1,720,962 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
A result on the existence of infinitely many solutions of a nonlinear elliptic boundary value problems at resonance
No full textLinear 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 textGlobal 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
Global continuation in euclidean spaces of the perturbed unit eigenvectors corresponding to a simple eigenvalue
No full textIn the Euclidean space Rk, we consider the perturbed eigenvalue problem Lx + εN(x) = λx, ||x|| = 1, where ε, λ are real parameters, L is a linear endomorphism of Rk, and N: Sk−1 → Rk is a continuous map defined on the unit sphere of Rk . We prove a global continuation result for the solutions (x, ε, λ) of this problem. Namely, under the assumption that x_* is one of the two unit eigenvectors of L corresponding to a simple eigenvalue λ_*, we show that, in the set of all the solutions, the connected component containing (x_*, 0, λ_*) is either unbounded or meets a solution (x*, 0, λ*) having x* ≠ x_*. Our result is inspired by a paper of R. Chiappinelli concerning the local persistence property of eigenvalues and eigenvectors of a perturbed self-adjoint operator in a real Hilbert space
Topological methods for the global controllability of nonlinear systems
No full textSufficient conditions for the local and global controllability of general nonlinear systems, by means of controls belonging to a fixed finite-dimensional subspace of the space of all admissible controls, are established with the aid of topological methods, such as homotopy invariance principles. Some applications to certain classes of nonlinear control processes are given, and various known results on the controllability of perturbed linear systems are also derived as particular cases
- …
