\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 $$x' = A(t)x + f(t) \label{e1}$$ 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 $$y'= A(t)y. \label{e2}$$ 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{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_{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_{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{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} \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{gathered} | \Psi (t)Y(t)P_{-} Y^{-1}(s)\Psi ^{-1}(s)| \leq K \quad \text{for }-\infty 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. 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