A nonlocal Sturm–Liouville eigenvalue problem

Abstract

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.

Spectral analysis of a class of nonlocal elliptic operators related to Brownian motion with random jumps

“…These papers study the location and multiplicities of the eigenvalues and the existence of a principal eigenvalue (this last point is automatic in our situation). See, for example, [4], [5], [3] and the references therein for results and applications. See also Remark 3 after Theorem 3 and Remark 1 after Theorem 4.…”

Existence of Positive Solutions due to Non-local Interactions in a Class of Nonlinear Boundary Value Problems

“…The spectral properties of this non-local operator have been extensively studied in certain cases: see [4,5,14,20] for the case n = 1; [23] for n ≥ 3, and; [1,15,16,24] for general n. It is shown in these papers that the eigenvalues, corresponding eigenfunctions and other spectral properties of the operator L(ǫ) are precisely those of its closed extensionL(ǫ) and hence in the following, we simply refer to the operator L(ǫ) as defined in (5).…”

“…an eigenvalue for which corresponding eigenfunctions can be chosen to be positive. See [14,15,16] for spectral results most relevant to the context here and also [20,24] for the general background to the spectral theory of these operators. Note that such results are not restricted to small ǫ.…”

Large amplitude solutions of spatially non-homogeneous non-local reaction diffusion equations

“…Introduction. Non-local reaction diffusion equations arise in a variety of physical applications [4,6,11,18,19]. In the past few years, several analytic and numeric techniques have been developed for non-local equations to help determine the possible types of solutions and their ensuing stability properties [1,2,4,6,7,[9][10][11][12][13][18][19][20][21].…”

“…One is to find a spatially homogeneous solution and determine whether any solutions bifurcate from it [4,9,10,20]. The second approach is to find a solution to a related local equation, and determine what impact the non-local term has on it [6,7,[10][11][12][13]15,21]. A crucial observation that has been noted by various researchers is that non-local equations possess a number of asymptotically stable solutions that local equations do not.…”