a$ for some real number $a$, then
$f(t)=0_{X}$ for all $t\in \mathbb{R}$;
\item[(viii)] If $A: X\to Y$ is continuous, where $Y$ is another $q$
-Fr\'echet space, $0 0$, or $x(t)=0, \forall t
\in {\mathbb{R}}$.
\item[(ii)] Let $x:\mathbb{R}_{+}\to X$ and $f:\mathbb{R}
\to X$ be two continuous functions and $T=(T(t))_{t\in \mathbb{R}
_{+}}$ be a strongly continuous semigroup of bounded linear operators on
$X$. Suppose that
\begin{equation*}
x(t)=T(t)(x(0))+\int_{0}^{t}T(t-s)(f(s))ds,t\in \mathbb{R}_{+}.
\end{equation*}
Then for given $\mathit{t}$ in $\mathbb{R}$ and $b>a>0$, $a+t>0$,
we have
\begin{equation*}
x(t+b)=T(t+a)(x(b-a))+\int_{-a}^{t}T(t-s)(f(s+b))ds.
\end{equation*}
\end{itemize}
\end{theorem}
\begin{proof}
(i) Let us suppose that we have $\inf_{t\in {\mathbb{R}}}\|x(t)\|=0$. Let
$(s_{n}')_{n}$ be a sequence of real numbers such that $\lim_{n\to
+\infty }\|x[s_{n}']\|=0$. Since, by hypothesis, as function of $t$,
the function $x(t)$ is almost automorphic, by Definition \ref{def3.2}, we
can extract a subsequence $(s_{n})_{n}$ of $(s_{n}')_{n}$ such that
for all $t\in \mathbb{R}$, there exists $y(t)\in X$ with the property
\begin{equation*}
\lim_{n\to +\infty }\|y(t)-x(t+s_{n})\|=\lim_{n\to +\infty
}\|y(t-s_{n})-x(t)\|=0,
\end{equation*}
with the above convergence on $\mathbb{R}$ being pointwise. Also, we can
easily deduce that
\begin{equation*}
x(t+s_{n})=T(t+s_{n})[x_{0}]=T(t)(T(s_{n})[x_{0}])=T(t)[x(s_{n})].
\end{equation*}
From the above limits, we obtain
\begin{equation*}
\|y(t)\|\leq \|y(t)-x(t+s_{n})\|+\|x(t+s_{n})\|\leq
\end{equation*}
\begin{equation*}
\|y(t)-x(t+s_{n})\|+\||T(t)\||\cdot \|x(s_{n})\|,
\end{equation*}
thus passing to the limit as $n\to +\infty $, it follows that $\|y(t)\|=0$,
that is, $y(t)=0_{X}$, for all $t\in \mathbb{R}$. This immediately implies
$x(t)=0$, for all $t\in \mathbb{R}$.
(ii) As in the proof of \cite[Theorem 2.4.7]{n1}, we obtain
\begin{equation*}
x(t+b)=T(t+a)\Big[ x(b-a)-\int_{0}^{b-a}T(b-a-s)(f(s))ds\Big] %
+\int_{0}^{t+b}T(t+b-s)(f(s))ds.
\end{equation*}
Then from the above relation we get
\begin{align*}
& x(t+b)+T(t+a)\Big[ \int_{0}^{b-a}T(b-a-s)(f(s))ds\Big] \\
&=T(t+a)[x(b-a)]+\int_{0}^{t+b}T(t+b-s)(f(s))ds.
\end{align*}
Taking into account that $T$ commutes with the integral (since it is linear
and continuous operator), by the property $T(u+v)=T(u)[T(v)],\forall u,v\in
\mathbb{R}_{+}$ and by the substitution $u=s-b$, we obtain
\begin{equation*}
x(t+b)+\int_{-b}^{-a}T(t-u)[f(u+b)]du=T(t+a)[x(b-a)]+
\int_{-b}^{t}T(t-u)[f(u+b)]du.
\end{equation*}
But because $t>-a$, we can write
\begin{equation*}
\int_{-b}^{t}T(t-u)[f(u+b)]du=\int_{-b}^{-a}T(t-u)[f(u+b)]du+
\int_{-a}^{t}T(t-u)[f(u+b)]du,
\end{equation*}
we immediately get the required relation from the statement of theorem. The
theorem is proved.
\end{proof}
In what follows, we will be concerned with the behavior of asymptotically
almost automorphic semigroups of linear operators $T=T(t),t\in \mathbb{R}
_{+} $ on $p$-Fr\'{e}chet spaces, $0 0$
and right-continuous at $t=0$, for each $x\in X$. The mapping $u(\cdot ,x)$
is called a motion originating at $x\in X$.
\item[(iii)] $u(t,\cdot ) :X\to X$ is continuous for each
$t\geq 0$ ;
\item[(iv)]
$u(t+s,x)=u(t,u(s,x)),\forall x\in X$, for all $t,s\in \mathbb{R}_{+}$.
\end{itemize}
\end{definition}
\begin{theorem} \label{thm4.5}
Let $(X,+,\cdot,\|\cdot\|)$ be a $p$-Fr\'echet
space, where $0 0 $, $\exists
l(\epsilon) >0 $ such that any interval of length $l(
\epsilon) $ of the real line contains at least one point $\xi$ with
\begin{equation*}
\|f(t+\xi) - f(t)\| <\epsilon,\quad \forall t\in
\mathbb{R}.
\end{equation*}
\end{definition}
\subsection*{Remarks}
(1) A set $E\subset \mathbb{R}$ is called relatively dense
(in $\mathbb{R}$), if there exists a number $l>0$ such that every interval
$(a,a+l)$
contains at least one point of $E$. By using this concept, we can
reformulate Definition \ref{def5.1} as follows: $f:\mathbb{R}\to X$ is
called B-almost periodic if for every $\varepsilon >0$, there exists a
relatively dense set $\{\tau \}_{\varepsilon }$, such that
\begin{equation*}
\sup_{t\in \mathbb{R}}\|f(t+\tau )-f(t)\|\leq \varepsilon ,\quad
\mbox{for
all }\tau \in \{\tau \}_{\varepsilon }.
\end{equation*}
Also, each $\tau \in \{\tau \}_{\varepsilon }$ is called
$\varepsilon $-almost period of $f$.
(2) It was proved in \cite[Theorem 3.6]{g6} that the range of an B-almost
periodic function with values in the $p$-Fr\'{e}chet space
$(X,+,\cdot ,\|\cdot \|)$ is relatively compact (r.c. for short) in the
complete metric space $(X,D)$, with $D(x,y)=\|x-y\|$.
Similar to the case of Banach spaces, we have developed a theory of
Bochner's transform for $p$-Fr\'{e}chet spaces (see \cite{g6}), as follows.
Let us denote $AP(X) =\{ f:\mathbb{R}\to X;\mbox{ }f
\mbox{ is B-almost
periodic}\} $ and for $f\in AP(X) $, let us define
$\| f\| _{b}=\sup \{ \|f(t) \| ; t\in \mathbb{R}\} $.
By \cite[Theorem 3.2]{g6}, we get $\|f\|_{b}<+\infty $.
It follows that $\|\cdot \|_{b}$ also is a $p$-norm on the
space
\begin{equation*}
C_{b}(\mathbb{R},X) =\{f:\mathbb{R}\to X;
\mbox{ is continuous and bounded on }\mathbb{R}\}.
\end{equation*}
In addition, since $(X,D)$ is a complete metric space, by standard
reasonings it follows that $C_{b}(\mathbb{R},X)$ becomes complete metric
space with respect to the metric $D_{b}(f,g)=\|f-g\|_{b}$, that is, $(C_{b}(
\mathbb{R},X),\|\cdot \|_{b})$ becomes a $p$-Fr\'{e}chet space. Then, the
result in \cite[Theorems 3.2 and 3.5]{g6} shows that $AP(X) $ is a closed
subset of $C_{b}(\mathbb{R},X) $, that is, $( AP(X) ,D_{b}) $ is complete
metric space and therefore $(AP(X),\|\cdot \|_{b})$ becomes $p$-Fr\'{e}chet
space.
The Bochner transform on $C_{b}(\mathbb{R},X)$ is defined as in the case of
Banach spaces, by
\begin{equation*}
\tilde{f}:\mathbb{R}\to C_{b}(\mathbb{R},X),\tilde{f}(s)(t)=f(t+s),
\end{equation*}
for all $t\in \mathbb{R}$ and we write $\tilde{f}=B(f)$. The properties of
Bochner's transform on $p$-Fr\'{e}chet spaces, $0 0$ independent of $n, m$.
\end{itemize}
Then, $f$ is B-almost periodic.
\end{theorem}
It is clear that $AP(X)\subset AA(X)$, and in general, the concepts of
B-almost periodicity and almost automorphy are not equivalent. However
Theorem \ref{thm5.4} allows us to prove the equivalence between the B-almost
periodicity and almost automorphy of the ``orbits'' of a group/semigroup. In
this sense, we present the following result.
\begin{theorem} \label{thm5.5}
Let $(T(t))_{t\in \mathbb{R}}$ be a family of
uniformly bounded group of bounded linear operators on a $p$-Fr\'echet
space $(X,+,\cdot,\|\cdot\|)$, $0 0$ such that $\|T(t)(x_0)\|\leq
M\|x_0\|$, for all $t\in \mathbb{R}$. Also, the range $R_{T(t)(x_0)}$ is
relatively compact since $T(t)(x_0)$ is almost automorphic as function of $t$
(see Theorem \ref{thm3.3}, (v)). Thus given an r.d. sequence of real numbers
$(s'_{n})$, we can find a subsequence $(s_{n})$ such that $%
(T(s_n)(x_0))_{n\in \mathbb{N}}$ is Cauchy. Now, in view of the following
inequality
\begin{equation*}
c\|[T(t+s_n)(x_{0})-T(t+s_m)(x_0)]\|\leq \|[T(s_n)(x_{0})-T(s_m)(x_0)]\|,
\end{equation*}
for all $t\in \mathbb{R}$, (where $c=\frac{1}{M}$) we conclude that
$T(t)(x_0)$ is B-almost periodic by Theorem \ref{thm5.4}.
\end{proof}
We remark that Theorem \ref{thm5.5} is an extension of a result \cite{c1} in
Banach spaces to $p$-Fr\'{e}chet spaces, $0 0$ there exists $\delta > 0$ such
that $\|x(t_{1})-x(t_{2})\|<\delta$ implies $\|x(t+t_{1})-x(t+t_{2})\|<
\epsilon$ for all $t\in \mathbb{R}$.
\end{definition}
\begin{example} \label{exa5.7} \rm
If $(T(t))_{t\in {\mathbb{R}}}$ is a family of
uniformly bounded group of continuous linear operators on $X$, then the
function $x(t):=T(t)(e)$ for some $e\in X$ is a strongly stable
motion in $X$.
\end{example}
\begin{theorem} \label{thm5.8}
If $x\in C({\mathbb{R}}, X)$ is a strongly
stable motion with a relatively compact range in $X$, then $x\in AP(X)$.
\end{theorem}
The proof of the above theorem is a direct consequence of Theorem \ref%
{thm5.3}. By Definition \ref{def3.6} we have introduced the concept of
asymptotically almost automorphic function with values in a $p$-Fr\'echet
space, $00$ be fixed. There exists $\delta >0$, such that
$\|h(t)\|<\varepsilon $, for all $t>\delta $. From the continuity of $Q$ on
$[0,\delta ]$, there exists $M>0$ such that $Q(t)\leq M$, for all
$t\in [0,\delta ]$. In conclusion, $0\leq Q(t)\leq M+\varepsilon ,\forall t\in
\mathbb{R_{+}}$, which implies the desired conclusion.
(iv) Let $f=g+h$ be the decomposition in Definition \ref{def3.6}. By Theorem
\ref{thm3.3}, (viii), $\phi \circ g:\mathbb{R}\to Y$ is almost automorphic
and also by hypothesis, $\phi \circ f$, $\phi \circ g$, are continuous on
$\mathbb{R_{+}}$. Denote now $\Gamma (t)=\phi (f(t))-\phi (g(t))$. Let
$\varepsilon >0$. By the uniform continuity of $\phi $ on the compact set $B$
, there exists $\delta >0$, such that $\|\phi (x)-\phi (y)\|_{2}<\varepsilon
$, for all $\|x-y\|_{1}<\delta $, $x,y\in B$. On the other hand, by
hypothesis, we have $\lim_{t\to +\infty }\|h(t)\|_{1}=0$, therefore there
exists $t_{0}$ (depending on $\delta $ ), such that
$\|h(t)\|_{1}=\|f(t)-g(t)\|_{1}<\delta $, for all $t>t_{0}$.
Then, for $t>t_{0}$ we
obtain,
\begin{equation*}
\|\Gamma (t)\|_{2}=\|\phi (f(t))-\phi (g(t))\|_{2}<\varepsilon ,
\end{equation*}
for all $t>t_{0}$, which means $\lim_{t\to +\infty }\|\Gamma (t)\|_{2}=0$.
(v) Let us suppose now that $f$ has two decompositions
$f=g_{1}+h_{1}=g_{2}+h_{2}$. For all $t\geq 0$ we get
$g_{1}(t)-g_{2}(t)=h_{2}(t)-h_{1}(t)$, which implies
\begin{equation*}
\lim_{t\to +\infty }\|g_{1}(t)-g_{2}(t)\|\leq \lim_{t\to +\infty
}\|h_{2}(t)\|+\lim_{t\to +\infty }\|h_{1}(t)\|=0.
\end{equation*}
Consider the sequence $(n)$. Since $g_{1}-g_{2}$ is almost automorphic,
there exists a subsequence $(n_{k})$ such that
\begin{equation*}
\lim_{k\to +\infty }\|[g_{1}(t+n_{k})-g_{2}(t+n_{k})]-F(t)\|=0
\end{equation*}
and
\begin{equation*}
\lim_{k\to +\infty }\|F(t-n_{k})-[g_{1}(t)-g_{2}(t)]\|=0,
\end{equation*}
with the convergence holding pointwise on $\mathbb{R}$. But
\begin{equation*}
\|F(t)\|\leq
\|F(t)-[g_{1}(t+n_{k})-g_{2}(t+n_{k})]\|+\|g_{1}(t+n_{k})-g_{2}(t+n_{k})\|.
\end{equation*}
Passing to the limit as $k\to +\infty $ and taking the above relations into
account, it follows $\|F(t)\|=0,\forall t\in \mathbb{R_{+}}$, which implies
$g_{1}(t)-g_{2}(t)=0,\forall t$. Therefore, $h_{2}(t)-h_{1}(t)=0$, for all
$t\in \mathbb{R_{+}}$, which proves the theorem.
\end{proof}
\subsection*{Remark}
Concerning the derivative and indefinite integral of asymptotically almost
automorphic functions, we have the same negative phenomenon as in the case
of almost automorphic functions (see the Remark after the proof of Theorem
\ref{thm3.3}).
We also have the following result.
\begin{theorem} \label{thm3.8}
If $(X,+,\cdot,\|\cdot\|)$ is a $p$-Fr\'echet
space with $0