Semigroup Forum

, Volume 87, Issue 1, pp 275–276 | Cite as

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

Erratum

Erratum to: Semigroup Forum (2011) 82:455–477 DOI10.1007/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 BC1−β(R,Xα)↪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.

A function \(f \in \mathit{BC}(\mathbb{R}, \mathbb{X})\) is said to belong to \(\mathit{AA}(\mathbb{X})\) [resp., \(\mathit{AA}_{u}(\mathbb{X})\)] if for every sequence of real numbers \((s'_{n})_{n \in\mathbb{N}}\) there exists a subsequence \((s_{n})_{n \in\mathbb{N}}\) such that
$$\lim_{n \to\infty} f(t + s_n) = g(t),\qquad \lim _{n \to\infty} g(t - s_n) = f(t) $$
pointwise on \(\mathbb{R}\) [resp., uniformly on compacts of \(\mathbb{R}\)].
In contrast with [1], here the space BCγ(R,Xα) will be viewed as a locally convex Fréchet space equipped with the following metric (see [3])
$$\Delta(f, g) = \sum_{n=1}^\infty2^{-n} \frac{\rho_n(f,g)}{1 + \rho_n(f,g)} $$
where, for h=fg,
$$\Delta_n (f, g) = \Delta_n(h, 0) = \|h \|_{C[-n,n]} + \gamma. \sup \biggl\{\frac{\|h(t) - h(s)\|_\alpha}{|t-s|^\gamma}: t, s \in[-n, n], t \not=s \biggr\}. $$

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
$$\mathit{id}: \mathrm{GN}_{1-\beta}(\mathbb{R}, \mathbb{X}_\alpha) \cap \mathit{AA}_u(\mathbb{X}_\alpha) \hookrightarrow \mathit{AA}_u(\mathbb{X}_\alpha) $$
is compact, too.

II. For the matrix Al(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.

III. Consider the following assumptions:
(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 tF(t,u) belongs to \(\mathit{AA}_{u}(\mathbb{X})\) for all \(u \in\mathbb{X}_{\alpha}\). The function uF(t,u) is uniformly continuous on any bounded subset K of \(\mathbb{X}\) for each \(t \in\mathbb{R}\). Finally,
$$\|F(t, u) \|_{\infty} \leq{\mathcal{M}} \bigl(\Vert u\Vert _{\alpha ,\infty}\bigr), $$
where \({\mathcal{M}}: \mathbb{R}^{+} \mapsto\mathbb{R}^{+}\) is a continuous, monotone increasing function satisfying
$$\lim_{r \to\infty} \frac{{\mathcal{M}}(r)}{r} = 0. $$

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. 1.
    Diagana, T.: Almost automorphic mild solutions to some classes of nonautonomous higher-order differential equations. Semigroup Forum 82(3), 455–477 (2011) MathSciNetMATHCrossRefGoogle Scholar
  2. 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. 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

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsHoward UniversityWashingtonUSA

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