Application of self-adjoint boundary value problems to investigation of stability of periodic delay systems

The problem on determining conditions for the asymptotic stability of linear periodic delay systems is considered. Solving this problem, we use the function space of states. Conditions for the asymptotic stability are determined in terms of the spectrum of the monodromy operator. To find the spectrum, we construct a special boundary value problem for ordinary differential equations. The motion of eigenvalues of this problem is studied as the parameter changes. Conditions of the stability of the linear periodic delay system change when an eigenvalue of the boundary value problem intersects the circumference of the unit disk. We assume that, at this moment, the boundary value problem is self-adjoint. Sufficient coefficient conditions for the asymptotic stability of linear periodic delay systems are given.     

Hyperbolicity of periodic solutions of functional differential equations with several delays

We study conditions for the hyperbolicity of periodic solutions to nonlinear functional differential equations in terms of the eigenvalues of the monodromy operator. The eigenvalue problem for the monodromy operator is reduced to a boundary value problem for a system of ordinary differential equations with a spectral parameter. This makes it possible to construct a characteristic function. We prove that the zeros of this function coincide with the eigenvalues of the monodromy operator and, under certain additional conditions, the multiplicity of a zero of the characteristic function coincides with the algebraic multiplicity of the corresponding eigenvalue.     

On the singular nonlinear self-adjoint eigenvalue problem for differential algebraic systems of equations

The general nonlinear self-adjoint eigenvalue problem for a differential algebraic system of equations on a half-line is examined. The boundary conditions are chosen so that the solution to this system is bounded at infinity. Under certain assumptions, the original problem can be reduced to a self-adjoint system of differential equations. After certain transformations, this system, combined with the boundary conditions, forms a nonlinear self-adjoint eigenvalue problem. Requirements for the appropriate boundary conditions are clarified. Under the additional assumption that the initial data are monotone functions of the spectral parameter, a method is proposed for calculating the number of eigenvalues of the original problem that lie on a prescribed interval of this parameter.     

Irregular Boundary Value Problems with Discontinuous Coefficients and the Eigenvalue Parameter

In this study, a Birkhoff-irregular boundary value problem for linear ordinary differential equations of the second order with discontinuous coefficients and the spectral parameter has been considered. Therefore, at the discontinuous point, two additional boundary conditions (called transmission conditions) have been added to the boundary conditions. The eigenvalue parameter is of the second degree in the differential equation and of the first degree in a boundary condition. The equation contains an abstract linear operator which is (usually) unbounded in the space Lq(−1, 1). Isomorphism and coerciveness with defects 1 and 2 are proved for this problem. The case of the biharmonic equation is also studied.     

Multiple bifurcations in a delayed predator–prey diffusion system with a functional response

The present paper is concerned with a delayed predator–prey diffusion system with a Beddington–DeAngelis functional response and homogeneous Neumann boundary conditions. If the positive constant steady state of the corresponding system without delay is stable, by choosing the delay as the bifurcation parameter, we can show that the increase of the delay can not only cause spatially homogeneous Hopf bifurcation at the positive constant steady state but also give rise to spatially heterogeneous ones. In particular, under appropriate conditions, we find that the system has a Bogdanov–Takens singularity at the positive constant steady state, whereas this singularity does not occur for the corresponding system without diffusion. In addition, by applying the normal form theory and center manifold theorem for partial functional differential equations, we give normal forms of Hopf bifurcation and Bogdanov–Takens bifurcation and the explicit formula for determining the properties of spatial Hopf bifurcations.     

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.     

Indefinite Boundary Eigenvalue Problems in a Pontrjagin Space Setting

We study eigenvalue problems Fy = λ Gy consisting of Hamiltonian systems of ordinary differential equations on a compact interval with symmetric λ-linear boundary conditions. The problems we are interested in are non-definite: neither left-nor right-definite. Instead of this, we give some weak condition on one coefficient of the Hamiltonian system which ensures that a hermitian form associated with the operator F has at most finitely many negative squares. This enables us to study the problem by the help of a compact self-adjoint operator in a Pontrjagin space and we obtain as a main result uniformly convergent eigenfunction expansions. In the final section, applications to formally self-adjoint differential equations of higher order are given.     

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.     

Existence of positive solutions for a class of higher-order nonlinear fractional differential equations with integral boundary conditions and a parameter

In this paper, we study the existence of positive solutions for a class of higher-order nonlinear fractional differential equations with integral boundary conditions and a parameter. By using the properties of the Green’s function, u ₀-positive function and the fixed point index theory, we obtain some existence results of positive solution under some conditions concerning the first eigenvalue with respect to the relevant linear operator. The method of this paper is a unified method for establishing the existence of multiple positive solutions for a large number of nonlinear differential equations of arbitrary order with any allowed number of non-local boundary conditions.     

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.     

Nonlinear Eigenvalue Problems in the Stability Analysis of Morphogen Gradients

This paper is concerned with several eigenvalue problems in the linear stability analysis of steady state morphogen gradients for several models of Drosophila wing imaginal discs including one not previously considered. These problems share several common difficulties including the following: (a) The steady state solution which appears in the coefficients of the relevant differential equations of the stability analysis is only known qualitatively and numerically. (b) Though the governing differential equations are linear, the eigenvalue parameter appears nonlinearly after reduction to a problem for one unknown. (c) The eigenvalues are determined not only as solutions of a homogeneous boundary value problem with homogeneous Dirichlet boundary conditions, but also by an alternative auxiliary condition to one of the Dirichlet conditions allowed by a boundary condition of the original problem. Regarding the stability of the steady state morphogen gradients, we prove that the eigenvalues must all be positive and hence the steady state morphogen gradients are asymptotically stable. The other principal finding is a novel result pertaining to the smallest (positive) eigenvalue that determines the slowest decay rate of transients and the time needed to reach steady state. Here we prove that the smallest eigenvalue does not come from the nonlinear Dirichlet eigenvalue problem but from the complementary auxiliary condition requiring only to find the smallest zero of a rational function. Keeping in mind that even the steady state solution needed for the stability analysis is only known numerically, not having to solve the nonlinear Dirichlet eigenvalue problem is both an attractive theoretical outcome and a significant computational simplification.     

Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary conditions

By taking as a “prototype problem” a one-delay linear autonomous system of delay differential equations we present the problem of computing the characteristic roots of a retarded functional differential equation as an eigenvalue problem for a derivative operator with non-local boundary conditions given by the particular system considered. This theory can be enlarged to more general classes of functional equations such as neutral delay equations, age-structured population models and mixed-type functional differential equations.It is thus relevant to have a numerical technique to approximate the eigenvalues of derivative operators under non-local boundary conditions. In this paper we propose to discretize such operators by pseudospectral techniques and turn the original eigenvalue problem into a matrix eigenvalue problem. This approach is shown to be particularly efficient due to the well-known “spectral accuracy” convergence of pseudospectral methods. Numerical examples are given.     

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.     

Spectra of discrete symplectic eigenvalue problems with separated boundary conditions

In this paper we consider a discrete symplectic eigenvalue problem with separated boundary conditions and obtain formulas for the number of eigenvalues on a given interval of the variation of the spectral parameter. In addition, we compare the spectra of two symplectic eigenvalue problems with different separated boundary conditions.     

The Fourier-finite element method for the Poisson problem on a non-convex polyhedral cylinder

We study the Poisson problem with zero boundary datum in a (finite) polyhedral cylinder with a non-convex edge. Applying the Fourier sine series to the equation along the edge and by a corner singularity expansion for the Poisson problem with parameter, we define the edge flux coefficient and the regular part of the solution on the polyhedral cylinder. We present a numerical method for approximating the edge flux coefficient and the regular part and show the stability. We derive an error estimate and give some numerical experiments.     

Sampling Integrodifferential Transforms Arising from Second Order Differential Operators

We give sampling theorems associated with boundary value problems whose differential equations are of the form M(y) = λS(y), where M and S are differential expressions of the second and first order respectively and the eigenvalue parameter may appear in the boundary conditions. The class of the sampled functions is not a class of integral transforms as is the case in the classical sampling theory, but it is a class of integrodifferential transforms. We use solutions of the problem as well as Green's function to derive two sampling theorems.     

Boundary value problems for higher order linear impulsive differential equations

In this paper higher order linear impulsive differential equations with fixed moments of impulses subject to linear boundary conditions are studied. Green's formula is defined for piecewise differentiable functions. Properties of Green's functions for higher order impulsive boundary value problems are introduced. An appropriate example of the Green's function for a boundary value problem is provided. Furthermore, eigenvalue problems and basic properties of eigensolutions are considered.     

An Optimal Control Method for Nonlinear Inverse Diffusion Coefficient Problem

This paper investigates the solution of a parameter identification problem associated with the two-dimensional heat equation with variable diffusion coefficient. The singularity of the diffusion coefficient results in a nonlinear inverse problem which makes theoretical analysis rather difficult. Using an optimal control method, we formulate the problem as a minimization problem and prove the existence and uniqueness of the solution in weighted Sobolev spaces. The necessary conditions for the existence of the minimizer are also given. The results can be extended to more general parabolic equations with singular coefficients.     

THE SINGULAR SECOND ORDER NONLINEAR EIGENVALUE PROBLEM WITH INFINITELY MANY POSITIVE SOLUTIONS

1 IntroductionIn this paper we consider positive solutions for second order non1ineareigenVaue problem of the fOrmwhere A > 0, a,P,7,b 2 0 and p = 7P a7 6cr > 0.Positive solutions of nonlinear eigenvalue problem such as (P) have beenstudied extensiveIy by many authors (see tll-[61 and the references therein).However, to my best knowledge, less attention has been given to infinitelymany positive solutions. It must be pointed out that in all existing literaturesconcerning multiple positiv…     

Two-Dimensional Trace-Normed Canonical Systems of Differential Equations and Selfadjoint Interface Conditions

The class of two-dimensional trace-normed canonical systems of differential equations on is considered with selfadjoint interface conditions at 0. If one or both of the intervals around 0 are H-indivisible the interface conditions which give rise to selfadjoint relations (multi-valued operators) are determined. It is shown that the corresponding generalized Fourier transforms are partially isometric. Compression to the halfline (0, ) results in boundary conditions which depend on the eigenvalue parameter involving a fractional linear transform of the Titchmarsh-Weyl coefficient of the halfline (–, 0). The corresponding generalized Fourier transforms are isometric except possibly on a one-dimensional subspace where they are contractive.