1,721,049 research outputs found
An infinite dimensional version of the Kronecker index and its relation with the Leray–Schauder degree
Full text linkLet f be a compact vector field of class C1 on a real Hilbert space H. Denote by B the open unit ball of H and by S = ∂B the unit sphere. Given a point q ∉ f(S), consider the self-map of S defined by (Equation Presented) If H is finite dimensional, then S is an orientable, connected, compact differentiable manifold. Therefore, the Brouwer degree, degBr(fq∂) is well defined, no matter what orientation of S is chosen, assuming it is the same for S as domain and codomain of fq∂. This degree may be considered as a modern reformulation of the Kronecker index of the map fq∂. Let degBr(f,B,q) denote the Brouwer degree of f on B with target q. It is known that one has the equality (Equation Presented) Our purpose is an extension of this formula to the infinite dimensional context. Namely, we will prove that (Equation Presented) where degLS(∙) denotes the Leray–Schauder degree and degbf(∙) is the degree earlier introduced by M. Furi and the first author, which extends, to the infinite dimensional case, the Brouwer degree and the Kronecker index. In other words, here, we extend to the Leray–Schauder degree the boundary dependence property which holds for the Brouwer degree in the finite dimensional context
Persistence of the normalized eigenvectors of a perturbed operator in the variational case
No full textLet H be a real Hilbert space and denote by S its unit sphere. Consider the nonlinear eigenvalue problem Ax+εB(x)=δx, where A:H→H is a bounded self-adjoint (linear) operator with nontrivial kernel KerA, and B:H→H is a (possibly) nonlinear perturbation term. A unit eigenvector x0∈S∩KerA of A (thus corresponding to the eigenvalue δ=0, which we assume to be isolated) is said to be persistent, or a bifurcation point (from the sphere S∩KerA), if it is close to solutions x∈S of the above equation for small values of the parameters δ∈R and ε≠0. In this paper, we prove that if B is a C1 gradient mapping and the eigenvalue δ=0 has finite multiplicity, then the sphere S∩KerA contains at least one bifurcation point, and at least two provided that a supplementary condition on the potential of B is satisfied. These results add to those already proved in the non-variational case, where the multiplicity of the eigenvalue is required to be odd
Multiplicity of forced oscillations for the spherical pendulum acted on by a retarded periodic force
Full text linkTopological persistence of the unit eigenvectors of a perturbed Fredholm operator of index zero
No full textLet A; C : E -> F be two bounded linear operators between real Banach spaces, and denote by S the unit sphere of E (or, more generally, let S = g(-1) (1), where g is any continuous norm in E). Assume that mu(0) is an eigenvalue of the problem Ax = mu Cx, that the operator L = A - mu C-0 is Fredholm of index zero, and that C satis fi es the transversality condition Img L + C (Ker L) = F, which implies that the eigenvalue mu(0) is isolated (and when F = E and C is the identity implies that the geometric and the algebraic multiplicities of mu(0) coincide).
We prove the following result about the persistence of the unit eigenvectors: Given an arbitrary C-1 map M : E -> F, if the (geometric) multiplicity of mu(0) is odd, then for any real epsilon sufficiently small there exists x(epsilon) is an element of S and mu(epsilon) near mu(0) such that Ax(epsilon) + epsilon M (x(epsilon)) = mu(epsilon)Cx(epsilon).
This result extends a previous one by the authors in which E is a real Hilbert space, F = E, A is selfadjoint and C is the identity. We provide an example showing that the assumption that the multiplicity of mu(0) is odd cannot be removed
On general properties of n-th order retarded functional differential equations
Full text linkConsider the second order RFDE (retarded functional differential equation) x’’ (t) = f(t, xt), where f is a continuous realvalued function defined on the Banach space R x C1([−r, 0],R). The weak assumption of continuity on f (due to the strong topology of C1([−r, 0],R)) makes not convenient to transform this equation into a first order RFDE of the type z’ (t) = g(t, zt). In fact, in this case, the associated R2-valued function g could be discontinuous (with the C0- topology) and, in addition, not necessarily defined on the whole space R x C([−r, 0],R2). Consequently, in spite of what happens for ODEs, the classical results regarding existence, uniqueness, and continuous dependence on data for first order RFDEs could not apply. Motivated by this obstruction, we provide results regarding general properties, such as existence, uniqueness, continuous dependence on data and continuation of solutions of RFDEs of the type x(n)(t) = f(t, xt), where f is an Rk-valued continuous function on the Banach space R x C(n−1)([−r, 0],Rk). Actually, for the sake of generality, our investigation will be carried out in the case of infinite delay
On the persistence of the eigenvalues of a perturbed fredholm operator of index zero under nonsmooth perturbations
Full text linkMultiplicity of forced oscillations for scalar retarded functional differential equations
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