# Erratum to: Almost automorphic mild solutions to some classes of nonautonomous higher-order differential equations

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DOI: 10.1007/s00233-012-9394-2

- Cite this article as:
- Diagana, T. Semigroup Forum (2013) 87: 275. doi:10.1007/s00233-012-9394-2

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**Erratum to: Semigroup Forum (2011) 82:455–477 DOI****10.1007/s00233-010-9261-y**

In this Note we correct some errors that occurred in our paper [T. Diagana, Almost automorphic mild solutions to some classes of nonautonomous higher-order differential equations. *Semigroup Forum*. **82**(3) (2011), 455–477].

I. It was recently found that Lemma 3.7 in Diagana [1], which is taken from the paper by Goldstein and N’Guérékata [2], contained an error. Indeed, the injection *BC*^{1−β}(**R**,**X**_{α})↪*BC*(**R**,**X**) as stated in both [1] and [2], is in fact not compact and that is crucial for the use of the Schauder fixed point to prove the existence of an almost automorphic solution to Eq. (3.1) appearing in [1]. The above-mentioned issue has been fixed in a recent Note by Goldstein and N’Guérékata [3]. The main objective of this paper is to correct Lemma 3.7 and slightly modify assumptions (H.4)–(H.5) to adapt Theorem 3.8 of [1] to this new setting.

*BC*

^{γ}(

**R**,

**X**

_{α}) will be viewed as a locally convex Fréchet space equipped with the following metric (see [3])

*h*=

*f*−

*g*,

Let \(\mathrm{GN}_{\gamma}(\mathbb{R}, \mathbb{X}_{\alpha})\) be the locally convex Fréchet space \((\mathit{BC}^{\gamma}(\mathbb{R}, \mathbb{X}_{\alpha}), \Delta)\).

Lemma 3.7 in [1] should be replaced with

### Lemma 0.1

*The set*\(\mathrm{GN}_{1-\beta}(\mathbb{R}, \mathbb{X}_{\alpha})\)

*is compactly contained in*\(\mathrm{GN}_{0}(\mathbb{R}, \mathbb{X}_{\alpha})\),

*that is*,

*the canonical injection*\(\mathit{id}: \mathrm{GN}_{1-\beta}(\mathbb{R}, \mathbb{X}_{\alpha}) \hookrightarrow\mathrm{GN}_{0}(\mathbb{R}, \mathbb{X}_{\alpha})\)

*is compact*,

*which yields*

*is compact*,

*too*.

II. For the matrix *A*_{l}(*t*) in Eq. (4.1) to be decomposed as \(A_{l}(t) = K_{l}^{-1} J_{l}(t) K_{l}(t)\) as stated in [1], one has to suppose that each root \(\rho_{k}^{l}\) (*k*=1,2,…,*n*) of the polynomial \(Q_{t}^{l}(\cdot)\) in page 457 is of multiplicity one.

- (h.4)
\(R(\omega, A(\cdot)) \in \mathit{AA}_{u}(B(\mathbb{X}_{\alpha}))\).

- (h.5)
- The function \(F: \mathbb{R}\times\mathbb{X}_{\alpha}\mapsto\mathbb{X}\) is such that
*t*↦*F*(*t*,*u*) belongs to \(\mathit{AA}_{u}(\mathbb{X})\) for all \(u \in\mathbb{X}_{\alpha}\). The function*u*↦*F*(*t*,*u*) is uniformly continuous on any bounded subset*K*of \(\mathbb{X}\) for each \(t \in\mathbb{R}\). Finally,where \({\mathcal{M}}: \mathbb{R}^{+} \mapsto\mathbb{R}^{+}\) is a continuous, monotone increasing function satisfying$$\|F(t, u) \|_{\infty} \leq{\mathcal{M}} \bigl(\Vert u\Vert _{\alpha ,\infty}\bigr), $$$$\lim_{r \to\infty} \frac{{\mathcal{M}}(r)}{r} = 0. $$

In view of the above, Theorem 3.8 in [1] should be replaced with

### Theorem 0.2

*Suppose assumptions* (H.1)*–*(H.2)*–*(H.3)*–*(h.4)*–*(h.5) *hold*, *then the nonautonomous differential equation* (3.1) *has a mild solution which belongs to*\(\mathit{AA}_{u} (\mathbb{X}_{\alpha})\).