Nonexistence of uniform exponential dichotomies for delay equations (Q1614745)

The aim of this paper is to show an error in the certain article by \textit{W. Zhang} and \textit{J. Wu} [J. Differ. Equation 165, No. 2, 414-429 (2000; Zbl 0967.34065)]. These authors claimed that for any \(\mu\in\mathbb{R}\) there exist constants \(\gamma_2> \gamma_1>0\) and \(K\), \(K\) which can be chosen independent of \(\mu\), such that \[ \bigl\|T(t)P_- \bigr\|\leq Ke^{(\mu-\gamma_2)t},\;t\geq 0,\quad\bigl\|T(t)P_+\bigr\|\leq Ke^{-(\mu-\gamma_1) t},\;t\geq 0, \] where \(P_+\) is the eigenprojection associated with the spectral set \(\{\lambda\in \sigma(A): \text{Re} \lambda \geq\mu\}\), \(P_-=I-P_+\) and \(A\) is the generator of \(\{T(t)\}_{t\geq 0}\). The author shows that the proof of this theorem contains two mistakes and \(K\) cannot be chosen independent of \(\mu\) as \(\mu\to-\infty\). Hence, the theorem is false. The result concerning the homoclinic solution \(x(t)=(sh1) (cht)^{-1}\) to \(\dot x(t)=(ch1)(sh1)^{-1}x(t)-(sh1)^{-1} (1+x^2(t)) x(t-1)\) is also false.