\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 142, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/142\hfil Perron-type theorem] {Perron-type theorem for fractional differential systems} \author[N. D. Cong, T. S. Doan, H. T. Tuan \hfil EJDE-2017/142\hfilneg] {Nguyen Dinh Cong, Thai Son Doan, Hoang The Tuan} \address{Nguyen Dinh Cong \newline Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, 10307 Ha Noi, Viet Nam} \email{ndcong@math.ac.vn} \address{Thai Son Doan \newline Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, 10307 Ha Noi, Viet Nam} \email{dtson@math.ac.vn} \address{Hoang The Tuan \newline Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, 10307 Ha Noi, Viet Nam} \email{httuan@math.ac.vn} \dedicatory{Communicated by Mokhtar Kirane} \thanks{Submitted March 17, 2017. Published Jun 17, 2017.} \subjclass[2010]{26A33, 34A08, 34A30, 34E10} \keywords{Fractional differential equations; linear systems; bounded solutions; \hfill\break\indent Perron-type theorem; asymptotic behavior} \begin{abstract} In this article, we prove a Perron-type theorem for fractional differential systems. More precisely, we obtain a necessary and sufficient condition for a system of linear inhomogeneous fractional differential equations to have at least one bounded solution for every bounded inhomogeneity. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} In recent years, fractional differential equations have attracted increasing interest due to their varied applications on various fields of science and engineering, see e.g., \cite{Bandyopadhyay, Kai, Podlubny, Samko}. Several results on asymptotic behavior of fractional differential equations are published: e.g., on Linear theory \cite{Matignon, Cong_1}, Stability theory for nonlinear systems \cite{Ahmed, Cong_4}, Stable manifolds \cite{Cong_6}, Stability theory for perturbed linear systems \cite{Cong_5}. However, the qualitative theory of fractional differential equations is still in its infancy. One of the reasons for this fact might be that these equations do not generate semigroups and the well-developed theory to ordinary differential equations cannot be applied directly. Consider the inhomogenneous system of the order $\alpha \in (0,1)$ involving Caputo derivative \begin{equation}\label{mainEq} ^{C\!}D^\alpha_{0+}x(t)=Ax(t)+f(t),\quad x(0)=x_0\in \mathbb{R}^d, \end{equation} where $t\in [0,\infty)$, $A\in\mathbb{R}^{d\times d}$ and $f:[0,\infty)\to \mathbb{R}^d$. Motivated by Perron's work, an interesting question arises here: what is the necessary and sufficient condition on $A$ for which \eqref{mainEq} has at least one bounded solution for any bounded continuous vector-valued function $f$? In the case of ordinary differential equations ($\alpha=1$), the answer is known: the matrix $A$ is hyperbolic; see \cite[Proposition 3, p. 22]{Coppel}. However, for the fractional case the question is still open. Note that Matignon \cite{Matignon} gives a necessary and sufficient condition of the matrix $A$ such that for any external force $f$ and any initial condition $x_0$, the solution $x$ of \eqref{mainEq} is bounded. In this article, we give a Perron-type theorem for fractional differential systems saying that the inhomogeneous system \eqref{mainEq} has at least one bounded solution for any bounded continuous function $f$ if and only if the matrix $A$ satisfies a (fractional) hyperbolic condition \begin{equation}\label{SpecCond} \sigma(A) \cap \big\{\lambda\in \mathbb{C}: \lambda=0\text{ or } |\arg{(\lambda)}|=\frac{\alpha \pi}{2}\big\}=\emptyset, \end{equation} where $\sigma(A)$ is the set of all eigenvalues of the matrix $A$. This result is a natural analog of the known theorem of the theory of ordinary differential equations. Our approach is as follows. First, we transform the matrix $A$ of the system \eqref{mainEq} into its Jordan normal form to obtain a simpler system. Next, using the variation of constants formula and a procedure of substitution, we describe explicitly bounded solutions. Finally, by estimating Mittag-Leffler functions, we show the asymptotic behavior of solutions which enable us to describe the set of unbounded solutions of \eqref{mainEq} when the matrix does not satisfy the hyperbolic condition. The paper is organized as follows. In Section \ref{sec.preliminaries}, we present some basics of fractional calculus and some preliminary results related to Mittag-Leffler functions. In Section~\ref{sec.main}, we state and prove the main result of the paper (Theorem~\ref{thm.main}). To conclude the introductory section, we fix some notation which will be used later. Let $\mathbb{R}$, $\mathbb{C}$ be the set of all real numbers and complex numbers, respectively. Denote by $\mathbb{R}_{\geq 0}$ the set of all nonnegative real numbers. For a Banach space $(X,\|\cdot\|)$, we define $(C_b(\mathbb{R}_{\geq 0};X),\|\cdot\|_\infty)$ as the space of all continuous function $\xi:\mathbb{R}_{\geq 0}\to X$ such that \[ \|\xi\|_\infty:=\sup_{t\geq 0}\|\xi(t)\|<\infty. \] For any $\lambda\in \mathbb{C}\setminus\{0\}$, we define its argument to be in the interval $-\pi < \arg{(\lambda)}\leq \pi$ and $\Re \lambda$, $\Im \lambda$ the real part, the imaginary part of the complex number $\lambda$, respectively. For $\alpha \in (0,1)$, we define the sets \begin{gather} \Lambda_{\alpha}^u :=\{\lambda \in\mathbb{C}\setminus\{0\}:|\arg{(\lambda)}| < \frac{\alpha \pi}{2}\},\label{SectorUnstable}\\ \Lambda_{\alpha}^s :=\{\lambda\in\mathbb{C}\setminus\{0\}:|\arg{(\lambda)}| > \frac{\alpha \pi}{2}\}. \label{SectorStable} \end{gather} \section{Preliminaries}\label{sec.preliminaries} \subsection{Inhomogeneous linear fractional differential equations} For $\alpha>0$, $[a,b]\subset \mathbb{R}$ and $x:[a,b]\to \mathbb{R}$ is a measurable function such that $\int_a^b|x(\tau)|\,d\tau<\infty$, the Riemann-Liouville integral operator of order $\alpha$ is defined by \[ (I_{a+}^{\alpha}x)(t):=\frac{1}{\Gamma(\alpha)} \int_a^t(t-\tau)^{\alpha-1}x(\tau)\,d\tau,\quad t\in (a,b], \] where the Gamma function $\Gamma:(0,\infty)\to \mathbb{R}$ is defined as \[ \Gamma(\alpha):=\int_0^\infty \tau^{\alpha-1}\exp(-\tau)\,d\tau. \] The \emph{Caputo fractional derivative} $^{C\!}D_{a+}^\alpha x$ of a function $x\in C^m([a,b])$ is defined by \[ (^{C\!}D_{a+}^\alpha x)(t):=(I_{a+}^{m-\alpha}D^mx)(t),\quad \forall t\in [a,b], \] where $D^m=\frac{d^m}{dt^m}$ is the usual $m^{th}$-order derivative and $m:=\lceil\alpha\rceil$ is the smallest integer larger or equal to $\alpha$, see, e.g., \cite[p.~79]{Podlubny}. While the Caputo fractional derivative of a $d$-dimensional vector function $x(t)=(x_1(t),\dots,x_d(t))^{\mathrm{T}}$ is defined component-wise as \[ (^{C\!}D_{a+}^\alpha x)(t):=(^{C\!}D_{a+}^\alpha x_1(t),\dots,^{C\!}D_{a+}^\alpha x_d(t))^{\mathrm{T}}. \] In this paper, we consider the initial value problem \begin{equation}\label{mainEq1} ^{C\!}D^\alpha_{0+}x(t)=Ax(t)+f(t), \quad x(0)=\xi\in\mathbb{R}^d \end{equation} with $\alpha\in (0,1)$ and $f:[0,\infty)\to \mathbb{R}^d$ is a continuous function. It is well known that the initial problem \eqref{mainEq1} has a unique solution defined on the whole interval $[0,\infty)$, see, e.g., \cite[Theorem 6.8]{Kai}. An explicit formula of this solution is given by using \emph{Mittag-Leffler functions} which are defined as \[ E_{\alpha,\beta}(M):=\sum_{k=0}^\infty\frac{A^k}{\Gamma(\alpha k+\beta)},\quad E_\alpha(M):=E_{\alpha,1}(M),\quad \forall M\in \mathbb{C}^{d\times d}, \] where $\beta\in \mathbb{R}$. Next we have a variation of constants formula for fractional differential equations. \begin{theorem} \label{Var_Const_Form} Let $\xi \in \mathbb{R}^d$ and $\varphi(\cdot,\xi)$ denote the solution of the initial problem \eqref{mainEq1}. Then the following (variation of constants) formula holds \[ \varphi(t,\xi)=E_\alpha(t^\alpha A) \xi +\int_0^t (t-\tau)^{\alpha-1}E_{\alpha,\alpha}((t-\tau)^\alpha A)f(\tau)\,d\tau, \quad \forall t\geq 0. \] \end{theorem} The proof of the above theorem uses the same arguments as in the proof of \cite[Lemma 2]{Cong_3}; see also \cite[Remark 4]{Cong_3}. \subsection{Some useful properties of Mittag-Leffler functions} To investigate the asymptotic behavior of the solutions to linear fractional differential equations, it is important to know the behavior of Mittag-Leffler functions. Hence, we next introduce some basic properties of these functions. To save the length of the paper we give only sketch of the proofs of the results presented in this subsection. \begin{lemma}\label{lemma3} Let $\lambda\in\mathbb{C}$ be arbitrary. There exist a positive real number $m(\alpha,\lambda)$ such that for every $t\geq 1$ the following estimations hold: \begin{itemize} \item [(i)] if $\lambda\in\Lambda_\alpha^u$ then \begin{gather*} \big|E_\alpha(\lambda t^{\alpha})-\frac{1}{\alpha}\exp{(\lambda^{1/\alpha}t)}\big| \le \frac{m(\alpha,\lambda)}{t^{\alpha}},\\ \big|t^{\alpha-1}E_{\alpha,\alpha}(\lambda t^{\alpha}) -\frac{1}{\alpha}\lambda^{\frac{1}{\alpha}-1}\exp{(\lambda^{1/\alpha}t)}\big| \le \frac{m(\alpha,\lambda)}{t^{\alpha+1}}; \end{gather*} \item [(ii)] if $\lambda\in\Lambda_\alpha^s$ then $$ |t^{\alpha-1}E_{\alpha,\alpha}(\lambda t^{\alpha})| \le \frac{m(\alpha,\lambda)}{t^{\alpha+1}}. $$ \end{itemize} \end{lemma} For a proof of this theorem one uses integral representations of Mittag-Leffler functions and the method of estimation of the integrals similar to that of the proofs of \cite[Theorem 1.3 and 1.4, pp. 32--34]{Podlubny}. \begin{lemma}\label{lemma4} Let $\lambda\in\mathbb{C}\setminus\{0\}$. There exists a positive constant $K(\alpha,\lambda)$ such that for all $t \ge 0$ the following estimates hold: \begin{itemize} \item [(i)] if $\lambda\in\Lambda_\alpha^u$, then \begin{gather*} \int_t^\infty |\lambda^{\frac{1}{\alpha}-1}E_\alpha(\lambda t^\alpha) \exp(-\lambda^{1/\alpha}\tau)|\,d\tau \leq K(\alpha,\lambda),\\ \int_0^t|(t-\tau)^{\alpha-1}E_{\alpha,\alpha}(\lambda(t-\tau)^{\alpha}) - \lambda^{\frac{1}{\alpha}-1}E_\alpha(\lambda t^\alpha) \exp(-\lambda^{1/\alpha}\tau)|\,d\tau \leq K(\alpha,\lambda); \end{gather*} \item [(ii)] if $\lambda\in\Lambda_\alpha^s$, then \[ \int_0^t|(t-\tau)^{\alpha-1}E_{\alpha,\alpha}(\lambda(t-\tau)^{\alpha})|\,d\tau \leq K(\alpha,\lambda). \] \end{itemize} \end{lemma} The proof of the above lemma follows easily by using Lemma \ref{lemma3} and repeating arguments used in the proof of \cite[Lemma 5]{Cong_2}. \begin{lemma}\label{LimitLemma} For any function $g\in C_b(\mathbb{R}_{\ge 0};\mathbb{R})$ and $\lambda\in\Lambda_\alpha^u$, we have \begin{equation} \label{Eq4} \begin{aligned} &\lim_{t\to\infty}\int_0^t (t-\tau)^{\alpha-1}\frac{E_{\alpha,\alpha} (\lambda(t-\tau)^\alpha)}{E_\alpha(\lambda t^\alpha)}g(\tau)\,d\tau\\ &=\lambda^{\frac{1}{\alpha}-1}\int_0^\infty\exp(-\lambda^{1/\alpha}\tau)g(\tau) \,d\tau. \end{aligned} \end{equation} \end{lemma} The proof of the above lemma uses Lemmas \ref{lemma3} and \ref{lemma4}, and arguments analogous to those used in the proof of \cite[Lemma 8]{Cong_2}. \section{A Perron type theorem for fractional differential equations}\label{sec.main} The main result of this section is a Perron-type theorem for fractional systems. \begin{theorem}\label{thm.main} Let $A\in\mathbb{R}^{d\times d}$ and $\alpha\in (0,1)$. The inhomogeneous system \begin{equation*}\label{mainEq1a} ^{C\!}D^\alpha_{0+}x(t)=Ax(t)+f(t) \end{equation*} has at least one bounded solution for any $f\in C_b(\mathbb{R}_{\geq 0};\mathbb{R}^d)$ if only if the matrix $A$ satisfies the condition $$ \sigma(A) \cap \{\lambda\in \mathbb{C}: \lambda=0\text{ or } |\arg{(\lambda)}|=\frac{\alpha \pi}{2}\}=\emptyset. $$ \end{theorem} The proof of Theorem \ref{thm.main} is divided into the sufficiency part (Proposition \ref{sufficiency part1}) and the necessity part (Proposition \ref{necessity part1}). Firstly, we show the sufficiency part. \begin{proposition}[Sufficient part of Theorem \ref{thm.main}]\label{sufficiency part1} Let $A\in\mathbb{R}^{d\times d}$ satisfy the hyperbolic condition \eqref{SpecCond} \[ \sigma (A)\cap \{\lambda\in \mathbb{C}: \lambda=0\text{ or }|\arg{(\lambda)}| =\frac{\alpha \pi}{2}\}=\emptyset. \] Then, for any $f\in C_b(\mathbb{R}_{\geq 0};\mathbb{R}^d)$, the corresponding inhomogeneous system \begin{equation}\label{mainEq2} ^{C\!}D^{\alpha}_{0+}x(t)=A x(t)+f(t), \end{equation} has at least one bounded solution. \end{proposition} Before proving Proposition \ref{sufficiency part1}, we transform the matrix $A$ of the system \eqref{mainEq2} into its Jordan normal form to obtain a simpler system. Let $T\in\mathbb{R}^{d\times d}$ be a nonsingular matrix transforming $A$ into its Jordan normal form, i.e., \[ T^{-1}A T=\operatorname{diag}(A_1,\dots,A_n), \] where for $j=1,\dots,n$, the block $A_j$ is of the form \[ A_j:=\begin{pmatrix} \lambda_j & 1 & 0 & \dots & 0 \\ 0 & \lambda_j & 1 & \dots & 0\\ \vdots &\vdots & \ddots & \ddots &\vdots\\ 0 & 0 &\dots & \lambda_j & 1 \\ 0& 0 &\dots &0 & \lambda_j \\ \end{pmatrix}_{d_j \times d_j}, \] with $\lambda_j \in \sigma(A)\cap \mathbb{R}$, or \[ A_j= \begin{pmatrix} D_j & I & 0 & \dots & 0 \\ 0 & D_j & I & \dots & 0\\ \vdots &\vdots & \ddots & \ddots &\vdots\\ 0 & 0 &\dots & D_j & I \\ 0& 0 &\dots &0 & D_j \\ \end{pmatrix}_{d_j\times d_j}, \] here \[ D_j=\begin{pmatrix} a_j & -b_j\\ b_j & a_j \end{pmatrix},\quad I=\begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix}, \quad a_j, b_j \in \mathbb{R},\quad b_j\neq 0, \] and $\lambda_j=a_j+ib_j\in \sigma(A)$. By the change of variable $x=Ty$, the system \eqref{mainEq2} is transformed into the equation \begin{equation}\label{eqn.jordan} ^{C\!}D^{\alpha}_{0+}y(t) = By(t)+g(t), \end{equation} where $B$ is the real Jordan normal form of $A$, i.e., \begin{equation*}\label{eqn.jordan1} B= T^{-1}AT = \operatorname{diag}(A_1,\dots,A_n), \quad \text{and} \quad g(t) = T^{-1} f(t). \end{equation*} On the other hand, without loss of generality, we may rewrite \eqref{eqn.jordan} in the form \begin{equation}\label{eq.tam} ^{C\!}D^{\alpha}_{0+}y(t) = \operatorname{diag}(B^s,B^u)y(t)+(g^s(t),g^u(t))^{\rm T}, \end{equation} where $B^{s/u}$ is the part of $B$ corresponding to the collection of all blocks with the eigenvalues belonging to $\Lambda_\alpha^{s/u}$. Note that system \eqref{mainEq2} has at least one bounded solution for any bounded continuous function $f$ if and only if the system \eqref{eq.tam} has at least one bounded solution for any bounded continuous function $g$. Thus, we only focus on the system \eqref{eq.tam}. We need the following preparatory lemmas for the proof of Proposition \ref{sufficiency part1}. \begin{lemma}\label{Stable} Let $\lambda\in \mathbb{C}\setminus\{0\}$. Consider the inhomogeneous equation \begin{equation}\label{ScalarEq} ^{C\!}D^{\alpha}_{0+}x(t)=\lambda x(t)+g(t), \end{equation} where $g\in C_b(\mathbb{R}_{\ge 0};\mathbb{C})$. Then, the following statements hold: \begin{itemize} \item [(i)] if $\lambda\in \Lambda_\alpha^s$, then all solutions of \eqref{ScalarEq} are bounded on $\mathbb{R}_{\ge 0}$; \item[(ii)] if %$\lambda:=r(\cos\varphi+i\sin\varphi)\in \Lambda_\alpha^u$ $\lambda \in \Lambda_\alpha^u$, then the equation \eqref{ScalarEq} has exactly one bounded solution. \end{itemize} \end{lemma} \begin{proof} (i) Using the variation of constants formula provided by Theorem~\ref{Var_Const_Form}, for any $\xi\in\mathbb{C}$, the solution $\varphi(\cdot,\xi)$ of \eqref{ScalarEq} has the representation \[ \varphi(t,\xi)=E_\alpha(\lambda t^\alpha)\xi +\int_0^t (t-\tau)^{\alpha-1}E_{\alpha,\alpha}(\lambda(t-\tau)^{\alpha})g(\tau) \,d\tau,\quad \forall t\geq 0. \] From \cite[Theorem 1.4, p. 33]{Podlubny}, we see that the quantity $E_\alpha(\lambda t^\alpha)\xi$ is bounded on $[0,\infty)$. On the other hand, by Lemma \ref{lemma4}(ii), there exists a positive constant $C$ such that \[ \int_0^t|(t-\tau)^{\alpha-1}E_{\alpha,\alpha} (\lambda(t-\tau)^{\alpha})g(\tau)|\,d\tau \leq C \sup_{\tau\geq 0}|g(\tau)|,\quad \forall t\geq 0. \] Thus $\varphi(\cdot,\xi)$ is bounded for any $\xi\in\mathbb{C}$. (ii) Let $$ \xi^*:=-\lambda^{\frac{1}{\alpha}-1}\int_0^\infty \exp{(-\lambda^{1/\alpha}\tau)}g(\tau)\,d\tau. $$ By the variation of constants formula provided by Theorem~\ref{Var_Const_Form}, we see that the function \begin{align*} \varphi(t,\xi^*) &= E_\alpha(\lambda t^\alpha)\Big(-\lambda^{\frac{1}{\alpha}-1} \int_0^\infty \exp{(-\lambda^{1/\alpha}\tau)}g(\tau)\,d\tau \Big)\\ &\quad +\int_0^t (t-\tau)^{\alpha-1}E_{\alpha,\alpha} (\lambda (t-\tau)^\alpha)g(\tau)\,d\tau,\quad \forall t\geq 0, \end{align*} is a solution of \eqref{ScalarEq}. We will prove that this function is the only bounded solution. Indeed, for any $t\geq 0$, we have \begin{align*}\label{scalarext} &|\varphi(t,\xi^*)| \\ &\leq \int_t^\infty \big|\lambda^{\frac{1}{\alpha}-1}E_\alpha(\lambda t^\alpha) \exp(-\lambda^{1/\alpha}\tau)\big|\,|g(\tau)|\,d\tau\\ &\quad +\int_0^t\big| (t-\tau)^{\alpha-1}E_{\alpha,\alpha}(\lambda(t-\tau)^{\alpha})- \lambda^{\frac{1}{\alpha}-1}E_\alpha(\lambda t^\alpha) \exp(-\lambda^{1/\alpha}\tau)\big|\,|g(\tau)|\,d\tau, \end{align*} which together with Lemma \ref{lemma4}(i) imply \[ |\varphi(t,\xi^*)|\leq 2 K(\alpha,\lambda)\sup_{\tau\geq 0}|g(\tau)|,\quad \forall t\geq 0. \] Thus, $\varphi(\cdot,\xi^*)$ is bounded on $[0,\infty)$. Now, assume that $\varphi(\cdot,\xi)$ is another bounded solution of \eqref{ScalarEq} for some $\xi\in\mathbb{C}$. Then, \[ \varphi(t,\xi^*)-\varphi(t,\xi) =E_\alpha(\lambda t^\alpha)(\xi^*-\xi),\quad \forall t\geq 0. \] Because $\lim_{t\to\infty}E_\alpha(\lambda t^\alpha)=\infty$, we have $\xi^*=\xi$. This implies \[ \varphi(t,\xi^*)=\varphi(t,\xi),\quad \forall t\geq 0. \] Hence, equation \eqref{ScalarEq} has exactly one bounded solution. The proof is complete. \end{proof} \begin{remark} \rm Consider the system \begin{gather} \label{extra1} ^{C}D^\alpha_{0+}x_1(t) =ax_1(t)-bx_2(t)+g_1(t),\\ \label{extra2} ^{C}D^\alpha_{0+}x_2(t) =bx_1(t)+ax_2(t)+g_2(t), \end{gather} where $a,b\in \mathbb{R}$ and $g_1,g_2\in C_b(\mathbb{R}_{\geq 0};\mathbb{R})$. In the light of Proposition \ref{Stable}, we obtain the following results: \begin{itemize} \item[(i)] if $a+ib\in \Lambda^s_\alpha$, then all solutions of system \eqref{extra1}-\eqref{extra2} are bounded; \item[(ii)] if $\lambda:=a+ib\in \Lambda^u_\alpha$, then system \eqref{extra1}-\eqref{extra2} has exactly one bounded solution as \begin{equation*} (x_1(t),x_2(t))^{\rm T}=(\Re u(t),\Im u(t))^{\rm T},\quad \forall t\geq 0, \end{equation*} where \begin{align*} u(t)&= E_\alpha(\lambda t^\alpha)\Big(-\lambda^{\frac{1}{\alpha}-1} \int_0^\infty \exp{(-\lambda^{1/\alpha}\tau)}(g_1(\tau)+ig_2(\tau))\,d\tau\Big)\\ &\quad +\int_0^t (t-\tau)^{\alpha-1}E_{\alpha,\alpha}(\lambda(t-\tau)^\alpha) (g_1(\tau)+ig_2(\tau))\,d\tau,\quad \forall t\geq 0. \end{align*} \end{itemize} Indeed, setting $u(t)=x_1(t)+ix_2(t)$ and $g(t)=g_1(t)+ig_2(t)$, system \eqref{extra1}-\eqref{extra2} is equivalent to the equation \[ ^{C}D^\alpha_{0+}u(t)=\lambda u(t)+g(t)\,. \] Then Lemma \ref{Stable} is applied. \end{remark} Next, we prove an analogous result of Lemma \ref{Stable} for inhomogeneous systems whose linear parts are of the form $J_{k,\lambda}:=\lambda \operatorname{Id}_{k}+ N_k$, where $k\in\mathbb N$, $\operatorname{Id}_{k}$ denotes the unit matrix in $\mathbb{R}^{k\times k}$ and \[ N_k:=\begin{pmatrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0\\ \vdots &\vdots & \ddots & \ddots &\vdots\\ 0 & 0 &\dots & 0 & 1 \\ 0& 0 &\dots &0 & 0 \\ \end{pmatrix}_{k \times k}. \] \begin{lemma}\label{propo1} Let $k\in\mathbb N$ and $\lambda\in \mathbb{C}\setminus\{0\}$. Consider the inhomogeneous system \begin{equation}\label{Eq5} ^{C\!}D^{\alpha}_{0+}x(t)=J_{k,\lambda} x(t)+g(t),\quad t\geq 0, \end{equation} where $g\in C_b(\mathbb{R}_{\ge 0};\mathbb{C}^k)$. Then, the following statements hold: \begin{itemize} \item [(i)] if $\lambda\in \Lambda_\alpha^s$, then all solutions of \eqref{Eq5} are bounded; \item[(ii)] if $\lambda\in \Lambda_\alpha^u$, then equation \eqref{Eq5} has exactly one bounded solution. \end{itemize} \end{lemma} \begin{proof} By the definition of $J_{k,\lambda}$, the system \eqref{Eq5} is rewritten in the form \begin{equation}\label{tam3} ^{C\!}D^{\alpha}_0 x_{i}(t)=\lambda\; x_{i}(t)+x_{i+1}(t)+g_{i}(t),\quad i=1,\dots, k-1 \end{equation} and \begin{equation}\label{tam1} ^{C\!}D^{\alpha}_0 x_{k}(t)=\lambda\; x_{k}(t)+g_{k}(t). \end{equation} (i) Assume that $g\in C_b(\mathbb{R}_{\geq 0};\mathbb{C}^k)$. Let $$x^0=(x^0_1,\dots,x^0_k)^{\mathrm{T}}\in\mathbb{C} ^k$$ be an arbitrary vector and $\varphi(t,x^0)=(\varphi_1(t),\dots,\varphi_k (t))^{\rm T}$ denote the solution of \eqref{Eq5} satisfying the initial condition $\varphi(t,x^0)=x^0$. From \eqref{tam1}, we have \begin{equation}\label{tam2} \varphi_{k}(t)=E_\alpha(\lambda t^\alpha)x^0_k +\int_0^t (t-s)^{\alpha-1}E_{\alpha,\alpha}(\lambda (t-s)^\alpha)g_k(s)\,ds. \end{equation} It follows from Lemma~\ref{Stable}(i) that $\varphi_k$ is bounded in $\mathbb{R}_{\ge 0}$. Substitute $\varphi_k$ into \eqref{tam3} and applying Lemma~\ref{Stable}(i) again we get that $\varphi_{k-1}$ is also bounded. Continue this process we will get that $\varphi_{k-2},\dots, \varphi_1$ are all bounded. (ii) Using arguments similar to that of the part (i) above and with the application of Lemma~\ref{Stable}(ii), we obtain the proof of Lemma~\ref{propo1}(ii). \end{proof} \begin{corollary}\label{realcase} For $\lambda:=a+ib\in \mathbb{C}\setminus\{0\}$, we consider the equation \begin{equation}\label{extra3} ^{C}D^\alpha_{0+}x(t)=J_{2k,\lambda} x(t)+f(t), \end{equation} where $f \in C_b(\mathbb{R}_{\geq 0};\mathbb{R}^{2k})$ and $J_{2k,\lambda}$ is a real Jordan block in the form \[ J_{2k,\lambda}:= \begin{pmatrix} D & I & 0 & \dots & 0 \\ 0 & D & I & \dots & 0\\ \vdots &\vdots & \ddots & \ddots &\vdots\\ 0 & 0 &\dots & D& I \\ 0& 0 &\dots &0 & D \\ \end{pmatrix}_{2k\times 2k}, \] with \[ D=\begin{pmatrix} a & -b\\ b & a \end{pmatrix},\quad I=\begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix}. \] Then, the following statements hold: \begin{itemize} \item[(i)] if $\lambda\in \Lambda_\alpha^s$, then all solutions of \eqref{extra3} are bounded; \item[(ii)] if $\lambda\in \Lambda_\alpha^u$, then \eqref{extra3} has exactly one bounded solution. \end{itemize} \end{corollary} \begin{proof} By using the change of variable $u_j(t)=x_{2j-1}(t)+ix_{2j}(t)$, for $j=1,\dots,k$, from the system \eqref{extra3}, we obtain the system \begin{gather} ^{C\!}D^{\alpha}_{0+} u_1(t)= \lambda u_1(t)+u_2(t)+g_1(t)\\ ^{C\!}D^{\alpha}_{0+} u_2(t)= \lambda u_2(t)+u_3(t)+g_2(t)\\ \notag \ldots \\ \label{tam3.1} ^{C\!}D^{\alpha}_{0+} u_{k-1}(t) = \lambda u_{k-1}(t)+u_{k}(t)+g_{k-1}(t)\\ \label{tam1.1} ^{C\!}D^{\alpha}_{0+} u_{k}(t)=\lambda u_{k}(t)+g_{k}(t), \end{gather} here $g_j(t)=f_{2j-1}(t)+if_{2j}(t)$, for $j=1,\dots, k$. If $\lambda\in \Lambda_\alpha^s$, according to Lemma \ref{propo1}(i), we see that all solutions of this system are bounded which implies that all solutions of the system \eqref{extra3} are bounded, too. In the case $\lambda\in \Lambda_\alpha^u$, from Lemma \ref{propo1}(ii), this system has exactly one bounded solution $u(t)=(u_1(t),\dots, u_{k})^{\rm T}$. Hence, system \eqref{extra3} has exactly one bounded solution $\varphi(t)=(\varphi_1(t),\dots,\varphi_{2k}(t))^{\rm T}$, where for all $t\geq 0$, \begin{gather*} \varphi_{2j-1}(t)=\Re u_j(t), \quad j=1,\dots,k; \\ \varphi_{2j}(t)=\Im u_j(t), \quad j=1,\dots, k. \end{gather*} \end{proof} We now prove the sufficiency part of Theorem \ref{thm.main}. \begin{proof}[Proof of Proposition \ref{sufficiency part1}] Because system \eqref{mainEq2} has at least one bounded solution for any bounded continuous function $f$ if and only if system \eqref{eq.tam} has at least one bounded solution for any bounded continuous function $g$. We now focus on system \eqref{eq.tam}. Applying Lemma~\ref{propo1}(ii) and Corollary \ref{realcase}(ii) to each Jordan block from $B^u$, we can find exactly one bounded solution $u(t)$ of the equation (the unstable part of equation \eqref{eq.tam}) $$ ^{C\!}D^{\alpha}_{0+}y^u(t) = B^u y^u(t)+g^u(t); $$ and applying Lemma~\ref{propo1}(i) and Corollary \ref{realcase}(i) to each Jordan block from $B^s$, we see that all solutions of equation (the stable part of equation \eqref{eq.tam}) $$ ^{C\!}D^{\alpha}_{0+}y^s(t) = B^s y^s(t)+g^s(t),\quad y^s(0) = y_0^s, $$ are bounded. Put $\varphi(t):=(u(t),v(t))^{\rm T}$ for any $t\geq 0$, where $v(t)$ is an arbitrary solution of \eqref{eq.tam}. Then, this function is a bounded solution of system \eqref{eq.tam}. The proof is complete. \end{proof} Finally, we prove the necessary condition in the statement of Theorem \ref{thm.main}. \begin{proposition}[Necessity part of Theorem \ref{thm.main}]\label{necessity part1} Consider the system \begin{equation}\label{mainEq3} ^{C\!}D^{\alpha}_{0+}x(t)=A x(t)+f(t), \end{equation} where $f:\mathbb{R}_{\geq 0}\to \mathbb{R}^d$. Assume that for any bounded continuous function $f$, the system \eqref{mainEq3} has at least one bounded solution. Then, the matrix $A$ has to satisfy the hyperbolic condition \eqref{SpecCond}: \[ \sigma (A)\cap \big\{\lambda\in \mathbb{C}: \lambda=0\text{ or } |\arg{(\lambda)}|=\frac{\alpha \pi}{2}\big\}=\emptyset. \] \end{proposition} Before going to the proof of this theorem, we need the following technical proposition. \begin{lemma}\label{revPropo} Consider the equation \begin{equation}\label{revEq} ^{C\!}D^{\alpha}_{0+}x(t)=\lambda x(t)+f(t), \end{equation} where $\lambda =0$ or $\arg{(\lambda)}=\frac{\alpha \pi}{2}$ and $f:\mathbb{R}_{\geq 0}\to \mathbb{C}$. Then, there exists a bounded continuous function $f$ such that every solutions of this equation are unbounded. \end{lemma} \begin{proof} First, we consider the case $\lambda=0$. In this case, we choose $f(t)=\Gamma(1+\alpha)$. It is obvious that for any $x_0\in \mathbb{R}$, equation \eqref{revEq} with the initial condition $x(0)=x_0$, has a unique solution as \[ \varphi(t,x_0)=x_0+t^\alpha,\quad \forall t\geq 0. \] This solution is unbounded for any $x_0\in R$. In the case $\arg{(\lambda)}=\frac{\alpha \pi}{2}$, we write $\lambda$ in the form $\lambda=r(\cos \frac{\alpha \pi}{2}+i\sin \frac{\alpha \pi}{2})$ and choose $f(t)=\exp{(ir^{1/\alpha}t)}$. By Theorem \ref{Var_Const_Form}, the solution $\varphi(\cdot,x_0)$ of \eqref{revEq} starting from $x_0$ satisfies \[ \varphi(t,x_0)=E_\alpha(\lambda t^\alpha)x_0 +\int_0^t (t-\tau)^{\alpha-1}E_{\alpha,\alpha}(\lambda (t-\tau)^\alpha) \exp{(ir^{1/\alpha}\tau)}\,d\tau. \] We claim that this solution is unbounded. Indeed, the quantity $x_0E_\alpha(\lambda t^\alpha)$ is bounded because of \cite[Theorem 1.1, p.~30]{Podlubny}, while the quantity $$ \int_{t-1}^t (t-\tau)^{\alpha-1}E_{\alpha,\alpha}(\lambda (t-\tau)^\alpha)\,d\tau $$ is bounded by the estimate $$ \big|\int_{t-1}^t (t-\tau)^{\alpha-1}E_{\alpha,\alpha} (\lambda (t-\tau)^\alpha)\,d\tau\big| \le \int_0^1 s^{\alpha-1}|E_{\alpha,\alpha}(\lambda s^\alpha)|\,ds. $$ Furthermore, using \cite[Theorem 1.1, p.~30]{Podlubny}, we show that the quantity \[ \int_{0}^{t-1} (t-\tau)^{\alpha-1}E_{\alpha,\alpha} (\lambda (t-\tau)^\alpha)\exp{(ir^{1/\alpha}\tau)}\,d\tau \] is unbounded. Indeed, for $\theta \in (\frac{\alpha \pi}{2},\alpha \pi)$ is arbitrary but fixed and $\varepsilon \in (0,\frac{|\lambda|}{2})$ satisfies \begin{equation}\label{unbouded_est} |\lambda| t^\alpha-\varepsilon \geq |\lambda| t^\alpha \sin (\theta-\frac{\alpha \pi}{2}) \end{equation} for all $t\geq 1$, we denote by $\gamma(\varepsilon,\theta)$ the contour consisting of the following three parts \begin{itemize} \item [(i)] $\arg (z)=-\theta$, $|z|\ge \varepsilon$; \item [(ii)] $-\theta\le \arg (z)\le \theta$, $|z|=\varepsilon$; \item [(iii)] $\arg (z)=\theta$, $|z|\ge \varepsilon$. \end{itemize} The contour $\gamma(\varepsilon,\theta)$ divides the complex plane $(z)$ into two domains, which we denote by $G^{-}(\varepsilon,\theta)$ and $G^{+}(\varepsilon,\theta)$. These domains lie correspondingly on the left and on the right side of the contour $\gamma(\varepsilon,\theta)$. According to \cite[Theorem 1.1, p.~30]{Podlubny}, we have \begin{align*} &\int_{0}^{t-1} (t-\tau)^{\alpha-1}E_{\alpha,\alpha} (\lambda (t-\tau)^\alpha)\exp{(ir^{1/\alpha}\tau)}\,d\tau\\ &=\int_0^{t-1} (t-\tau)^{\alpha-1}\frac{1}{\alpha}\lambda^{\frac{1-\alpha}{\alpha}} (t-\tau)^{1-\alpha}\exp(\lambda^{1/\alpha}(t-\tau))\exp(i r^{1/\alpha}\tau)\,d\tau\\ &\quad +\int_0^{t-1} (t-\tau)^{\alpha-1} \frac{1}{2\alpha \pi i}\int_{\gamma(\varepsilon,\theta)} \frac{\exp(\xi^{1/\alpha})\xi^{\frac{1-\alpha}{\alpha}}} {\xi-\lambda (t-\tau)^\alpha}\,d\xi \exp(i r^{1/\alpha}\tau)\,d\tau\\ &=I_4(t)+I_5(t). \end{align*} Clearly, we see that \begin{equation} \begin{aligned} I_4(t) &=\int_0^{t-1} (t-\tau)^{\alpha-1}\frac{1}{\alpha} \lambda^{\frac{1-\alpha}{\alpha}}(t-\tau)^{1-\alpha} \exp(\lambda^{1/\alpha}(t-\tau))\exp(i r^{1/\alpha}\tau)\,d\tau\\ &=\frac{\lambda^{\frac{1-\alpha}{\alpha}}}{\alpha}(t-1) \exp(\lambda^{1/\alpha}t). \end{aligned}\label{unbouded_est1} \end{equation} On the other hand, by \eqref{unbouded_est}, we obtain \begin{equation} \begin{aligned} |I_5(t)| &\leq \frac{\int_{\gamma(\varepsilon,\theta)}|\exp(\xi^{1/\alpha}) \xi^{\frac{1-\alpha}{\alpha}}|\,d\xi}{2\alpha \pi \sin (\theta-\frac{\alpha \pi}{2})} \int_0^{t-1}\frac{(t-\tau)^{\alpha-1}}{|\lambda|(t-\tau)^\alpha}\,d\tau\\ &\leq \frac{\int_{\gamma(\varepsilon,\theta)}|\exp(\xi^{1/\alpha}) \xi^{\frac{1-\alpha}{\alpha}}|\,d\xi}{2\alpha \pi |\lambda| \sin (\theta-\frac{\alpha \pi}{2})}\log t. \end{aligned} \label{unbounded_est2} \end{equation} From \eqref{unbouded_est1} and \eqref{unbounded_est2}, this implies that the quantity \[ \int_{0}^{t-1} (t-\tau)^{\alpha-1}E_{\alpha,\alpha} (\lambda (t-\tau)^\alpha)\exp{(ir^{1/\alpha}\tau)}\,d\tau \] is unbounded. So, the solution $\varphi(\cdot,x_0)$ is unbounded for any $x_0\in \mathbb{C}$. The proof is complete. \end{proof} \begin{corollary}\label{unbounded3} Assume that $\lambda=a+ib $ is a complex number satisfying $\arg{(\lambda)}=\frac{\alpha \pi}{2}$. Consider the fractional differential equation \begin{gather} \label{unbounded1} ^{C}D^\alpha_{0+}x_1(t)=ax_1(t)-bx_2(t)+f_1(t),\\ \label{unbounded2} ^{C}D^\alpha_{0+}x_2(t)=bx_1(t)+ax_2(t)+f_2(t). \end{gather} Then, we can find $f_1,f_2\in C_b(\mathbb{R}_{\geq 0};\mathbb{R})$ such that all solutions of this system are unbounded. \end{corollary} \begin{proof} Consider the equation \begin{equation*} ^{C}D^\alpha_{0+}u(t)=\lambda u(t)+f(t),\quad t>0, \end{equation*} where $f(t)=f_1(t)+if_2(t)$. From the proof of Lemma \ref{revPropo}, choosing the function $f$ as $f(t)=\exp{(ir^{1/\alpha}t)}$ (where $r=\sqrt{a^2+b^2}$), we see that all solutions of this equation are unbounded. Hence, by choosing $f_1(t)=\cos (r^{1/\alpha}t)$ and $f_2(t)=\sin (r^{1/\alpha}t)$, all solutions of the system \eqref{unbounded1}-\eqref{unbounded2} are unbounded. The proof is complete. \end{proof} \begin{proof}[Proof of Proposition \ref{necessity part1}] First, we consider the case $0\in \sigma(A)$. Without loss of generality (transform $A$ to the Jordan form and change the order of coordinates if necessary), we can write the matrix $A$ in the equation \eqref{mainEq3} in the form \[ A=\begin{pmatrix} \hat{A} & A_1\\ 0 & 0 \end{pmatrix}. \] Choosing $f=(\hat{f}(t),\Gamma(1+\alpha))^{\rm T}$ with $\hat{f}\in C_b(\mathbb{R}_{\geq 0};\mathbb{R}^{d-1})$. Due to Lemma \ref{revPropo}, the last coordinate of any solution of \eqref{mainEq3} is unbounded. Hence, all solutions of \eqref{mainEq3} are unbounded. Next, in the case the spectrum $\sigma(A)$ has at least one eigenvalue $\lambda$ such that $\arg{(\lambda)}=\frac{\alpha \pi}{2}$, we can assume that \[ A=\begin{pmatrix} \hat{A} & A_2\\ 0 & D \end{pmatrix} \] where \begin{equation*} D=\begin{pmatrix} a & -b\\ b & a \end{pmatrix}, \end{equation*} with $a=\Re \lambda$, $b=\Im \lambda$. Due to Corollary \ref{unbounded3}, by choosing the function $f=(\hat{f},f_1,f_2)^{\rm T}$, where $\hat{f}\in C_b(\mathbb{R}_{\geq 0};\mathbb{R}^{d-2})$, and \[ f_1(t)=\cos (r^{1/\alpha}t),\quad f_2(t)=\sin (r^{1/\alpha}t), \] with $r=|\lambda|$, we see that at least one of the last two coordinates of any solution of \eqref{mainEq3} is unbounded. Hence, all solutions of \eqref{mainEq3} are unbounded. 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