# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations. (English) Zbl 1194.47047
The present paper discusses the existence and uniqueness of an almost automorphic (a weighted pseudo almost automorphic) mild solution to a class of semilinear evolution equations ${x}^{\text{'}}\left(t\right)=A\left(t\right)x\left(t\right)+f\left(t,x\left(t\right)\right)$ in a Banach space. The main results are Theorems 3.2 and 4.2. However, Theorem 3.2 can be seen from [H.-S. Ding, W. Long and G. M. N’Guérékata, Nonlinear Anal., Theory Methods Appl. 70, No. 12 (A), 4158–4164 (2009; Zbl 1161.43301)]. Moreover, the authors use the Banach contraction mapping principle to obtain the conclusion in the proof of Theorem 4.2; thus, the completeness of the space $WPAA\left(R,\rho \right)$ is needed. But from Lemma 2.10 in [J. Blot, G. M. Mophou, G. M. N’Guérékata and D. Pennequin, Nonlinear Anal., Theory Methods Appl. 71, No. 3–4 (A), 903–909 (2009, Zbl 1177.34077)], one only knows that the space $WPAA\left(R,\rho \right)$ is a Banach space if $\rho \in {U}_{b}$. Actually, to the best of the reviewer’s knowledge, there is no proof in the literature that says that $WPAA\left(R,\rho \right)$ is complete in the case when $\rho$ is not necessarily bounded. On the other hand, when $\rho$ is bounded, Theorem 4.2 is known from [T.-J. Xiao, X.-X. Zhu and J. Liang, Nonlinear Anal., Theory Methods Appl. 70, No. 11 (A), 4079–4085 (2009, Zbl 1175.34076)] since $WPAA\left(R,\rho \right)=PAA\left(X\right)$ in this case.
##### MSC:
 47D06 One-parameter semigroups and linear evolution equations 34G10 Linear ODE in abstract spaces