\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 123, pp. 1--5.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/123\hfil Almost periodic solutions] {A note on almost periodic solutions of semilinear equations in Banach spaces} \author[L. Nguyen\hfil EJDE-2006/123\hfilneg] {Lan Nguyen} \address{Lan Nguyen \newline Department of Mathematics, Western Kentucky University, Bowling Green, KY 42101, USA} \email{Lan.Nguyen@wku.edu} \date{} \thanks{Submitted December 28, 2005. Published October 6, 2006.} \subjclass[2000]{34G10, 34K06, 47D06} \keywords{Abstract semilinear differential equations; $C_0$-semigroups} \begin{abstract} In this article, we generalize the main result obtained by Bahaj \cite{bs}. Also our proof is shorter than the original proof. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} This article concerns the semilinear equation $$\label{se} u'(t)+Au(t)= f(t, u(t)), \quad t\in\mathbb{R},$$ where $-A$ generates a $C_0$-semigroup on a Banach space $E$ and $f$ is a continuous function from $\mathbb{R}\times E$ to $E$. In \cite [Theorem 3.1]{bs}, the existence and uniqueness of almost periodic solutions to (\ref{se}) was established under the following conditions: \begin{itemize} \item[(i) ]$-A$ generates an analytic semigroup $(S(t))_{t\ge 0}$ on $X$ satisfying $\|T(t)\|\le e^{-\beta t}$ for some $\beta >0$; \item[(ii)]$f(t, x):\mathbb{R}\times D(A^{\alpha})\mapsto E$ satisfying \\ (A1) $f$ is uniformly almost periodic; \\ (A2) There are numbers $L>0$ (sufficiently small) and $0\le \theta \le 1$ such that $$\|f(t_1, x_1)-f(t_2, x_2)\|\le L(|t_1-t_2|^{\theta}+\|x_1-x_2\|_{\alpha})$$ for $t_1, t_2$ in $\mathbb{R}$ and $x_1, x_2$ in $D(A^{\alpha})$, where $D(A^{\alpha})$ ($\alpha \ge 0$) is the domain of the fractional power $A^{\alpha}$ with the norm $\|x\|_{\alpha}=\|A^{\alpha}x\|$. \end{itemize} % In this note we generalize that result to an operator $-A$, which generates a $C_0$ semigroup admitting an exponential dichotomy and to some subspaces of $BC(\mathbb{R})$, the Banach space of bounded, continuous function from $\mathbb{R}$ to $E$ with the sup-norm. Namely, we consider the following subspaces: $BUC(\mathbb{R})$, the space of bounded, uniformly continuous functions on $\mathbb{R}$; $AP(\mathbb{R})$, the space of almost periodic functions on $\mathbb{R}$; $P(\omega)$, the space of $\omega$-periodic functions; $C_1:=\{f\in BC(\mathbb{R}): \lim_{t\to \pm \infty}f(t)$ exists $\}$; $C_0: = \{f\in BC(\mathbb{R}): \lim_{t\to \pm \infty}f(t) =0$. \smallskip Recall, that a function $f(t):\mathbb{R}\mapsto E$ is called almost periodic if the set $\{f_s: s\in \mathbb{R}\}$ is relatively compact in $BC(\mathbb{R})$, where $f_s(\cdot):= f(s+\cdot)$ is the $s$-translation of $f$. Note that all above subspaces are Banach spaces with the sup-norm. In this note we prove the following theorem. \begin{theorem}\label{main} Let $-A$ generate an analytic $C_0$-semigroup $T(t)$ satisfying $\{i\lambda : \lambda \in \mathbb{R}\} \subset \varrho(-A)$, let $\mathcal{M}$ be one of the above mentioned subspaces of $BC(\mathbb{R})$, and let $f(t, x): \mathbb{R}\times D(A^{\alpha})\mapsto E$, where $0\le \alpha <1$, satisfy the following conditions \begin{itemize} \item[(B1)] For each $u\in \mathcal{M}$, the function $t\mapsto f(t, u(t))$ is in $\mathcal{M}$; \item[(B2)] For $u$ and $v$ in $D(A^{\alpha})$ we have $$\label{d} \|f(t, u)-f(t, v)\|\le L\|A^{\alpha}u-A^{\alpha}v\|.$$ \end{itemize} Then Equation \eqref{se} has a unique mild solution (defined below) in $\mathcal{M}$ for a sufficiently small $L$. Moreover, if $f(t, x)$ satisfies condition {\rm (B1)} and {\rm (A2)}, then this solution is a classical solution. \end{theorem} It is easy to see that the main result in \cite{bs} is a particular case of Theorem \ref{main}, when $\mathcal{M}=AP(\mathbb{R})$ and $\sigma(-A) \subset \{\lambda \in \mathbb{C}: Re\lambda <-\beta\}$ for some $\beta >0$. \section{Preparation and Proof of Theorem \ref{main}} To prove Theorem \ref{main}, we first consider the linear equation $$\label{li} u'(t)+Au(t)= f(t), \quad t\in \mathbb{R},$$ where $-A$ generates a semigroup $(T(t))_{t\ge 0}$. A continuous function $u$ is called a mild solution to (\ref{li}) if it satisfies $$u(t)=T(t-s)u(s) +\int_s^tT(t-\tau)f(\tau)d\tau, \quad t\ge s.$$ Similarly, a mild solution to (\ref{se}) is of the form $$u(t)=T(t-s)u(s) +\int_s^tT(t-\tau)f(\tau, u(\tau))d\tau, \quad t\ge s.$$ Suppose $\mathcal{M}$ is a closed subspace of $BC(\mathbb{R})$. We say that $\mathcal{M}$ is admissible with respect to (w.r.t. for short) Equation (\ref{li}) if for each function $f \in \mathcal{M}$, Equation (\ref{li}) has a unique mild solution $u\in \mathcal{M}$. Over the last two decades, the study of the admissibility of $BC(\mathbb{R})$ and the above mentioned subspaces w.r.t. Equation (\ref{li}) has been of increasing interest (see e.g. \cite{pruss} and \cite{vusc1}). Recently, to the nonautonomous equation $$\label{nonli} u'(t)+A(t)u(t)= f(t), \quad t\in \mathbb{R} ,$$ the admissibility of several spaces, such as $BUC(\mathbb{R})$, $L_p(\mathbb{R})$ and $AP(\mathbb{R})$ has also been intensively investigated (see e.g. \cite{hutter,larasc,masc} and references therein). In both cases, it is involved with the concept so-called \emph{exponential dichotomy} of a $C_0$-semigroup (or of an evolution family, in nonautonomous case). Recall, a $C_0$-semigroup $T(t)$ has an exponential dichotomy if there exist a projection operator $P \in B(E)$ and two numbers $M>0$, $\delta >0$ such that \begin{itemize} \item[(i)] $PT(t)=T(t)P$ for all $t \ge 0$; \item[(ii)] $\|T(t)Px\|\leq Me^{-\delta t}\|Px\|$ for all $x\in E$ and $t\ge 0$; \item[(iii)] $T(t)(I-P)$ extends to a $C_0$-group on $N(P)$, the nullspace of $P$, and $\|T(t)(I-P)x\|\leq Me^{\delta t}\|(I-P)x\|$ for all $x\in E$ and $t\le 0.$ \end{itemize} We have the following result (\cite[Theorem 4]{pruss}). \begin{theorem} \label{pruss} The following three statements are equivalent. \begin{itemize} \item[(i)] Operator $-A$ generates a $C_0$-semigroup, which admits an exponential dichotomy. \item[(ii)] For each function $f\in BC(\mathbb{R})$, Equation \eqref{li} has a unique mild solution in $BC(\mathbb{R})$. \item[(iii)] $S=\{\mu \in \mathbb{C}: |\mu|=1\} \subset \varrho(T(t))$ for one (all) $t>0$. \end{itemize} \end{theorem} In this case, the mild solution of Equation (\ref{li}) has the form $$u(t):= \int_{-\infty}^{\infty}G(t-s)f(s)ds, \quad \mbox{where } G(t):= \begin{cases}T(t)P &\mbox{for } t>0, \\ -T(t)(I-P) &\mbox{for } t<0 \end{cases}$$ which is the Green's kernel. Moreover, $u\in \mathcal{M}$ whenever $f\in \mathcal{M}$, where $\mathcal{M}$ is one of the above mentioned subspaces of $BC(\mathbb{R})$ (\cite[Theorem 5] {pruss}). If $A$ now generates an analytic semigroup, then $A^{\alpha}$ is given by $$A^{\alpha}x=A^{\alpha}Px+e^{\alpha \pi i}(-A)^{\alpha}(I-P)x.$$ We have the following lemma. \begin{lemma}\label{analytic} If $-A$ generates an analytic semigroup, then $u(t) \in D(A^{\alpha})$ for $0<\alpha <1$, and $\|\tilde{u}\|\le C\|f\|$, where $\tilde{u}(t):= A^{\alpha}u(t)$, for some $C>0$. \end{lemma} \begin{proof} First note that for each $t>0$, $A^{\alpha}T(t)$ is a bounded operator and $\|A^{\alpha}T(t)\|\le Mt^{-\alpha} e^{-\beta t}$ for some positive $M$ and $\beta$ (\cite[Theorem 2.6.13]{pazy}). Hence, $\int_0^{\infty}\|A^{\alpha}T(t)\|dt \leq M_1 <\infty$. Using this fact we have \begin{align*} \|A^{\alpha}u(t)\| &= \| \int_{-\infty}^{\infty}A^{\alpha}G(t-s)f(s)ds\|\\ &\leq \| \int_{-\infty}^tA^{\alpha}T(t-s)Pf(s)ds\| +\|\int_t^{\infty}(-A)^{\alpha}T(t-s)(I-P)f(s)ds\| \\ &= I_1+I_2, \end{align*} where $$I_1 \le \int_{-\infty}^t\|A^{\alpha}T(t-s)\|\cdot \|f\|ds =\int_0^{\infty}\|A^{\alpha}T(s)\|ds\cdot \|f\| \le M_1\|f\|$$ and \begin{align*} I_2 &\le \int_t^{\infty}\|(-A)^{\alpha}T(t-s)(I-P)f(s)\|ds\\ &= \int_0^{\infty}\|(-A)^{\alpha}T(-s')(I-P)f(s'+t)\|ds'\\ &\le \int_0^{\infty}\|(-A)^{\alpha}T(-s')\| \cdot \|(I-P)f(s'+t)\|ds'\\ &\le M_1 \|f\|. \end{align*} Hence, $\|A^{\alpha}u(t)\|\le 2 M_1\|f\|$ for each $t\in \mathbb{R}$. \end{proof} We now turn to (\ref{se}). First, we state a preliminary result. \begin{lemma}\label{preli} Let $-A$ generate a $C_0$-semigroup and $B$ be an invertible operator on $E$, and $\mathcal{M}$ be a closed subspace of $BC(\mathbb{R})$ with the property: $\mathcal{M}$ is admissible w.r.t. \eqref{li} and $\tilde{u}(\cdot):= Bu(\cdot)\in \mathcal{M}$ and $\|\tilde{u}\|\le C\|f\|$ for each $f\in \mathcal{M}$. Moreover, suppose $f(t, x):\mathbb{R}\times D(B)\mapsto E$ satisfying \begin{itemize} \item[(B1)] For every $u\in \mathcal{M}$, the function $t\mapsto f(t, u(t))$ is in $\mathcal{M}$; \item[(B2)] For $u$ and $v$ in $D(B)$ we have $$\label{d2} \|f(t, u)-f(t, v)\|\le L\|Bu-Bv\|.$$ \end{itemize} Then Equation \eqref{se} has a unique mild solution in $\mathcal{M}$ for $L$ small enough. \end{lemma} \begin{proof} Let $K: \mathcal{M}\mapsto \mathcal{M}$ be the operator defined as follows: For each $f\in \mathcal{M}$, $Kf$ is the unique mild solution to (\ref{li}). Then $K$ is a linear and bounded operator on $\mathcal{M}$. For each $u\in \mathcal{M}$ put $\tilde{u}(t):= f(t, B^{-1}u(t))$. Define the map $\tilde{K}:\mathcal{M}\mapsto \mathcal{M}$ by $$(\tilde{K}u)(t):= B(K\tilde{u})(t).$$ By the assumption, $B(K\tilde{u})(\cdot)$ also belongs to $\mathcal{M}$. Hence $\tilde{K}$ is well defined. If $u$ and $v$ are in $\mathcal{M}$, we have \begin{align*} \|(\tilde{K}u)(t)-(\tilde{K}v)(t)\| &= \|B(K\tilde{u})(t)-B(K\tilde{v})(t)\| \\ &= \|B[(K\tilde{u})(t)-(K\tilde{v})(t)]\| \\ &\le C\cdot \mbox{sup}_{t\in \mathbb{R}}\|f(t, B^{-1}u(t))-f(t,B^{-1} v(t)\| \\ &\le C \cdot L \|u-v\|. \end{align*} Hence $\|\tilde{K}u-\tilde{K}v\| \le C \cdot L\|u-v\|$. So $\tilde{K}$ is a contraction map for sufficiently small $L$. Let $\phi(t)$ be the unique fixed point of $\tilde{K}$, then it is easy to see that $u(t)=B^{-1}\phi(t)$ is the unique mild solution of (\ref{se}) in $\mathcal{M}$. \end{proof} \begin{proof}[Proof of Theorem \ref{main}] Since $-A$ generates an analytic semigroup, the spectral mapping theorem holds, i.e., $\sigma(T(t))= e^{t \sigma (-A)}$ (\cite[Corollary III.3.12]{nagel}). Hence, condition $\{i\lambda : \lambda \in \mathbb{R}\} \subset \varrho(-A)$ implies that $(T(t))$ admits an exponential dichotomy, and hence, space $BC(\mathbb{R})$ is admissible w.r.t. Equation (\ref{li}). Define the operator $\bar{K}: BC(\mathbb{R})\mapsto BC(\mathbb{R})$ by follows: for each $f\in BC(\mathbb{R})$, $(\bar{K}f)(t):=A^{\alpha}u(t)$, where $u(t)$ is the unique solution to (\ref{li}).Then $\bar{K}$ is a linear and, by Lemma \ref{analytic}, bounded operator. We now apply Lemma \ref{preli} with $B=A^{\alpha}$, and it suffices us to complete the proof by showing that $\bar{K}$ leaves all above mentioned subspaces of $BC(\mathbb{R})$ invariant. Let $f_t(\cdot):=f(\cdot +t)$ be the left translation of a function $f$. It is easy to see that $K(f_t)=(Kf)_t$, and this yields $\bar{K}(f_t) = (\bar{K}f)_t$. Hence, $P(\omega)$ and $AP(\mathbb{R})$ are invariant w.r.t. $\bar{K}$. Moreover, $\|(\bar{K}f)_t-\bar{K}f\|=\|\bar{K}f_t-\bar{K}f\| \le \|\bar{K}\|\cdot\|f_t-f\|$, which shows that $BUC(\mathbb{R})$ is also invariant w.r.t. $\bar{K}$. Finally, since $Kf(\pm \infty)=A^{-1}f(\pm \infty)$, we have $\bar{K}f(\pm\infty) =A^{\alpha -1}f(\pm\infty)$, and this proves that $\bar{K}$ leaves $C_1$ and $C_0$ invariant. \end{proof} \begin{thebibliography}{99} \bibitem{bs} M. Bahaj, O. Sidki: \emph{Almost periodic solutions of semilinear equations with analytic semigroups in Banach spaces}. 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