Exponential dichotomies by Ekeland's variational principle (Q2665538)

Exponential dichotomy, which is a type of hyperbolicity in the context of linear time-varying systems, is the central focus of this paper. First, two well known characterizations of the exponential dichotomy in \(\mathbb{R}^+\) are presented: \begin{itemize} \item [(i)] If the system \(\dot{v}=A(t)v+f(t)\) in \(\mathbb{R}^N\), admits a bounded solution for any continuous and bounded function \(f\), then the linear system \(\dot{v}=A(t)v\) has exponential dichotomy in \(\mathbb{R}^+\). \item[(ii)] If the system \(\dot{v}=A(t)v+f^k(t)\) in \(\mathbb{R}^N\), for \(k\in\{1,\dots,N\}\) satisfies the same conditions as in (i) then its linear system \(\dot{v}=A(t)v\) has exponential dichotomy in \(\mathbb{R}^+\), where in this case \(f^k\) is sufficiently close to the function \(\alpha_k(t)u^k(t)\), where \(\{\alpha_1,\dots,\alpha_N\}\) are continuous and bounded scalar functions with appropriate properties with respect to their sign and \(\{u^1,\dots,u^N\}\) are functions obtained by means of the Gram-Schmidt process from a fundamental system \(\{x^1,\dots,x^N\}\) of solutions of the linear system. \end{itemize} Item (i) corresponds to a classical result in the context of the exponential dichotomy and is also used to prove result (ii), but in this paper the authors want to prove a characterization (ii) without using statement (i). To achieve this, the proof is divided into the analysis of the scalar case and the vector case; for the scalar case the Ekeland's Variational Principle is used to assign appropriate sign properties to the bounded solutions of \(\dot{v}=A(t)v+f^1(t)\) when \(f^1\) also has them, and for the vector case a triangularization argument is used to achieve the goal. On the other hand and in order to conclude, when the exponential dichotomy is studied on the whole line, the concept of weakly recurrent matrix is used in order to characterize it.