Stepanov-like almost automorphic solutions for nonautonomous evolution equations. (English) Zbl 1146.47029

Authors’ abstract: “We study the convolution of Stepanov-like almost automorphic functions and \(L^1\) functions. Also we consider nonautonomous evolution equations, with a periodic operator coefficient and Stepanov-like almost automorphic forcing, and show that, under certain assumptions, any bounded mild solution is almost automorphic.”
A function \(f\in L_{\text{loc}}^p(\mathbb R,X)\), where \(X\) is a Banach space and \(1\leq p<\infty\), is said to be \(S^p\)-almost automorphic (=: \(S^p\) aa) or \(\in AS^p(X)\) if its Bochner transform \(f^b:\mathbb R\to L^p([0,1],X)\) is almost automorphic (in the sense of Bochner, =: aa), with \((f^b(t))(s):= f(t+s)\), \(t\in\mathbb R\), \(0\leq s\leq 1\). Several properties of \(AS^p\) and convolution \(\varphi\times f\) are given, \(y=g+\varphi\times y\) for \(g\in AS^p(\mathbb R)\) and \(\varphi\in L^1(\mathbb R,\mathbb R)\) with \(\int_{\mathbb R}|\varphi|\,dt<1\) has a unique solution \(\in AS^p(\mathbb R)\).
Theorem. Every bounded mild solution \(x\) of
\[ x'(t)= A(t)x(t)+ h(t) \tag \(*\) \]
on \(\mathbb R\) is aa, provided that \(h\in AS^p(X)\cap C(\mathbb R,X)\), \(A(t)\) generates an exponentially bounded one-periodic strongly continuous evolutionary process \((U(t,s))_{s\leq t}\) (\(U(t+1,s+1)= U(t,s)\), \(\|U(t,s)\|\leq Ke^{\omega(t-s)}\) if \(s\leq t\), \(\omega\), \(K\) positive constants), the spectrum of \(U(1,0)\) on the unit circle is countable, and \(X\) does not contain an isomorphic copy of \(c_0\).
This is shown with the help of the following Theorem. If \(h\in AS^p(X)\cap C(\mathbb R,X)\), then a bounded mild solution of \((*)\) is aa if and only if \((x(n))_{n\in\mathbb Z}\) is aa.


47D06 One-parameter semigroups and linear evolution equations
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
34G10 Linear differential equations in abstract spaces
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