On Exponential Decay and the Markus–Yamabe Conjecture in Infinite Dimensions with Applications to the Cima System

Abstract

In this work we extend to infinite dimensions a previous result of Lyascenko given in finite dimensions for a class of Lipschitz functions. In particular, this extension substitutes the class of Lipschitz functions by a class of Holder functions. Then we consider applications of these results to analyze its relation with the Markus–Yamabe Conjecture in infinite dimensions. We discuss in detail a celebrated example of Cima et al. ( Adv Math 131(2): 453–457, 1997) of a nonlinear system in dimension three whose Jacobian has a unique eigenvalue \(-1\) of multiplicity 3, and yet an explicit unbounded solution as \(t \rightarrow \infty \) exists. We also present explicit solutions of the same equation that tends to 0 as \(t \rightarrow \infty \). Then we look at this conjecture by considering not the properties of the whole system, but instead the properties of some solutions. Finally we present an application in an infinite dimensional Hilbert space, where we use different techniques to study the local and global asymptotic stabilities.