An extension of the concept of exponential dichotomy in Fréchet spaces which is stable under perturbation

Éder Rítis Aragão Costa

doi:10.3934/cpaa.2019041

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2019, Volume 18, Issue 2: 845-868. Doi: 10.3934/cpaa.2019041

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An extension of the concept of exponential dichotomy in Fréchet spaces which is stable under perturbation

Éder Rítis Aragão Costa

Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo - Campus de São Carlos, São Carlos, SP, Brazil

Received: February 28, 2018

Revised: April 30, 2018

Published: October 2018

The author was partially supported by Grant: 2014/02899-3, São Paulo Research Foundation (FAPESP), Brazil

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Abstract

In this paper we prove versions, in Fréchet spaces, of the classical theorems related to exponential dichotomy for a sequence of continuous linear operators on Banach spaces. To be more specific, here we define a kind of exponential dichotomy in Fréchet spaces, which extends the former one in Banach spaces, establish necessary conditions for its existence and provide sufficient conditions for its stability under perturbation.

We apply the conclusions by providing an example of a semigroup of bounded linear operators, on a Fréchet space, which has this new exponential dichotomy but does not in Banach spaces, namely, $\{e^{mΔ}:\; m∈ \mathbb{N}\}$, where $Δ$ is the Laplace operator on the unbounded domain $\mathbb{R}^{n}\setminus \{0\}$.

Also, we show how these new concepts allow us to study a hyperbolic equilibrium point of a backwards heat equation with nonlinearity involving convolution products, which cannot be obtained from the knowledge of exponential dichotomy in Banach spaces.

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Mathematics Subject Classification: Primary: 34D09, 37B55, 46F05; Secondary: 35S05, 37D05.

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