\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 152, pp. 1--15.\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-2017/152\hfil Finite time blow-up of solutions] {Finite time blow-up of solutions for a nonlinear system of fractional differential equations} \author[A. Mennouni, A. Youkana \hfil EJDE-2017/152\hfilneg] {Abdelaziz Mennouni, Abderrahmane Youkana} \address{Abdelaziz Mennouni \newline Department of Mathematics, University of Batna 2, 05078 Batna, Algeria} \email{aziz.mennouni@yahoo.fr} \address{Abderrahmane Youkana \newline Department of Mathematics, University of Batna 2, 05078 Batna, Algeria} \email{abder.youkana@yahoo.fr} \dedicatory{Communicated by Mokhtar Kirane} \thanks{Submitted March 14, 2017. Published June 25, 2017.} \subjclass[2010]{33E12, 34K37} \keywords{Fractional differential equation; Caputo fractional derivative; \hfill\break\indent blow-up in finite time} \begin{abstract} In this article we study the blow-up in finite time of solutions for the Cauchy problem of fractional ordinary equations \begin{gather*} u_{t} +a_1\,^{c}D_{0_{+}}^{\alpha_1} u +a_2\,^{c}D_{0_{+}}^{\alpha_2} u+\dots +a_{n}\,^{c}D_{0_{+}}^{\alpha_n} u =\int_0^{t} \frac{(t-s)^{-\gamma_1}}{ \Gamma(1-\gamma_1) }f(u(s),v(s))ds,\\ v_{t} +b_1\,^{c}D_{0_{+}}^{\beta_1} v+ b_2\,^{c}D_{0_{+}}^{\beta_2} v+\dots +b_{n}\,^{c}D_{0_{+}}^{\beta_n} v = \int_0^{t} \frac{(t-s)^{-\gamma_2}}{ \Gamma(1-\gamma_2) }g(u(s),v(s))ds, \end{gather*} for $t>0$, where the derivatives are Caputo fractional derivatives of order $\alpha_i, \beta_i$, and $f$ and $g$ are two continuously differentiable functions with polynomial growth. First, we prove the existence and uniqueness of local solutions for the above system supplemented with initial conditions, then we establish that they blow-up in finite time. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \allowdisplaybreaks \section{Introduction} In this work, we study the system of ordinary fractional differential equations \begin{equation} \label{11} \begin{aligned} &u_{t} +a_1\,^{c}D_{0_{+}}^{\alpha_1} u +a_2\,^{c}D_{0_{+}}^{\alpha_2} u+\dots +a_{n}\,^{c}D_{0_{+}}^{\alpha_n} u \\ &=\int_0^{t} \frac{(t-s)^{-\gamma_1}}{ \Gamma(1-\gamma_1) }f(u(s),v(s))ds,\\ &v_{t} +b_1\,^{c}D_{0_{+}}^{\beta_1} v+ b_2\,^{c}D_{0_{+}}^{\beta_2} v+\dots +b_{n}\,^{c}D_{0_{+}}^{\beta_n} v \\ &= \int_0^{t} \frac{(t-s)^{-\gamma_2}}{ \Gamma(1-\gamma_2) }g(u(s),v(s))ds, \end{aligned} \end{equation} for $t>0$, with initial data \begin{equation} \label{12} u(0)=u_0>0,\quad v(0)=v_0>0, \end{equation} and where $ 0<\alpha_i <1$, $ 0< \beta_i <1$, $i=1,\dots,n$, $ 0<\gamma_j <1 $, $ j=1,2$, $f$ and $g$ are two real continuous differentiable functions defined on $ \mathbb{R}\times \mathbb{R} $, $ a_{i}$, $b_{i}$ $i=1,\dots,n$ are positive constants, $ \Gamma$ is the Euler function and $ D_{0^{+}}^{\alpha_i}$, $ D_{0^{+}}^{\beta_i}$, $i=1,\dots,n$, are Caputo fractional derivatives. In recent years, fractional differential equations have played an important role in the study of models for many phenomena in various fields of physics, biology and engineering, such as aerodynamics, viscoelasticity, control of dynamic systems, electrochemistry, porous media, etc (see \cite{Bushell,CapDong,Hilfer, Samko} and the references therein); their study attracted the attention of many researchers (see for instance \cite{KirKa, KirOlmRob, Liu, Mydlar} and the references therein). In addition, a particular attention was given for the study of the local existence and uniqueness of solutions for these systems and their properties like the blow-up in finite time, the global existence, the asymptotic behavior, etc. (see \cite{CapDong, KirOlmRob,Liu, Mydlar}). In \cite{KirMal}, the profile of the blowing-up solutions has been investigated for the following nonlinear nonlocal system \begin{gather*} u_{t}(t)+ D_{0_{+}}^{\alpha}(u-u_0)(t)=|v(t)|^q ,\quad t>0, \; q >1,\\ v_{t}(t)+ D_{0_{+}}^{\beta}(v-v_0)(t)=|u(t)|^p, \quad t>0,\; p >1,\\ u(0)=u_0>0, \quad v(0)=v_0>0, \end{gather*} as well as for solutions of systems obtained by dropping either the usual derivatives or the fractional derivatives. In \cite{KadKir}, some results on the blow-up of the solutions and lower bounds of the maximal time have been established for the system \begin{gather*} u_{t}(t)+\rho D_{0_{+}}^{\alpha}(u-u_0)(t)=e^{v(t)} ,\quad t>0, \; \rho >0,\\ v_{t}(t)+\sigma D_{0_{+}}^{\beta}(v-v_0)(t)=e^{u(t)}, \quad t>0,\; \sigma >0,\\ u(0)=u_0>0,\quad v(0)=v_0>0, \end{gather*} and the subsystem obtained by dropping the usual derivatives. In the spirit of the interesting works \cite{Furati, KadKir, KirMal}, we prove that the non global existence of solutions to \eqref{11}-\eqref{12} holds for polynomial nonlinearities. For the existence of solutions for the system \eqref{11}-\eqref{12}, we will use the Schauder theorem. Our paper is organized as follows: In Section 2, we give some preliminary results for fractional derivatives. In Section 3, we will prove the local existence and uniqueness of the solutions. In Section 4, we will state and prove our main result on the blow- up in finite time of solutions for system \eqref{11}-\eqref{12}. \section{Preliminaries and mathematical background} For the convenience of the reader, we shall recall some known results concerning fractional integrals and derivatives that will be useful in the sequel. The Riemann-Liouville fractional integral of order $0<\alpha<1$ with lower limit $0$ is defined for a locally integrable function $ \varphi : \mathbb{R}_{+} \to \mathbb{R} $ by \[ J_{0_{+}}^{\alpha}\varphi(t)=\frac{1}{\Gamma(\alpha)} \int_0^{t}\frac{\varphi(s)}{(t-s)^{1-\alpha}}ds,\quad t>0, \] where $\Gamma$ is the Euler Gamma function. The left-handed and right-handed Riemann-Liouville fractional derivatives of order $ \alpha$ with $0<\alpha<1$ of a continuous function $\psi(t)$ are defined by \[ D^{\alpha}_{0_{+}}\psi(t)=\frac{1}{\Gamma(1-\alpha)}\frac{d}{dt} \int_0^{t}\frac{\psi(s)}{(t-s)^{\alpha}}ds,\quad t>0, \] and \[ D^{\alpha}_{T^{-}}\psi(t)=-\frac{1}{\Gamma(1-\alpha)}\frac{d}{dt} \int_{t}^{T}\frac{\psi(s)}{(s-t)^{\alpha}}ds,\quad t>0, \] respectively. One can see that $$ \frac{d}{dt}J_{0^+}^{1-\alpha}\psi(t)=D_{0{+}}^{\alpha}\psi(t),\quad t>0. $$ The integration by parts formula (see \cite{Samko}) in $ [0,T]$ reads \begin{equation*} \int_0^{T} h(t) D^{\alpha}_{0_{+}}k(t) dt = \int_0^{T} (D^{\alpha}_{T_{-}} h(t))k(t)dt, \end{equation*} for functions $h, k$ in $C([0,T]) $ such that $D_{0^{+}}^{\alpha} k$ and $ D_{T^{-}}^{\alpha} h$ are continuous. The Caputo fractional derivative of order $0<\alpha<1$ of an absolutely continuous function $\phi(t)$ of order $ 0 < \alpha < 1$ is defined by \[ ^{c}D^{\alpha}_{0_{+}} \phi(t) = J^{1-\alpha}_{0_{+}} \frac{d}{dt} \phi(t)=\frac{1}{\Gamma(1-\alpha)} \int_0^{t}(s-t)^{-\alpha} \phi'(s) ds. \] The relation between the Riemann-Liouville and the Caputo fractional derivatives for an absolutely continuous function $\phi(t)$ is given by \begin{equation*} ^{c}D^{\alpha}_{0_{+}} \phi(t) = D^{\alpha}_{0_{+}}(\phi(t)-\phi(0)), \quad 0 <\alpha <1. \end{equation*} \section{Existence and uniqueness of solutions} In this section, we deal with the existence and uniqueness of local solutions for problem \eqref{11}-\eqref{12}. We say that $(u,v)$ is a local classical solution if it satisfies equations \eqref{11}-\eqref{12} on some interval $ (0,T^*)$. Our main result in this section reads as follows. \begin{theorem} \label{thm3.1} Assume that the functions $f$ and $g$ are of class $C^{1}(\mathbb{R}\times\mathbb{R},\mathbb{R})$. Then system \eqref{11}-\eqref{12} admits a unique local classical solution on a maximal interval $( 0, T_{\rm max})$ with the alternative: either\ $T_{\rm max}=+\infty$ and the solution is global; or $$ T_{\rm max}<+\infty \quad\text{and}\quad \lim_ {t\to T_{\rm max}}(| u(t)|+| v(t)|)=+\infty. $$ \end{theorem} \begin{proof} For the sake of completeness, we give the proof of the existence of solutions of \eqref{11}-\eqref{12}. Let $ k>0$ be a positive constant and \begin{equation}\label{31} h:= \min\{ \sigma_1,\ \sigma_2\}>0, \end{equation} where \begin{gather*} \sigma_1:= \min\Big\{ \min_{1\leq i\leq n} \Big( \frac{1}{2n^2\bar{a}\max_{1\leq i \leq n} ( \frac{1}{\Gamma(2-\alpha_{i})})}\Big)^{\frac{1}{1-\alpha_{i}}}, \Big( \frac{k\Gamma(2-\gamma_1)}{2M}\Big) ^\frac{1}{1-\gamma_1} \Big\}, \\ \sigma_2:= \min\Big\{ \min_{1\leq i\leq n} \Big( \frac{1}{2n^2\bar{b}\max_{1\leq i \leq n} \frac{1}{\Gamma(2-\beta_{i})})} \Big)^{\frac{1}{1-\beta_{i}}}, \Big( \frac{k\Gamma(2-\gamma_2)}{2M}\Big) ^\frac{1}{1-\gamma_2}\Big\}, \\ \bar{a}=\max_{1\leq i\leq n}\{ a_i\},\quad \bar{b} =\max_{1\leq i\leq n}\{ b_i\}. \end{gather*} Let $ C([0,h]) \times C([0,h])$ be the space of all continuous functions $(\chi,\psi)$ on $ [0,h] $ equipped with the norm $$ \| (\chi,\psi) \|_{\infty}= \max ( \| \chi \|_{\infty},\; \| \psi \|_{\infty}), $$ where $$ \|\chi\|_{\infty}=\max_{0\leq t\leq h}| \chi(t)|,\quad \|\psi\|_{\infty}=\max_{0\leq t\leq h}| \psi(t)|. $$ For simplicity, we assume $ \alpha_1\leq \alpha_2 \leq\dots \leq \alpha_n$ and $ \beta_1\leq \beta_2 \leq\dots \leq \beta_n$. Now, in order to prove the existence of solutions for problem \eqref{11}-\eqref{12}, we rewrite it as a system of integral equations in $ C([0,h])\times C([0,h])$, \begin{equation} \label{32} \begin{aligned} x(t)& =-a_1J^{1-\alpha_1}_{0_{+}}x(t)-a_2J^{1-\alpha_2}_{0_{+}}x(t) -\dots -a_{n}J^{1-\alpha_n}_{0_{+}}x(t) + J_{0^{+}}^{1-\gamma_1} f (u_0 \\ &\quad +\int_0^{t} x(s)ds, v_0+\int_0^{t} y(s)ds) \\ y(t) &=-b_1J^{1-\beta_1}_{0_{+}}y(t)-b_2J^{1-\beta_2}_{0_{+}}y(t) -\dots -b_{n}J^{1-\beta_n}_{0_{+}}y(t)+J_{0^{+}}^{1-\gamma_2} g (u_0 \\ &\quad+\int_0^{t} x(s)ds, v_0+\int_0^{t} y(s)ds), \end{aligned} \end{equation} via the transformation \[ u(t)=u_0+\int_0^t x(s)ds,\quad v(t)=v_0+\int_0^t y(s)ds, \] and the relation $^{c}D_{0^+}^{\alpha}\psi(t)=J_{0^{+}}^{1-\alpha}\frac{d}{dt}\psi(t)$, and we shall prove the existence of local solutions for \eqref{32}. So, let us define the operator $A : C([0,h]) \times C([0,h])\to C([0,h])\times C([0,h]) $ by $$ A(x,y)= (A_1(x,y),\ A_2(x,y)), $$ where \begin{equation} \label{33} \begin{aligned} A_1(x(t),y(t)) &=-\sum_{i=1}^{n}a_{i}J^{1-\alpha_i}_{0_{+}}x(t)\\ &\quad +J_{0^{+}}^{1-\gamma_1} f \Big(u_0+\int_0^{t} x(s)ds, v_0+\int_0^{t} y(s)ds\Big), \\ A_2(x(t),y(t))&= -\sum_{i=1}^{n}b_{i}J^{1-\beta_i}_{0_{+}}y(t)\\ &\quad +J_{0^{+}}^{1-\gamma_2} g \Big(u_0+\int_0^{t} x(s)ds, v_0+\int_0^{t} y(s)ds\Big). \end{aligned} \end{equation} Let us define the set $$ D :=\big\{ (x,y) \in C([0,h]) \times C([0,h]),\; \| (x,y) \|_{\infty}= \sup( \| x \|_{\infty},\; \| y \|_{\infty}) \leq k \}, $$ as a domain of the operator $A$, which is a convex, bounded, and closed subset of the Banach space $ C([0,h])\times C([0,h])$. Since $f$ and $g$ are continuously differentiable on $ [u_0-kh,u_0+kh]\times [v_0-kh,v_0+kh]$, there exists a positive constant $ M$ such that for any $ t$ in $ [0,h]$ and any $(x,y)$ in $D$, \begin{gather} \label{34} \big| f(u_0+\int_0^t x(s)ds,\ v_0+\int_0^t y(s))ds \big| \leq M, \\ \label{35} \big| g((u_0+\int_0^t x(s)ds,\ v_0+\int_0^t y(s))ds \big| \leq M, \end{gather} and for any $(u_j,v_j)$ in $ [u_0-kh,u_0+kh]\times [v_0-kh,v_0+kh]$, $j=1,2,$ and any $t$ in $ [0,h]$, there exist two positive constants $L_1$ and $L_2$ depending on $u_0, v_0, k, h $ and on $f$ and $g$ respectively such that \begin{gather} \label{36} | f(u_1(t),v_1(t))-f(u_2(t),v_2(t))| \leq L_1\| (u_1(t)-u_2(t),v_1(t)-v_2(t))\|,\\ \label{37} | g(u_1(t),v_1(t))-g(u_2(t),v_2(t))| \leq L_2\| (u_1(t)-u_2(t),u_1(t)-u_2(t))\|, \end{gather} where $\| (u_1(t)-u_2(t),v_1(t)-v_2(t))\|$=$| u_1(t)-u_2(t)|+| v_1(t)-v_2(t)|$. Now, by using \eqref{31} and \eqref{36} and \eqref{37}, for all $z_1=( x_1,y_1)\in D$ and $z_2=(x_2,y_2) \in D$ satisfying $ \| z_1-z_2\|_\infty\ <\delta$, where $\delta$ is a positive constant which will be defined later, we obtain \begin{equation} \begin{aligned} &\| A_1(z_1)-A_1(z_2) \|_{\infty} \\ &= \sup_{ 0\leq t \leq h} | -\sum_{i=1}^{n} a_{i}J^{1-\alpha_i}_{0_{+}}x_1(t) + J_{0^{+}}^{1-\gamma_1} f(u_0 + \int_0^{t} x_1(s)ds,\ v_0 + \int_0^{t} y_1(s)ds) \\ &\quad + \sum_{i=1}^{n}a_{i}J^{1-\alpha_i}_{0_{+}}x_2(t) - J_{0^{+}}^{1-\gamma_1}f(u_0 + \int_0^{t} x_2(s)ds,\ v_0 + \int_0^{t} y_2(s)\,ds) \ | \\ &\leq \sup_{ 0\leq t \leq h} | -\sum_{i=1}^{n} a_{i}J^{1-\alpha_i}_{0_{+}} (x_1(t)-x_2(t)) \\ &\quad + J_{0^{+}}^{1-\gamma_1}\{ f(u_0 + \int_0^{t} x_1(s)ds, v_0 + \int_0^{t} y_1(s) ds) \\ &\quad - f(u_0 + \int_0^{t} x_2(s)ds, v_0 + \int_0^{t} y_2(s)ds)\}| \\ &\leq \sum_{i=1}^{n} \frac{a_{i}}{\Gamma(1-\alpha_i)} \int_0^{h} (t-s)^{-\alpha_i} \| z_1-z_2\|_\infty ds + \frac{L_1}{\Gamma(2-\gamma_1)} h^{2-\gamma_1}\| z_1-z_2\|_\infty \\ &\leq \Big( n\bar{a}\max_{1\leq\ i \leq n}\{\frac{1}{\Gamma(2-\alpha_i)}\} \sum_{i=1}^{n}{h^{1-\alpha_i} }+ \frac{L_1}{\Gamma(2-\gamma_1)} \ h^{2-\gamma_1} \Big) \delta, \end{aligned} \end{equation} and in the same way, we obtain \begin{equation} \| A_2(z_1)-A_2(z_2) \|_{\infty}\leq \Big( n\bar{b}\max_{1\leq\ i \leq n}\{\frac{1}{\Gamma(2-\beta_i)}\} \sum_{i=1}^{n}{h^{1-\beta_i} }+\frac{L_2}{\Gamma(2-\gamma_2)} h^{2-\gamma_2}\Big)\delta. \end{equation} Now, given an $\varepsilon >0$, pick $\delta = \min\big\{\frac{\varepsilon}{ \omega_1},\frac{\varepsilon}{ \omega_2 } \big\}$, where \begin{gather*} \omega_1 : = n\bar{a}\max_{1\leq\ i \leq n} \big\{\frac{1}{\Gamma(2-\alpha_i)}\big\} \sum_{i=1}^{n}{h_{i}^{1-\alpha_i}} + \frac{L_1}{\Gamma(2-\gamma_1)} h^{2-\gamma_1}\, , \\ \omega_2 : = n\bar{b}\max_{1\leq\ i \leq n} \big\{\frac{1}{\Gamma(2-\beta_i)}\big\} \sum_{i=1}^{n}{h_{i}^{1-\beta_i}} +\frac{L_2}{\Gamma(2-\gamma_2)} h^{2-\gamma_2}\,. \end{gather*} One can see that $\| A(z_1)-A(z_2) \|_{\infty}< \varepsilon$, consequently, $ A $ is a continuous operator on $D $. Next, from \eqref{33}, \eqref{34}, \eqref{35} and \eqref{31}, for all $z= (x,y) \in D $ we have \begin{equation}\label{310} \begin{aligned} &\| A_1(z) \|_{\infty}\\ & \leq \sup_{0\leq t\leq h} \Big| \sum_{i=1}^{n} \frac{ a_{i}}{\Gamma(1-\alpha_i)}\int_0^{t} (t-s)^{-\alpha_i} x(s) \,ds \\ &\quad + \frac{1}{\Gamma(1-\gamma_1)}\int_0^{t} (t-s)^{-\gamma_1} f( u_0 +\int_0^{t}x(s)ds,v_0+\int_0^{t}y(s))ds \Big| \\ &\leq \sum_{i=1}^{n} \frac{a_{i}}{\Gamma(1-\alpha_i)} \| z \|_\infty \int_0^{h} (t-s)^{-\alpha_i} ds + \frac{1}{\Gamma(1-\gamma_1)}\int_0^{t} (t-s)^{-\gamma_1} M\,ds \\ &\leq nk\bar{a} \max_{1\leq i \leq n}\big\{ \frac{1}{\Gamma(2-\alpha_i)}\big\} \sum_{i=1}^{n} {h^{1-\alpha_i}} + \frac{1}{\Gamma(2-\gamma_1)}Mh^{1-\gamma_1} \leq k. \end{aligned} \end{equation} and \begin{equation}\label{311} \begin{aligned} &\| A_2(z) \|_{\infty}\\ &\leq \sup_{0 \leq t \leq h } \big| \sum_{i=1}^{n} \frac{ b_{i}}{\Gamma(1-\beta_i)}\int_0^{t} (t-s)^{-\beta_i} x(s) \,ds + \frac{1}{\Gamma(2-\gamma_2)}Mh^{1-\gamma_1} \big| \\ &\leq \sum_{i=1}^{n} \frac{a_{i}}{\Gamma(1-\beta_i)} \| z \|_{\infty} \int_0^{h} (t-s)^{-\beta_i} ds + \frac{1}{\Gamma(2-\gamma_2)}Mh^{1-\gamma_2} \\ &\leq nk\bar{b} \max_{1\leq i \leq n}\big\{ \frac{1}{\Gamma(2-\beta_i)}\Big\} \sum_{i=1}^{n} {h^{1-\beta_i}} + \frac{1}{\Gamma(2-\gamma_2)}Mh^{1-\gamma_2} \leq k. \end{aligned} \end{equation} Inequalities \eqref{310} and \eqref{311} assert that $ A(D) \subset D$. Thus, the set $ A(D )$ is uniformly bounded. Now, for all $ 0\leq t_1\leq t_2\leq h $ with $ | t_1-t_2 | < \eta$, and all $ z=(x,y) \in C([0,h])\times C([0,h]) $, from \eqref{36} we have \begin{align} &| A_1( \ z(t_1))-A_1( \ z(t_2) \ ) | \nonumber \\ &= \Big| -\sum_{i=1}^{n}\frac{a_{i}}{\Gamma(1-\alpha_i)} \int_0^{t_1} (t_1-s)^{-\alpha_i}x(s)ds \nonumber \\ &\quad + \frac{1}{\Gamma(1-\gamma_1)}\int_0^{t_1}(t_1-s)^{-\gamma_1} f( u_0+\int_0^{s}x(\tau)d\tau,v_0+\int_0^{s}y(\tau)d\tau)ds \nonumber \\ &\quad + \sum_{i=1}^{n} \frac{a_{i}}{\Gamma(1-\alpha_i)}\int_0^{t2} (t_2-s)^{-\alpha_i}x(s)ds \nonumber \\ &\quad - \frac{1}{\Gamma(1-\gamma_1)}\int_0^{t_2}(t_2-s)^{-\gamma_1}f(u_0 +\int_0^{s}x(\tau)d\tau, \ v_0+\int_0^{s}y(\tau)d\tau) ds\Big| \nonumber \\ &\leq \sum_{i=1}^{n}\frac{ a_{i}}{\Gamma(1-\alpha_i)} \int_0^{t_1} \left( (t_1-s )^{-\alpha_i}-(t_2-s)^{-\alpha_i}\right) | x(s) | ds \nonumber \\ &\quad + \sum_{i=1}^{n}\frac{a_{i}}{\Gamma(1-\alpha_i)} \int_{t_1}^{t_2} (t_2-s)^{-\alpha_i} | x(s) | ds \nonumber \\ &\quad + \frac{1}{\Gamma(1-\gamma_1)} \int_0^{t_1} \left( (t_1-s )^{-\gamma_1}-(t_2-s)^{-\gamma_1} \right) \nonumber \\ &\quad\times \Big| f(u_0 +\int_0^{s}x(\tau)d\tau,v_0+\int_0^{s}y(\tau)d\tau)\Big| ds \nonumber \\ &\quad + \frac{1}{\Gamma(1-\gamma_1)}\int_{t_1}^{t_2}(t_2-s)^{-\gamma_1} \Big| f( u_0+\int_0^{s}x(\tau)d\tau, v_0+\int_0^{s}y(\tau)d\tau) \Big| ds \nonumber \\ &\leq k\bar{a}\sum_{i=1}^{n}\frac{1}{\Gamma(2-\alpha_i)} (t_2-t_1) ^{1-\alpha_i}+\frac{2M}{\Gamma(2-\gamma_1)} (t_2-t_1) ^{1-\gamma_1}. \label{312} \end{align} Similarly, we obtain \begin{equation}\label{313} \begin{aligned} &| A_2( z(t_1) ) - A_2( z(t_2) ) |\\ &\leq k\bar{b}\sum_{i=1}^{n}\frac{1}{\Gamma(2-\beta_i)} ( t_2-t_1) ^{1-\beta_i}+\frac{2M}{\Gamma(2-\gamma_2)} (t_2-t_1) ^{1-\gamma_2}. \end{aligned} \end{equation} From \eqref{312} and \eqref{313} it yields that $ A(D) $ is equicontinuous, and so by using Arzela-Ascoli theorem, we find that $ A(D) $ is relatively compact in $ C([0,h])\times C([0,h])$. Finally, by Schauder theorem, we conclude that the operator $ A $ has at least one fixed point, this means that the system of integral equations \eqref{32} has at least one local continuous solution $(x,y) $ defined on $ [0,h] $. Now, since for all $t\in [0,h]$, \begin{equation} \label{314} u(t)= u_0 +\int_0^{t} x(s)ds, \quad v(t)= v_0 + \int_0^{t} y(s)ds, \end{equation} where $x $ and $y$ are solutions of system \eqref{32} of integral equations, it follows that $ u'(t)=x(t)$, $v'(t)=y(t)$ for any $t$ in $(0,h)$. Using the definition of Caputo fractional derivative, we find for all $t$ in $ (0,h)$, \begin{equation} \label{315 } \begin{aligned} {}^cD^{{\alpha}_{i}}_{0_+} u(t)= J^{1-\alpha_i}_{0_+} x(t) = \frac{1}{\Gamma(1-\alpha_i)} \int_0^T (t-s)^{-\alpha_i} x(s) \,ds, \quad i=1,\dots,n, \\ {}^cD^{{\beta}_{i}}_{0_+} v(t)\ = J^{1-\beta_i}_{0_+} y(t) = \frac{1}{\Gamma(1-\beta_i)} \int_0^T (t-s)^{-\beta_i} y(s) \,ds, \quad i=1,\dots,n. \end{aligned} \end{equation} Combining \eqref{314}, \eqref{315 } and \eqref{32}, for all $t$ in $(0,h)$ we obtain \begin{equation} \begin{gathered} u'(t)+\sum_{i=1}^{n}a_iJ^{1-\alpha_i}_{0_+}\frac{du(t)}{dt} = J_{0_+}^{1-\gamma_1} f(u(s),v(s)) \\ v'(t)+\sum_{i=1}^{n} b_iJ^{1-\beta_i}_{0_+}\frac{dv(t)}{dt} = J_{0_+}^{1-\gamma_2}g(u(s),v(s)). \end{gathered} \end{equation} Since $(u(0),v(0))=(u_0,v_0)$, we conclude that $ (u,v)$ is a classical solution for \eqref{11}-\eqref{12} on $(0,h)$, and this solution may be extended (see \cite {Carvalho}) to a maximal interval $( 0, T_{\rm max})$ with the alternative: either $T_{\rm max}=+\infty$ and the solution is global; or $$ T_{\rm max}<+\infty \quad \text{and} \quad \lim_ {t\to T_{\rm max}}(| u(t)|+| v(t)|)=+\infty. $$ Next, we shall prove uniqueness. Assume that the Cauchy problem \eqref{11}-\eqref{12} admits two classical solutions $ (u_1,v_1) $ and $ (u_2,v_2)$ with the same initial data $(u_0,v_0)$ on $ (0, T_{\rm max})$. Observe that for all $ t \in (0,\rho)$ with $\rhoT. \end{cases} \end{equation} Then by multiplying the first equation in \eqref{11} by $ \phi $ and integrating over $ {(0,T)} $, we obtain \begin{equation}\label{42} \begin{aligned} &\int_0^{T} u_{t}(t) \phi(t) dt+ \int_0^{T} \sum_{i=1}^{n} a_{i} ( D^{\alpha_i}_{0_{+}} (u(t)-u_0)) \phi(t) dt \\ &= \int_0^{T}( J_{0_+}^{1-\gamma_1}f(u(t),v(t))) \phi(t) dt. \end{aligned} \end{equation} Let \[ \psi(t):=\int_0^{t} \phi(s)ds= -\frac{1 }{\lambda+1}T^{-\lambda}(T-t)^{\lambda+1}\quad t\in [0,T]. \] Integrating by parts, and since $\psi(T)=0$, yields \begin{equation}\label{43} \begin{aligned} \int_0^{T}( J_{0_+}^{1-\gamma_1}f(u(t),v(t))) \phi(t) dt &=-\int_0^{T}\frac{d}{dt}( J_{0_+}^{1-\gamma_1}f(u(t),v(t)))\psi(t) dt \\ &=-\int_0^{T}( D_{_{0_+}}^{\gamma_1}f(u(t),v(t)))\psi(t) dt \\ &=-\int_0^{T}( D_{T_{-}}^{\gamma_1}\psi (t))f((u(t),v(t))dt. \end{aligned} \end{equation} Recall (see \cite{Furati}) the formulas \[ D_{T^{-}}^{\gamma_j}\phi(t)= C_{\lambda, \gamma_j} T^{-\lambda}(T-t)^{\lambda-\gamma_j},\quad \text{where} C_{\lambda, \gamma_j}=\frac{\lambda\Gamma(\lambda-\gamma_j)} {\Gamma(\lambda-2\gamma_j+1)}, \] and \begin{equation}\label{44} D_{T^{-}}^{\gamma_j}\psi(t) = -\dfrac{1}{\lambda+1}C_{\lambda+1, \gamma_j} T^{-\lambda}(T-t)^{\lambda+1-\gamma_j} =-C_{\lambda,\gamma_j}'\phi(t)(T-t)^{1-\gamma_j}, \end{equation} for $j=1,2$, where $C_{\lambda,\gamma_j}'=\frac{1}{\lambda+1}C_{\lambda+1,\gamma_j}$, $j=1,2$. Then \begin{equation}\label{45} -\int_0^{T}( D_{_{0_+}}^{\gamma_1}\psi(t))f(u(t),v(t)) dt =\int_0^{T}C_{\lambda,\gamma_1}'\phi(t)(T-t)^{ 1-\gamma_1}f(u(t),v(t)) dt. \end{equation} From \eqref{42}, \eqref{43} and \eqref{45} and since $u_0$ is positive and $\phi$ is in $C^{1}([0,T])$, thanks to \eqref{41}, an integration by parts yields \begin{equation}\label{46} \begin{aligned} &C_{\lambda,\gamma_1}'\int_0^{T}\phi(t)(T-t)^{ 1-\gamma_1}f(u(t),v(t)) dt\\ &\leq-\int_0^{T} u(t)\phi'(t) dt + \sum_{i=1}^{n} \int_0^{T} u(t)D^{\alpha_i}_{T_{-}} ( a_{i} \phi(t)) dt. \end{aligned} \end{equation} Observe that if \ $p{'}$ is the conjugate of \ $p$, then \begin{equation}\label{47} \begin{aligned} &\int_0^{T} u(t) (-\phi'(t)) dt\\ &= \int_0^{T} u(t) ( \phi (t)) ^{\frac{1}{p}} ( \phi(t)) ^{-1/p} (T-t)^\frac{{ 1-\gamma_2}}{p}(T-t)^\frac{ -(1-\gamma_2)}{p}(- \phi'(t)) dt \\ &\leq C_{\lambda,\gamma_2}'\frac{b}{4} \int_0^{T} | u(t)| ^{p} \phi(t)(T-t)^{1-\gamma_2} \,dt \\ &\quad + \Big(\frac{4}{bC_{\lambda,\gamma_2}'}\Big) ^{p'/p} \int_0^{T} ( \phi(t) )^{-p'/p} (T-t)^{-(1-\gamma_2)\frac{p{'}}{p}} | (\phi'(t)) | ^{p'} dt \\ &\leq C_{\lambda,\gamma_2}'\frac{1}{4 } \int_0^{T} g(u(t),v(t))\phi(t) (T-t)^{1-\gamma_2} dt \\ &\quad + \Big(\frac{4}{b C_{\lambda,\gamma_2}'}\Big) ^{p'/p} \int_0^{T} (\phi(t) ) ^{-p'/p}(T-t)^{-(1-\gamma_2)\frac{p{'}}{p}} | \phi'(t)| ^{p'} dt, \end{aligned} \end{equation} and for all $ 1\leq i \leq n$, \begin{equation}\label{48} \begin{aligned} &\int_0^{T} u(t) (D^{\alpha_i}_{T_{-}} (a_{i} \phi(t) )dt \\ &= \int_0^{T} u (t)( \phi(t) )^{\frac{1}{p}} (\phi(t) ) ^{-\frac{1}{p}} (T-t)^\frac{{ 1-\gamma_2}}{p}(T-t)^\frac{-(1-\gamma_2)}{p} D^{\alpha_i}_{T_{-}}( a_{i} \phi(t)) dt \\ &\leq C_{\lambda,\gamma_2}'\frac{b}{4n} \int_0^{T} | u(t) |^{p} \phi(t)(T-t)^{1-\gamma_2} dt \\ &\quad + \Big(\frac{4n}{bC_{\lambda,\gamma_2}'}\Big) ^{p'/p} a_{i}^{p'} \int_0^{T} (\phi(t)) ^{-p'/p}(T-t)^{-(1-\gamma_2)\frac{p{'}}{p}} | (D^{\alpha_i}_{T_{-}}\phi(t)) | ^{p'}dt \\ &\leq C_{\lambda,\gamma_2}' \frac{1}{4n} \int_0^{T} g(u(t),v(t)) \phi(t)(T-t)^{1-\gamma_2} dt \\ &\quad + \Big( \frac{4n}{bC_{\lambda,\gamma_2}'}\Big) ^{p'/p} \bar{a}^{p'}\int_0^{T} (\phi(t)) ^{-p'/p}(T-t) ^{-(1-\gamma_2)p'/p} | (D^{\alpha_i}_{T_{-}}\phi(t)) | ^{p'}dt. \end{aligned} \end{equation} Furthermore, \begin{equation}\label{49} \begin{aligned} &C_{\lambda,\gamma_1}'\int_0^{T}f(u(t),v(t)) \phi(t)(T-t)^{1-\gamma_1}dt \\ &\leq \frac{1}{2} C_{\lambda,\gamma_2}'\int_0^{T} g(u(t),v(t)) \phi(t)(T-t)^{1-\gamma_2}\,dt \\ &\quad + \Big(\frac{4}{bC_{\lambda,\gamma_2}'}\Big) ^{p'/p} \int_0^{T} ( \phi(t) ) ^{-p'/p} (T-t)^{-(1-\gamma_2)\frac{p{'}}{p}} | \phi'(t) | ^{p'} dt \\ &\quad + \bar{a}^{p'}\Big( \frac{4n}{bC_{\lambda,\gamma_2}'}\Big) ^{p'/p} \sum_{i=1}^{n} \int_0^{T} ( \phi(t) ) ^{-p'/p} (T-t)^{-(1-\gamma_2)\frac{p{'}}{p}} \ | D^{\alpha_i}_{T_{-}}\phi(t)) | ^{p'}dt. \end{aligned} \end{equation} Analogously, if $q'$ is the conjugate of $q$, we obtain \begin{equation}\label{410} \begin{aligned} &C_{\lambda,\gamma_2}'\int_0^{T} g(u(t),v(t)) \phi(t)(T-t)^{1-\gamma_2} dt \\ &\leq -\int_0^{T} v(t)\phi'(t) dt + \sum_{i=1}^{n}\int_0^{T} v(t) D^{\beta_i}_{T_{-}} ( \ b_{i}(t)\phi(t) ) dt \\ &\leq \frac{1}{2 } C_{\lambda,\gamma_1}'\int_0^{T} f(u(t),v(t)) \phi(t)(T-t)^{1-\gamma_1} dt \\ &\quad + \Big( \frac{4}{aC_{\lambda,\gamma_1}'}\Big) ^{q'/q} \int_0^{T} ( \phi(t) ) ^{-q'/q}(T-t)^{-(1-\gamma_1)q'/q} | \phi'(t) | ^{q'} dt \\ &\quad + \bar{b}^{q'}\Big( \frac{4n}{aC_{\lambda,\gamma_1}'}\Big) ^{q'/q} \sum_{i=1}^{n} \int_0^{T} ( \phi(t) ) ^{-q'/q} (T-t)^{-(1-\gamma_1)q'/q} | D^{\beta_i}_{T-}\phi(t)|^{q'}dt. \end{aligned} \end{equation} Denote \begin{gather*} A := C_{\lambda,\gamma_1}'\int_0^{T} f(u(t),v(t)) \phi(t) (T-t)^{1-\gamma_1} dt,\\ B : = C_{\lambda,\gamma_2}'\int_ {0}^{T} g(u(t),v(t)) \phi(t) (T-t)^{1-\gamma_2} dt, \\ C : = \int_0^{T}( \phi(t) ) ^{-p'/p} (T-t)^{-(1-\gamma_2)\frac{p'}{p}} | \phi'(t) \ | ^{p'} dt, \\ D := \int_0^{T} ( \phi(t) )^{-q'/q}(T-t)^{-(1-\gamma_1)q'/q} | \phi'(t) | ^{q'} dt, \\ E : = \int_0^{T} ( \phi(t) ) ^{-p'/p}(T-t)^{-(1-\gamma_2)\frac{p'}{p}} \sum_{i=1}^{n} | D^{\alpha_i}_{T-}\phi(t) | ^{p'}dt,\\ F := \int_0^{T} ( \phi(t) ) ^{-q'/q}(T-t)^{-(1-\gamma_1)q'/q} \sum_{i=1}^{n} | D^{\beta_i }_{T-}\phi(t)| ^{q'} dt. \end{gather*} From \eqref{49} and \eqref{410} we have \begin{gather*} A \leq \frac{1}{2} B + \Big( \frac{4}{bC_{\lambda,\gamma_2}'}\Big) ^{p'/p} (C+ n^{p'/p}\bar{a}^{p'}E), \\ B \leq \frac{1}{2} A+ \Big( \frac{4}{aC_{\lambda,\gamma_1}'}\Big) ^{q'/q} (D+ n^{q'/q}\bar{b}^{q'}F), \end{gather*} then \begin{align*} A&\leq \frac{1}{2} \Big( \frac{1}{2} A + \Big( \frac{4}{aC_{\lambda,\gamma_1}'}\Big) ^{q'/q} (D+ n^{q'/q}\bar{b}^{q'}F)\Big) + \Big( \frac{4}{bC_{\lambda,\gamma_2}'}\Big) ^{p'/p} (C+n^\frac{p'}{p} \bar{a}^{p'}E) \\ &= \frac{1}{4 } A + \frac{1}{2} \Big( \frac{4}{aC_{\lambda,\gamma_1}'}\Big) ^{q'/q}(D+n^{q'/q}\bar{b}^{q'}F) +\Big( \frac{4}{bC_{\lambda,\gamma_2}'}\Big) ^{p'/p} (C+n^\frac{p'}{p}\bar{a}^{p'}E); \end{align*} thus \[ A\leq \frac{2}{3} \Big( \frac{4}{aC_{\lambda,\gamma_1}'}\Big) ^{q'/q} (D+ n^{q'/q}\bar{b}^{q'}F) + \frac{4}{3} \Big( \frac{4}{bC_{\lambda,\gamma_2}'}\Big) ^{p'/p} (C+ n^\frac{p'}{p}\bar{a}^{p'}E) \] and \begin{align*} B&\leq \frac{1}{2} \Big( \frac{2}{3} \Big( \frac{4}{aC_{\lambda,\gamma_1}'} \Big) ^{q'/q}(D+ n^{q'/q}\bar{b}^{q'}F) + \frac{4}{3}\Big( \frac{4}{bC_{\lambda,\gamma_2}'}\Big) ^{p'/p} (C+n^\frac{p'}{p} \bar{a}^{p'}E) \Big) \\ &\quad + \Big( \frac{4}{aC_{\lambda,\gamma_1}'}\Big) ^{q'/q} (D+ n^{q'/q}\bar{b}^{q'}F)\\ &\leq \frac{4}{3} \Big( \frac{4}{aC_{\lambda,\gamma_1}'}\Big) ^{q'/q} (D+n^{q'/q}\bar{b}^{q'}F) + \frac{2}{3} \Big( \frac{4}{bC_{\lambda,\gamma_2}'} \Big) ^{p'/p} ( C +n^\frac{p'}{p}\bar{a}^{p'}E ). \end{align*} Taking into account \eqref{42}, \eqref{47} and \eqref{48}, we deduce that \begin{align*} &u_0 \int_0^{T} D^{\alpha_1}_{T_{-}} \phi(t) dt \\ &= \frac{u_0 }{a_1} \int_0^{T} D^{\alpha_1}_{T_{-}} (a_1\phi(t) ) dt \\ &\leq \frac{ 1}{a_1} \Big( -\int_0^{T} u(t) \ \phi'(t) dt + \int_0^{T} \sum_{i=1}^{n}u(t) \ D^{\alpha_i}_{T_{-}} (a_{i}\phi(t)) \,dt \Big) \\ &\leq \frac{ 1}{ a_1} \Big( \frac{1}{2}B +\Big( \frac{4}{bC_{\lambda,\gamma_2}'}\Big) ^{p'/p} (C+ n^\frac{p'}{p}\bar{a}^{p'}E)\Big) \\ &\leq \frac{1}{a_1} \Big( \frac{2}{3} \Big( \frac{4}{aC_{\lambda,\gamma_1}'} \Big) ^{q'/q} (D + n^{q'/q}\bar{b}^{q'} F ) + \frac{4}{3} \Big( \frac{4}{bC_{\lambda,\gamma_2}'}\Big) ^{p'/p} (C+ n^\frac{p'}{p}\bar{a}^{p'}E)\Big) . \end{align*} For $ \lambda>\max\{\frac{p'}{p}+p{'}-1, \frac{q'}{q}+q{'}-1\} $, it holds \begin{equation} \label{411} \int_0^{T} D^{\alpha_i}_{T_{-}} \phi(t)dt = C_{\alpha_i,\lambda} T^{1-\alpha_i}, \end{equation} where \[ C_{\alpha_i,\lambda}= \frac{\Gamma(\lambda+1)}{\Gamma(\lambda - \alpha_i +2)}, \quad \forall 1\leq i \leq n\,. \] Also there exists a positive constant $K$ such that \begin{gather} C \leq K T^{(\gamma_2-1)\frac{p'}{p}+1-p'}, \quad D \leq K T^{(\gamma_1-1)\frac{q'}{q}+1-q'}, \nonumber\\ \label{412} E\leq K \sum_{i=1}^{n}T^{(\gamma_2-1)\frac{p{'}}{p}+1-p' \alpha_i}, \quad F\leq K \sum_{i=1}^{n}T^{(\gamma_1-1)\frac{q{'}}{q}+1-q'\beta_i}, \quad \forall 1\leq i \leq n. \end{gather} Consequently, \begin{equation}\label{413} \begin{aligned} &u_0 \int_0^{T}D^{\alpha_1}_{T_{-}}\phi(t) dt \\ &\leq \frac{ 2}{3 a_1} \Big( \frac{4}{aC_{\lambda,\gamma_1}'}\Big) ^{q'/q} K\Big( T^{(\gamma_1-1)\frac{q'}{q}+1-q'} +n^{q'/q}\bar{b}^{q'}\sum_{i=1}^{n} T^{(\gamma_1-1)\frac{q{'}}{q}+1-q'\beta_i}\Big) \\ &\quad + \frac{4}{3a_1}\Big( \frac{4}{bC_{\lambda,\gamma_2}'}\Big) ^{p'/p }K\Big( T^{(\gamma_2-1)\frac{p'}{p}+1-p'} + n^\frac{p'}{p}\bar{a}^{p'}\sum_{i=1}^{n} T^{(\gamma_2-1)\frac{p{'}}{p} +1-p' \alpha_i}\Big). \end{aligned} \end{equation} Using \eqref{411} and \eqref{413}, we obtain \begin{equation}\label{414} \begin{aligned} u_0 &\leq C_{\alpha_1, \lambda}^{-1}K \Big\{ \frac{ 2}{3 a_1} \Big( \frac{4}{aC_{\lambda,\gamma_1}'}\Big) ^{q'/q} \Big( T^{(\gamma_1-1)\frac{q'}{q}+\alpha_1-q'} \\ &\quad +n^{q'/q}\bar{b}^{q'}\sum_{i=1}^{n} T^{(\gamma_1-1) \frac{q{'}}{q}+\alpha_1-q'\beta_i}\Big) \Big\} \\ &\quad + C_{\alpha_1, \lambda}^{-1}K\Big\{ \frac{4}{3a_1} \Big( \frac{4}{bC_{\lambda,\gamma_1}'}\Big) ^{p'/p} \Big( T^{(\gamma_2-1)\frac{p'}{p}+\alpha_1-p' }\\ &\quad +n^\frac{p'}{p}\bar{a}^{p'}\sum_{i=1}^{n} T^{(\gamma_2-1)\frac{p{'}}{p}+\alpha_1-p' \alpha_i}\Big) \Big\}. \end{aligned} \end{equation} Similarly we obtain \begin{align*} &v_0 \int_0^{T}D^{\beta_1}_{T_{-}}\phi(t) dt \\ &\leq \frac{1}{b_1}\Big( \frac{1}{2}A + \Big( \frac{4}{aC_{\lambda,\gamma_1}'}\Big) ^{q'/q}(D +n^{q'/q}\bar{b}^{q'}F)\Big) \\ &\leq C_{\beta, \lambda}^{-1}\Big( \frac{4}{3b_1} \Big( \frac{4}{aC_{\lambda,\gamma_1}'}\Big) ^{q'/q}( D + n^{q'/q}\bar{b}^{q'} F ) + \frac{2}{3 b_1} \Big( \frac{4}{bC_{\lambda,\gamma_2}'}\Big)^{p'/p} (C+ n^\frac{p'}{p}\bar{a}^{p'} E)\Big) , \end{align*} which yields \begin{equation}\label{415} \begin{aligned} v_0 &\leq C_{\beta_1, \lambda}^{-1}K\Big\{ \frac{4}{3b_1} \Big( \frac{4}{aC_{\lambda,\gamma_1}'}\Big) ^{q'/q} \Big( T^{(\gamma_1-1)\frac{q'}{q}+\beta_1-q'} \\ &\quad + n^{q'/q}{\bar{b}^{q'}} \sum_ {i=1}^{n} T^{(\gamma_1-1)\frac{q{'}}{q}+\beta_1-q'\beta_i}\Big) \Big\} \\ &\quad + C_{\beta_1, \lambda}^{-1}K\Big\{ \frac{2}{3b_1} \Big( \frac{4}{bC_{\lambda,\gamma_2}'}\Big) ^{p'/p} \Big( T^{(\gamma_2-1)\frac{p'}{p}+\beta_1-p'}\\ &\quad + n^\frac{p'}{p}\bar{a}^{p'} \sum_{i=1}^{n} T^{(\gamma_2-1)\frac{p{'}}{p} +\beta_1-p' \alpha_i}\Big) \Big\}. \end{aligned} \end{equation} One can observe that \begin{equation}\label{416} \begin{gathered} (\gamma_1-1)\frac{q'}{q}+\alpha_1-q'<0,\quad (\gamma_2-1)\frac{p'}{p}+\alpha_1-p'<0, \\ (\gamma_1-1)\frac{q'}{q}+\beta_1-q'<0, \quad (\gamma_2-1)\frac{p'}{p}+\beta_1-p'<0,\\ (\gamma_2-1)\frac{p'}{p}+\alpha_1-p'\alpha_i<0,\quad (\gamma_1-1)\frac{q{'}}{q}+\beta_1-q'\beta_i<0,\quad \forall 1\leq i\leq n, \\ (\gamma_2-1)\frac{p{'}}{p}+\beta_1-p' \alpha_i<\beta_1-\alpha_1, \quad (\gamma_1-1)\frac{q'}{q}+\alpha_1-q'\beta_i<\alpha_1-\beta_1,\\ \forall 1\leq i\leq n. \end{gathered} \end{equation} Inequalities \eqref{416} reduce to \[ (\gamma_2-1)\frac{p{'}}{p}+\beta_1-p' \alpha_i<0,\quad \forall 1\leq i\leq n, \] or \[ (\gamma_1-1)\frac{q'}{q}+\alpha_1-q'\beta_i<0,\quad \forall 1\leq i\leq n. \] Taking the limit when $T$ approaches infinity in \eqref{414} and \eqref{415}, we find \begin{equation} 0