Axiomatization of the degree of Fitzpatrick, Pejsachowicz and Rabier

AbstractIn this paper, we prove an analogue of the uniqueness theorems of Führer [15] and Amann and Weiss [1] to cover the degree of Fredholm operators of index zero constructed by Fitzpatrick, Pejsachowicz and Rabier [13], whose range of applicability is substantially wider than for the most classical degrees of Brouwer [5] and Leray–Schauder [22]. A crucial step towards the axiomatization of this degree is provided by the generalized algebraic multiplicity of Esquinas and López-Gómez [8, 9, 25], $$\chi $$ χ , and the axiomatization theorem of Mora-Corral [28, 32]. The latest result facilitates the axiomatization of the parity of Fitzpatrick and Pejsachowicz [12], $$\sigma (\cdot ,[a,b])$$ σ ( · , [ a , b ] ) , which provides the key step for establishing the uniqueness of the degree for Fredholm maps.

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Uniqueness of the algebraic multiplicity

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500003589 ◽

2004 ◽

Vol 134 (5) ◽

pp. 985-990

Author(s):

Carlos Mora-Corral

Keyword(s):

Banach Space ◽

Complex Variable ◽

Smooth Function ◽

Algebraic Multiplicity ◽

Fredholm Operators ◽

Index Zero ◽

The Family

Given a smooth function 𝔏 of a real or complex variable and taking its values in the class of Fredholm operators of index zero in a Banach space, there are some available definitions in the literature of algebraic multiplicity of the family 𝔏 at a point x0 of the parameter at which the operator 𝔏(x0) becomes non-invertible. The purpose of this paper is to suggest an axiomatic for the multiplicity and to prove that the algebraic multiplicity is uniquely determined by a few of its properties, independently of its construction.

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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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Nonlinear Fredholm maps of index zero and their singularities

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-1978-0464283-2 ◽

1978 ◽

Vol 68 (3) ◽

pp. 317-317 ◽

Cited By ~ 12

Author(s):

R. A. Plastock

Keyword(s):

Index Zero ◽

Fredholm Maps

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GLOBAL BRANCH OF SOLUTIONS FOR NON-LINEAR SCHRÖDINGER EQUATIONS WITH DEEPENING POTENTIAL WELL

Proceedings of the London Mathematical Society ◽

10.1017/s0024611505015637 ◽

2006 ◽

Vol 92 (3) ◽

pp. 655-681 ◽

Cited By ~ 27

Author(s):

C. A. STUART ◽

HUAN-SONG ZHOU

Keyword(s):

Potential Well ◽

Index Zero ◽

Topological Methods ◽

Open Set ◽

Connected Sets ◽

Non Linear ◽

Asymptotic Linearization ◽

Positive Eigenfunction ◽

Fredholm Maps ◽

Global Branch

We consider the stationary non-linear Schrödinger equation\begin{equation*}\Delta u + \{1 + \lambda g(x)\} u = f(u)\mbox{with}u \in H^{1} (\mathbb{R}^{N}), u \not\equiv 0,\end{equation*} where $\lambda >0$ and the functions $f$ and $g$ are such that\begin{equation*} \lim_{s \rightarrow 0}\frac{f(s)}{s} = 0 \mbox{and} 1 < \alpha + 1 = \lim _{|s| \rightarrow \infty}\frac{f(s)}{s} < \infty\end{equation*} and \begin{equation*} g(x)\equiv 0 \mbox{on} \bar{\Omega}, g(x)\in (0, 1] \mbox{on} {\mathbb{R}^{N}} \setminus {\overline{\Omega}} \mbox{and} \lim_{|x| \rightarrow + \infty} g(x) = 1 \end{equation*} for some bounded open set $\Omega \in \mathbb{R}^{N}$. We use topological methods to establish the existence of two connected sets $\mathcal{D}^{\pm}$ of positive/negative solutions in $\mathbb{R} \times W^{2, p} (\mathbb{R}^{N})$ where $p \in [2, \infty) \cap (\frac{N}{2},\infty)$ that cover the interval $(\alpha,\Lambda(\alpha))$ in the sense that \begin{align*} P \mathcal{D}^{\pm} & = (\alpha, \Lambda(\alpha)) \text{where}P(\lambda, u) = \lambda \text{and furthermore,} \\ \lim_{\lambda \rightarrow \Lambda(\alpha)-}\left\Vert u_{\lambda} \right\Vert _{L^{\infty} (\mathbb{R}^{N})} & = \lim_{\lambda \rightarrow \Lambda (\alpha )-} \left\Vert u_{\lambda} \right\Vert _{W^{2, p}(\mathbb{R}^{N})} = \infty \text{ for }(\lambda, u_{\lambda}) \in \mathcal{D}^{\pm}. \end{align*} The number $\Lambda(\alpha)$ is characterized as the unique value of $\lambda$ in the interval $(\alpha, \infty)$ for which the asymptotic linearization has a positive eigenfunction. Our work uses a degree for Fredholm maps of index zero.

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