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New almost periodic type functions and solutions of differential equations. (English) Zbl 0880.43009

The authors consider the classes $A{{\Pi }}_{c}:={{\Pi }}_{c}+{{\Pi }}_{0}$ and $A{{\Pi }}_{r}:={{\Pi }}_{r}+{{\Pi }}_{0}$, where ${{\Pi }}_{0}$ is either ${C}_{0}\left(J,X\right)=\left\{\varphi :J\to X$ continuous with $\varphi \left(t\right)\to 0$ as $|t|\to \infty \right\}$, $PA{P}_{0}\left(J,X\right):=\left\{\varphi \in {C}_{b}\left(J,X\right):M\left(|\varphi |\right):={lim}_{t\to \infty }{\int }_{0}^{t}|\varphi \left(s\right)|ds=0\right\}$ or $WA{P}_{0}\left(J,X\right):=\left\{\varphi$ weakly Eberlein almost periodic $\in {C}_{b}\left(J,X\right):M\left(|\varphi |\right)=0\right\}$, with ${C}_{b}=$ bounded continuous functions, $J=ℝ$ or $\left[0,\infty \right)$, $X$ Banach space; ${{\Pi }}_{c}\left(ℝ,X\right)$ denotes any translation invariant closed linear subspace of ${C}_{b}\left(ℝ,X\right)$ containing all constants and with $\varphi$ also ${\gamma }_{\lambda }·\varphi$, ${\gamma }_{\lambda }\left(t\right):={e}^{i\lambda t}$, $\lambda \in ℝ$, and such that $\varphi \to \varphi \mid \left[0,\infty \right)$ is isometric; ${{\Pi }}_{r}$ denotes a ${{\Pi }}_{0}$ space containing only recurrent functions; ${{\Pi }}_{c}\left(J,X\right):={{\Pi }}_{c}\left(ℝ,X\right)\mid J$. These classes subsume various generalized almost periodic and almost automorphic $\left(aa\right)$ functions, e.g. ${{\Pi }}_{r}=$ almost periodic functions or $=AA:=$ uniformly continuous bounded Levitan - a.p. functions. It is shown that the $+$ in $A{\Pi }$ is direct, $A{{\Pi }}_{r}$ and ${{\Pi }}_{c}+{C}_{0}$ are closed subspaces of ${C}_{b}$ and closed with respect to convolution with ${L}^{1}$. If $\varphi \in A{{\Pi }}_{r}$ and $P\varphi \left(t\right):={\int }_{0}^{t}\varphi ds$ is ergodic, then $P\varphi \in A{{\Pi }}_{r}$. When the Beurling spectrum of a uniformly continuous bounded $\varphi$ has positive distance from 0, then if $\varphi \in A{{\Pi }}_{r}\left(J,X\right)$ [resp. all differences $\varphi \left(·+s\right)-\varphi \left(·\right)\in A{{\Pi }}_{r}\right]$, then $P\varphi \in A{{\Pi }}_{r}$ [resp. $\varphi \in A{{\Pi }}_{r}$].

As an application, if $y$ is a bounded solution on $J$ of ${y}^{\text{'}\text{'}}+{a}_{1}{y}^{\text{'}}+{a}_{0}y=\varphi \in A{\Pi }\left(J,X\right)$ with ${\left(i\mu \right)}^{2}+i\mu {a}_{1}+{a}_{0}\ne 0$ for all real $\mu$, then $y\in A{{\Pi }}_{r}\left(J,X\right)$; for systems ${Y}^{\text{'}}=AY+\varphi +\mu G\left(t,Y\left(t\right)\right)$ where the matrix $A$ has no purely imaginary eigenvalues and $\varphi$, $G$ are suitably asymptotic $aa$, the existence of a unique $aaa$ solution for sufficiently small $\mu$ is shown. Similarly, a bounded solution $u$ of the quasilinear heat equation ${u}_{xx}={u}_{t}+\psi \left(x,t,u\right)$ with initial data ${u|}_{t=0}$ $aaa$ is again $aaa$. (In theorem 4.6, p. 1146, instead of “(4.3)” it should read “(4.4) with $\lambda =i\nu$ and $\psi$ instead of $\varphi$”).

##### MSC:
 43A60 Almost periodic functions on groups, etc.; almost automorphic functions 28B05 Vector-valued set functions, measures and integrals (measure theory) 43A45 Spectral synthesis on groups, semigroups, etc. 34A12 Initial value problems for ODE, existence, uniqueness, etc. of solutions 35K55 Nonlinear parabolic equations