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.     

Error estimate of eigenvalues of discrete linear Hamiltonian systems with small perturbation

In this paper, eigenvalues of perturbed discrete linear Hamiltonian systems are considered. A new variational formula of eigenvalues is first established. Based on it, error estimates of eigenvalues of systems with small perturbation are given under certain non-singularity conditions. Small perturbations of the coefficient functions, the weight function and the coefficients of the boundary condition are all involved. As a direct consequence, continuous dependence of eigenvalues on boundary value problems is obtained under the non-singularity conditions. In addition, two examples are presented to illustrate the necessity of the non-singularity conditions and the complexity of the problem in the singularity case.     

Variational formulation of Hamiltonian boundary value problems

A variational formulation of Hamiltonian boundary value problems is given. The results are illustrated by Dirichlet problems for linear and nonlinear equations.     

Nonlinear boundary value problems with concave nonlinearities near the origin

In this paper we consider the effect of concave nonlinearities for the solution structure of nonlinear boundary value problems such as Dirichlet and Neumann boundary value problems of elliptic equations and periodic boundary value problems for Hamiltonian systems and nonlinear wave equations.     

Discrete spectrum of cranked quantum and elastic waveguides

The spectrum of quantum and elastic waveguides in the form of a cranked strip is studied. In the Dirichlet spectral problem for the Laplacian (quantum waveguide), in addition to well-known results on the existence of isolated eigenvalues for any angle α at the corner, a priori lower bounds are established for these eigenvalues. It is explained why methods developed in the scalar case are frequently inapplicable to vector problems. For an elastic isotropic waveguide with a clamped boundary, the discrete spectrum is proved to be nonempty only for small or close-to-π angles α. The asymptotics of some eigenvalues are constructed. Elastic waveguides of other shapes 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.     

Spectral Bounds for Tricomi Problems and Application to Semilinear Existence and Existence with Uniqueness Results

For the linear Tricomi problem, it is shown that real eigenvalues corresponding to generalized eigenfunctions must be positive and that the energy integral methods used to prove solvability results can give lower bounds on the spectrum. Exploiting the linear solvability theory and spectral information, standard nonlinear analysis tools are employed to yield results on existence and uniqueness for semilinear problems. In particular, using the Leray-Schauder principle, existence of generalized solutions with sublinear nonlinearities is established. For sublinear or asymptotically linear nonlinearities that satisfy a Lipschitz condition, the contraction mapping principle is employed to give results on existence with uniqueness. The Lipschitz constant depends on lower bounds for the spectrum of the linear problem. For certain superlinear problems, maximum principles for the linear problem are used via the method of upper and lower solutions to give results on existence.     

Spectral theory of discrete linear Hamiltonian systems

This paper is concerned with spectral problems for a class of discrete linear Hamiltonian systems with self-adjoint boundary conditions, where the existence and uniqueness of solutions of initial value problems may not hold. A suitable admissible function space and a difference operator are constructed so that the operator is self-adjoint in the space. Then a series of spectral results are obtained: the reality of eigenvalues, the completeness of the orthogonal normalized eigenfunction system, Rayleigh's principle, the minimax theorem and the dual orthogonality. Especially, the number of eigenvalues including multiplicities and the number of linearly independent eigenfunctions are calculated.     

Parameter-uniform numerical methods for some linear and nonlinear singularly perturbed convection diffusion boundary turning point problems

Both linear and nonlinear singularly perturbed two point boundary value problems are examined in this paper. In both cases, the problems have a boundary turning point and are of convection-diffusion type. Parameter-uniform numerical methods composed of monotone finite difference operators and piecewise-uniform Shishkin meshes, are constructed and analyzed for both the linear and the nonlinear class of problems. Numerical results are presented to illustrate the theoretical parameter-uniform error bounds established.     

Numerical error bounds for fourth order boundary value problems,simultaneous estimation ofu(x) andu″(x)

Summary For certain nonlinear two-point boundary value problems of the fourth order an estimation theory is developed which yields simultaneous estimates of the solution and its second derivative. Methods for computing numerical error bounds for approximate solutions are described and tested. The theory provides also uniqueness and existence statements. The results can be applied to many problems for which a corresponding theory on two-sided bounds is not suitable.     

Rigorous convex underestimators for general twice-differentiable problems

In order to generate valid convex lower bounding problems for nonconvex twice-differentiable optimization problems, a method that is based on second-order information of general twice-differentiable functions is presented. Using interval Hessian matrices, valid lower bounds on the eigenvalues of such functions are obtained and used in constructing convex underestimators. By solving several nonlinear example problems, it is shown that the lower bounds are sufficiently tight to ensure satisfactory convergence of the BB, a branch and bound algorithm which relies on this underestimation procedure [3].     

Dual extremum principles and error bounds for a class of boundary value problems

Error bounds for a wide class of linear and nonlinear boundary value problems are derived from the theory of dual extremum principles. The results are illustrated by two examples arising in the theory of heat transfer, which involve mixed boundary conditions.     

Positive solutions of a nonlinear impulsive equation with periodic boundary conditions

In this paper, we investigate nonlinear second order differential equations subject to linear impulse conditions and periodic boundary conditions. Sign properties of an associated Green’s function are exploited to get existence results for positive solutions of the nonlinear boundary value problem with impulse. Upper and lower bounds for positive solutions are also given. The results obtained yield periodic positive solutions of the corresponding periodic impulsive nonlinear differential equation on the whole real axis.     

On the Nonlinear Schrödinger Equation on the Half Line with Homogeneous Robin Boundary Conditions

Boundary value problems for the nonlinear Schrödinger equations on the half line with homogeneous Robin boundary conditions are revisited using Bäcklund transformations. In particular: relations are obtained among the norming constants associated with symmetric eigenvalues; a linearizing transformation is derived for the Bäcklund transformation; the reflection‐induced soliton position shift is evaluated and the solution behavior is discussed. The results are illustrated by discussing several exact soliton solutions, which describe the soliton reflection at the boundary with or without the presence of self‐symmetric eigenvalues.     

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.     

Eigenvalues of second-order difference equations with coupled boundary conditions

This paper is concerned with coupled boundary value problems for self-adjoint second-order difference equations. Existence of eigenvalues is proved, numbers of eigenvalues are calculated, and relationships between the eigenvalues of a self-adjoint second-order difference equation with three different coupled boundary conditions are established. These results extend the relevant existing results of periodic and antiperiodic boundary value problems.     

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.     

On positive solutions for a nonlinear boundary value problem with impulse

In this paper we study nonlinear second order differential equations subject to separated linear boundary conditions and to linear impulse conditions. Sign properties of an associated Green’s function are investigated and existence results for positive solutions of the nonlinear boundary value problem with impulse are established. Upper and lower bounds for positive solutions are also given.     

Geometrical formulation of Hamiltonian boundary value problems

A geometrical formulation of nonlinear Hamiltonian boundary value problems is presented. It involves a distance geometry which is a generalization of hypercircle geometry for linear boundary value problems. The connection with dual extremum principles is also exhibited.     

Error bounds via a new extremum variational principle,mean square residual and weighted mean square residual

Error bounds for a wide class of nonlinear one-dimensional boundary value problems are derived from a new extremum variational principle. A new least-squares approximate technique, based on a weighted mean square residual, is established. Also, the value of the weighted mean square residual and value of the classical mean square residual are used for error estimate. The results are illustrated by four examples.