Nonlinear discrete Sturm-Liouville problems

In this paper we study nonlinear boundary value problems of the form

     

Singular left-definite Sturm-Liouville problems

We study singular left-definite Sturm-Liouville problems with an indefinite weight function. The existence of eigenvalues is established based on the existence of eigenvalues of corresponding right-definite problems. Furthermore, for each singular left-definite problem with limit-circle non-oscillatory endpoints we construct a regular left-definite problem with the same eigenvalues and use it to obtain properties of eigenvalues and eigenfunctions. Inequalities among eigenvalues recently established for regular left-definite problems are extended to the singular case.     

Nonuniformly definite multiparameter spectral problems

We establish theorems about the expansion of nonuniformly right-definite multiparameter problems in generalized eigenfunctions. We also give an abstract criterion of existence of commuting self-adjoint extensions of a family of symmetric operators.To Yurii Makarovich BerezanskiiTranslated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 5, pp. 659–670, May, 1995.This research was supported by the Ukrainian State Committee on Science and Technology.     

Discretization of multiparameter eigenvalue problems

Summary Although multiparameter eigenvalue problems, as for example Mathieu's differential equation, have been known for a long time, so far no work has been done on the numerical treatment of these problems. So in this paper we extend the spectral theory for one parameter (cf. [7, II, VII]) to multiparameter eigenvalue problmes, formulate in the framework of discrete approximation a convergent numerical treatment, establish algebraic bifurcation equations for the intersection points of the eigenvalue curves and illustrate this with some numerical examples.     

On regularization in multiparameter eigenvalue problems

We consider the computation of simple and multiple eigenvalues and the corresponding eigenelements of a multiparameter eigenvalue problem in the sense of Atkinson by the reduction pseudoperturbation method. Namely, we regularize the original operator functions by finite-dimensional linear operators, thus reducing the case of multiple eigenvalues to that of simple ones.     

Eigenvector expansion of multiparameter spectral problems

Expansion theorems for abstract multiparameter spectral problems are proved. The theorems include results for general operators with continuous spectrum and elliptic differential operators.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 7, pp. 901–913, July, 1992.     

Comparison cones for multiparameter eigenvalue problems

We consider the multiparameter eigenvalue problem (T_r + ∑_{s = 1}^kλ_sV_rs) x_r = 0, x_r ≠ 0, 1 ? r ? k, where T_r and V_rs are self-adjoint linear operators on Hilbert spaces H_r, the V_rs being bounded. The problem may be posed in either ⊕_{r = 1}^kH_r or ⊕_{r = 1}^kH_r and we develop variational approaches for both settings. We explore the rôles played in both settings by $\text{C ={λ ∈ R}{}^{\mathrm{k}}\text{|}\text{∑}\text{k}\text{s=1}\text{λ}{}_{\mathrm{s}}\text{(V}{}^{\mathrm{rs}}\text{x}{}_{\mathrm{r}}\text{,x}{}_{\mathrm{r}}\text{? 0}$ for some nonozero and related cones in $\text{R}$^k. We also compare certain geometrical conditions on C with analytical definiteness conditions already in the literature.     

Eigenfunction expansion for symmetric multiparameter problems

An abstract theorem on the expansion in generalized eigenvectors of multiparameter problems for Hermitian operators is established. The applications to differential operators are considered.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 8, pp. 1166–1169, August, 1993.     

Banach空间中非线性Sturm-Liouville问题的解

利用M(o)nch不动点定理,在更广泛的的条件下,证明了Banach空间中Sturm-Liouville问题解的存在性.     

一类Sturm-Liouville奇异边值的全局结构

本文运用正则锥上的增算子的不动点的存在性的结果,讨论了一般非线性Sturm-Liouville奇异边值问题,得出了有关解的存在性的新的结论.     

Asymptotic Analysis of Forced Nonlinear Sturm-Liouville Systems

A generalized WKB method is used to construct formal asymptotic approximations of solutions of certain forced nonlinear Sturm-Liouville systems. By means of three connected expansions it is possible to obtain a fairly complete picture of the global behavior of the small-norm solution branches. Results are presented for both slowly varying and rapidly varying forcing functions.     

Bernstein approximations of nonlinear Sturm-Liouville problems

In this paper we deal with Sturm-Liouville boundary value problems

(∗)     

Left-definite spaces of singular Sturm-Liouville problems

The left-definite Hilbert spaces for singular Sturm-Liouville problems of the limit-circle type are explicitly constructed. The construction only uses an explicit form of the left-definite boundary conditions, together with a principle solution and a non-principle solution of the differential equation involved.     

Half-range solutions of indefinite Sturm-Liouville problems

For diffusion equations of indefinite Sturm-Liouville type, we develop two equivalent methods of constructing explicit representations for the solutions. The first method is based on an eigenfunction expansion whereas the second uses a Wiener-Hopf-type integral equation and factorization. Some illustrative examples are worked out.     

Regular approximations of singular Sturm-Liouville problems

Given any self-adjoint realization S of a singular Sturm-Liouville (S-L) problem, it is possible to construct a sequence {S_r{ of regular S-L problems with the properties (i) every point of the spectrum of S is the limit of a sequence of eigenvalues from the spectrum of the individual members of {S_r{ (ii) in the case when 5 is regular or limit-circle at each endpoint, a convergent sequence of eigenvalues from the individual members of {S_r{ has to converge to an eigenvalue of S (iii) in the general case when S is bounded below, property (ii) holds for all eigenvalues below the essential spectrum of S.