**5. Conclusions**

We have given a new proof of a roughness result for linear systems with an exponential dichotomy different than the one in [8]. This new proof has the advantage that it is is more direct, can be easily extended to system having an exponential dichotomy on the whole line and gives a precise estimate on the norm of the difference of the projections of the dichotomies of the perturbed and the unperturbed system. Moreover it extends also to more general situations. Indeed the assumptions that sup*<sup>t</sup>*∈*<sup>I</sup>*|*B*(*t*)| < *ε* is used just to prove that the map

$$\mathbf{x}(t) \mapsto X(t)P\xi + \int\_0^\infty \Gamma(t,s)B(s)\mathbf{x}(s)ds\omega$$

where

$$\Gamma(t,s) = \begin{cases} X(t)PX(s)^{-1} & \text{if } 0 \le s \le t \\\ -X(t)(\mathbb{I}-P)X(s)^{-1} & \text{if } 0 \le t < s\_\tau \end{cases}$$

is a contraction on *C*0*b*. According to [7] this holds also under the weaker assumption that

$$\inf\_{t,T>0} \sup\_{t\geq 0} \int\_{t}^{t+T} |B(s)| ds \left(\frac{k}{1 - e^{-aT}} + \frac{k}{1 - e^{-\beta T}}\right) < 1\tag{10}$$

and the fixed point *<sup>x</sup>*(*<sup>t</sup>*,*<sup>s</sup>*) satisfies again *x* ≤ *<sup>C</sup>*|*ξ*|, for a suitable constant *C*. The remaining part of the proof showing that this fixed point indeed belongs to *C*0*δ* just depends on the fact that |*B*(*t*)| ∈ *<sup>L</sup>*<sup>1</sup>(R+). Hence Theorem 3 holds also under the weaker condition (10) instead of sup*t*≥0|*B*(*t*)| < *ε*.

**Author Contributions:** Investigation, M.F.; Methodology, F.B. The contributions of all authors are equal. All authors have read and agreed to the published version of the manuscript .

**Funding:** Partially supported by the Slovak Research and Development Agency under the contract No. APVV-18-0308 and by the Slovak Grant Agency VEGA No. 1/0358/20 and No. 2/0127/20.

**Conflicts of Interest:** The authors declare no conflict of interest.
