\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 241, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/241\hfil Existence of bounded solutions]
{Existence of bounded solutions for nonlinear hyperbolic partial
 differential equations}

\author[T. Diagana, M. M. Mbaye \hfil EJDE-2015/241\hfilneg]
{Toka Diagana, Mamadou Moustapha Mbaye}

\address{Toka Diagana \newline
Department of Mathematics, Howard University,
2441 6th Street N.W., Washington, DC 20059, USA}
\email{tdiagana@howard.edu}

\address{Mamadou Moustapha Mbaye \newline
Universit\'e Gaston Berger de Saint-Louis,
UFR SAT, D\'epartement de Math\'ematiques, B.P. 234, Saint-Louis, S\'en\'egal}
\email{tafffmbaye@yahoo.fr}


\thanks{Submitted July 4, 2015. Published September 21, 2015.}
\subjclass[2010]{43A60, 34B05, 34C27, 42A75, 47D06, 35L90}
\keywords{Hyperbolic partial differential equations; bounded solutions;
\hfill\break\indent almost automorphic; pseudo-almost automorphic}

\begin{abstract}
 In this article we first establish a new representation formula for
 bounded solutions to a class of nonlinear second-order hyperbolic partial
 differential equations. Next, we use of our newly-established
 representation formula to establish the existence of bounded solutions
 to these nonlinear partial differential equations.
\end{abstract}

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

\section{Introduction}

Aziz and Meyers \cite{AM} established the existence, uniqueness, and continuous
dependence on the initial data of periodic solutions to the class
of nonlinear second-order hyperbolic partial
differential equations
\begin{equation}\label{LL}
\begin{gathered}
 \frac{\partial^2 u}{\partial x \partial t} +
a(t,x)\frac{\partial u}{\partial x} + b(t,x) \frac{\partial
u}{\partial t} + c(t,x) u = f(t,x, u), \quad \text{in }\mathbb{R} \times [0,T],\\
 u(t,0) = \theta(t), \quad \text{for all }t \in \mathbb{R},
\end{gathered}
\end{equation}
where $a, b, c: \mathbb{R} \times [0, T] \to \mathbb{R}$ and
$f: \mathbb{R} \times [0, T] \times \mathbb{R} \to \mathbb{R}$ are $p$-periodic functions
and $\theta: \mathbb{R} \to \mathbb{R}$ is a $p$-periodic continuously differentiable function.
The main tool utilized by Aziz and Meyers is a representation formula presented by
 Picone \cite{PI}.
Some years ago, Al-Islam \cite{N} used the same representation formula to
study the existence and uniqueness of pseudo-almost periodic solutions to
 \eqref{L} under some appropriate conditions.

The use of Picone's representation formula is somewhat tedious
as it is expressed in terms of three functions
$\alpha, \beta$, and $\gamma$, which are solutions to some other
partial differential equations.
The first objective of this paper consists of using
operator theory tools to establish a new representation formula
for bounded solutions to \eqref{LL} in the special case $\theta(t) \equiv 0$;
that is,
\begin{equation}\label{L}
\begin{gathered}
 \frac{\partial^2 u}{\partial x \partial t} +
a(t,x)\frac{\partial u}{\partial x} + b(t,x) \frac{\partial
u}{\partial t} + c(t,x) u = f(t,x, u), \quad \text{in }\mathbb{R} \times [0, T],\\
 u(t,0) = 0, \quad \text{for all }t \in \mathbb{R}.
\end{gathered}
\end{equation}

Our second objective consists of using our newly-established
representation formula to study the existence of bounded
(respectively, pseudo-almost automorphic) solutions to \eqref{L}
 when the coefficients $a, b, c, a_x: \mathbb{R} \times [0, T] \to \mathbb{R}$ are bounded
 (respectively, almost automorphic) and the forcing term $f: \mathbb{R} \times [0, T]
\times \mathbb{R} \to \mathbb{R}$ is bounded (respectively, pseudo-almost automorphic in $t \in \mathbb{R}$
uniformly with respect to the two other variables).

One should point out that other slightly different versions of \eqref{L}
have been considered in the literature. In particular, Poorkarimi and Wiener
\cite{PW} studied bounded and almost periodic solutions to a
slightly modified version of \eqref{L}, which in fact represents a
mathematical model for the dynamics of gas absorption. However, the study
of pseudo-almost automorphic solutions to \eqref{L} is an untreated
original question, which constitutes the
main motivation of this article.

The study of periodic, almost periodic, almost automorphic,
pseudo-almost periodic, weighted pseudo-almost periodic, and
pseudo-almost automorphic solutions to differential differential equations
constitutes one of the most relevant topics in qualitative
theory of differential equations mainly due to their applications. Some
contributions on pseudo-almost automorphic solutions to
differential and partial differential equations have recently been
made in \cite{CE, TDbook, L, LLL, LL, XJ}.
Here we study the existence of bounded
(respectively, pseudo-almost automorphic) solutions to \eqref{L}
under some appropriate assumptions. One should point out
that the case $\theta \not\equiv 0$ makes the operators involved in our
study nonlinear. Such a case will be left for future investigations.


The article is organized as follows:
Section 2 is devoted to preliminaries and notations from operator theory
as well as from the concept of pseudo-almost automorphy.
In Section 3, we establish a representation formula.
Section 4 is devoted to the main result.
In Section 5, we give an example to illustrate our main result.

\section{Preliminaries}

\subsection*{Notation}
Let $(\mathbb{X}, \|\cdot\|)$ and $(\mathbb{Y}, \|\cdot\|_{\mathbb{Y}})$ be Banach spaces.
Let $BC(\mathbb{R} , \mathbb{X})$
(respectively, $BC(\mathbb{R} \times \mathbb{Y}, \mathbb{X})$) denote the collection of
all $\mathbb{X}$-valued bounded continuous functions (respectively, the
class of jointly bounded continuous functions $F: \mathbb{R} \times \mathbb{Y}
\to \mathbb{X}$). The space $BC(\mathbb{R}, \mathbb{X})$ equipped with its natural
norm, that is, the sup norm defined by
$$ \|u\|_\infty = \sup_{t \in \mathbb{R}} \|u(t)\|,$$ is a
Banach space. Furthermore, $C(\mathbb{R}, \mathbb{Y})$ (respectively, $C(\mathbb{R} \times
\mathbb{Y}, \mathbb{X})$) denotes the class of continuous functions from $\mathbb{R}$ into
$\mathbb{Y}$ (respectively, the class of jointly continuous functions $F:
\mathbb{R} \times \mathbb{Y} \to \mathbb{X}$).

If $A$ is a linear operator upon $\mathbb{X}$, then the notations $D(A)$ and $\rho(A)$
stand respectively for the domain and the resolvent of $A$.
The space $B(\mathbb{X}, \mathbb{Y})$ denotes the collection of all bounded
linear operators from $\mathbb{X}$ into $\mathbb{Y}$ equipped with its natural
uniform operator topology $\|\cdot\|$. We also set $B(\mathbb{Y}) = (\mathbb{Y}, \mathbb{Y})$
 whose corresponding norm will be denoted $\|\cdot\|$.

\subsection*{Pseudo-Almost Automorphic Functions}

\begin{definition}\label{DDD}
A function $f\in C(\mathbb{R},\mathbb{X})$ is said to be almost automorphic if for
every sequence of real numbers $(s'_n)_{n \in \mathbb{N}}$, there
 exists a subsequence $(s_n)_{n \in \mathbb{N}}$ such that
 $$ g(t):=\lim_{n\to\infty}f(t+s_n)$$
 is well defined for each $t\in\mathbb{R}$, and
 $$ \lim_{n\to\infty}g(t-s_n)=f(t)$$
 for each $t\in \mathbb{R}$.
\end{definition}

If the convergence above is uniform in $t\in \mathbb{R}$, then $f$ is
almost periodic. Denote by
$AA(\mathbb{X})$ the collection of such almost automorphic functions.
Note that $AA(\mathbb{X})$ equipped with the sup-norm
$\|\cdot\|_\infty$ is a Banach space.


\begin{definition}\label{KKK} \rm
A jointly continuous function $F: \mathbb{R} \times \mathbb{Y} \to \mathbb{X}$ is said
to be almost automorphic in $t \in \mathbb{R}$ if $t \to F(t,x)$ is
almost automorphic for all $u \in K$ ($K \subset \mathbb{Y}$ being any
bounded subset). Equivalently, for every sequence of real numbers
$(s'_n)_{n \in \mathbb{N}}$,
there exists a subsequence $(s_n)_{n \in \mathbb{N}}$ such that
 $$
 H(t, u):=\lim_{n\to\infty}F(t+s_n, u)
 $$
is well defined in $t\in\mathbb{R}$ and for each $u \in K$, and
 $$ \lim_{n\to\infty}H(t-s_n, u)=F(t, u)$$
for all $t\in \mathbb{R}$ and $u \in K$.
The collection of such functions will be denoted by $AA(\mathbb{Y}, \mathbb{X})$.
\end{definition}

Define
$$
PAP_0(\mathbb{R}, \mathbb{X}) := \big\{ f \in BC(\mathbb{R}, \mathbb{X}): \lim_{T \to \infty}
{\frac{1}{2T}} \int_{-T}^T \| f(s)\| ds
= 0\big\}.
$$
Similarly, $PAP_0(\mathbb{Y}, \mathbb{X})$ will denote the collection of all
bounded continuous functions $F: \mathbb{R} \times \mathbb{Y} \to \mathbb{X}$ such
that
$$
\lim_{T \to \infty} {\frac{1}{2T}} \int_{-T}^T \| F(s, x)\| ds =0
$$ uniformly in $x \in K$, where $K \subset \mathbb{Y}$ is any bounded
subset.


\begin{definition}[\cite{L, LL}]\label{DEF} \rm
A function $f \in BC(\mathbb{R}, \mathbb{X})$ is called pseudo-almost automorphic
if it can be expressed as $f = g + \phi$, where $g \in AA(\mathbb{X})$ and
$\phi \in PAP_0(\mathbb{X})$. The collection of such functions will be
denoted by $PAA({\mathbb X})$.
\end{definition}

The functions $g$ and $\phi$ appearing in Definition \ref{DEF} are
respectively called the {\it almost automorphic} and the {\it
ergodic perturbation} components of $f$.

\begin{definition} \rm
A bounded continuous function $F: \mathbb{R} \times \mathbb{Y} \to \mathbb{X}$ is said
to be pseudo-almost automorphic whenever it can be expressed as
$F= G + \Phi$, where $G\in AA(\mathbb{Y},\mathbb{X})$ and $\Phi \in PAP_0(\mathbb{Y}, \mathbb{X})$.
The collection of such functions will be denoted by $PAA(\mathbb{Y}, \mathbb{X})$.
\end{definition}

\begin{theorem}[\cite{LL}] \label{MN}
The space $PAA(\mathbb{X})$ equipped with the supremum
norm $\|\cdot\|_\infty$ is a Banach space.
\end{theorem}

\begin{theorem}\label{thm2.6} % LI
Suppose $F: \mathbb{R} \times \mathbb{Y} \to \mathbb{X}$ belongs to $PAA(\mathbb{Y}, \mathbb{X})$; $F = G+H$,
with $u \to G(t,u)$ being uniformly continuous on any bounded subset
 $K$ of $\mathbb{Y}$ uniformly in $t \in \mathbb{R}$. Furthermore, we suppose that
 there exists $L > 0$ such that
$$
\|F(t,u) - F(t,v)\| \leq L \|u-v\|_{\mathbb{Y}}
$$
for all $u,v \in \mathbb{Y}$ and $t \in \mathbb{R}$.
Then the function defined by $h(t) = F(t, \varphi(t))$ belongs to
$PAA(\mathbb{X})$ provided $\varphi \in PAA(\mathbb{Y})$.
\end{theorem}

\begin{theorem}[\cite{LL}]
 If $F: \mathbb{R} \times \mathbb{Y} \to \mathbb{X}$ belongs to $PAA(\mathbb{Y}, \mathbb{X})$
and if $u \to F(t,u)$ is uniformly continuous on any bounded
subset $K$ of $\mathbb{Y}$ for each $t \in \mathbb{R}$, then the function defined
by $h(t) = F(t, \varphi(t))$ belongs to $PAA(\mathbb{X})$ provided
$\varphi \in PAA(\mathbb{Y})$.
\end{theorem}

For more on pseudo-almost automorphic functions and related issues,
we refer the reader to the  book by Diagana \cite{TDbook}.

\section{Representation formula for bounded solutions of \eqref{L}}

Let $\mathcal{C}_T = C[0, T]$ be the Banach space of all continuous
functions from $[0, T]$ to $\mathbb{R}$ equipped with the sup norm defined
by
$$
\|\varphi\|_{T} := \sup_{x \in [0, T]} |\varphi(x)|
$$
for all $\varphi \in \mathcal{C}_T$.

To study \eqref{L} our first task consists of using
operator theory tools to establish a new representation formula.
For that, if $q: [0, T] \to \mathbb{R}$ is a measurable function, we
consider the linear operators $A$ and $B$ defined on $\mathcal{C}_T$ by
\begin{gather*}
D(A) = \big\{\varphi \in \mathcal{C}_T: \varphi_x
= \frac{d\varphi}{dx} \in \mathcal{C}_T \text{ and }  \varphi(0) = 0\big\}, \quad
 A\varphi = \frac{d\varphi}{dx},  \text{ for all }  \varphi \in D(A), \\
D(B_q) = \big\{\varphi \in \mathcal{C}_T: q \varphi\in \mathcal{C}_T \big\}, \quad
 B_q \varphi = q \varphi.
\end{gather*}

 Obviously, if $q \in \mathcal{C}_T$, then $D(B_q) = \mathcal{C}_T$.
Moreover, using the above-mentioned operators, one can easily see
that \eqref{L} can be rewritten as follows
\begin{equation}\label{MM}
(A + B_b) \frac{\partial u}{\partial t} + (B_a A + B_c) u = f.
\end{equation}

To study \eqref{MM}, we consider the differential equation
\begin{equation}\label{M}
L \frac{dv}{dt} + M v = g,
\end{equation}
where $L = A + B_\beta$ and $M =B_\alpha A + B_\gamma$ with
$\alpha, \beta, \gamma: [0, T] \to \mathbb{R}$ being continuous
functions.
Notice that $L$ and $M$ are respectively defined by
$$
D(L) = D(A) \cap D(B_\beta) = D(A) \quad\text{and} \quad
Lv = \frac{dv}{dx} + \beta v , \quad \text{for all } v \in D(A)
$$
and
$$
D(M) = D(B_\alpha A) \cap D(B_\gamma) =D(A) \quad \text{and} \quad
Mv = \alpha \frac{dv}{dx} + \gamma v , \quad \text{for all } v \in D(A).
$$
The next lemma shows that \eqref{M} in
fact is not a singular differential equation ($0 \in \rho(L)$), which makes our
computations less tedious.

\begin{lemma}\label{I}
If the function $\beta: [0, T] \to \mathbb{R}$ is continuous, then the
operator $L$ is invertible and its inverse $L^{-1}$ is given for all
$w \in \mathcal{C}_T$ by
$$
L^{-1} w(x): = \int_{0}^x K(x,y) w(y) dy,
$$
where the kernel $K$ is defined by
$$
K(x,y) := e^{- \int_{y}^x \beta(r) dr}
$$
for all $0 \leq y \leq x \leq T$.
Furthermore, if
$ \beta_{\ast} : = \inf_{y \in [0, T]} \beta(y) > 0$,
then $\|L^{-1}\| \leq T$.
\end{lemma}

\begin{proof}
First of all, we need to solve the differential equation
\begin{equation}\label{LI}
\frac{du}{dy} + \beta u = v
\end{equation}
where $u \in D(A)$ and $v \in \mathcal{C}_T$.
For that, multiplying both sides of \eqref{LI} by the function
$ R(y) = e^{ \int_{0}^y \beta(r)dr}$ and
integrating on $[0, x]$, we obtain
\begin{align*}
u(x) &= e^{- \int_{0}^x \beta(r)dr}  \int_{0}^x e^{
\int_{0}^y \beta(r)dr} v(y) dy \\
&= \int_{0}^x K(x,y) v(y) dy
\end{align*}
where $K(x,y) = e^{- \int_{y}^x \beta(r) dr}$ for all $0 \leq y \leq x \leq T$.
Therefore,
$$
L^{-1} v(x): = \int_{0}^x K(x,y) v(y) dy
$$
for all $v \in \mathcal{C}_T$.

Now, using the fact $K(x,y) \leq e^{-\beta_{\ast} (x-y)} \leq 1$
for $0 \leq y \leq x \leq T$, one can easily see that
\[
\|L^{-1} v(x) \|
\leq  \|v\|_T \int_0^x |K(x,y)| dy
\leq T \|v\|_T
\]
and hence
$\|L^{-1}\| \leq T$.
\end{proof}


Let $\mathbb{Z}_T$ (respectively $\mathbb{Y}_T$) be the Banach space of all
bounded (jointly) continuous functions from $\mathbb{R} \times [0, T]$ to
$\mathbb{R}$ (respectively, from $[0, T] \times \mathbb{R}$ to $\mathbb{R}$) equipped
with the sup norm defined for each $u \in \mathbb{Z}_T$ (respectively,
$u \in \mathbb{Y}_T$) by
$$
\|u\|_{T, \infty} := \sup_{t \in \mathbb{R}, x \in [0, T]}
| u(t,x)|.
$$
 Moreover, we set
\begin{gather*}
K_t(x,y) := e^{- \int_y^x b (t,r) dr}, \\
H(t,x) = \frac{\partial a}{\partial x} (t,x) + a(t,x) b(t,x) - c(t,x)
\end{gather*}
for all $t \in \mathbb{R}$ and $x, y \in [0, T]$.
Let us point out that the quantity $H$ given above is also known as
the Euler's invariant, see for instance Ibragimov \cite{I}.

The proof of the main results of this paper requires the following
assumptions:
\begin{itemize}
\item[(H1)] There exists $\delta > 0$ such that $a(t,x) \geq
\delta$ for all $t \in \mathbb{R}$ and $x \in [0, T]$.

\item[(H2)] The function $f: \mathbb{R} \times [0, T] \times \mathbb{R} \to
\mathbb{R}$ is Lipschitz in the third variable uniformly in the first and
second variables; that is, there exists $C > 0$ such that
\begin{equation}\label{LIP}
\big| f(t,x, u) - f(t,x,v)\big| \leq C |u - v|
\end{equation}
for all $u, v \in \mathbb{R}$ uniformly in $t \in \mathbb{R}$ and $x \in [0, T]$.

\item[(H3)] The function $f = g + h \in PAA(\mathbb{Y}_T, \mathbb{R})$ ($g$ being
the almost automorphic component while $h$ represents the ergodic
part). Moreover, $g: \mathbb{Y}_T \to \mathbb{R}$, $(x, u) \to g(t,x,u)$
is uniformly continuous on bounded subset of $\mathbb{Y}_T$ uniformly in
$t \in \mathbb{R}$.

\item[(H4)] The functions $ (t,x) \to a(t,x),
\frac{\partial a}{\partial x} (t,x), b(t,x), c(t,x)$ are jointly
continuous and almost automorphic in $t \in \mathbb{R}$ uniformly in $x
\in [0, T]$.
\end{itemize}
Under (H4), we set
\begin{gather*}
C_\infty := \sup_{t \in \mathbb{R}, x \in
[0, T]} |H(t,x)|= \sup_{t\in \mathbb{R}, x \in [0, T]} \Big|
\frac{\partial a}{\partial x} (t,x) + a(t,x) b(t,x) -c(t,x)\Big|,
\\
B_\infty := \sup_{s\in \mathbb{R}, x \in [0,
T]} \Big (\int_0^x e^{- \int_y^x b(s, r)dr} dy\Big).
\end{gather*}
We have the following representation formula for bounded solutions of \eqref{L}.

\begin{theorem}\label{R}
 Assume {\rm (H1)--(H2)} and the functions $a, b, c: \mathbb{R} \times [0, T]
\to \mathbb{R}$ are jointly bounded continuous. Then \eqref{L} has a
unique bounded continuous solution $\widetilde u$ whenever
$C +C_\infty<\delta B_{\infty}^{-1}$.
Furthermore, $\widetilde u$ is given by
the new representation formula
\begin{equation}\label{RE}
\widetilde u(t,x) = \int_{-\infty}^t e^{ -
\int_s^t a(\sigma,x) d\sigma} G \widetilde u (s, x) ds
\end{equation}
where
\begin{align*}
G \widetilde u(t, x)
&= \int_0^x \Big[\frac{\partial a}{\partial y} (t,y) + a(t,y) b(t,y)
- c(t,y)\Big] K_t(x, y) \widetilde u(t,y) dy \\
&\quad+ \int_0^x K_t (x, y) f(t, y, \widetilde u(t,y)) dy.
\end{align*}
\end{theorem}

\begin{proof}
Replacing $\alpha$ by $a$, $\beta$ by $b$, and
$\gamma$ by $c$, in the previous setting and using the fact
$L^{-1}$ exists (Lemma \ref{I}), it follows that the solvability
of \eqref{L} is equivalent to that of the following
first-order partial differential equation
\begin{equation}\label{X}
\frac{\partial u}{\partial t} = - L^{-1} M u + L^{-1} f.
\end{equation}
Notice that the operator $L^{-1} M$ can be explicitly
computed. Indeed, for each $v \in D(A)$, we have
\begin{align*}
L^{-1} M v(x)
&=  L^{-1} \Big(a \frac{dv}{dx} + c v\Big) (x)\\
&=  \int_0^x K_t(x,y) a(t,y) \frac{dv}{dy} dy + \int_{0}^x
K_t(x,y) c(t,y) v(y) dy \\
&=  \Big[ a(t,y) K_t(x,y) v(y) \Big]_{0}^x - \int_0^x
\frac{\partial}{\partial y} [a(t,y) K_t (x,y)] v(y) dy \\
&\quad+ \int_{0}^x K_t(x,y) c(t,y) v(y) dy \\
&=  a(t,x) v(x) - \int_0^x \Big[\frac{\partial a}{\partial y}(t,y)
+ a(t,y) b(t,y) - c(t,y) \Big] K_t(x,y) v(y) dy.
\end{align*}
Using the expression of $L^{-1} M$, one can easily see that \eqref{X}
is equivalent to
\begin{equation}\label{XX}
\frac{\partial u}{\partial t} = - a(t,x) u + Gu(t,x)
\end{equation}
where
\begin{align*}
Gu(t, x)
&=  \int_0^x \Big[\frac{\partial a}{\partial y} (t,y)
+ a(t,y) b(t,y) - c(t,y)\Big] K_t(x, y) u(t,y) dy \\
&\quad + \int_0^x K_t (x, y) f(t, y, u(t,y)) dy.
\end{align*}
Clearly, bounded solutions to \eqref{XX} are given by
$$
u(t,x) = \int_{-\infty}^t \exp\Big\{ - \int_s^t a(\sigma,x)
d\sigma \Big\} G u (s, x) ds.
$$
Setting
$$
\Gamma u (t,x) : = \int_{-\infty}^t e^{ - \int_s^t a(\sigma,x)
d\sigma} G u (s, x) ds,
$$
one can easily see that $\Gamma$ maps $\mathbb{Z}_T$ into itself.

In addition, it is easy to see that
$$
\| \Gamma u - \Gamma v\|_{T, \infty}
\leq B_\infty \delta^{-1} \big(C + C_\infty\big) \| u - v\|_{T, \infty}.
$$
Therefore, the nonlinear integral operator $\Gamma$
has a unique fixed point $\widetilde u \in \mathbb{Z}_T$ whenever $C + C_\infty <
\delta B_{\infty}^{-1}$.
In this event, the function $\widetilde u$ is the only bounded continuous
solution to \eqref{L}.
\end{proof}


\section{Existence of pseudo-almost automorphic solutions}

\begin{theorem}\label{O}
Assume {\rm (H1)--(H4)} and that
$ b_\ast:= \inf_{t \in \mathbb{R}, x \in [0, T]} b(t,x) > 0$.
Then \eqref{L} has a unique pseudo almost automorphic
solution $\widetilde u$ whenever
$C+\mathcal{C}_\infty < \delta B_{\infty}^{-1}$.
\end{theorem}

\begin{proof}
Let $u = u_1 + u_2 \in PAA(\mathbb{Z}_T)$ and let
$f = g + h \in PAA(\mathbb{Y}_T, \mathbb{R})$ where $u_1$ and $g$ are
the almost automorphic components while $u_2$ and $h$ represent the ergodic
part. Consequently, $G$ can be rewritten as $G u = G_1 u + G_2 u$, where
\begin{align*}
G_1 u(t,x)
&=  \int_0^x \Big[\frac{\partial a}{\partial y} (t,y) + a(t,y) b(t,y) -
c(t,y)\Big] K_t(x, y) u_1(t,y) dy \\
&\quad + \int_0^x K_t (x, y) g(t, y, u(t,y)) dy
\end{align*}
and
\begin{align*}
 G_2 u(t,x)
&=  \int_0^x \Big[\frac{\partial a}{\partial y} (t,y) + a(t,y) b(t,y) -
c(t,y)\Big] K_t(x, y) u_2(t,y) dy \\
&\quad + \int_0^x K_t (x, y) h(t, y, u(t,y)) dy
\end{align*}
Since $t \to b(t,x)$ is almost automorphic uniformly in
$x \in [0, T]$, then for every sequence of real numbers
$(s'_n)_{n \in \mathbb{N}}$ there exists a subsequence $(s_n)_{n \in \mathbb{N}}$ such that
$$
b_1(t,r):=\lim_{n\to\infty}b(t+s_n, r)
$$
is well defined for each $t\in\mathbb{R}$ uniformly in $r \in [0, T]$, and
$$
b(t,r) = \lim_{n\to\infty}b_1(t-s_n, r)
$$
for each $t\in \mathbb{R}$ uniformly in $r \in [0, T]$.

Now
$$
-\int_{y}^x b_1(t,r)dr =-\int_{y}^x \lim_{n\to\infty}b(t+s_n, r) dr
 = -\lim_{n \to \infty} \int_{y}^x b(t+s_n, r) dr
$$
 is well defined for each $t\in\mathbb{R}$ uniformly in $x,y \in [0, T]$, and
 $$
-\int_{y}^x b(t,r) dr= -\int_{y}^x \lim_{n\to\infty}b_1(t-s_n, r)dr
= -\lim_{n\to\infty}\int_{y}^x b_1(t-s_n, r)dr
$$
for each $t\in \mathbb{R}$ uniformly in $x,y \in [0, T]$.

 Using the continuity of the exponential function it follows that
$$
K_{t}^1(x,y):=\lim_{n\to\infty} K_{t+s_n} (x,y)
$$
is well defined for each $t\in\mathbb{R}$ uniformly in $x, y \in [0, T]$,
and
 $$
K_{t}(x,y) = \lim_{n\to\infty} K_{t-s_n}^1 (x,y)
$$
 for each $t\in \mathbb{R}$ uniformly in $x,y \in [0, T]$,
and hence $t \to K_t (x,y)$ is
almost automorphic uniformly in $x, y \in [0, T]$.

Clearly, $t \to H(t,y) K_t(x, y)u_1(t,y)$ and $t \to K_t (x, y)
g(t, y, u(t,y))$ are almost automorphic functions for all $x, y
\in [0, T]$ as products of almost automorphic functions. It easily
follows that $t \to G_1 u(t,x)$ is almost automorphic
uniformly in $x \in [0, T]$.

Now
\begin{align*}
&\frac{1}{2r} \int_{-r}^r |G_2 u(t,x)| dt \\
&=  \frac{1}{2r}
\int_{-r}^r \Big|\int_0^x H(t,y)K_t(x, y) u_2(t,y)
 + \int_0^x K_t (x, y) h(t, y, u(t,y)) dy \Big| dt \\
&\leq \frac{C_\infty e^{Tb_{\ast}}}{2r}\int_{-r}^r \int_0^x |u_2(t,y))| \,dy\,dt
 + \frac{e^{Tb_{\ast}}}{2r} \int_{-r}^r \int_0^x |h(t, y, u(t,y))| \,dy\,dt \\
&\leq C_\infty e^{Tb_{\ast}}\int_{0}^x \Big(\frac{1}{2r}
 \int_{-r}^r |u_2(t,y))| dt\Big) dy
 + e^{Tb_{\ast}}\int_{0}^x \Big(\frac{1}{2r} \int_{-r}^r
 |h(t, y, u(t,y))| dt\Big) dy,
\end{align*}
and thus
$$
\lim_{T \to \infty} \frac{1}{2r} \int_{-r}^r |G_2 u(t,x)|
dt= 0
$$
uniformly in $x \in [0, T]$. Therefore $t \to Gu(t,x)
\in PAA(\mathbb{Z}_T)$ uniformly in $x \in [0, T]$.

Now
$$
\Gamma u (t,x) : = \int_{-\infty}^t e^{
- \int_s^t a(\sigma,x) d\sigma} G u (s, x) ds
= \Gamma_1 u(t,x) + \Gamma_2 u(t,x),
$$
where
$$
\Gamma_j u (t,x) : = \int_{-\infty}^t e^{ - \int_s^t a(\sigma,x)
d\sigma} G_j u (s, x) ds, \quad j = 1, 2.
$$
Since
$s \to e^{ - \int_s^t a(\sigma,x) d\sigma} G_1 u (s, x)$
is almost automorphic and that
$$
\| \Gamma_1 u\|_{T, \infty} \leq \| G_1 u\|_{T, \infty}\delta^{-1} < \infty
$$
it follows that $t \mapsto \Gamma_1 u(t,x)$ is almost
automorphic uniformly in $x \in [0, T]$.

Now
\begin{align*}
\frac{1}{2r} \int_{-r}^r |\Gamma_2 u(t,x)| dt
&\leq \frac{1}{2r} \int_{-r}^r \int_{-\infty}^t e^{-\delta (t-s)} |G_2 u(s,x)|
 \,ds\,dt \\
&=  \int_{0}^\infty e^{-\delta \sigma}\Big(\frac{1}{2r}
\int_{-r}^r |G_2 u(t-\sigma, x))| dt\Big) d\sigma.
\end{align*}
Since $PAP_0(\mathbb{Z}_T)$ is translation invariant and  $G_2 \in
PAP_0(\mathbb{Z}_T)$ it follows that
$$
\lim_{r \to \infty} \frac{1}{2r}
\int_{-r}^r |G_2 u(t-\sigma, x))| dt = 0
$$
for each $\sigma \in \mathbb{R}$, uniformly in $x \in [0, T]$.

Using the Lebesgue's Dominated Convergence Theorem it follows
that
$$
\lim_{r \to \infty} \frac{1}{2r} \int_{-r}^r |\Gamma_2 u(t,x)| dt = 0
$$
uniformly in $x \in [0, T]$.


In view of the above, it follows that $t \to \Gamma u(t,x)$
is pseudo-almost automorphic uniformly in $x \in [0, T]$. Therefore,
$\Gamma$ maps $PAA(\mathbb{Z}_T)$ into itself. Moreover, from Theorem
\ref{R}, we have
$$
\| \Gamma u - \Gamma v \|_{T, \infty} \leq
B_\infty (C + C_\infty)\delta^{-1} \| u - v \|_{T, \infty}.
$$
Therefore  $\Gamma$ has a unique fixed point
$\widetilde u \in \mathbb{Z}_T$ whenever
$C+C_\infty < \delta B_{\infty}^{-1}$.
In this event, the function $\widetilde u$ is the only
pseudo-almost automorphic solution to \eqref{L}.
\end{proof}

\section{An example}

Fix $\delta_0 > 0$. Consider the system of nonlinear hyperbolic partial
differential equations \eqref{L} in which
\begin{gather*}
 a(t,x)=\delta_0(2 + \sin t)(2+\cos \frac{x}{\delta_0}), \quad
 b(t,x)=2 - \sin t, \\
c(t,x)= \delta_0(4 - \sin^{2} t)(2+\cos \frac{x}{\delta_0}), \quad
f(t,x,u)=\frac{1}{2}\big(u \sin t+ e^{-|t|}\sin u\big)
\end{gather*}
 for all $t \in \mathbb{R}$, $x \in [0, 1]$, and $u \in \mathbb{R}$.
For all $u, v \in \mathbb{R}$, $t \in \mathbb{R}$ and $x \in [0, 1]$, we have
\begin{gather*}
| f(t,x, u) - f(t,x,v)|
 =\frac{1}{2}|(u - v) \sin t+ e^{-|t|}(\sin u - \sin v)|
\leq |u - v|, \\
a(t,x)=\delta_0(2 + \sin t) (2 + \cos \frac{x}{\delta_0})\geq \delta_0>0, \\
 b_\ast:= \inf_{t \in \mathbb{R}, x \in [0, 1]} b(t,x)
= \inf_{t \in \mathbb{R}, x \in [0, 1]} (2 - \sin t)=1 >0.
\end{gather*}
Clearly, assumptions (H1)--(H4) are satisfied with $\delta=\delta_0$ and $C=1$.
From
\begin{align*}
 \int_0^x e^{- \int_y^x b(s, r)dr} dy
&=\int_0^x e^{- \int_y^x(2 - \sin s)dr} dy\\
 &=\int_0^x e^{- (2 - \sin s)( x - y)} dy\\
 &= \frac{1 - e^{ x(\sin s - 2)}}{2 - \sin s},
\end{align*}
we deduce that
\begin{equation*}
 B_\infty = \sup_{s \in \mathbb{R},x \in [0,1]}
\Big(\frac{1 - e^{ x(\sin s - 2)}}{2 - \sin s}\Big)\leq 1 - e^{-3}.
\end{equation*}
Similarly,
\begin{align*}
 C_\infty=\sup_{t \in \mathbb{R},x \in [0,1]}|H(t,x)|
=\sup_{t \in \mathbb{R},x \in [0,1]} \Big|-&(2 + \sin t)\sin \frac{x}{\delta_0}\Big|\leq 3.
\end{align*}
In view of the above,
$B_{\infty}(C + C_\infty)\leq 4(1 - e^{-3})$.
Therefore, using Theorem \ref{O}, it follows that \eqref{L} with the
above-mentioned coefficients has a unique pseudo-almost automorphic
solution whenever $\delta_0$ is chosen so $\delta_0> 4(1 - e^{-3})$.

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\end{document}