Journal ArticleDOI
4.—A Cauchy Problem for an Ordinary Integro-differential Equation
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In this paper, an integro-differential equation (IDE) on a finite closed interval is studied, and sufficient conditions for the Cauchy problem to be solvable for arbitrary right-hand sides.Abstract:
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.read more
Citations
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Journal ArticleDOI
A nonlocal Sturm–Liouville eigenvalue problem
TL;DR: In this article, a nonlocal eigenvalue problem 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.
Journal ArticleDOI
Spectral properties of non-local differential operators
Fordyce A. Davidson,Niall Dodds +1 more
TL;DR: In this article, the spectral properties of a class of non-local operators arising from the study of nonlocal reaction-diffusion equations have been investigated and the stability of steady states of these operators is discussed.
Journal ArticleDOI
Positivity of solutions of elliptic equations with nonlocal terms
W. Allegretto,A. Barabanova +1 more
TL;DR: In this paper, the authors studied a nonlocal problem for a second-order partial differential equation which depends on a parameter n, and proved the existence of critical values 0 and 0 > such that for all ≦∆≦ and for all non-negative right-hand sides, their problem has nonnegative solutions.
Book ChapterDOI
One-dimensional Perturbations, Asymptotic Expansions, and Spectral Gaps
TL;DR: In this article, the spectral properties of the self-adjoint extension A(τ) of S can be determined via the analytic properties of Weyl function (Q-function) Qτ(z) corresponding to S and A(t), and conversely, local analogs of the Kac-Donoghue classes of Nevanlinna functions are introduced.
Journal Article
One-dimensional perturbations of selfadjoint operators with finite or discrete spectrum
TL;DR: In this paper, the authors studied the asymptotic behaviour of eigenvalues under rank one perturbations when such perturbation become infinitely large, and they used extension theory and associated Q-functions to study the properties of selfadjoint limiting functions under such large perturbs.
References
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Book
Perturbation theory for linear operators
TL;DR: The monograph by T Kato as discussed by the authors is an excellent reference work in the theory of linear operators in Banach and Hilbert spaces and is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory.
Book
Theory of Ordinary Differential Equations
TL;DR: The prerequisite for the study of this book is a knowledge of matrices and the essentials of functions of a complex variable as discussed by the authors, which is a useful text in the application of differential equations as well as for the pure mathematician.
Book
Theory of Functions of a Complex Variable
TL;DR: In this paper, the Laurent series is used for expanding functions in Taylor series, and the calculus of residues is used to expand functions in Laurent series volumes II, III, and IV.
Book
Boundary value problems of mathematical physics.
TL;DR: In this paper, the Green's function is used to describe the behavior of cylinder functions at zero and at infinity in the context of second-order differential operators, and the modified Bessel function is considered.