Erratum to: Almost automorphic mild solutions to some classes of nonautonomous higherorder differential equations
Erratum to: Semigroup Forum (2011) 82:455–477 DOI 10.1007/s002330109261y
In this Note we correct some errors that occurred in our paper [T. Diagana, Almost automorphic mild solutions to some classes of nonautonomous higherorder 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 abovementioned 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.
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
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 \(\mathit{AA}_{u} (\mathbb{X}_{\alpha})\).
References
 1.Diagana, T.: Almost automorphic mild solutions to some classes of nonautonomous higherorder differential equations. Semigroup Forum 82(3), 455–477 (2011) MathSciNetMATHCrossRefGoogle Scholar
 2.Goldstein, J.A., N’Guérékata, G.M.: Almost automorphic solutions of semilinear evolution equations. Proc. Am. Math. Soc. 133(8), 2401–2408 (2005) MATHCrossRefGoogle Scholar
 3.Goldstein, J.A., N’Guérékata, G.M.: Corrigendum on “Almost automorphic solutions of semilinear evolution equations”. Proc. Am. Math. Soc. 140, 1111–1112 (2012) MATHCrossRefGoogle Scholar