Coerciveness Estimate for Ventcel Boundary Value Problem for a Differential Equation

We give some simple algebraic conditions on the coefficients of a boundary value problem for a differential equations of Ventcel type, depending on a spectral parameter, which guarantee the existence, uniqueness of a solution and coerciveness estimate, for a spectral parameter lying in some sector.     

Nonlinear spectral problem for a self-adjoint vector differential equation

We consider a spectral problem that is nonlinear in the spectral parameter for a self-adjoint vector differential equation of order 2n. The boundary conditions depend on the spectral parameter and are self-adjoint as well. Under some conditions of monotonicity of the input data with respect to the spectral parameter, we present a method for counting the eigenvalues of the problem in a given interval. If the boundary conditions are independent of the spectral parameter, then we define the notion of number of an eigenvalue and give a method for computing this number as well as the set of numbers of all eigenvalues in a given interval. For an equation considered on an unbounded interval, under some additional assumptions, we present a method for approximating the original singular problem by a problem on a finite interval.     

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 general nonlinear self-adjoint spectral problem for differential-algebraic systems

We consider a general self-adjoint spectral problem, nonlinear with respect to the spectral parameter, for linear differential-algebraic systems of equations. Under some assumptions, we present a method for reducing such a problem to a general self-adjoint nonlinear spectral problem for a system of differential equations. In turn, this permits one to pass to a problem for a Hamiltonian system of ordinary differential equations. In particular, in this way, one can obtain a method for computing the number of eigenvalues of the original problem lying in a given range of the spectral parameter.     

Integral equations of a two-dimensional lightguide

An integral equation describing oscillations of a two-dimensional periodic lightguide is studied. The integral operator is an analytic function of a spectral parameter. The homogenous integral equation has only the zero solution for all values of the spectral parameter expect for some isolated values. Bibliography: 9 titles. Translated fromProblemy Matematicheskogo Analiza, No. 18, 1998, pp. 78–84.     

Comparison theorems for self‐adjoint linear Hamiltonian eigenvalue problems

In this work we derive new comparison results for (finite) eigenvalues of two self‐adjoint linear Hamiltonian eigenvalue problems. The coefficient matrices depend on the spectral parameter nonlinearly and the spectral parameter is present also in the boundary conditions. We do not impose any controllability or strict normality assumptions. Our method is based on a generalization of the Sturmian comparison theorem for such systems. The results are new even for the Dirichlet boundary conditions, for linear Hamiltonian systems depending linearly on the spectral parameter, and for Sturm–Liouville eigenvalue problems with nonlinear dependence on the spectral parameter.     

On optimality of two adaptive choices for the parameter of Dai–Liao method

Orthonormal matrices are a class of well-conditioned matrices with the least spectral condition number. Here, at first it is shown that a recently proposed choice for parameter of the Dai–Liao nonlinear conjugate gradient method makes the search direction matrix as close as possible to an orthonormal matrix in the Frobenius norm. Then, conducting a brief singular value analysis, it is shown that another recently proposed choice for the Dai–Liao parameter improves spectral condition number of the search direction matrix. Thus, theoretical justifications of the two choices for the Dai–Liao parameter are enhanced. Finally, some comparative numerical results are reported.     

Uniqueness Theorems for Sturm–Liouville Operators with Boundary Conditions Polynomially Dependent on the Eigenparameter from Spectral Data

In this paper, we discuss the inverse problems for Sturm–Liouville operators with boundary conditions polynomially dependent on the spectral parameter. We establish some uniqueness theorems on the potential q(x) for the half inverse problem and the interior inverse problem from spectral data, respectively.     

On Some Integral Equations in Hilbert Space with an Application to the Theory of Elasticity

A Volterra type integral equation in a Hilbert space with an additional linear operator L and a spectral parameter depending on time is considered. If the parameter does not belong to the spectrum of L unconditional solvability of the considered problem is proved. In the case where the initial value of the parameter coincides with some isolated point of the spectrum of the operator L sufficient conditions for solvability are established. The obtained results are applied to the partial integral equations associated with a contact problem of the theory of elasticity.     

Acceleration of Convergence of Static and Dynamic Iterations

A new technique for acceleration of convergence of static and dynamic iterations for systems of linear equations and systems of linear differential equations is proposed. This technique is based on splitting the matrix of the system in such a way that the resulting iteration matrix has a minimal spectral radius for linear systems and a minimal spectral radius for some value of a parameter in Laplace transform domain for linear differential systems.This revised version was published online in October 2005 with corrections to the Cover Date.     

Estimates for a spectral parameter in elliptic boundary value problems with discontinuous nonlinearities

Under study are the two classes of elliptic spectral problems with homogeneous Dirichlet conditions and discontinuous nonlinearities (the parameter occurs in the nonlinearity multiplicatively). In the former case the nonlinearity is nonnegative and vanishes for the values of the phase variable not exceeding some positive number c; it has linear growth at infinity in the phase variable u and the only discontinuity at u = c. We prove that for every spectral parameter greater than the minimal eigenvalue of the differential part of the equation with the homogeneous Dirichlet condition, the corresponding boundary value problem has a nontrivial strong solution. The corresponding free boundary in this case is of zero measure. A lower estimate for the spectral parameter is established as well. In the latter case the differential part of the equation is formally selfadjoint and the nonlinearity has sublinear growth at infinity. Some upper estimate for the spectral parameter is given in this case.     

Inverse problems for the boundary value problem with the interior nodal subsets

In this paper, inverse nodal problems for Sturm–Liouville equations with boundary conditions polynomially dependent on the spectral parameter were studied. The authors showed that some uniqueness theorems on the potential function hold by the Weyl function, respectively.     

The band spectrum of the periodic Airy–Schrödinger operator on the real line

We introduce the periodic Airy–Schrödinger operator and we describe its band spectrum. This is an example of solvable model with a periodic potential which is not differentiable at its extrema. We prove that there exists a sequence of explicit constants giving upper bounds of the semiclassical parameter for which explicit estimates are valid. We completely determine the behaviour of the edges of the first spectral band with respect to the semiclassical parameter. Then, we investigate the spectral bands and gaps situated in the range of the potential. We prove precise estimates on the widths of these spectral bands and these spectral gaps and we determine an upper bound on the integrated spectral density in this range. Finally, we get estimates of the edges of spectral bands and thus of the widths of spectral bands and spectral gaps which are stated for values of the semiclassical parameter in fixed intervals.     

Finding Root Spaces for a Linear Algebraic Spectral Problem

Computational Mathematics and Mathematical Physics - For an algebraic spectral problem that is linear with respect to the spectral parameter, some numerical methods are considered to find the root...     

Two-fold completeness of root vectors of a system of quadratic pencils

We consider a spectral problem for a system of second order (in the spectral parameter) abstract pencils in a Hilbert space and prove the completeness and the Abel basis property of a system of eigenvectors and associated vectors. In some special cases, we obtain the expansion of vectors with respect to eigenvectors. Further, it is considered a relevant application of these abstract results to boundary-value problems for second and fourth order ordinary differential equations with a quadratic spectral parameter both in the equation and in boundary-value conditions.     

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.     

On a spectral problem for the bessel equation of zero order

In the present paper, for part of the eigenfunction system corresponding to a spectral problem with spectral parameter in the boundary condition, we write out a spectral problem for the same differential equation with a nonlocal boundary condition that does not contain the spectral parameter. In addition, we construct the biorthogonal system.     

STURM-LIOUVILLE PROBLEMS WITH EIGENDEPENDENT BOUNDARY AND TRANSMISSIONS CONDITIONS

1IntroductionIt is well-known that the Sturmian Theory is an important aid in solving many problemsin mathematical physics.Therefore this theory is one of the most actual and extensivelydeveloped field in spectral analysis of boundary-value problems of St…     

An Improved Spectral Conjugate Gradient Algorithm for Nonconvex Unconstrained Optimization Problems

In this paper, an improved spectral conjugate gradient algorithm is developed for solving nonconvex unconstrained optimization problems. Different from the existent methods, the spectral and conjugate parameters are chosen such that the obtained search direction is always sufficiently descent as well as being close to the quasi-Newton direction. With these suitable choices, the additional assumption in the method proposed by Andrei on the boundedness of the spectral parameter is removed. Under some mild conditions, global convergence is established. Numerical experiments are employed to demonstrate the efficiency of the algorithm for solving large-scale benchmark test problems, particularly in comparison with the existent state-of-the-art algorithms available in the literature.     

Spectral properties of a fourth-order differential operator with integrable coefficients

The aim of this paper is to study spectral properties of differential operators with integrable coefficients and a constant weight function. We analyze the asymptotic behavior of solutions to a differential equation with integrable coefficients for large values of the spectral parameter. To find the asymptotic behavior of solutions, we reduce the differential equation to a Volterra integral equation. We also obtain asymptotic formulas for the eigenvalues of some boundary value problems related to the differential operator under consideration.