On the Persistence of the Eigenvalues of a Perturbed Fredholm Operator of Index Zero under Nonsmooth Perturbations

We consider a second-order algebro-differential equation. Equations and systems of second-order differential equations describe the operation of an electronic triode circuit with feedback, rotation of a rigid body with a cavity, reading information from a disk, etc. The highest derivative is preceded by an irreversible operator. This is a Fredholm operator with index zero, kernel of arbitrary dimension, and Jordan chains of arbitrary lengths. Equations with irreversible operators at the highest derivative are called algebro-differential. In this case, the solution to the problem exists under certain conditions on the components of the desired function. To solve the equation with respect to the derivative, the method of cascade splitting of the equation is used, which consists in the stepwise splitting of the equation into equations in subspaces of decreasing dimensions. Cases of one-step and two-step splitting are considered. The splitting uses the result on the solution of a linear equation with Fredholm operator. In each case, the corresponding result is formulated as a theorem. To illustrate the result obtained in the case of one-step splitting, an illustrative example of the Cauchy problem is given.

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Topological Persistence of the Unit Eigenvectors of a Perturbed Fredholm Operator of Index Zero

Zeitschrift für Analysis und ihre Anwendungen ◽

10.4171/zaa/1516 ◽

2014 ◽

Vol 33 (3) ◽

pp. 347-367 ◽

Cited By ~ 4

Author(s):

Raffaele Chiappinelli ◽

Massimo Furi ◽

Maria Patrizia Pera

Keyword(s):

Fredholm Operator ◽

Index Zero ◽

Topological Persistence

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Global Perturbation of Nonlinear Eigenvalues

Advanced Nonlinear Studies ◽

10.1515/ans-2021-2127 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Julián López-Gómez ◽

Juan Carlos Sampedro

Keyword(s):

Perturbation Theory ◽

Spectral Parameter ◽

Classical Theory ◽

General Setting ◽

Perturbation Parameter ◽

Linear Operators ◽

Fredholm Operators ◽

Index Zero ◽

Finite Dimensional ◽

Nonlinear Eigenvalues

Abstract This paper generalizes the classical theory of perturbation of eigenvalues up to cover the most general setting where the operator surface 𝔏 : [ a , b ] × [ c , d ] → Φ 0 ⁢ ( U , V ) {\mathfrak{L}:[a,b]\times[c,d]\to\Phi_{0}(U,V)} , ( λ , μ ) ↦ 𝔏 ⁢ ( λ , μ ) {(\lambda,\mu)\mapsto\mathfrak{L}(\lambda,\mu)} , depends continuously on the perturbation parameter, μ, and holomorphically, as well as nonlinearly, on the spectral parameter, λ, where Φ 0 ⁢ ( U , V ) {\Phi_{0}(U,V)} stands for the set of Fredholm operators of index zero between U and V. The main result is a substantial extension of a classical finite-dimensional theorem of T. Kato (see [T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Class. Math., Springer, Berlin, 1995, Chapter 2, Section 5]).

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Rigorous Formulation of the Lasing Eigenvalue Problem as a Spectral Problem for a Fredholm Operator Function

Lobachevskii Journal of Mathematics ◽

10.1134/s1995080218080127 ◽

2018 ◽

Vol 39 (8) ◽

pp. 1148-1157 ◽

Cited By ~ 10

Author(s):

A. O. Spiridonov ◽

E. M. Karchevskii ◽

A. I. Nosich

Keyword(s):

Eigenvalue Problem ◽

Spectral Problem ◽

Fredholm Operator ◽

Operator Function ◽

Rigorous Formulation

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On Matrices of Index Zero or One

SIAM Journal on Applied Mathematics ◽

10.1137/0117102 ◽

1969 ◽

Vol 17 (6) ◽

pp. 1118-1121 ◽

Cited By ~ 18

Author(s):

Adi Ben-Israel

Keyword(s):

Index Zero

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On Fredholm properties of Lu = u′ − A(t)u for paths of sectorial operators

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500003759 ◽

2005 ◽

Vol 135 (1) ◽

pp. 39-60 ◽

Cited By ~ 3

Author(s):

Davide di Giorgio ◽

Alessandra Lunardi

Keyword(s):

Banach Space ◽

Explicit Formula ◽

Fredholm Operator ◽

Finite Dimension ◽

Sufficient Conditions ◽

Spectral Flow ◽

Exponential Dichotomies ◽

Sectorial Operators ◽

Common Domain ◽

General Banach Space

We consider a path of sectorial operators t ↦ A (t) ∈ Cα (R, L (D, X)), 0 < α < 1, in general Banach space X, with common domain D (A (t)) = D and with hyperbolic limits at ±∞. We prove that there exist exponential dichotomies in the half-lines (−∞, −T] and [T, +∞) for large T, and we study the operator (Lu)(t) = u′(t) − A(t)u(t) in the space Cα (R, D) ∩ C1+α (R, X). In particular, we give sufficient conditions in order that L is a Fredholm operator. In this case, the index of L is given by an explicit formula, which coincides to the well-known spectral flow formula in finite dimension. Such sufficient conditions are satisfied, for instance, if the embedding D ↪ X is compact.

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