Robustness of exponential behavior in Banach spaces

We consider a generalization of the notion of exponential dichotomy in which the exponential behaviors on \(\mathbb {R}^+\) and \(\mathbb {R}^-\) need not agree at the origin, although they still satisfy a certain compatibility condition. A nontrivial example is the change from stable to unstable behavior in a given direction, such as in a saddle-node bifurcation. Our main aim is to show that this exponential behavior is robust, in the sense that it persists under sufficiently small linear perturbations. We emphasize that we consider arbitrary evolution families in Banach spaces. This includes any differentiable evolution family obtained from a nonautonomous linear equation \(x^{\prime } =A(t) x\) possibly with A(t) unbounded, although in general we do not require the evolution families to be differentiable.