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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 05, pp. 1--12.\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 2009 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2009/05\hfil $\Psi$-bounded solutions]
{$\Psi$-bounded solutions for linear differential
systems with  Lebesgue $\Psi$-integrable functions on $\mathbb{R}$
as right-hand sides}

\author[A. Diamandescu\hfil EJDE-2009/05\hfilneg]
{Aurel Diamandescu}

\address{Aurel Diamandescu \newline
University of Craiova,
Department of Applied Mathematics,
13, ``Al. I. Cuza'' st., 200585 Craiova, Romania}
\email{adiamandescu@central.ucv.ro}

\thanks{Submitted October 9, 2008. Published January 6, 2009.}
\subjclass[2000]{34D05, 34C11}
\keywords{$\Psi$-bounded; $\Psi$-integrable}

\begin{abstract}
 In this paper we give a characterization for the existence
 of $\Psi$-bounded solutions on $\mathbb{R}$ for the system
 $x'=A(t)x + f(t)$, assuming that $f$ is a Lebesgue $\Psi$-integrable
 function on $\mathbb{R}$. In addition, we give a result in connection
 with the asymptotic behavior of the $\Psi$-bounded solutions of this system.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

This work is concerned with linear differential system
\begin{equation}
x' = A(t)x + f(t)  \label{e1}
\end{equation}
where $x(t)$, $f(t)$ are in $\mathbb{R}^d$ and $A$ is a continuous
$d\times d$ matrix-valued function.
The basic problem under consideration is the determination of necessary and
sufficient conditions for the existence of a solution with some specified
boundedness condition. A clasic result in this type of problems
is given by Coppel \cite[Theorem 2, Chapter V]{c2}.

The problem of $\Psi$-boundedness of the solutions for systems of ordinary
differential equations has been studied in many papers,
\cite{a1,b1,c1,d1,d2,d3,d4,h1,h2,m1}.
In \cite{d1,d2,d3}, the author proposes the novel concept of
$\Psi$-boundedness of solutions, $\Psi $ being a continuous matrix-valued
function, allows a better identification of various types of asymptotic
behavior of the solutions on $\mathbb{R}_{+}$.

Similarly, we can consider solutions of \eqref{e1} which are $\Psi$-bounded
not only $\mathbb{R}_{+}$ but on  $\mathbb{R}$.
In this case, the conditions for the existence of at least one
$\Psi$-bounded solution are rather complicated, as shown in \cite{d4}
and below.
In \cite{d4}, it is given a necessary and sufficient condition so that
the system \eqref{e1} has at least one $\Psi$-bounded solution on
$\mathbb{R}$ for every continuous and $\Psi$-bounded function $f$
on $\mathbb{R}$.

The aim of present paper is to give a necessary and sufficient condition so
that the nonhomogeneous system of ordinary differential equations \eqref{e1}
has at least one $\Psi$-bounded solution on $\mathbb{R}$ for every
Lebesgue $\Psi$-integrable function $f$ on $\mathbb{R}$.
The introduction of the matrix function $\Psi $ permits to obtain a mixed
asymptotic behavior of the components of the solutions.
Here, $\Psi $ is a continuous matrix-valued function on $\mathbb{R}$.

\section{Definitions, Notations and hypotheses}

Let $\mathbb{R}^d$ be the Euclidean $d$-space.
For $x = (x_{1}, x_{2}, x_{3}, \dots ,x_d)^T\in \mathbb{R}^d$,
let $\| x\|  = \max \{| x_{1}| , |x_{2}|, |x_{3}| ,\dots ,
| x_d| \}$ be the norm of $x$. For a $d\times d$ real matrix
$A = (a_{ij})$, we define the norm
$|A|  = \sup_{\| x\| \leq 1} \| Ax\| $. It is well-known that
\[
|A|  = \max_{1\leq i\leq d} \{\sum_{j=1}^d| a_{ij}| \}.
\]
Let $\Psi _{i} : \mathbb{R} \to  (0,\infty )$,
$i = 1,2,\dots d$, be continuous functions and
\begin{equation*}
\Psi =\mathop{\rm diag}[\Psi _{1},\Psi _{2},\dots \Psi _d].
\end{equation*}



\textbf{Definition.} A function
$\varphi  : \mathbb{R}\to \mathbb{R}^d$ is said to be $\Psi$-bounded on
$\mathbb{R}$ if $\Psi \varphi $ is bounded on $\mathbb{R}$.

\textbf{Definition.} A function
$\varphi  : \mathbb{R} \to \mathbb{R}^d$ is said to be Lebesgue
$\Psi$-integrable on $\mathbb{R}$ if $\varphi $ is measurable and
$\Psi \varphi $ is Lebesgue integrable on $\mathbb{R}$.

By a solution of \eqref{e1}, we mean an absolutely continuous function
satisfying  \eqref{e1} for almost all $t \in \mathbb{R}$.

Let $A$ be a continuous $d\times d$ real matrix and let the associated linear
differential system be
\begin{equation}
y'= A(t)y.  \label{e2}
\end{equation}
Let $Y$ be the fundamental matrix of \eqref{e2} for which
$Y(0) = I_d$ (identity $d\times d$ matrix).

Let the vector space $\mathbb{R}^d$ be represented as a direct sum
of three subspaces $X_{-}$, $X_{0}$, $X_{+}$ such that a solution
$y(t)$ of \eqref{e2} is $\Psi$-bounded on $\mathbb{R}$ if and only
if $y(0) \in  X_{0}$
and $\Psi$-bounded on $\mathbb{R}_{+} = [0,\infty )$ if and only if
$y(0) \in  X_{-} \oplus  X_{0}$. Also, let
$ P_{-}, P_{0}, P_{+}$ denote
the corresponding projection of $\mathbb{R}^d$ onto
$X_{-}$, $X_{0}$, $X_{+}$ respectively.

\section{Main result}

\begin{theorem} \label{thm1}
 If $A$ is a continuous $d\times d$ real matrix on
$\mathbb{R}$, then  \eqref{e1} has at least one $\Psi$-bounded solution on
$\mathbb{R}$ for every Lebesgue $\Psi$-integrable function
$f : \mathbb{R}\to \mathbb{R}^d$ on $\mathbb{R}$ if and only if
there exists a positive constant $K$ such that
\begin{equation}
\begin{gathered}
| \Psi (t)Y(t)P_{-} Y^{-1}(s)\Psi ^{-1}(s)|
 \leq   K  \quad \text{for }t > 0,\; s \leq 0 \\
| \Psi (t)Y(t)(P_{0}+P_{-})Y^{-1}(s)\Psi ^{-1}
(s)|  \leq   K  \quad \text{for }t > 0,\; s >0, s < t \\
| \Psi (t)Y(t)P_{+} Y^{-1}(s)\Psi ^{-1}(s)|
 \leq   K  \quad \text{for }t > 0, s > 0, \;s \geq  t \\
| \Psi (t)Y(t)P_{-} Y^{-1}(s)\Psi ^{-1}(s)|
 \leq   K  \quad \text{for }t \leq  0,\; s <t  \\
| \Psi (t)Y(t)(P_{0}+P_{+})Y^{-1}(s)\Psi ^{-1}
(s)|  \leq   K  \quad \text{for } t \leq  0, \; s \geq t,\; s <0 \\
| \Psi (t)Y(t)P_{+} Y^{-1}(s)\Psi ^{-1}(s)|
 \leq   K  \quad \text{for }t \leq  0,\; s \geq  t,\; s \geq 0
\end{gathered}   \label{e3}
\end{equation}
\end{theorem}

\begin{proof}
 First, we prove the ``only if'' part.
Thus, suppose that the system \eqref{e1} has at least one
$\Psi$-bounded solution on $\mathbb{R}$ for every Lebesgue
$\Psi$-integrable function f : $\mathbb{R}\to \mathbb{R}^d$ on
$\mathbb{R}$.

We shall denote by $C_{\Psi }$ the Banach space of all $\Psi$-bounded and
continuous functions $x : \mathbb{R}\to \mathbb{R}^{d}$ with the norm
$\| x\| _{C_{\Psi }} = \sup_{t\in \mathbb{R}} \| \Psi (t)x(t)\| $
and by $B$ the Banach space of all Lebesgue $\Psi$-integrable functions
$x :\mathbb{R}\to \mathbb{R}^d$ with the norm
$\| x\| _{B} = \int_{-\infty }^{+\infty}\| \Psi (t)x(t)\| dt$.

We shall denote by $D$ the set of all functions $x :\mathbb{R}
\to \mathbb{R}^d$ which are absolutely continuous on
all intervals $J\subset \mathbb{R}$, $\Psi$-bounded on $\mathbb{R}$,
$x(0) \in X\_\oplus X_{+}$ and $x'-Ax \in  B$.

Obviously, $D$ is a vector space and
$x \to \| x\| _{D} = \| x\| _{C_{\Psi }} + \| x' - Ax\| _{B}$
is a norm on $D$.

\textbf{Step 1.} $(D, \| \cdot \| _{D})$ is a Banach
space.
Let $(x_{n})_{n\in \mathbb{N}}$ be a fundamental sequence of
elements of $D$. Then, it is a fundamental sequence in $C_{\Psi }$. Therefore,
there exists a continuous and $\Psi$-bounded function $x : \mathbb{R}
\to \mathbb{R}^d$ such that
$\lim_{n\to \infty } \Psi (t)x_{n}(t) = \Psi (t)x(t)$,
uniformly on $\mathbb{R}$. From the inequality
\begin{equation*}
\| x_{n}(t) - x(t)\| \leq  | \Psi ^{-1}(t)| \| \Psi (t)x_{n}
(t) - \Psi (t)x(t)\| , \quad t\in \mathbb{R},
\end{equation*}
it follows that $\lim_{n\to \infty } x_{n} (t) = x(t)$, uniformly
on every compact of $\mathbb{R}$.
Thus, $x(0) \in  X_{-}\oplus  X_{+}$.

On the other hand, the sequence $(f_{n})_{n\in \mathbb{N}}$,
where $f_{n}(t) = x_{n}'(t)- A(t)x_{n}(t)$, is a fundamental sequence
in the Banach space $B$. Thus, there exists $f\in  B$ such that
\begin{equation*}
\lim_{n\to \infty }\int_{-\infty}^{+\infty }\| \Psi (t)(f_{n}(t)
- f(t))\| dt = 0.
\end{equation*}
For a fixed, but arbitrary, $t \in  \mathbb{R}$, we have
\begin{align*}
x(t) -\ x(0)
&= \lim_{n\to \infty }\big( x_{n}(t) -x_{n}(0)\big) \\
& = \lim_{n\to \infty }\int_{0}^{t}x_{n}'(s)ds \\
&= \lim_{n\to \infty } \int_{0}^{t}
[ \Psi ^{-1}(s)( \Psi (s)(f_{n}(s)-f(s)) + f(s) + A(s)x_{n}(s)] ds \\
&= \int_{0}^{t}\big( f(s) + A(s)x(s)\big) ds.
\end{align*}
It follows that $x'- Ax = f \in  B$ and $x$ is absolutely
continuous on all intervals $J \subset \mathbb{R}$.
Thus, $x \in  D$.

Now, from
\begin{equation*}
\lim_{n\to \infty } \Psi (t)x_{n} (t) = \Psi(t)x(t), \quad
\text{uniformly on }\mathbb{R}
\end{equation*}
and
\begin{equation*}
\lim_{n\to \infty } \int_{-\infty}^{+\infty }
\| \Psi (t) [ (x_{n}' (t) - A(t)x_{n}(t)) - (x'(t)-A(t)x(t))] \|dt = 0,
\end{equation*}
it follows that $\lim_{n\to \infty }\| x_{n}-x\| _{D} = 0$. Thus,
$(D, \| \cdot\| _{D})$ is a Banach space.



\textbf{Step 2.} There exists a positive constant $K$ such that, for every
$f\in  B$ and for corresponding solution $x \in  D$ of \eqref{e1}, we have
\begin{equation}
\sup_{t\in \mathbb{R}} \| \Psi (t)x(t)\|  \leq
 K \int_{-\infty }^{+\infty}\| \Psi (t)f(t)\| dt,  \label{e4}
\end{equation}
For this, define the mapping $T : D \to  B$, $Tx = x'- Ax$.
This mapping is obviously linear and bounded, with $\| T\|\leq  1$.

Let $Tx = 0$. Then, $x'= Ax$, $x \in  D$. This shows that $x$ is
a $\Psi$-bounded solution on $\mathbb{R}$ of \eqref{e2}.
Then, $x(0) \in  X_{0}\cap \big( X_{-}\oplus X_{+}\big)  = \{0\}$.
Thus, $x =0$, such that the mapping $T$ is ``one-to-one'' .

Now, let $f\in  B$ and let $x$ be the $\Psi$-bounded solution on
$\mathbb{R}$ of the system \eqref{e1} which exists by assumption.
Let $z$ be the solution of the Cauchy problem
$$
 x' = A(t)x + f(t), \quad z(0) = (P_{-} + P_{+})x(0).
$$
Then $u = x - z$ is a solution of \eqref{e2} with
$u(0) = x(0)- (P_{-} + P_{+})x(0) = P_{0}x(0)$.
From the Definition of $X_{0}$, it follows that $u$ is
$\Psi$-bounded on $\mathbb{R}$. Thus, $z$ is $\Psi$-bounded on
$\mathbb{R}$. Therefore, $z$ belongs to $D$ and $Tz = f$.
Consequently, the mapping $T$ is ``onto'' .

From a fundamental result of  Banach: ``If $T$ is a bounded
one-to-one linear operator of one Banach space onto another, then the
inverse operator $T^{-1}$ is also bounded'' , we have
$\| T^{-1}f\| _{D}\leq \| T^{-1\!\!}\| \| f\| _{B}$, for all
$f \in  B$.

For a given $f \in  B$, let $x = T^{-1}f$ be the corresponding solution
$x \in  D$ of \eqref{e1}. We have
\begin{equation*}
\| x\| _{D}=\| x\|_{C_{\Psi }}+\| x' -\text{ Ax}\|
_{B}=\| x\| _{C_{\Psi }}+\|f\| _{B} \leq  \| T^{-1\!\!}\| \| f\| _{B}
\end{equation*}
or
\begin{equation*}
\| x\| _{C_{\Psi }}\leq \big( \| T^{-1}\| -1\big) \| f\| _{B} =K\| f\| _{B}.
\end{equation*}
This inequality is equivalent to \eqref{e4}.

\textbf{Step 3.} The end of the proof.
Let $T_{1} < 0 < T_{2}$ be a fixed points but
arbitrarily, and let $f : \mathbb{R}\to \mathbb{R}^d$ a function in
$B$ which vanishes on $(-\infty , T_{1}]\cup  [T_{2},+\infty )$.
It is easy to see that the function $x : \mathbb{R}\to \mathbb{R}^d$
defined by
\[
x(t) = \begin{cases}
-\int_{T_{1}}^{0}Y(t)P_{0} Y^{-1}(s)f(s)ds
 -\int_{T_{1}}^{T_{2}}Y(t)P_{+}Y^{-1}(s)f(s)ds, & t < T_{1}
\\
\int_{T_{1}}^{t}Y(t)P_{-} Y^{-1}(s)f(s)ds
 + \int_{0}^{t}Y(t)P_{0} Y^{-1}(s)f(s)ds  \\
-\int_{t}^{T_{2}}Y(t)P_{+} Y^{-1}(s)f(s)ds,
& T_{1} \leq  t \leq  T_{2}
\\
\int_{T_{1}}^{T_{2}}Y(t)P_{-} Y^{-1} (s)f(s)ds
 + \int_{0}^{T_{2}}Y(t)P_{0} Y^{-1} (s)f(s)ds,
& t > T_{2}
\end{cases}
\]
is the solution in $D$ of the system \eqref{e1}.
Now, we put
\[
 G(t,s) = \begin{cases}
Y(t)P_{-} Y^{-1}(s),        & s \leq 0 <t,\\
Y(t)(P_{0}+ P_{-})Y^{-1}(s), & 0<s<t, \\
-Y(t)P_{+} Y^{-1}(s),       & 0<t \leq s, \\
Y(t)P_{-} Y^{-1}(s),        & s<t\leq 0, \\
-Y(t)(P_{0}+ P_{+})Y^{-1}(s), & t\leq s <0, \\
-Y(t)P_{+} Y^{-1}(s), & t\leq 0 \leq s\,.
\end{cases}
\]
This function is continuous on $\mathbb{R}^{2}$ except on the line
$t = s$, where it has a jump discontinuity. Then, we have that
$x(t) = \int_{T_{1}}^{T_{2}}G(t,s)f(s)ds$,
$t \in \mathbb{R}$. Indeed,\\
$\bullet $ for $t < T_{1}$, we have
\begin{align*}
 & \int_{T_{1}}^{T_{2}} G(t,s)f(s)ds\\
&= - \int_{T_{1}}^{0}Y(t)(P_{0} + P_{+})Y^{-1}(s)f(s)ds
-\int_{0}^{T_{2}}Y(t)P_{+}Y^{-1}(s)f(s)ds \\
&= -\int_{T_{1}}^{0}Y(t)P_{0}Y^{-1}(s)f(s)ds
 - \int_{T_{1}}^{T_{2}}Y(t)P_{+}Y^{-1}(s)f(s)ds \\
&= x(t)
\end{align*}
$\bullet $ for $t \in  [T_{1},0]$, we have
\begin{align*}
\int_{T_{1}}^{T_{2}}G(t,s)f(s)ds
&= \int_{T_{1}}^{t}Y(t)P_{-}Y^{-1}(s)f(s)ds
 -\int_{t}^{0}Y(t)(P_{0} + P_{+} )Y^{-1}(s)f(s)ds\\
&\quad -\int_{0}^{T_{2}}Y(t)P_{+}Y^{-1}(s)f(s)ds \\
&=\int_{T_{1}}^{t}Y(t)P_{-}Y^{-1}(s)f(s)ds
 + \int_{0}^{t}Y(t)P_{0}Y^{-1}(s)f(s)ds\\
&\quad -\int_{t}^{T_{2}}Y(t)P_{+}Y^{-1}(s)f(s)ds\\
&= x(t),
\end{align*}
$\bullet $ for $t \in  (0,T_{2}]$, we have
\begin{align*}
 \int_{T_{1}}^{T_{2}}G(t,s)f(s)ds
&= \int_{T_{1}}^{0}Y(t)P_{-}Y^{-1}(s)f(s)ds
 +\int_{0}^{t}Y(t)(P_{0} + P_{-})Y^{-1}(s)f(s)ds\\
&\quad -\int_{t}^{T_{2}}Y(t)P_{+}Y^{-1}(s)f(s)ds \\
&=\int_{T_{1}}^{t}Y(t)P_{-}Y^{-1}(s)f(s)ds
 + \int_{0}^{t}Y(t)P_{0}Y^{-1}(s)f(s)ds\\
&\quad - \int_{t}^{T_{2}}Y(t)P_{+}Y^{-1}(s)f(s)ds\\
&= x(t),
\end{align*}
$\bullet $ for $t > T_{2}$, we have
\begin{align*}
 \int_{T_{1}}^{T_{2}}G(t,s)f(s)ds
&= \int_{T_{1}}^{0}Y(t)P_{-}Y^{-1}(s)f(s)ds
 + \int_{0}^{T_{2}}Y(t)(P_{0} + P_{-})Y^{-1}(s)f(s)ds \\
& = \int_{T_{1}}^{T_{2}}Y(t)P_{-}Y^{-1}(s)f(s)ds
 + \int_{0}^{T_{2}}Y(t)P_{0}Y^{-1}(s)f(s)ds \\
&= x(t).
\end{align*}
Now, the inequality \eqref{e4} becomes
\begin{equation*}
\sup_{t\in \mathbb{R}} \| \Psi (t)\int_{T_{1}}^{T_{2}} G(t,s)f(s)ds\|
 \leq K\int_{T_{1}}^{T_{2}}\|\Psi \text{(t)f(t)}\| dt.
\end{equation*}
For a fixed points $s \in \mathbb{R}$, $\delta >0$ and
$\xi \in \mathbb{R}^d$, but arbitrarily, let $f$ the function defined by
\begin{equation*}
f(t) = \begin{cases}
\Psi ^{-1}(t)\xi , & \text{for }s \leq t\leq  s+ \delta \\
0, & \text{elsewhere}.
\end{cases}
\end{equation*}
Clearly, $f \in  B$, $\| f\| _{B} = \delta\| \xi \| $. The above
inequality becomes
\begin{equation*}
\| \int_{s}^{s + \delta }\Psi (t)G(t,u)
\Psi ^{-1}(u)\xi du\|
\leq K\delta \| \xi \| ,\quad \text{for all t }\in  \mathbb{R}.
\end{equation*}
Dividing by $\delta $ and letting $\delta \to 0$, we obtain for
any $t \neq  s$,
\begin{equation*}
\| \Psi (t)G(t,s)\Psi ^{-1}(s)\xi \|
\leq K\| \xi \|,\quad  \text{for all } t \in  \mathbb{R},\;
\xi \in \mathbb{R}^d.
\end{equation*}
Hence, $| \Psi (t)G(t,s)\Psi ^{-1}(s)| \leq  K$, which is
equivalent to \eqref{e3}. By continuity, \eqref{e3} remains
valid also in the excepted case $t = s$.


Now, we prove the ``if'' part.
Suppose that the fundamental matrix $Y$ of \eqref{e2} satisfies the condition
\eqref{e3} for some $K > 0$.
Let $f : \mathbb{R}\to \mathbb{R}^d$ be a Lebesgue $\Psi$-integrable
function on $\mathbb{R}$. We  consider the function
$u : \mathbb{R}\to \mathbb{R}^d$ defined by
\begin{equation}
\begin{aligned}
u(t)
& = \int_{-\infty }^{t} Y(t)P_{-} Y^{-1}(s)f(s)ds
+ \int_{0}^{t}Y(t)P_{0} Y^{-1}(s)f(s)ds\\
&\quad - \int_{t}^{\infty }Y(t)P_{+} Y^{-1}(s)f(s)ds.  \label{e5}
\end{aligned}
\end{equation}

\textbf{Step 4.}
The function $u$ is well-defined on $\mathbb{R}$.
Indeed, for $v < t \leq  0$, we have
\begin{align*}
 \int_{v}^{t}\| Y(t)P_{-}Y^{-1}(s)f(s)\| ds
&=\int_{v}^{t}\| \Psi ^{-1}(t)\Psi (t)Y(t)P_{-}Y^{-1}(s)\Psi ^{-1}(s)
\Psi (s)f(s)\| ds\\
&\leq | \Psi ^{-1}(t)| \int_{v}^{t} \!| \Psi (t)Y(t)P_{-}Y^{-1}(s)
\Psi ^{-1}(s)| \|\Psi (s)f(s)\| ds \\
&\leq K| \Psi ^{-1}(t)| \int_{v}^{t}\| \Psi (s)f(s)\| ds,
\end{align*}
 which shows that the integral
$\int_{-\infty }^{t} Y(t)P_{-}Y^{-1}(s)f(s)ds$ is absolutely convergent.
For $t > 0$, we have the same result.

Similarly, the integral $\int_{t}^{\infty }Y(t)P_{+}Y^{-1}(s)f(s)ds$
is absolutely convergent.
Thus, the function $u$ is well-defined and is an absolutely continuous
function on all intervals $J \subset \mathbb{R}$.



\textbf{Step 5.} The function $u$ is a solution of  \eqref{e1}.
Indeed, for almost all $t \in  \mathbb{R}$, we have
\begin{align*}
u'(t) &= \int_{-\infty }^{t} A(t)Y(t)P_{-}Y^{-1}(s)f(s)ds
+ Y(t)P_{-}Y^{-1}(t)f(t) \\
&\quad +\int_{0}^{t} A(t)Y(t)P_{0}Y^{-1}
(s)f(s)ds + Y(t)P_{0}Y^{-1}(t)f(t) \\
&\quad -\int_{t}^{\infty }A(t)Y(t)P_{+}Y
^{-1}(s)f(s)ds + Y(t)P_{+}Y^{-1}(t)f(t) \\
& = A(t)u(t) + Y(t)(P_{-} + P_{0} + P_{+})Y^{-1}
(t)f(t) = A(t)u(t) + f(t).
\end{align*}
This shows that the function $u$ is a solution of  \eqref{e1}.



\textbf{Step 6.}
The solution $u$ is $\Psi$-bounded on $\mathbb{R}$.
Indeed, for $t < 0$, we have
\begin{align*}
\Psi (t)u(t) &= \int_{-\infty }^{t}\Psi
(t)Y(t)P_{-}Y^{-1}(s)\Psi ^{-1}(s)\Psi (s)f(s)ds \\
&\quad + \int_{0}^{t}\Psi (t)Y(t)P
_{0}Y^{-1}(s)\Psi ^{-1}(s)\Psi (s)f(s)ds \\
&\quad  - \int_{t}^{\infty }\Psi
(t)Y(t)P_{+}Y^{-1}(s)\Psi ^{-1}(s)\Psi (s)f(s)ds\\
& =\int_{-\infty }^{t}\Psi (t)Y(t)P_{-}Y^{-1}(s)\Psi ^{-1}(s)\Psi
(s)f(s)ds \\
&\quad -\int_{t}^{0}\Psi
(t)Y(t)(P_{0} + P_{+})Y^{-1}(s)\Psi ^{-1}(s)\Psi (s)f(s)ds \\
&\quad  - \int_{0}^{\infty }\Psi
(t)Y(t)P_{+}Y^{-1}(s)\Psi ^{-1}(s)\Psi (s)f(s)ds\,.
\end{align*}
Then
\[
\| \Psi (t)u(t)\|  \leq K\cdot \int_{-\infty }^{\infty }\| \Psi (s)f(s)\|
ds.
\]
For $t \geq  0$, we have
\begin{align*}
\Psi (t)u(t) &= \int_{-\infty }^{t}\Psi
(t)Y(t)P_{-}Y^{-1}(s)\Psi ^{-1}(s)\Psi (s)f(s)ds \\
&\quad+ \int_{0}^{t}\Psi (t)Y(t)P_{0}Y^{-1}(s)\Psi ^{-1}(s)\Psi (s)f(s)ds\\
&\quad-\int_{t}^{\infty }\Psi
(t)Y(t)P_{+}Y^{-1}(s)\Psi ^{-1}(s)\Psi (s)f(s)ds \\
&= \int_{-\infty }^{0}\Psi (t)Y(t)P_{-}Y^{-1}(s)\Psi ^{-1}(s)\Psi (s)f(s)ds \\
&\quad+ \int_{0}^{t}\Psi (t)Y(t)(P_{0} + P_{-})Y^{-1}(s)\Psi ^{-1}(s)
\Psi (s)f(s)ds \\
&\quad -\int_{t}^{\infty }\Psi
(t)Y(t)P_{+}Y^{-1}(s)\Psi ^{-1}(s)\Psi (s)f(s)ds\,.
\end{align*}
Then
\begin{equation*}
\| \Psi (t)u(t)\|  \leq K\cdot
\int_{-\infty }^{\infty }\| \Psi (s)f(s)\| ds.
\end{equation*}
Hence,
\begin{equation*}
\sup_{t\in \mathbb{R}} \| \Psi (t)u(t)
\|  \leq K\cdot \int_{-\infty }^{\infty
}\| \Psi (s)f(s)\| ds,
\end{equation*}
which shows that the solution $u$ is $\Psi$-bounded on $\mathbb{R}$.
The proof is now complete.
\end{proof}

In a particular case, we have the following result.

\begin{theorem} \label{thm2}
If the homogeneous equation \eqref{e2} has no nontrivial
$\Psi$-bounded solution on $\mathbb{R}$, then the  \eqref{e1} has a unique
$\Psi $-bounded solution on $\mathbb{R}$ for every Lebesgue
$\Psi$-integrable function $f : \mathbb{R}\to \mathbb{R}^{d}$ on
$\mathbb{R}$ if and only if there exists a positive constant $K$ such
that
\begin{equation}
\begin{gathered}
| \Psi (t)Y(t)P_{-} Y^{-1}(s)\Psi ^{-1}(s)|
 \leq   K  \quad \text{for }-\infty  <s<t< +\infty \\
| \Psi (t)Y(t)P_{+} Y^{-1}(s)\Psi ^{-1}(s)|
 \leq   K  \quad \text{for }-\infty  <t\leq s< +\infty
\end{gathered}  \label{e6}
\end{equation}
\end{theorem}

In this case, $P_{0} = 0$ and the proof is as above.

Next, we prove a theorem in which we will see that the asymptotic
behavior of solutions to \eqref{e1} is determined completely by the
asymptotic behavior of the fundamental matrix $Y$.

\begin{theorem} \label{thm3} Suppose that:

\noindent (1) the fundamental matrix $Y(t)$ of \eqref{e2} satisfies:
\begin{itemize}
\item[(a)] condition \eqref{e3} is satisfied for some $K >0$;

\item[(b)] the following conditions are satisfied:
 \begin{itemize}
 \item[(i)] $\lim_{t\to \pm \infty }|\Psi (t)Y(t)P_{0}|  = 0$;
 \item[(ii)] $\lim_{t\to -\infty }|\Psi (t)Y(t)P_{+}|  = 0$;
 \item[(iii)] $\lim_{t\to +\infty } |\Psi (t)Y(t)P_{-}|  = 0$;
 \end{itemize}
\end{itemize}

\noindent(2) the function $f : \mathbb{R}\to \mathbb{R}^{d}$ is
Lebesgue $\Psi$-integrable on $\mathbb{R}$.

Then, every $\Psi$-bounded solution $x$ of \eqref{e1} is such that
\begin{equation*}
\lim_{t\to \pm \infty } \| \Psi (t)x(t)\|  = 0.
\end{equation*}
\end{theorem}

\begin{proof} By Theorem \ref{thm1}, for every Lebesgue
$\Psi$-integrable function $f : \mathbb{R}\to \mathbb{R}^{d}$,
the equation \eqref{e1} has at least one $\Psi$-bounded solution on
$\mathbb{R}$.

Let $x$ be a $\Psi$-bounded solution on $\mathbb{R}$ of \eqref{e1}.
Let $u$ be defined by \eqref{e5}. The function $u$ is a $\Psi$-bounded
solution on $\mathbb{R}$ of \eqref{e1}.

Now, let the function $y(t) = x(t) - u(t) - Y(t)P_{0}(x(0) - u(0))$,
$t\in \mathbb{R}$.
Obviously, $y$ is a solution on $\mathbb{R}$ of \eqref{e2}.
Because $\Psi (t)Y(t)P_{0}$ is bounded on $\mathbb{R}$, $y$ is
$\Psi$-bounded on $\mathbb{R}$.
Thus, $y(0) \in  X_{0}$.
On the other hand,
\begin{align*}
y(0) &= x(0) - u(0) - Y(0)P_{0}(x(0) - u(0)) \\
& = (P_{-} + P_{+})(x(0) - u(0)) \in  X_{-} \oplus  X_{+}.
\end{align*}
Therefore, $y(0) \in  X_{0}\cap  (X_{-}\oplus  X_{+}) = \{0\}$
and then, $y = 0$.
It follows that
\begin{equation*}
x(t) = Y(t)P_{0}(x(0) - u(0)) + u(t), t \in \mathbb{R}.
\end{equation*}
Now, we prove that $\lim_{t\to \pm \infty }\| \Psi (t)u(t)\|  = 0$.
For $t \geq  0$, we write again
\begin{align*}
\Psi (t)u(t) &= \int_{-\infty }^{0}\Psi (t)Y(t)P_{-}Y^{-1}(s)
\Psi ^{-1}(s)\Psi (s)f(s)ds \\
& \quad+\int_{0}^{t}\Psi (t)Y(t)(P_{0}
+ P_{-})Y^{-1}(s)\Psi ^{-1}(s)\Psi (s)f(s)ds \\
&\quad -\int_{t}^{\infty }\Psi (t)Y(t)P_{+}Y^{-1}(s)
\Psi ^{-1}(s)\Psi (s)f(s)ds.
\end{align*}
Let $\varepsilon >0$. From the hypotheses:
There exists $t_{0}<0$ such that
$$
\int_{-\infty }^{t_{0}}\| \Psi (s)f(s)\| ds < \frac{\varepsilon }{5K};
$$
there exists $t_{1}>0$ such that, for all $t \geq  t_{1}$,
$$
 | \Psi (t)Y(t)P_{-}| < \frac{\varepsilon
}{5}(1+\int_{t_{0}}^{0}\| Y^{-1}(s)f(s)\| ds)^{-1};
$$
there exists $t_{2}> t_{1}$ such that, for all $t \geq  t_{2}$,
$$
\int_{t}^{\infty }\| \Psi (s)f(s)\| ds < \frac{\varepsilon }{5K};
$$
there exists $t_{3}> t_{2}$ such that, for all $t \geq  t_{3}$,
$$
| \Psi (t)Y(t)( P_{0} + P_{-})|  < \frac{\varepsilon }{5}(1+\int_{0}^{t
_{2}}\| Y^{-1}(s)f(s)\| ds )^{-1}.
$$
Then, for $t \geq  t_{3}$, we have
\begin{align*}
&\| \Psi (t)u(t)\|\\
&\leq \int_{-\infty }^{t_{0}}| \Psi (t)Y(t)P_{-}Y^{-1}(s)
\Psi ^{-1}(s)| \| \Psi (s)f(s)\| ds \\
&\quad+ \int_{t_{0}}^{0}| \Psi (t)Y(t)P_{-}| \| Y^{-1}(s)f(s)\| ds
 + \int_{0}^{t_{2}}| \Psi (t)Y(t)( P_{0}\\
&\quad  + P_{-})| \| Y^{-1}(s)f(s) \| ds
  + \int_{t_{2}}^{t}| \Psi (t)Y(t)( P_{0} + P_{-})Y^{-1}(s)
\Psi ^{-1}(s)| \| \Psi (s)f(s)\| ds \\
&\quad + \int_{t}^{\infty }| \Psi (t)Y(t)P_{+} Y^{-1}(s)\Psi ^{-1}(s)| \|
\Psi (s)f(s)\| ds \\
& < K\int_{-\infty }^{t_{0}}\| \Psi
(s)f(s)\| ds + \frac{\varepsilon }{5(1 + \int_{t_{0}}^{0}\|  Y^{-1}(s)f(s)
\| ds )}\int_{t_{0}}^{0}\| Y^{-1}(s)f(s)\| ds \\
&\quad + \frac{\varepsilon }{5(1+\int_{0}^{t_{2}}\|  Y^{-1}(s)f(s)\|
ds)}\int_{0}^{t_{2}}\| Y^{-1}(s)f(s)\| ds \\
&\quad + K\int_{t_{2}}^{t}\| \Psi
(s)f(s)\| ds + K\int_{t}^{\infty }\| \Psi (s)f(s)\| ds \\
& < K \frac{\varepsilon }{5K}
+ \frac{\varepsilon }{5} + \frac{\varepsilon }{5}
+ K(\int_{t_{2}}^{t}\| \Psi (s)f(s)\| ds
+ \int_{t}^{\infty }\| \Psi (s)f(s)\| ds) \\
& < \frac{3\varepsilon }{5} + K\frac{\varepsilon }{
5K} < \varepsilon .
\end{align*}
This shows that
$\lim_{t\to +\infty } \| \Psi (t)u(t)\|  = 0$.

For $t < 0$, we write again
\begin{align*}
\Psi (t)u(t)& = \int_{-\infty }^{t}\Psi
(t)Y(t)P_{-}Y^{-1}(s)\Psi ^{-1}(s)\Psi (s)f(s)ds \\
&\quad - \int_{t}^{0}\Psi (t)Y(t)(P
_{0} + P_{+})Y^{-1}(s)\Psi ^{-1}(s)\Psi (s)f(s)ds \\
&\quad  -\int_{0}^{\infty }\Psi (t)Y(t)P_{+}Y^{-1}(s)\Psi ^{-1}(s)\Psi
(s)f(s)ds.
\end{align*}
Let $\varepsilon >0$. From the hypotheses, we have:
There exists $t^{0}>0$ such that
\[
\int_{t^{0}}^{+\infty }\| \Psi (s)f(s)\| ds < \frac{\varepsilon }{5K};
\]
 there exists $t_{4}<0$ such that, for all $t < t_{4}$,
\[
 | \Psi (t)Y(t)P_{+}|  < \frac{\varepsilon
}{5}(1+\int_{0}^{t^{0}}\| Y^{-1}
(s)f(s)\| ds )^{-1};
\]
there exists $t_{5}< t_{4}$ such that, for all t $\leq  t_{5}$,
\[
\int_{-\infty }^{t}\| \Psi
(s)f(s)\| ds < \frac{\varepsilon }{5K};
\]
 there exists $t_{6}< t_{5}$ such that, for all $t \leq  t_{6}$,
\[
| \Psi (t)Y(t)( P_{0} + P_{+})|  <
\frac{\varepsilon }{5}(1+\int_{t_{5}}^{0
}\| Y^{-1}(s)f(s)\| ds )^{-1}.
\]
Then, for $t \leq  t_{6}$, we have
\begin{align*}
& \| \Psi (t)u(t)\|\\
& \leq  \int_{-\infty }^{t}| \Psi (t)Y(t)P_{-}Y^{-1}(s)
\Psi ^{-1}(s)| \| \Psi (s)f(s)\| ds \\
&\quad+ \int_{t}^{t_{5}}| \Psi (t)Y(t)(P_{0} + P_{+})Y^{-1}(s)
\Psi ^{-1}(s)| \| \Psi (s)f(s)\| ds \\
&\quad + \int_{t_{5}}^{0}| \Psi
(t)Y(t)(P_{0} + P_{+})| \| Y^{-1}(s)f(s)\| ds
 + \int_{0}^{t^{0}}| \Psi (t)Y(t)P_{+}| \| Y^{-1}(s)f(s)\| ds \\
&\quad + \int_{t^{0}}^{+\infty }| \Psi
(t)Y(t)P_{+}Y^{-1}(s)\Psi ^{-1}(s)| \| \Psi (s)f(s)
\| ds \\
&<  K \int_{-\infty }^{t}\| \Psi (s)f(s)\| ds + K
\int_{t}^{t_{5}}\|\Psi (s)f(s)\| ds \\
&\quad + \frac{\varepsilon }{5(1 +\int_{t
_{5}}^{0}\|  Y^{-1}(s)f(s)\| ds)}
\int_{t_{5}}^{0}\| Y^{-1}(s)f(s)\| ds\\
&\quad + \frac{\varepsilon }{5(1 +\int_{0
}^{t^{0}}\|  Y^{-1}(s)f(s)\|ds)}
\int_{0}^{t^{0}}\| Y^{-1}(s)f(s)\|
ds + K\int_{t^{0}}^{+\infty }\| \Psi (s)f(s)\| ds \\
&<  K(\int_{-\infty }^{t}\| \Psi
(s)f(s)\| ds + \int_{t}^{t_{5}}\|
\Psi (s)f(s)\| ds) + \frac{\varepsilon }{5} +
\frac{\varepsilon }{5} + K\frac{\varepsilon }{5K} \\
&<  K\frac{\varepsilon }{5K} + \frac{3\varepsilon
}{5} < \varepsilon .
\end{align*}
This shows that $\lim_{t\to -\infty }\|\Psi (t)u(t)\|  = 0$.

Now, it is easy to see that $\lim_{t\to \pm \infty }\| \Psi (t)x(t)\|  = 0$.
The proof is now complete.
\end{proof}

The next result follows from Theorems \ref{thm2} and \ref{thm3}.

\begin{corollary} \label{coro1}
 Suppose that
\begin{enumerate}
\item the homogeneous equation \eqref{e2} has no nontrivial
$\Psi$-bounded solution on $\mathbb{R}$;

\item the fundamental matrix $Y(t)$ of \eqref{e2} satisfies:\\
(i) the condition \eqref{e6} for some $K > 0$.\\
(ii) $\lim_{t\to -\infty }| \Psi (t)Y(t)P_{+}|  = 0$;\\
(iii) $\lim_{t\to +\infty } | \Psi (t)Y(t)P_{-}|  = 0$;

\item the function $f : \mathbb{R}\to \mathbb{R}^{d}$ is Lebesgue
$\Psi$-integrable on $\mathbb{R}$.
\end{enumerate}
Then \eqref{e1} has a unique solution $x$ on $\mathbb{R}$ such that
\begin{equation*}
\lim_{t\to \pm \infty } \| \Psi (t)x(t)\|  = 0.
\end{equation*}
\end{corollary}

Note that Theorem \ref{thm3} is no longer true if we require that the function
$f$ be $\Psi$-bounded on $\mathbb{R}$
(more, even $\lim_{t\to \pm \infty }\| \Psi (t)f(t)\|  = 0$),
instead of the condition (2) in the above the Theorem.
This is shown next.

\subsection*{Example}
Consider  \eqref{e1} with $A(t) = O_{2}$ and
$f(t) = (\sqrt{1+| t| },1)^{T}$.
Then, $Y(t) = I_{2}$ is a fundamental matrix for \eqref{e2}.
Consider
\begin{equation*}
\Psi (t) = \begin{pmatrix}
\frac{1}{1+| t| } & 0 \\
0 & \frac{1}{(1 + | t|)^{2}}
\end{pmatrix}.
\end{equation*}

The solutions of \eqref{e2} are $y(t) = (c_{1}, c_{2})^{T}$,
 where $c_{1}, c_{2} \in  \mathbb{R}$. Then
\begin{equation*}
\Psi (t)y(t) = (\frac{c_{1}}{1+| t| }
,\frac{c_{2}}{(1+| t| )^{2}})^{T}.
\end{equation*}
Therefore, $P_{-} = O_{2}$, $P_{+} = O_{2}$ and $P_{0} = I_{2}$.
The conditions \eqref{e3} are satisfied with $K = 1$.
In addition, the hypothesis (1b) of  Theorem \ref{thm3} is satisfied.
Because
\[
\Psi (t)f(t) = \Big(\frac{1}{\sqrt{1+| t| }},
\frac{1}{(1+| t| )^{2}}\Big)^{T},
\]
 the function $f$ is not Lebesgue $\Psi$-integrable on $\mathbb{R}$,
but it is $\Psi$-bounded on $\mathbb{R}$, with
$\lim_{t\to \pm \infty } \| \Psi (t)f(t)\|  = 0$.
The solutions of the system \eqref{e1} are
$x(t) = (F(t) + c_{1}, t + c_{2})^{T}$, where
\begin{equation*}
F(t) = \begin{cases}
-\frac{2}{3}(1 - t)^{3/2}  + \frac{4}{3}, & t<0 \\
\frac{2}{3}(1 + t)^{3/2}, & t\geq 0\,.
\end{cases}
\end{equation*}
It is easy to see that $\lim_{t\to \pm \infty }\| \Psi (t)x(t)\|  = +\infty $,
for all $c_{1}, c_{2} \in  \mathbb{R}$.
It follows that the all solutions of the system \eqref{e1}
are $\Psi$-unbounded on $\mathbb{R}$.


\subsection*{Remark}
If in the above example, $f(t) = (\frac{1}{1+| t| }, 0)^{T}$,
then $\int_{-\infty }^{+\infty }\| \Psi (t)f(t)\| dt =2$.
On the other hand, the solutions of  \eqref{e1} are
$ x(t) = (u(t) + c_{1}, c_{2})^{T}$,
 where
\[
 u(t) = \begin{cases}
-\ln(1 - t), & t<0 \\
\ln (1 + t), & t\geq 0\,.
\end{cases}
\]
We observe that the asymptotic properties of the components of the
solutions are not the same: The first component is unbounded and the
second is bounded on $\mathbb{R}$.
However, all solutions of \eqref{e1} are $\Psi$-bounded on $\mathbb{R}$ and
$\lim_{t\to \pm \infty }\| \Psi (t)x(t)\| = 0$.
This shows that the asymptotic properties of the components of the solutions
are the same, via the matrix function $\Psi $. This is obtained by
using a matrix function $\Psi $ rather than a scalar function.



\begin{thebibliography}{00}

\bibitem{a1} Akinyele, O.;
\emph{On partial stability and boundedness of degree
k,} Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., (8), 65(1978),
259 - 264.

\bibitem{b1} Boi, P. N.;
 \emph{Existence of $\Psi $-bounded
solutions on $R$ for nonhomogeneous linear differential equations,}
Electron. J. Diff. Eqns., vol. 2007 (2007), No. 52. pp. 1--10.

\bibitem{c1} Constantin, A.;
\emph{Asymptotic Properties of Solutions of
Differential Equations,} Analele Universit\u{a}\c{t}ii din Timi\c{s}oara,
Vol. XXX, fasc. 2-3, 1992, Seria \c{S}tiin\c{t}e Matematice, 183 - 225.

\bibitem{c2} Coppel, W. A;
 \emph{Stability and Asymptotic Behavior of
Differential Equations,} Heath, Boston, 1965.

\bibitem{d1} Diamandescu, A.;
\emph{Existence of $\Psi $-bounded
solutions for a system of differential equations,}
Electron. J. Diff. Eqns., Vol. 2004(2004), No. 63, pp. 1 - 6,

\bibitem{d2} Diamandescu, A.;
\emph{Note on the $\Psi $-boundedness
of the solutions of a system of differential equations,} Acta. Math. Univ.
Comenianae, Vol. LXXIII, 2(2004), pp. 223 - 233

\bibitem{d3} Diamandescu, A.;
\emph{A Note on the $\Psi $-boundedness for differential systems,}
Bull. Math. Soc. Sc. Math. Roumanie,
Tome 48(96), No. 1, 2005, pp. 33 - 43.

\bibitem{d4} Diamandescu, A.;
\emph{A note on the existence of $\Psi$-bounded solutions for a system
of differential equations on }$\mathbb{R}$,
Electron. J. Diff. Eqns., Vol. 2008(2008), No. 128, pp. 1 - 11.

\bibitem{h1} Hallam, T. G.;
 \emph{On asymptotic equivalence of the bounded
solutions of two systems of differential equations,} Mich. math. Journal,
Vol. 16(1969), 353-363.

\bibitem{h2} Han, Y., Hong, J.;
\emph{Existence of $\Psi$-bounded solutions for linear difference equations,}
Applied mathematics Letters 20 (2007) 301-305.

\bibitem{m1} Morchalo, J.;
\emph{On $( \Psi -L_{p})$-stability of nonlinear systems of differential
equations,} Analele Universit\v{a}\c{t}ii ``Al. I. Cuza'',
Ia\c{s}i, XXXVI, I, Matematic\u{a}, (1990), 4, 353-360.

\end{thebibliography}

\end{document}