Error Estimates Of Eigenvalues Of Perturbed Discrete Boundary Value Problems

With the rapid development of information technology and the wide applications of digital computer, many difference systems have appeared and been paid attention by more and more scholars (cf. [1-3, 9-23, 25-26, 30-50] and their references). The appearance of difference systems plays an important role in the practical applications. As we all know, continuous systems can be described by differential systems, while some systems (such as sample systems) can not be described by differential systems and can only be described by discrete systems. On the other hand, it is impossible to obtain an accurate solution of a general nonlinear differential system. So, we often compute its approximate solution by applying some discretization methods. In addition, discrete Hamiltonian systems originated from the discretization of continuous Hamiltonian systems and from discrete processes acting in accordance with the Hamiltonian principle such as discrete physical problems, discrete control problems and so on.In the past forty years, spectral theories for second-order difference equations have attracted a great deal of interest (cf. [3, 17, 23, 25, 41-43, 49] and their references). F. V. Atkinson [3] first studied second-order scalar and vector discrete Sturm-Liouville problems and converted the vector problem with separated boundary conditions into an equivalent spectral problem of a certain Hermitian matrix. In 1995, A. Jirari [25] studied second-order scalar discrete Sturm-Liouville problems with a more general boundary condition which is still separated and extended partial results of [3]. D. T. Smith [42] discussed the spectrum of self-adjoint operator of second-order difference equations by using oscillation of solutions. In [41], Shi and Chen studied second-order vector discrete Sturm-Liouville problems and obtained a series of spectral results by introducing a self-adjoint operator on a suitable admissible function space.With the further research of spectral theory of second-order difference equations, discrete linear Hamiltonian systems have attracted a lot of attention and some good results have been obtained (cf. [2, 9, 10, 12, 18, 21, 34, 36, 38, 39, 45, 46] and their references). M. Bohner [10] studied eigenvalue problems of a class of discrete linear Hamiltonian systems and obtained isolatedness and lower boundedness of eigenvalues by introducing the notion of strict controllability and applying an index theorem, Reid roundabout theorems and comparison theorems. Shi [38] studied the eigenvalue prob-lems of discrete linear Hamiltonian systems. By constructing a self-adjoint operator on a certain admissible function space, she obtained a series of results, including a vari-ational priciple of eigenvalues. In addition, S. L. Clark and F. Gesztesy [18] studied the Wely-Titchmarsh theory for singular finite difference Hamiltonian systems with a separated boundary condition. Shi [39] established the Wely-Titchmarsh theory for discrete linear Hamiltonian systems with a singular end-point. Subsequently, Sun and Shi [45] established several strong limit point criteria for a class of singular discrete linear Hamiltonian systems.Higher-order discrete linear problems also have been investigated by some scholars besides second-order discrete Sturm-Liouville problems and discrete linear Hamiltonian systems. Zhou [50], G. Grzegorczyk and J. Werbowski [22] studied a higher-order linear difference equation in which the leading coefficient is equal to 1, and established some criteria for the oscillation of solutions. Shi and Chen [40] investigated higher-order discrete linear boundary value problems and obtained some spectral results. However, the characteristics of higher-order difference equations lead to a certain complexity. So, compared with the research of second-order difference equations and discrete Hamilto-nian systems, it is more difficult to study higher-order difference equations. Thus, there are a few references in higher-order difference equations. For more information about higher-order discrete linear problems, the reader is referred to [19, 30, 32].Continuous dependence of eigenvalues on problems has been concerned in the past years. Obviously, it is of theoretical significance. In addition, there must appear some errors in coefficients of equations and data of boundary conditions in deriving these mathematical models. So it is very important in applications. Further, it is also funda-mental from the numerical calculation perspective for eigenvalues and eigenfunctions. Based on continuous dependence of eigenvalues on problems, the codes SLEIGN in [8] and SLEIGN2 in [4-7] were designed to compute the eigenvalues and eigenfunctions of second-order continuous Sturm-Liouville problems. In 1996, Kong and A. Zettl [27] considered regular second-order continuous Sturm-Liouville problems and obtained an expression for the derivative of an eigenvalue with respect to the parameters: endpoints, boundary condition, coefficients of the equation, and the weight function. Their results show that for each fixed eigenvalue, there exists a continuous eigenvalue branch through it. In 1999, Kong, Wu, and A. Zettl [28] more deeply studied the continuous dependence of eigenvalues of regular second-order continuous Sturm-Liouville problems. They con-structed a jump set that consists of all the discontinuity points of the eigenvalues. Their results show that for a fixed k, the k-th. eigenvalueλ_k is not a continuous function of the boundary condition. More recently, Sun, Shi, and Wu [47, 48] studied second-order scalar regular discrete Sturm-Liouville problems. They proved that perturbed discrete Sturm-Liouville problems have eigenvalues near the isolated eigenvalues of the original problem. The continuous eigenvalue branches and the jump set J, consisting of all the discontinuity points of the eigenvalues, were further studied. By their results, there are some similar properties and some different properties of continuous dependence of eigenvalues between the continuous and the discrete cases. More detailed discussions are referred to [28, 48].A natural question arises: when a Sturm-Liouville problem is perturbed, how is to estimate the error of eigenvalues between the perturbed and original problems? This problem is very important since there must be some errors of data of each model as described above. However, to the best of our knowledge, there have been no results about this problem in either the continuous or discrete cases in the existing literature.We mentioned above that F. V. Atkinson [3] converted a second-order vector prob-lem with separated boundary conditions into an equivalent spectral problem of a certain Hermitian matrix. In [29], A. M. Ostrowski studied the relationship of eigenvalues be-tween two matrices and obtained the following result:Assume that A = (a_ij), B = (b_ij)∈C^n×n, and the sets of eigenvalues of A and B are denoted byσ(A)={λ_i} andσ(B)={μ_i}, respectively. Then for anyμ∈σ(B), there existsλ_i(μ)∈σ(A) such thatand there is a permutation {Ï€(1),…,Ï€(n)} of {1,…, n} such thatwhereBy the above results, we can give an error estimate of eigenvalues of perturbed second-order eigenvalue problems with separated boundary conditions. For example, consider the following second-order vector discrete Sturm-Liouville problem with a Dirichlet boundary condition:where (?) andâ–³are the backward and forward difference operators, respectively, namely, (?)x(t)=x(t)-x(t-1)andΔx(t)=x(t+1)-x(t);C(t)(t∈[0,N]),B(t),and w(t)(t∈[1,N]) are d×d Hermitian matrices, w(t) > 0 for t∈[1,N]. We can convert the above vector problem (0.1) into an equivalent spectral problem of the following iVd-order Hermitian matrix H:where Perturbations of the coefficient functions and the weight function of problem (0.1) correspond to those of the elements of matrix H.Denote that H = (h_ij), its perturbation (?) = (?),σ(H) = {λ_i}, andσ(?) = {μ_i}. Then for anyμ∈σ(?), there existsλ_i(μ)∈σ(H) such thatand there is a permutation {Ï€(1),…,Ï€(Nd)} of {1,..., Nd} such thatwhereIt follows from (0.2) that the error estimate of eigenvalues depends on both 1/(Nd) power of perturbation amplitude of matrix and the length N of the interval discussed. In addition, from (0.3) we can see that the error estimate of eigenvalues is related to both 1/(Nd) power of perturbation amplitude of matrix and N². If the perturbation amplitude is very small ((?)1) and the length N of the interval is very large, then the error estimates in (0.2) and (0.3) will be very large. Therefore, the error estimate of eigenvalues of problem (0.1), given by this method, is very crude.In this paper, we discuss eigenvalues of discrete linear boundary value problems with general boundary conditions and obtain error estimates of eigenvalues of perturbed discrete boundary value problems by employing variational priciples of eigenvalues. This dissertation consists of three chapters, which are devoted to the error estimates of eigenvalues of perturbed second-order discrete Sturm-Liouville problems, perturbed discrete linear Hamiltonian systems eigenvalue problems, and perturbed higher-order discrete vector eigenvalue problems, respectively.In our study, we need to first investigate the perturbation problem of invertible matrix. It is well known that the invertible matrix with a small perturbation is still invertible. How small is such the perturbation? In Chapter 1, we answer this question and establish an inequality about the perturbation in Section 2. Moreover, under a certain non-singularity condition, we introduce a new admissible function space and establish a new variational formula in this space. Applying this variational formula, we give error estimate of eigenvalues of perturbed problems, sufficiently close to a given second-order vector discrete Sturm-Liouville problem. As a direct consequence, continuous dependence of eigenvalues on problems is obtained from the error estimate.In addition, an example is presented to illustrate the necessity of the non-singularity condition.In Chapter 2, we study error estimate of eigenvalues of discrete linear Hamilto-nian systems with small perturbation. In particular, we discuss two special cases of perturbation. When the leading coefficient of a second-order vector difference equation is nonsingular, the equation can be converted into a discrete linear Hamiltonian system. However, it is only required that the leading coefficient of second-order difference equation is nonsingular on certain subinterval in Chapter 1. So the results obtained in Chapter 2 can not contain completely the results obtained in Chapter 1.In Chapter 3, motivated by the ideas and methods used in Chapter 1, we extend the results in Chapter 1 to 2n-order discrete vector eigenvalue problems and obtain error estimate of eigenvalues of perturbed higher-order discrete vector boundary value problems. Although the method is similar, the problems we investigate in Chapter 3 are more comlex, since they are not only higher-dimentional but also higher-order.In addition, if the leading coefficient of a 2n-order vector difference equation is non-singular, then the equation can be converted into a discrete linear Hamiltonian system in Chapter 2. But the coefficient and the weight functions of the corresponding discrete linear Hamiltonian system do not satisfy some non-singularity conditions required in Chapter 2. Furthermore, it is only required that the leading coefficient of 2n-order difference equation is nonsingular on some subintervals in Chapter 3. So the results obtained in Chapter 2 can not contain completely the results obtained in Chapter 3.