A Degree Associated to Linear Eigenvalue Problems in Hilbert Spaces and Applications to Nonlinear Spectral Theory

Benevieri, Pierluigi; Calamai, Alessandro; Furi, Massimo; Pera, Maria Patrizia

doi:10.1007/s10884-020-09921-9

Cited by 4 publications

(19 citation statements)

References 19 publications

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“…The present paper generalizes the "global persistence" property of solution triples which, either in finite-dimensional or infinite-dimensional case, has been studied in [4][5][6][7] in the case of a simple eigenvalue. Since it is known that the persistence property need not hold if λ * is an eigenvalue of even multiplicity, it is natural to investigate the odd-multiplicity case.…”

Section: Below Asserts Thatmentioning

confidence: 88%

“…Instead, we use a notion of topological degree for oriented Fredholm maps of index zero between real differentiable Banach manifolds, developed by two authors of this paper, and whose construction and properties are summarized in Section 3 for the reader's convenience. Such a notion of degree has been introduced in [8] (see also [7,9,10] for additional details).…”

Section: Below Asserts Thatmentioning

confidence: 99%

“…The study of the local [2,13,[15][16][17][18][19] as well as global [3][4][5][6][7] persistence property when the eigenvalue λ * is not necessarily simple has been performed in recent papers by the authors, also in collaboration with R. Chiappinelli. In particular a first pioneering result in this sense is due to Chiappinelli [12], who proved the existence of the local persistence of eigenvalues and eigenvectors, in Hilbert spaces, in the case of a simple isolated eigenvalue.…”

Section: Below Asserts Thatmentioning

confidence: 99%

“…In this section we recall some notions that will be used in the sequel. We mainly summarize some concepts which are needed for the construction of the topological degree for oriented Fredholm maps of index zero between real differentiable Banach manifolds introduced in [8], here called bf -degree to distinguish it from the Leray-Schauder degree, called LS-degree (see [7,9,10] for additional details).…”

Section: Preliminariesmentioning

confidence: 99%

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Global persistence of the unit eigenvectors of perturbed eigenvalue problems in Hilbert spaces: the odd multiplicity case

Benevieri¹,

Calamai²,

Furi³

et al. 2021

Preprint

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Section: Below Asserts Thatmentioning

confidence: 88%

Section: Below Asserts Thatmentioning

confidence: 99%

Section: Below Asserts Thatmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

See 2 more Smart Citations

Global persistence of the unit eigenvectors of perturbed eigenvalue problems in Hilbert spaces: the odd multiplicity case

Benevieri¹,

Calamai²,

Furi³

et al. 2021

Preprint

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“…Usually, it pays to place the problems in a wisely chosen space. A case in point is [30] where the classical Brouwer degree is applied to the study of the eigenvalue problem for square real matrices yielding a result about global continuation in nonlinear spectral theory that, in turn, can be applied to a Rabinowitz-type global continuation property of the solutions of a perturbed motion equation with friction.…”

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confidence: 99%

Introduction to: Topological degree and fixed point theories in differential and difference equations

Pera

Spadini

2021

Phil. Trans. R. Soc. A.

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An infinite dimensional version of the intermediate value theorem

Benevieri

Calamai

Furi

et al. 2023

J. Fixed Point Theory Appl.

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Let $$\mathfrak {f}= I-k$$ f = I - k be a compact vector field of class $$C^1$$ C 1 on a real Hilbert space $$\mathbb {H}$$ H . In the spirit of Bolzano’s Theorem on the existence of zeros in a bounded real interval, as well as the extensions due to Cauchy (in $$\mathbb {R}^2$$ R 2 ) and Kronecker (in $$\mathbb {R}^k$$ R k ), we prove an existence result for the zeros of $$\mathfrak {f}$$ f in the open unit ball $$\mathbb {B}$$ B of $$\mathbb {H}$$ H . Similarly to the classical finite dimensional results, the existence of zeros is deduced exclusively from the restriction $$\mathfrak {f}|_\mathbb {S}$$ f | S of $$\mathfrak {f}$$ f to the boundary $$\mathbb {S}$$ S of $$\mathbb {B}$$ B . As an extension of this, but not as a consequence, we obtain as well an Intermediate Value Theorem whose statement needs the topological degree. Such a result implies the following easily comprehensible, nontrivial, generalization of the classical Intermediate Value Theorem: If a half-line with extreme$$q \notin \mathfrak {f}(\mathbb {S})$$ q ∉ f ( S ) intersects transversally the function$$\mathfrak {f}|_\mathbb {S}$$ f | S for only one point of $$\mathbb {S}$$ S , then any value of the connected component of$$\mathbb {H}{\setminus }\mathfrak {f}(\mathbb {S})$$ H \ f ( S ) containingqis assumed by$$\mathfrak {f}$$ f in$$\mathbb {B}$$ B . In particular, such a component is bounded.

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A Degree Associated to Linear Eigenvalue Problems in Hilbert Spaces and Applications to Nonlinear Spectral Theory

Cited by 4 publications

References 19 publications

Global persistence of the unit eigenvectors of perturbed eigenvalue problems in Hilbert spaces: the odd multiplicity case

Global persistence of the unit eigenvectors of perturbed eigenvalue problems in Hilbert spaces: the odd multiplicity case

Introduction to: Topological degree and fixed point theories in differential and difference equations

An infinite dimensional version of the intermediate value theorem

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