Topics in almost automorphy. (English) Zbl 1073.43004

New York, NY: Springer (ISBN 0-387-22846-2/hbk). xii, 168 p. (2005).
The present book is a sequel to the author’s previous book [“Almost automorphic and almost periodic functions in abstract spaces” (Kluwer Academic/Plenum Publishers, New York) (2001; Zbl 1001.43001)]. The contents of the present book are as follows.
Chapter 1: Bochner integral, Sobolev spaces, (semi)groups of linear operators and infinitesimal generator \(A\), evolution equations (*) \(x'=Ax+f\), mild and classical solutions, almost automorphic \((=aa)\) Banach space valued functions (Bochner’s definition), asymptotically \(aa\) functions and dynamical systems, almost periodic \((=ap)\) \(f:\mathbb{R}\to\) locally convex Fréchet space \(E\) (the “perfect Fréchet spaces” of Definition 1.65, p. 36, are exactly those Fréchet spaces that do not contain \(c_0\) by Kadets and B. Basit and M. Emam [Ann. Pol. Math. 41, 193–201 (1983; Zbl 0536.43014)], Theorem 3.3, p. 200).
Chapter 2: \(Aa\) solutions of (*), of \(x'=(A+B)x\) with \(A\), \(B\) infinitesimal generators with a subspace \(S\) of range space = Hilbert space which reduces \(A\) and \(B\), and of \(x''+2Bx'+Ax=0\). Existence of an \(aa\) mild solution of \(x'=Ax+f(t,x)\), \(A\) generates an exponentially stable \(C_0\)-semigroup. Weak almost periodicity of an optimal \((\|x\|_\infty\) is minimal) solution \(x\) of (*) with \(f\) \(ap\), similarly for \(aa\). A correspondence between solutions of (*) and those of \(x'=Ax+f+g(t,x)\) if the group for \(A\) is uniformly bounded and \(g\) has with respect to \(x\) a sufficiently small Lipschitz constant.
Chapter 3: Fuzzy-set valued \(ap\) functions on \(\mathbb{R}\), Bohr’s = Bochner’s definition, existence of a mean, Fourier series, approximation theorem, (**) \(y'+y=c\cdot f\) in this context, \(c\) fuzzy value.
Chapter 4: Differentiability for functions with fuzzy values, \(aa\) and asymptotic \(aa\) function with fuzzy values, \(aa\) solutions of (**), and of the one-dimensional wave equation with fuzzy Cauchy data.
Missing definitions, phrases like “recall”, “it is well known” instead of references, incomplete proofs and misprints make much of the text understandable only for specialists, and sometimes (e.g., the proof of Theorem 2.25, p. 75) not even for those.


43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
47D06 One-parameter semigroups and linear evolution equations
03E72 Theory of fuzzy sets, etc.
34G10 Linear differential equations in abstract spaces
42A75 Classical almost periodic functions, mean periodic functions
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