Existence of periodic orbits for vector fields via Fuller index and the averaging method

We prove a generalization of a theorem proved by Seifert and Fuller concerning the existence of periodic orbits of vector fields via the averaging method. Also we show applications of these results to Kepler motion and to geodesic flows on spheres

Topology and Homoclinic Trajectories of Discrete Dynamical Systems

We show that nontrivial homoclinic trajectories of a family of discrete, nonautonomous, asymptotically hyperbolic systems parametrized by a circle bifurcate from a stationary solution if the asymptotic stable bundles Es(+{\infty}) and Es(-{\infty}) of the linearization at the stationary branch are twisted in different ways.Comment: 19 pages, canceled the appendix (Properties of the index bundle) in order to avoid any text overlap with arXiv:1005.207

Atypical bifurcation without compactness

Abstract. We prove a global bifurcation result for an abstract equation of the type Lx + λh(λ, x) = 0, where L : E → F is a linear Fredholm operator of index zero between Banach spaces and h : R × E → F is a C 1 (not necessarily compact) map. We assume that L is not invertible and, under suitable conditions, we prove the existence of an unbounded connected set of nontrivial solutions of the above equation (i.e. solutions (λ, x) with λ = 0) whose closure contains a trivial solution (0,x). This result extends previous ones in which the compactness of h was required. The proof is based on a degree theory for Fredholm maps of index zero developed by the first two authors

A Borsuk-type theorem for some classes of perturbed Fredholm maps

We prove an odd mapping theorem of Borsuk type for locally compact perturbations of Fredholm maps of index zero between Banach spaces. We extend this result to a more general class of perturbations of Fredholm maps, defined in terms of measure of noncompactness

A degree theory for a class of perturbed Fredholm maps II

Art. ID 2715

On the degree for oriented quasi-Fredholm maps: its uniqueness and its effective extension of the Leray–Schauder degree

In a previous paper, the first and third author developed a~degree theory for oriented locally compact perturbations of C\sp{1} Fredholm maps of index zero between real Banach spaces. In the spirit of a celebrated Amann--Weiss paper, we prove that this degree is unique if it is assumed to satisfy three axioms: Normalization, Additivity and Homotopy invariance. Taking into account that any compact vector field has a canonical orientation, from our uniqueness result we shall deduce that the above degree provides an effective extension of the Leray--Schauder degree

Global persistence of the unit eigenvectors of perturbed eigenvalue problems in Hilbert spaces

We consider the nonlinear eigenvalue problem where are real parameters, L,C : G H are bounded linear operators between separable real Hilbert spaces, and N : S H is a continuous map defined on the unit sphere of G. We prove a global persistence result regarding the set of the solutions (x,) SR R of this problem. Namely, if the operators N and C are compact, under suitable assumptions on a solution p= (x, 0, ) of the unperturbed problem, we prove that the connected component of containing pis either unbounded or meets a triple p= (x, 0,) with p6= p. When C is the identity and G = H is finite dimensional, the assumptions on (x, 0,) mean that xis an eigenvector of L whose corresponding eigenvalue,is simple. Therefore, we extend a previous result obtained by the authors in the finite dimensional setting. Our work is inspired by a paper of R. Chiappinelli concerning the local persistence property of the unit eigenvectors of perturbed self-adjoint operators in a real Hilbert space

Differential Topology and General Equilibrium with Complete and Incomplete Markets

The goal of this publication is to provide basic tools of differential topology to study systems of nonlinear equations and to apply them to the analysis of general equilibrium models with complete and incomplete markets. The main content of general equilibrium analysis is to study existence, (local) uniqueness and efficiency of equilibria. To study existence Differential Topology and General Equilibrium with Complete and Incomplete Markets combines two features. First order conditions (of agents’ maximization problems) and market clearing conditions, instead of aggregate excess demand function. Then, the application to that “extended system” of a homotopy argument, which is stated and proved in a relatively elementary manner. Local uniqueness and smooth dependence of the endogenous variables from the exogenous ones are studied using a version of a so-called parametric transversality theorem. In a standard general equilibrium model, all equilibria are efficient, but that is not the case if some imperfection, like incomplete markets, asymmetric information, strategic interaction, is added. Then, for almost all economies, equilibria are inefficient, and an outside institution can Pareto Improve upon the market outcome. Those results are proved showing that a well-chosen system of equations has no solutions. The target audience of Differential Topology and General Equilibrium with Complete and Incomplete Markets consists of researchers interested in Economic Theory. The needed background is multivariate analysis, basic linear algebra and basic general topology