Linearization of class C1 for contractions on Banach Spaces

In this work we prove a C1-linearization result for contraction diffeomorphisms, near a fixed point, valid in infinite dimensional Banach spaces. As an intermediate step, we prove a specific result of existence of invariant manifolds, which can be interesting by itself and that was needed on the proof of our main theorem. Our results essentially generalize some classical results by P. Hartman in finite dimensions, and a result of X. Mora-J. Sola-Morales in the infinite dimensional case. It is shown that the result can be applied to some abstract systems of semilinear damped wave equations

Une contraction inversible qui n'est pas C1-linéarisable

We present an example of a smooth invertible contraction in aninfinite-dimensional Hilbert space that is not locally {\mathcalC}^{1}-linearizable near its fixed point.Nous présentons un exemple de contraction inversible et régulière dans un espace de Hilbert de dimension infinie qui n'est pas localement C1-linéarisable autour de son point fixe

Smooth linearization for a saddle on Banach spaces

As a continuation of a previous work on linearization of class C1 of diffeomorphisms and flows in infinite dimensions near a fixed point, in this work we deal with the case of a saddle point with some non-resonance restrictions for the linear part. Our result can be seen as an extension of results by P. Hartman [2] and Aronson, Belitskii and Zhuzhoma [1] in dimension two. We also present an application to a system of nonlinear wave equations.Peer Reviewe

A note on the relationship between spectral radius and norms of bounded linear operators

Let X be a Banach space and L ( X ) be the Banach algebra of bounded operators on X . In this note we prove that if we have a compact subset K of a commutative sub-algebra of L ( X ), and given " > 0, then it is possible to de ne a new norm in X , equivalent to its given norm, in such a way that inside a neighborhood U " of this compact set in the sub- algebra, the norms of all the operators di er from their spectral radius in less than " . If X is a Hilbert space then it is possible to de ne this new norm as an Hilbertian norm.Postprint (published version

Known results and open problems on C1 linearization in Banach spaces

The purpose of this paper is to review the results obtained by the authors on linearization of dynamical systems in infinite dimen- sional Banach spaces, especially in the C 1 case, and also to present some open problems that we believe that are still important for the understanding of the theory.Postprint (published version

Dynamics of a class of ODEs via wavelets

The objective of this paper is to study a perturbed linear hyperbolic differential equation. The first part of this work is dedicated to study perturbation of the equilibrium (special solution) of a perturbed hyperbolic system. On the second part we analyze the stable and the unstable manifolds of a perturbed semilinear differential equation. We assume that the perturbed forcing function belongs to an L2 class and that it is developed in a series of wavelets. Then we analyze the effect of this development on the special solution of the perturbed equation. Similar study is provided for the stable and unstable manifolds of this special solutions.Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo RegionalJunta de AndalucíaFundação de Amparo à Pesquisa do Estado de São PauloConselho Nacional de Desenvolvimento Científico e Tecnológico (Brasil

[Book of abstracts]

USPCAPESCNPqFAPESPICMC Summer Meeting on Differential Equations (2016 São Carlos

[Book of abstracts]

USPFAPESPCAPESICMC Summer Meeting on Differential Equations (2015 São Carlos

Book of Abstracts

USPCAPESFAPESPCNPqINCTMatICMC Summer Meeting on Differentail Equations.\ud São Carlos, Brasil. 3-7 february 2014