Eigenvalues of second-order linear equations with coupled boundary conditions on time scales

In this paper we consider coupled boundary value problems for second-order linear equations on time scales. By properties of eigenvalues of the Dirichlet boundary value problem and some oscillation results, existence of eigenvalues of this boundary value problem is proved, numbers of their eigenvalues are calculated, and their relationships are obtained. These results not only extend the existing ones of coupled boundary value problems for second-order difference equations but also contain more complicated time scales.     

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.     

带谱参数边界条件的四阶边值问题的矩阵表示

研究了一类具有有限谱的带有谱参数边界条件的四阶微分方程边值问题及其矩阵表示,证明了对任意正整数m,所考虑的问题至多有2m+6个特征值,进一步给出这类带有谱参数边条件的四阶边值问题与一类矩阵特征值问题之间在具有相同特征值的意义下是等价的.     

Boundary-value problem for a class of second-order parameter-dependent dynamic equations on a time scale

In this study, we consider a boundary value problem generated by a second-order dynamic equation on a time scale and boundary conditions depending on the spectral parameter. We give some properties of the solutions and obtain a formulation of the number of eigenvalues of the problem.     

Convergence of solutions and eigenelements of Steklov type boundary value problems with boundary conditions of rapidly varying type

We consider boundary value problems for the Laplace operator in a domain with boundary conditions of rapidly varying type: the Dirichlet homogeneous condition and the third (Fourier) boundary condition or a Steklov type condition. We construct the limit (homogenized) problem and prove that solutions, eigenvalues, and eigenfunctions of the original problem converge respectively to solutions, eigenvalues, and eigenfunctions of the limit problem. Bibliography: 47 titles. Illustrations: 2 figures.     

非线性V时标动态系统解的存在性与唯一性

引入V时标的概念,并对一类非线性V时标动态系统非共振边值问题,在非线性项不跨相应线性问题的特征值时,运用临界点理论,获得了问题解的存在性和唯一性定理.     

Using Time Scales to Study Multi-Interval Sturm–Liouville Problems with Interface Conditions

We consider a Sturm–Liouville problem defined on multiple intervals with interface conditions. The existence of a sequence of eigenvalues is established and the zero counts of associated eigenfunctions are determined. Moreover, we reveal the continuous and discontinuous nature of the eigenvalues on the boundary condition. The approach in this paper is different from those in the literature: We transfer the Sturm–Liouville problem with interface conditions to a Sturm–Liouville problem on a time scale without interface conditions and then apply the Sturm–Liouville theory for equations on time scales. In this way, we are able to investigate the problem in a global view. Consequently, our results cover the cases when the potential function in the equation is not strictly greater than zero and when the domain consists of an infinite number of intervals.     

The asymptotics of the eigenvalues of a fourth order differential operator with summable coefficients

A fourth order differential operator with summable coefficients and some boundary conditions is considered. Asymptotics of solutions to a fourth order differential equation is studied. The equation for eigenvalues is also studied and an asymptotics of the eigenvalues of the considered boundary value problem is obtained.     

Discrete Sturm-Liouville Problems With Parameter in the Boundary Conditions

This paper deals with discrete second order Sturm-Liouville problems in which the parameter that is part of the Sturm-Liouville difference equation also appears linearly in the boundary conditions. An appropriate Green's formula is developed for this problem, which leads to the fact that the eigenvalues are simple, and that they are real under appropriate restrictions. A boundary value problem can be expressed by a system of equations, and finding solutions to a boundary value problem is equivalent to finding the eigenvalues and eigenvectors of the coefficient matrix of a related linear system. Thus, the behavior of eigenvalues and eigenvectors is investigated using techniques in linear algebra, and a linear-algebraic proof is given that the eigenvalues are distinct under appropriate restrictions. The operator is extended to a self-adjoint operator and an expansion theorem is proved.     

Existence and Uniqueness of Solutions to Singular Higher Order Two-point BVPs on Time Scales

This paper investigates the existence and uniqueness of solutions for singular higher order boundary value problems on time scales by using mixed monotone method.The theorems obtained are very general.For the different time scale,the problem may be the corresponding continuous or discrete boundary value problem.     

时间测度上奇异高阶两点边值问题解的存在性与唯一性（英文）

This paper investigates the existence and uniqueness of solutions for singular higher order boundary value problems on time scales by using mixed monotone method.The theorems obtained are very general. For the different time scale, the problem may be the corresponding continuous or discrete boundary value problem.     

On the Uniqueness of the Solution of the Inverse Sturm–Liouville Problem with Nonseparated Boundary Conditions on a Geometric Graph

For the first time, the inverse Sturm–Liouville problem with nonseparated boundary conditions is studied on a star-shaped geometric graph with three edges. It is shown that the Sturm–Liouville problem with general boundary conditions cannot be uniquely reconstructed from four spectra. Nonseparated boundary conditions are found for which a uniqueness theorem for the solution of the inverse Sturm–Liouville problem is proved. The spectrum of the boundary value problem itself and the spectra of three auxiliary problems are used as reconstruction data. It is also shown that the Sturm–Liouville problem with these nonseparated boundary conditions can be uniquely recovered if three spectra of auxiliary problems are used as reconstruction data and only five of its eigenvalues are used instead of the entire spectrum of the problem.     

On the number of eigenvalues for parameter‐dependent diffusion problem on time scales

In this study, we consider parameter‐dependent diffusion eigenvalue problem on time scales. An upper bound on the number of eigenvalues for this problem on a finite time scale is given.     

三阶奇摄动非线性边值问题

利用微分不等式理论,研究了某一类三阶奇摄动非线性边值问题。以二阶边值问题的已知结果为基础,引入Volterra型积分算子,建立了三阶非线性边值问题的上下解方法。在适当条件下,构造出具体的上下解,得出解的存在性和渐进估计。结果表明这种技巧也为三阶奇摄动边值问题的研究提出了崭新的思路。最后举例验证文中定理的正确性。     

Symmetric positive solutions for second order boundary value problems with integral boundary conditions on time scales

This paper investigates the existence of symmetric positive solutions for a class of nonlinear boundary value problem of second order dynamic equations with integral boundary conditions on time scales. Under suitable conditions, the existence of symmetric positive solution is established by using monotone iterative technique     

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.     

On the convergence of solutions and eigenelements of a boundary value problem in a domain perforated along the boundary

We consider a boundary value problem in a model domain periodically perforated along the boundary. We assume that the homogeneous Neumann condition is posed on the external boundary and the homogeneous Dirichlet condition is posed on the boundary of the cavities. A limit (homogenized) problem is obtained. We prove the convergence of the solutions, eigenvalues, and eigenfunctions of the original problem to the solutions, eigenvalues, and eigenfunctions, respectively, of the limit problem.     

The generalized quasilinearization method and three point boundary value problems on time scales

In this paper, we study the convergence of monotone sequences of iterates for nonlinear second order dynamic equations with three point boundary conditions on time scales. We prove that it is possible to construct two sequences converging to the unique solution of the three point boundary value problem from above and below with high rate of convergence.     

高阶拟线性双曲型方程的精确边界能控性

By means of the existence and uniqueness of semi-global C^1 solution to the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigenvalues ,the local exact boundary controllability for higher order quasilinear hyperbolic equations is established.     

Nonlinear spectral problem for a singular system of ordinary differential equations with a nonlocal condition

We consider a nonlinear spectral problem for a system of ordinary differential equations defined on an unbounded half-line and supplemented with a nonlocal condition specified by a Stieltjes integral. We suggest a numerically stable method for finding the number of eigenvalues lying in a given bounded domain of the complex plane and for the computation of these eigenvalues and the corresponding eigenfunctions. Our approach uses a simpler (with uncoupled boundary conditions) auxiliary boundary value problem for the same equation.