A nonlocal eigenvalue problem of the form u ″ + a ( x ) u + Bu = λu with homogeneous Dirichlet boundary conditions is considered, where B is a rank-one bounded linear operator and x belongs to some bounded interval on the real line. The behaviour of the eigenvalues is studied using methods of linear perturbation theory. In particular, some results are given which ensure that the spectrum remains real. A Sturm-type comparison result is obtained. Finally, these results are applied to the study of some nonlocal reaction–diffusion equations.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0308210500029279

A nonlocal Sturm–Liouville eigenvalue problem

In this article we consider the spectral properties of a class of non-local operators that arise from the study of non-local reaction-diffusion equations. Such equations are used to model a variety of physical and biological systems with examples ranging from Ohmic heating to population dynamics. The operators studied here are bounded perturbations of linear (local) differential operators. The non-local perturbation is in the form of an integral term. It is shown here that the spectral properties of these non-local operators can differ considerably from those of their local counterpart. Multiplicities of eigenvalues are studied and new oscillation results for the associated eigenfunctions are presented. These results highlight problems with certain similar results and provide an alternative formulation. Finally, the stability of steady states of associated non-local reaction-diffusion equations is discussed.

Spectral properties of non-local differential operators

In this paper we study a nonlocal problem for a second-order partial differential equation which depends on a parameter n . We prove the existence of critical values 0 and 0 > such that for all ≦ɳ≦ and for all non-negative right-hand sides, our problem has nonnegative solutions. We obtain a formula for ɳ 0 , the maximal possible value of , and find the exact value of ɳ for spherical ɳ. We also study the corresponding eigenvalue problem. At the end of the paper, we consider the application of our results to the nonlinear system describing the distribution of temperature and potential in a microsensor.

Positivity of solutions of elliptic equations with nonlocal terms

Let S be a closed symmetric operator or relation with defect numbers (1, 1) and let A be a self-adjoint extension of S. The self-adjoint extensions A(τ), τ ∈ ℝ ∪ {∞}, of S, when parametrized by means of Kreĭn’s formula, can be seen as one-dimensional (graph) perturbations of A. The spectral properties of the self-adjoint extension A(τ) of (the completely nonself-adjoint part of) S can be determined via the analytic properties of the Weyl function (Q-function) Qτ(z) corresponding to S and A(t), and conversely. In order to study the limiting properties of these functions at spectral points, local analogs of the Kac-Donoghue classes of Nevanlinna functions are introduced, giving rise to asymptotic expansions at real points. In the case where the self-adjoint extension A has a (maximal) gap in its spectrum, all the perturbations A(τ) have the same gap in their spectrum with the possible exception of an isolated eigenvalue λ(τ), τ ∈ ℝ ∪{∞}. By means of the Weyl function Q τ(z) the (analytic) properties of this eigenvalue are established.

One-dimensional Perturbations, Asymptotic Expansions, and Spectral Gaps

There are many classical results concerning the spectrum of perturbations of selfadjoint operators; in particular the behaviour under "small perturbations" has been investigated throughoutly. In the present paper attention is paid to the asymptotic behaviour of eigenvalues under rank one perturbations when such perturbations become infinitely large. This leads in a natural way to the study of the eigenvalues of a selfadjoint limiting perturbation, which is necessarily a selfadjoint relation (multivalued linear operator). The use of extension theory and associated Q-functions facilitates the study of the asymptotic properties under such "large perturbations" and leads to similar interpretations as are known in the case of "small perturbations".

One-dimensional perturbations of selfadjoint operators with finite or discrete spectrum

In this paper we study an ordinary second-order integro-differential equation (IDE) on a finite closed interval. We demonstrate the equivalence of this equation to a certain integral equation, and deduce that the homogeneous IDE may have either 2 or 3 linearly independent solutions, depending on the value of a parameter λ. We study a Cauchy problem for the IDE, both by this integral equation approach and by an independent approach, based on the perturbation theory for linear operators. We give necessary and sufficient conditions for the Cauchy problem to be solvable for arbitrary right-hand sides—these conditions again depend on λ—and specify the behaviour of the IDE when these conditions are not satisfied. At the end of the paper some examples are given of the type of behaviour described.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0080454100009390

E. A. Catchpole

Papers

4.—A Cauchy Problem for an Ordinary Integro-differential Equation