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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 $C_0(J,X)=\{\varphi: J\to X$ continuous with $\varphi (t)\to 0$ as $|t|\to\infty\}$, $PAP_0 (J,X):=\{\varphi\in C_b(J,X): M(|\varphi|):=\lim_{t\to\infty}\int^t_0|\varphi(s)|ds=0\}$ or $WAP_0(J,X):=\{\varphi$ weakly Eberlein almost periodic $\in C_b (J,X): M(|\varphi|)=0\}$, with $C_b=$ bounded continuous functions, $J=\bbfR$ or $[0,\infty)$, $X$ Banach space; $\Pi_c(\bbfR,X)$ denotes any translation invariant closed linear subspace of $C_b(\bbfR,X)$ containing all constants and with $\varphi$ also $\gamma_\lambda\cdot\varphi$, $\gamma_\lambda(t):=e^{i\lambda t}$, $\lambda\in\bbfR$, and such that $\varphi\to\varphi\mid[0,\infty)$ is isometric; $\Pi_r$ denotes a $\Pi_0$ space containing only recurrent functions; $\Pi_c(J,X):=\Pi_c(\bbfR,X)\mid J$. These classes subsume various generalized almost periodic and almost automorphic $(aa)$ 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 $P\varphi(t):=\int^t_0\varphi ds$ 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 $\varphi\in A\Pi_r(J,X)$ [resp. all differences $\varphi (\cdot+s)-\varphi(\cdot)\in A\Pi_r]$, 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 $y''+a_1y'+a_0 y=\varphi\in A\Pi(J,X)$ with $(i\mu)^2+i\mu a_1+a_0\ne 0$ for all real $\mu$, then $y\in A\Pi_r(J,X)$; for systems $Y'=AY+\varphi+\mu G(t,Y(t))$ 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 $u_{xx}=u_t+\psi(x,t,u)$ 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 $\varphi$”).

43A60Almost periodic functions on groups, etc.; almost automorphic functions
28B05Vector-valued set functions, measures and integrals (measure theory)
43A45Spectral synthesis on groups, semigroups, etc.
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
35K55Nonlinear parabolic equations
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