Almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations
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 \(x'(t)=A(t)x(t)+f(t,x(t))\) in a Banach space. The main results are Theorems 3.2 and 4.2. However, Theorem 3.2 can be seen from [\textit{H.-S.\thinspace Ding, W.\,Long} and \textit{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 \(WPAA(R,\rho)\) is needed. But from Lemma 2.10 in [\textit{J.\,Blot, G.\,M.\thinspace Mophou, G.\,M.\,N'Guérékata} and \textit{D.\,Pennequin}, Nonlinear Anal., Theory Methods Appl.\ 71, No.\,3--4 (A), 903--909 (2009, Zbl 1177.34077)], one only knows that the space \(WPAA(R,\rho)\) 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 \(WPAA(R,\rho)\) 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 [\textit{T.-J.\thinspace Xiao, X.-X.\thinspace Zhu} and \textit{J.\,Liang}, Nonlinear Anal., Theory Methods Appl.\ 70, No.\,11 (A), 4079--4085 (2009, Zbl 1175.34076)] since \(WPAA(R,\rho) = PAA(X)\) in this case.
scientific article; zbMATH DE number 5655674