\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 35, pp. 1--7.\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 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/35\hfil Boundeness and Stepanov's almost periodicity]
{Boundeness and Stepanov's almost periodicity of solutions}

\author[Z. Hu\hfil EJDE-2005/35\hfilneg]
{Zuosheng Hu}

\address{School of Mathematics and Statistics, Carleton University \\
Ottawa, Ontario, K1S 5B6, Canada}
\email{zshhu@yahoo.com}

\date{}
\thanks{Submitted January 17, 2005. Published March 24, 2005.}
\subjclass[2000]{34C27, 34A30}
\keywords{Boundedness; Stepanov's almost periodicity}

\begin{abstract}
 In this paper, we establish a necessary condition of Stepanov's almost
 periodicity of solutions for general linear almost periodic systems,
 and then we construct an example of linear system in which all
 solutions are bounded, but any non-trivial solution is not Stepanov's
 almost periodic.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}

\section{Introduction}

It is well-known that for (Bohr) almost periodic differential equations
\begin{equation}
x'=A(t)x+f(t),  \label{equation}
\end{equation}
boundedness of solutions does not imply their almost periodicity. Conley and
Miller \cite{ref2} gave an example of an equation (\ref{equation}) with
$n=1 $ where a bounded solution is not almost periodic. In \cite{ref5},
Mingarelli, Pu and Zheng constructed an example, for each $n>1$, of an
equation (\ref{equation}) with almost periodic coefficients in which there
exists a bounded solution which is not almost periodic. In \cite{ref4}, Hu
and Mingarelli constructed a class of linear almost periodic systems in
which all solutions are bounded but there still exists no any non-trivial
solution which is almost periodic. The question arising naturally is whether
boundedness of solutions can imply their Stepanov's almost periodicity which
is a weaker almost periodicity defined by Stepanov (see \cite{ref1} for the
details). As far as we know, this is an open problem. In this paper, in
order to solve this open problem, we establish a necessary condition of
Stepanov's almost periodicity of solutions for general linear almost
periodic systems, and then we construct an example of linear system in which
all solutions are bounded, but any non-trivial solution is not Stepanov's
almost periodic. So, the answer of this problem is negative.

For completeness, we recall the definition of the Stepanov norm $S_l(f)$ of
a function $f$ $\in $ $L_1^{\rm loc}(\mathbb{R},X)$. The quantity
\[
S_l(f)=\sup_{t\in \mathbb{R}}\frac 1l\int_t^{t+l}\|f(s)\|ds\,
\]
where $l>0$ is some constant, is the Stepanov norm (or S$_l$-norm) of $f$.

Replacing the supremum norm by the S$_{l}$-norm in the definition of
continuity (respectively, uniform continuity, boundedness) of $f$, we can
introduce the concept of S$_{l}$-continuity (respectively, S$_{l}$-uniform
continuity, S$_{l}$-boundedness) of $f$. For example, we call $f\in
L_{1}^{\rm loc}(\mathbb{R},X)$ S$_{l}$-bounded if there exists a constant $M>0$ such that
$S_{l}(f)\leq M$. It is easy to show that $S_{l}$-boundedness ($S_{l}$%
-continuity, S$_{l}$-uniform continuity) is not dependent on the constant $l$
(see \cite{ref1}). So, we simply call such functions $S$-bounded, $S$%
-continuous, and $S$-uniformly continuous whenever these notions apply.

We define $S_{l}(t,f)$ as follows:
\begin{equation}
S_{l}(t,f)=\frac{1}{l}\int_{t}^{t+l}\|f(s)\|ds\quad \text{for all $t\in \mathbb{R}$}.
\label{snorm}
\end{equation}
 From (\ref{snorm}) we have that for any $t,s\in \mathbb{R}$,
\[
S_{l}(t,f_{s})=S_{l}(t+s,f).
\]
where $f_{s}$ is the translate of $f$. We use $S_{l}C(\mathbb{R},X)$ to denote the
set of all S$_{l}$-continuous functions. Obviously, $C(\mathbb{R},X)\subset $ $%
S_{l}C(\mathbb{R},X)$. Because $l$ can be taken any positive real number, we simply
denote $S_{l}C(\mathbb{R},X)$ by $SC(\mathbb{R},X)$. As in the case of Bohr almost periodic
functions, we introduce the definition of a Stepanov almost periodic
function.

\begin{definition} \label{de21} \rm
Let $f\in S_lC(\mathbb{R},X)$. If for any sequence $\{ \alpha
_n\} \subset \mathbb{R}$, there exist a subsequence $\{ \alpha _n'\} $ of $\{ \alpha _n\} $
and a function $g\in S_lC(\mathbb{R},X)$ such that
\[
\lim_{n\to \infty }S_l(t,f_{\alpha _n'}-g)=0,\quad \text{uniformly on } \mathbb{R},
\]
then $f$ is called S$_l$-almost periodic on $\mathbb{R}$.
\end{definition}

We use the notation in \cite{ref3}. $\alpha =\{\alpha _{n}\}$ is a
sequence of real numbers. $\alpha '\subset \alpha $ means that
$\alpha '=\{\alpha _{n}'\}\subset \{\alpha _{n}\}$ is a
subsequence of $\alpha $. For $f,g\in \mathbb{R}, ST_{\alpha }f=g$
means that there exist a sequence $\alpha $ and a real number
$l>0$ such that
\begin{equation}
\lim_{n\to \infty }\frac{1}{l}\int_{t}^{t+l}\|f(s+\alpha
_{n})-g(s)\|ds=0,  \label{hullcodition}
\end{equation}
pointwise for $t\in \mathbb{R}$ and $UST_{\alpha }f=g$ means that
(\ref{hullcodition}) holds uniformly on $t\in \mathbb{R}$.

Now we give the definition of the \textit{uniform Stepanov hull} of a
Stepanov almost periodic function.

\begin{definition} \rm
Let $f\in SC(\mathbb{R},X)$. The set
\[
\{g\in SC(\mathbb{R},X):\text{ there exists a sequence }\{ \alpha _n\}
\subset \mathbb{R}\text{ such that }UST_\alpha f=g\}
\]
is called the \emph{uniform Stepanov hull}, or simply \emph{uniform S-hull},
and is denoted by $\mathcal{SH}(f)$.
\end{definition}

Obviously, for any $f\in SC(\mathbb{R},X)$, $\mathcal{SH}(f)$ is not empty since $f\in
\mathcal{SH}(f)$.

\section{Necessary Condition}

\ Consider the general system of linear differential equations
\begin{equation}
x'=A(t)x+f(t),  \label{equation1}
\end{equation}%
where $x\in \mathbb{R}^{n}$, $A(t)$ is an $n\times n$ matrix function, and $f(t)$ is
an $n$-dimensional vector function, defined on $\mathbb{R}$. Throughout this paper,
we assume that $A(t)$ and $f(t)$ are all Bohr's almost periodic on $\mathbb{R}$.

\begin{lemma} \label{lem1}
Suppose that $\phi (t)$ is a solution of \eqref{equation1} and that
there exist a sequence $\alpha =\{\alpha _{n}\}$,
$g\in SH(f)$, $B\in SH(A)$ and $\varphi \in SH(\phi )$ such that
$UST_{\alpha}A=B,\ UST_{\alpha }f=g$ and $UST_{\alpha }\phi =\varphi $.
Then there exist a subsequence $\alpha '$ of $\alpha $ and a solution
$\tilde{\varphi}(t)$ of
\begin{equation}
x'=B(t)x+g(t),  \label{limitequation}
\end{equation}
such that $UST_{\alpha '}\phi =\tilde{\varphi}$ on $\mathbb{R}$. If $\phi $
is bounded, so is $\tilde{\varphi}$ with the same bound.
\end{lemma}

\begin{proof} By the assumption, there is a constant $l>0$ such
that
\begin{gather*}
\lim_{n\to \infty }\frac{1}{l}\int_{t}^{t+l}|A(s+\alpha
_{n})-B(s)|ds=0, \\
\lim_{n\to \infty }\frac{1}{l}\int_{t}^{t+l}|f(s+\alpha
_{n})-g(s)|ds=0, \\
\lim_{n\to \infty }\frac{1}{l}\int_{t}^{t+l}|\phi (s+\alpha
_{n})-\varphi (s)|ds=0,
\end{gather*}
uniformly on $\mathbb{R}$. In particular, for each $t\in \mathbb{R}$ it follows that
$\lim_{n\to \infty }\int_{0}^{l}|\phi (s+t+\alpha _{n})-\varphi
(t+s)|ds=0$; in other words, the sequence $\{\phi (t+s+\alpha _{n})\}$ in
$L^{1}[0,l]$ converges to $\varphi (t+s)$, and hence the sequence converges
to $\varphi (t+s)$ in the sense of measure on $[0,l]$. Since $t\in \mathbb{R}$ is
arbitrary, the sequence of measurable functions
$\{\phi (\tau +\alpha_{n})\} $ converges to $\varphi (\tau )$ in the sense
of measure on $\mathbb{R}$;
hence from a well known result in the theory of Lebesgue measure we know
that there is a subsequence $\alpha '$ of $\alpha $ such that
$\lim_{n\to \infty }\phi (\tau +\alpha _{n}')=\varphi (\tau )$ a.e. on $\mathbb{R}$.
Take a point $t_{0}$ in $\mathbb{R}$ such that
$\lim_{n\to \infty }\phi (t_{0}+\alpha _{n}')=\varphi (t_{0})$. Notice that
\begin{gather*}
\lim_{n\to \infty }\int_{t_{0}}^{t}|A(s+\alpha _{n}')-B(s)|ds=0, \\
\lim_{n\to \infty }\int_{t_{0}}^{t}|f(s+\alpha _{n}')-g(s)|ds=0, \\
\lim_{n\to \infty }\int_{t_{0}}^{t}|\phi (s+\alpha _{n}')-\varphi (s)|ds=0, \\
\end{gather*}
locally uniformly for $t\in \mathbb{R}$ (which means the uniformity of convergence on
any finite interval in $\mathbb{R}$). Since $\phi $ is a solution of \eqref{equation1},
we get the following relations:
\begin{equation}
\phi (t+\alpha _{n}')=\phi (t_{0}+\alpha _{n}')+\int_{t_{0}}^{t}
\{A(s+\alpha _{n}')\phi (s+\alpha _{n}')+f(s+\alpha _{n}')\}ds,
\label{solution}
\end{equation}
for $t\in \mathbb{R}$, $n=1,2,\dots $.
Note that $A(t)$ and $f(t)$ are bound on $\mathbb{R}$ because of Bohr's almost
periodicity of $A(t)$ and $f(t)$. Then, by Gronwall's inquality one can see
that $\{\phi (t+\alpha _{n}')\}$ is uniformly bounded on each
finite interval in $\mathbb{R}$. From these facts we see that
\[
\lim_{n\to \infty }\int_{t_{0}}^{t}\{A(s+\alpha _{n}')\phi
(s+\alpha _{n}')+f(s+\alpha _{n}')\}ds=\int_{t_{0}}^{t}\{B(s)\varphi (s)+g(s)\}ds,
\]
locally uniformly for $t\in \mathbb{R}$. Define a (continuous) function
$\tilde{\varphi}(t)$ by
\[
\tilde{\varphi}(t)=\varphi (t_{0})+\int_{t_{0}}^{t}\{B(s)\varphi
(s)+g(s)\}ds,\quad t\in \mathbb{R}.
\]
 From (\ref{solution}) we see that $\lim_{n\to \infty }\phi (t+\alpha
_{n}')=\tilde{\varphi}(t)$, locally uniformly for $t\in \mathbb{R}$.
Consequently, it follows that $\varphi (t)\equiv \tilde{\varphi}(t)$ a.e. on
$\mathbb{R}$. Hence
\[
\lim_{n\to \infty }\frac{1}{l}\int_{t}^{t+l}|\phi (s+\alpha
_{n}')-\tilde{\varphi}(s)|ds=\lim_{n\to \infty }\frac{1}{l}%
\int_{t}^{t+l}|\phi (s+\alpha _{n}')-{\varphi }(s)|ds=0,
\]
uniformly on $\mathbb{R}$, which shows $UST_{\alpha '}\phi =\tilde{\varphi}$.
Furthermore, we get
\begin{align*}
\tilde{\varphi}(t) &=\varphi (t_{0})+\int_{t_{0}}^{t}\{B(s)\varphi
(s)+g(s)\}ds \\
&=\tilde{\varphi}(t_{0})+\int_{t_{0}}^{t}\{B(s)\tilde{\varphi}%
(s)+g(s)\}ds,\quad t\in \mathbb{R},
\end{align*}%
which shows that $\tilde{\varphi}$ is a solution of (\ref{limitequation}).
The last conclusion is obvious. This completes the proof of Lemma.
\end{proof}

\noindent\textbf{Remark.} According to Lemma \ref{lem1}, if $\phi $ is a solution
of (\ref{equation1}) and assumptions are satisfied, we can simply say that
$UST_{\alpha }\phi $ is a solution of (\ref{limitequation}).

\begin{theorem} \label{thm1}
 Let $A(t)$ be Bohr's almost periodic on $\mathbb{R}$. If $x(t)$ is a non-trivial
Stepanov's almost periodic solution of the equation
\begin{equation}
x'=A(t)x,  \label{equation4}
\end{equation}%
then for any $l>0$,
\begin{equation}
\inf_{t\in \mathbb{R}}\frac{1}{l}\int_{t}^{t+l}\|x(s)\|ds>0  \label{equation5}
\end{equation}
\end{theorem}

\begin{proof} On the contrary, suppose that there exists a
real number $l>0$ such that (\ref{equation5}) does not hold, i. e.
\begin{equation}
\lim_{t\in \mathbb{R}}\frac{1}{l}\int_{t}^{t+l}\|x(s)\|ds=0.  \label{equation6}
\end{equation}
Then we will find a contradiction. From (\ref{equation6}), we can pick up a
sequence $\alpha =\{\alpha _{n}\}$ such that
\[
\lim_{n\to \infty }\frac{1}{l}\int_{\alpha _{n}}^{\alpha
_{n}+l}\|x(s)\|ds=0
\]
or
\begin{equation}
\lim_{n\to \infty }\frac{1}{l}\int_{0}^{l}\|x(s+\alpha _{n})\|ds=0.
\label{equation7}
\end{equation}
Since $A(t)$ is almost periodic on $\mathbb{R}$ and $x(t)$ is Stepanov's almost
periodic on $\mathbb{R}$, we can extract a subsequence
$\alpha '\subset\alpha $, $B(t)\in \mathcal{H}(A)$ and
$y(t)\in \mathcal{SH}(x)$ such that $y=UST_{\alpha '}x$ and
$B=UT_{\alpha '}A$. By Lemma \ref{lem1}, there
exist a subsequence $\alpha ''\subset \alpha '$
and a solution $\tilde{y}$ of the equation
\begin{equation}
y'=B(t)y  \label{equation8}
\end{equation}
such that
\begin{equation}
\tilde{y}=UST_{\alpha ''}x  \label{equation9}
\end{equation}
on $\mathbb{R}$. On the other hand, we have
\begin{equation}
\frac{1}{l}\int_{o}^{l}|\tilde{y}(s)|ds\leq \frac{1}{l}\int_{0}^{l}
|\tilde{y}(s)-x(s+\alpha _{n}'')|ds+\frac{1}{l}\int_{0}^{l}
|x(s+\alpha_{n}'')|ds  \label{quation10}
\end{equation}
Let $n\to \infty $, we obtain that
\[
\frac{1}{l}\int_{0}^{l}|\tilde{y}(s)|ds=0
\]
from (\ref{equation7}) and (\ref{equation9}). And thus, there exists at
least one $t_{0}\in [ 0,l]$ such that $\tilde{y}(t_{0})=0$. Since
$\tilde{y}(t)$ is a solution of (\ref{equation8}), we have
$\tilde{y}(t)=0$ for all $t\in \mathbb{R}$.  Then  (\ref{equation9}) implies
\[
\lim_{n\to \infty } \frac{1}{l} \int_{\tau }^{\tau +l}
\|x(s+\alpha_{n}'')\|ds=0
\]
uniformly for $\tau \in \mathbb{R}$; hence
\begin{align*}
\frac{1}{l}\int_{0}^{l}\|x(t+s)\|ds
&=\lim_{n\to \infty }\frac{1}{l}%
\int_{0}^{l}\|x(t-\alpha ''+s+\alpha _{n}'')\|ds \\
&=\lim_{n\to \infty }\frac{1}{l}\int_{t-\alpha ''}^{t-\alpha ''+l}
\|x(s+\alpha _{n}'')\|ds
=0
\end{align*}
and consequently, $x(t+s)=0$ on $[0,\;l]$ for any $t\in \mathbb{R}$. Since $t\in \mathbb{R}$
is arbitrary, we must have $x(t)\equiv 0$ on $\mathbb{R}$, which is a contradiction
to the fact that $x(t)$ is a non-trivial solution of (\ref{equation8}). This
completes the proof of this theorem.
\end{proof}

\begin{corollary} \label{coro1}
Let $a(t)$ be a scalar almost periodic function defined on $\mathbb{R}$. If each
bounded solution of the equation
\begin{equation}
x'=a(t)x  \label{equation10}
\end{equation}
is Stepanov's almost periodic on $\mathbb{R}$, then
\begin{equation}
\sup_{t\in \mathbb{R}}\int_0^ta(s)ds<\infty   \label{equation11}
\end{equation}
implies that for any real number $l>0$,
\begin{equation}
\inf_{t\in \mathbb{R}}\Big( \sup_{s\in [t,t+l]}\int_0^sa(\tau )d\tau \Big)
>-\infty .  \label{equation12}
\end{equation}
\end{corollary}

\begin{proof} Suppose that (\ref{equation11}) holds. Then
$x(t)=\exp\big(\int_{0}^{t}a(s)ds\big)$
is a non-trivial bounded solution of (\ref{equation10}), so it is Stepanov's
almost periodic. By Theorem \ref{thm1}, for any $l>0$,
\begin{equation}
\inf_{t\in \mathbb{R}}\frac{1}{l}\int_{t}^{t+l}e^{\int_{0}^{s}a(\tau )d\tau }ds>0.
\label{equation14}\/,.
\end{equation}
Now we show that for any $l>0$,
\[
\inf_{t\in \mathbb{R}}\sup_{s\in [t,t+l]}\int_0^sa(\tau )d\tau >-\infty .
\]
Otherwise, we can pick up a sequence $\{t_n\}\subset \mathbb{R}$ such that
\[
\sup_{s\in [t_n,t_n+l]}\int_0^sa(\tau )d\tau \le -n,\quad n=1,2,\dots .
\]
So,
\[
\int_0^sa(\tau )d\tau \le -n
\]
for all $s\in [t_n,t_n+l],n=1,2,\dots $. Hence,
\[
\frac 1l\int_{t_n}^{t_n+l}e^{\int_0^sa(\tau )d\tau }ds\le \frac
1l\int_{t_n}^{t_n+l}e^{-n}ds=e^{-n},\quad n=1,2,\dots .
\]
This implies that
\[
\lim_{n\to \infty }\frac 1l\int_{t_n}^{t_n+l}e^{\int_0^sa(\tau )d\tau }ds=0
\]
This contradicts (\ref{equation14}) and the proof of this corollary is
completed.
\end{proof}

\begin{corollary} \label{coro2}
Let $a(t)$ be an almost periodic function defined on $\mathbb{R}$. If there exists a
positive constant $M$ such that $\int_0^ta(s)ds\leq M$ and
\[
\inf_{t\in \mathbb{R}}\frac 1l\int_t^{t+l}e^{\int_0^sa(\tau )ds}ds=0
\]
then the function $\exp\big(\int_0^ta(s)ds\big)$ is not Stepanov's almost
periodic on $\mathbb{R}$.
\end{corollary}

\subsection*{Example}

According to Corollary \ref{coro2}, we can construct many examples of equations
whose solutions are bounded, but not Stepanov's almost periodic on $\mathbb{R}$.
 To construct such an example, let $n\geq 3$ and define
\[
g_{n}(t)=\begin{cases}
0 & t\in [ 0,1]\cup [ 2^{n-1}-1,2^{n-1}] \\
-n/(2^{n-1}-1) & t\in [ 2,2^{n-1}-2] \\
\text{linear} & t\in [ 1,2]\cup [ 2^{n-1}-2,2^{n-1}-1].
\end{cases}
\]
Now, extend $g_n(t)$ to be odd and periodic with period $2^n$.
Then, $g_n(t)$ satisfies
\begin{gather}
\int_0^tg_n(s)ds\le 0, \quad \text{for all } t\in \mathbb{R};  \\
\sup_{t\in \mathbb{R}}|g_n(t)|=\frac n{2^{n-1}-1}; \\
\int_0^tg_n(s)ds=-n\frac{2^{n-1}-3}{2^{n-1}-1}, \quad \text{for all }
 t\in [2^{n-1}-1,2^{n-1}]
\end{gather}
for each $n\in Z^{+}$. Since
\[
\sum_{n=3}^\infty \sup_{t\in \mathbb{R}}|g_n(t)|=\sum_{n=3}^\infty \frac
n{2^{n-1}-1}<\infty \,,
\]
the function
\[
g(t)=\sum_{n=3}^\infty g_n(t)
\]
is almost periodic on $\mathbb{R}$ and
\begin{equation}
\int_0^tg(s)ds=\sum_{n=3}^\infty \int_0^tg_n(s)ds\le 0,\quad t\in \mathbb{R}.
\label{equation18}
\end{equation}

 Now, let $l=1$, $t_{n}=2^{n-1}-1$, then
\begin{equation}
\begin{aligned}
\frac{1}{l}\int_{t_{n}}^{t_{n}+l}e^{\int_{0}^{s}g(\tau )d\tau }ds
&=\int_{2^{n-1}-1}^{2^{n-1}}e^{\int_{0}^{s}g(\tau )d\tau }ds \\
&\leq \int_{2^{n-1}-1}^{2^{n-1}}e^{\int_{0}^{s}g_{n}(\tau )d\tau }ds
\\
&=\int_{2^{n-1}-1}^{2^{n-1}}e^{-n\frac{2^{n-1}-3}{2^{n-1}-1}}ds   \\
&= e^{-n\frac{2^{n-1}-3}{2^{n-1}-1}}
\end{aligned}
\end{equation}
for each $n\in Z^{+}$. So,
\[
\lim_{n\to \infty }\frac{1}{l}\int_{t_{n}}^{t_{n}+l}e^{%
\int_{0}^{s}g(\tau )d\tau }ds=0.
\]
This implies
\[
\inf_{t\in \mathbb{R}}\frac{1}{l}\int_{t}^{t+l}e^{\int_{0}^{s}g(\tau )d\tau }ds=0.
\]
By Corollary \ref{coro2}, the function
$\exp\big(\int_{0}^{t}g(s)ds\big)$ is not Stepanov,s almost periodic on $\mathbb{R}$.
Therefore, all non-trivial
solutions of equation
\[
x'=g(t)x
\]
are not Stepanov's almost periodic on $\mathbb{R}$, but they are all bounded on $\mathbb{R}$
because (\ref{equation18}) holds.

\subsection*{Acknowledgement}
The author thanks the referee for his/her critical review of the original
manuscript and for the changes suggested; in particular, for the proof
of Lemma \ref{lem1}.  The author also wants to thank Professor Mingarelli,
at Carleton University, for his help and suggestions.

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\bibitem{ref3} A. M. Fink, \emph{Almost periodic differential
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\bibitem{ref4} Z. Hu and A. B. Mingarelli;
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\bibitem{ref5} Angelo B. Mingarelli, F. Q. Pu , L. Zheng;
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\end{thebibliography}

\end{document}