctions and associated functions.     

Conditions for the existence of positive solutions covering a class of boundary value problems in a uniform way

In this paper, we will give conditions which will guarantee the existence of positive solutions for a variety of second order boundary value problems, using the well-known Krasnoselskii fixed point theorem. We will deal with a specific differential equation meeting a specific initial condition and use a general boundary condition, involving a not necessarily linear functional. Our purpose is to pose conditions on that functional, which will guarantee that the Krasnoselskii fixed point theorem can be applied. It is important to notice that only the values of this functional on two specific functions are involved in the conditions we pose. This paper unifies the way we deal with a wide variety of boundary value problems and provides results which, to the best of our knowledge, are new.     

一类边界带特征参数的不连续高阶微分算子的自共轭性

考虑了一类具有特殊转移条件且边界条件中均带有特征参数的高阶边值问题,建立了一个与其相关的新空间H与新算子A,讨论了算子A在H中的自共轭性.     

A NEW ALGORITHM FOR COMPUTING LARGEST REAL PART EIGENVALUE OF MATRICES：COLLATZ ＆ PERRON-FROBERNIUS＇ APPROACH

This paper describes a new method and algorithm for the numerical solution of eigenvalues with the largest real part of positive matrices.The method is based on a numerical implementation of Collatz’s eigenvalue inclusion theorem for non-negative irreducible matrices.Eigenvalues are analyzed for the studies of the stability of linear systems.Finally, a numerical discussion is given to derive the required number of mathematical operations of the new algorithm. Comparisons between the new algorithm and several well known ones, such as Power, and QR methods, are discussed.     

Determination of the number of an eigenvalue of a singular nonlinear self-adjoint spectral problem for a linear Hamiltonian system of differential equations

We suggest a method for determining the number of an eigenvalue of a self-adjoint spectral problem nonlinear with respect to the spectral parameter, for some class of Hamiltonian systems of ordinary differential equations on the half-line. The standard boundary conditions are posed at zero, and the solution boundedness condition is posed at infinity. We assume that the matrix of the system is monotone with respect to the spectral parameter. The number of an eigenvalue is determined by the properties of the corresponding nontrivially solvable homogeneous boundary value problem. For the considered class of systems, it becomes possible to compute the numbers of eigenvalues lying in a given range of the spectral parameter without finding the eigenvalues themselves.     

Computation of the eigenvalues of Sturm-Liouville problems with parameter dependent boundary conditions using the regularized sampling method

The purpose in this paper is to compute the eigenvalues of Sturm-Liouville problems with quite general separated boundary conditions nonlinear in the eigenvalue parameter using the regularized sampling method, an improvement on the method based on Shannon sampling theory, which does not involve any multiple integration and provides higher order estimates of the eigenvalues at a very low cost. A few examples shall be presented to illustrate the power of the method and a comparison made with the the exact eigenvalues obtained as squares of the zeros of the exact characteristic functions.

     

Positive solutions of discrete third-order three-point right focal boundary value problems

We consider linear and nonlinear boundary value problems for third-order difference equations with three-point right focal boundary conditions. In the linear case, the existence of positive solutions corresponding to the first eigenvalue of the problem is established and an interval estimate for the first eigenvalue is obtained. In the nonlinear case, sufficient conditions for the existence and nonexistence of positive solutions are obtained. An example is included at the end of the paper to illustrate the sharpness of our results.     

A CLASS OF NONLOCAL BOUNDARY VALUE PROBLEMS OF NONLINEAR ELLIPTIC SYSTEMS IN UNBOUNDED DOMAINS

We consider the following boundary value problem ill the unbounded donain Liui = fi(x,u, Tu), i = 1, 2,' ! N,x E fl, (1) olLi "i0n Pi(x)t'i = gi(x,u), i = l, 2,',N,x E 0fl, (2) where x = (x i,', x.), u = (u1,' f uN), Th = (T1tti,', TNi'N) and [ n. 1 L, = -- I Z ajk(X)the i0j(X)C], Li,k=1' j=1 J] l Ltti = / K(x,y)ui(y)dy, x E n. jn K(x, y)ui(y)dy, x E n. Q denotes an unbounded dolllain in R", including the exterior of a boullded doinain and 0…     

Sampling expansions associated with Kamke problems

The present paper is devoted to the derivation of sampling expansions for entire functions which are represented as integral transforms where a differential operator is acting on the kernels. The situation generalizes the results obtained in sampling theory associated with boundary value problems to the case when the differential equation has the form where N and P are two differential expressions of orders n and p respectively, and is the eigenvalue parameter. Both self adjoint and non self adjoint cases will be considered with examples in which the boundary conditions are strongly regular. Received February 16, 1998; in final form March 15, 1999     

A new type of full multigrid method for the elasticity eigenvalue problem

This paper introduces a new type of full multigrid method for the elasticity eigenvalue problem. The main idea is to avoid solving large scale elasticity eigenvalue problem directly by transforming the solution of the elasticity eigenvalue problem into a series of solutions of linear boundary value problems defined on a multilevel finite element space sequence and some small scale elasticity eigenvalue problems defined on the coarsest correction space. The involved linear boundary value problems will be solved by performing some multigrid iterations. Besides, some efficient techniques such as parallel computing and adaptive mesh refinement can also be absorbed in our algorithm. The efficiency and validity of the multigrid methods are verified by several numerical experiments.     

一类奇摄动非线性椭圆型系统

本文讨论了一类在带形区域中的非线性椭圆型系统的奇轻边值问题,在适当的条件下 比较定理台相应边值问题解的存在性和斩近性态作了研究。