1,720,999 research outputs found
Normalized eigenvectors of a perturbed linear operator via general bifurcation
No full textLet X be a real Banach space, A:X → X a bounded linear operator, and B:X → X a (possibly nonlinear) continuous operator. Assume that λ = 0 is an eigenvalue of A and consider the family of perturbed operators A + εB, where ε is a real parameter. Denote by S the unit sphere of X and let SA = S ∩ KerA be the set of unit 0-eigenvectors of A. We say that a vector x0 ∈ SA is a bifurcation point for the unit eigenvectors of A + εB if any neighborhood of (0, 0, x0) ∈ × × X contains a triple (ε, λ, x) with ε = 0 and x a unit λ-eigenvector of A + εB, i.e. x ∈ S and (A + εB)x = λx.
We give necessary as well as sufficient conditions for a unit 0-eigenvector of A to be a bifurcation point for the unit eigenvectors of A + εB. These conditions turn out to be particularly meaningful when the perturbing operator B is linear. Moreover, since our sufficient condition is trivially satisfied when KerA is one-dimensional, we extend a result of the first author, under the additional assumption that B is of class C
A new theme in nonlinear analysis: continuation and bifurcation of the unit eigenvectors of a perturbed linear operator
No full textWe review some recent results concerning nonlinear eigenvalue problems of the form (*) Au + eB(u) =cu, where A is a linear Fredholm operator of index zero (with nontrivial kernel KerA) acting in a real Banach space X, and B from X to X is a (possibly) nonlinear perturbation term. We seek solutions u of (*) in the unit sphere S of X, and the emphasis is put on the existence - under appropriate conditions on B - of points u0 in S \ KerA (thus satisfying (*) for e = c = 0) which either can be continued as solutions of (*) for e different from 0 or - more generally - are bifurcation points for solutions of that kind
Topological persistence of the normalized eigenvectors of a perturbed self-adjoint operator
No full textLet T be a self-adjoint bounded operator acting in a real Hilbert space H, and denote by S the unit sphere of H. Assume that is an isolated eigenvalue of T of odd multiplicity greater than 1. Given an arbitrary operator B:H ! H of class C1, we prove that for any sufficiently small there exists such that Tx" C "B.x".
This result was conjectured, but not proved, in a previous article by the authors.We provide an example showing that the assumption that the multiplicity of 0 is odd
cannot be removed
An overview on spectral theory for nonlinear operators
No full text"We compare different spectral theories for nonlinear operators, focusing in particular on the notion of spectrum at a point recently introduced by the authors. We discuss the main properties of the nonlinear spectrum and present illustrating applications and examples.'
A new spectrum for nonlinear operators in Banach spaces
No full textGiven any continuous self-map f of a Banach space E over K (where K is R or C)
and given any point p of E, we define a subset (f, p) of K, called the ‘spectrum of f at p’, which
coincides with the usual spectrum (f) of f in the linear case. More generally, we show that
(f, p) is always closed and, when f is C1, coincides with the spectrum (f0(p)) of the Fr´echet
derivative of f at p. Some applications to bifurcation theory are given and some peculiar examples
of spectra are provided
Global persistence of the unit eigenvectors of perturbed eigenvalue problems in hilbert spaces: The odd multiplicity case
No full textWe study the persistence of eigenvalues and eigenvectors of perturbed eigenvalue problems in Hilbert spaces. We assume that the unperturbed problem has a nontrivial kernel of odd dimension and we prove a Rabinowitz-type global continuation result. The approach is topological, based on a notion of degree for oriented Fredholm maps of index zero between real differentiable Banach manifolds
THE BROUWER DEGREE ASSOCIATED TO CLASSICAL EIGENVALUE PROBLEMS AND APPLICATIONS TO NONLINEAR SPECTRAL THEORY
No full textThanks to a connection between two completely different topics, the classical eigenvalue problem in a finite dimensional real vector space and the Brouwer degree for maps between oriented differentiable real manifolds, we are able to solve, at least in the finite dimensional context, a conjecture regarding global continuation in nonlinear spectral theory that we formulated in some recent papers. The infinite dimensional case seems nontrivial, and is still unsolved
Linear controllability by piecewise constant controls with assigned switching times
No full textFor a linear time-dependent control process inRn, we prove that the complete controllability by means of the space of all the admissible controls is equivalent to the complete controllability by means of a suitablen-dimensional space of piecewise constant controls with at mostn preassigned switching times. An analogous result is also established for more general controllability problems
On the existence of forced oscillations for the spherical pendulum acted on by a retarded periodic force
No full textAn infinite dimensional version of the intermediate value theorem
Full text linkLet f = I - k be a compact vector field of class C1 on a real Hilbert space 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 R2) and Kronecker (in Rk), we prove an existence result for the zeros of f in the open unit ball B of H. Similarly to the classical finite dimensional results, the existence of zeros is deduced exclusively from the restriction f|S of f to the boundary S of 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 ?/ f(S) intersects transversally the function f|S for only one point of S, then any value of the connected component of H\f(S) containing q is assumed by f in B. In particular, such a component is bounded
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