\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 10, pp. 1--11.\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{9mm}} \begin{document} \title[\hfilneg EJDE-2009/10\hfil Impulsive fractional differential equations] {Existence and uniqueness of solutions to impulsive fractional differential equations} \author[M. Benchohra, B. A. Slimani\hfil EJDE-2009/10\hfilneg] {Mouffak Benchohra, Boualem Attou Slimani} % in alphabetical order \address{Mouffak Benchohra \newline Laboratoire de Math\'ematiques, Universit\'e de Sidi Bel-Abb\es, B.P. 89, 22000, Sidi Bel-Abb\es, Alg\'erie} \email{benchohra@univ-sba.dz} \address{Boualem Attou Slimani \newline Facult\'e des Sciences de l'Ing\'enieur, Universit\'e de Tlemcen, B.P. 119, 13000, Tlemcen, Alg\'erie} \email{ba\_slimani@yahoo.fr} \thanks{Submitted October 22, 2008. Published January 9, 2009.} \subjclass[2000]{26A33, 34A37} \keywords{Fractional derivative; impulses; Initial value problem; \hfill\break\indent Caputo fractional integral; nonlocal conditions; existence; uniqueness; fixed point} \begin{abstract} In this article, we establish sufficient conditions for the existence of solutions for a class of initial value problem for impulsive fractional differential equations involving the Caputo fractional derivative. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} This article studies the existence and uniqueness of solutions for the initial value problems (IVP for short), for fractional order differential equations \begin{gather}\label{e1} ^{c}D^{\alpha}y(t)= f(t,y), \quad t\in J=[0, T], \;t\neq t_{k}, \\ \label{e2} \Delta y\big|_{t=t_{k}}= I_{k}(y(t_{k}^{-})), \\ \label{e3} y(0)= y_{0}, \end{gather} where $k=1,\dots,m$, $0<\alpha\leq 1$, $^{c}D^{\alpha}$ is the Caputo fractional derivative, $f : J\times \mathbb{R}\to \mathbb{R}$ is a given function, $I_{k}:\mathbb{R}\to\mathbb{R}$, and $y_{0}\in\mathbb{R}$, $0=t_{0}0$, and $\varphi_{\alpha}(t)=0$ for $t\leq 0$, and $\varphi_{\alpha}\to \delta(t)$\ as $\alpha\to 0$, where $\delta$ is the delta function. \end{definition} \begin{definition}[\cite{KST,Pod}] \rm For a function $h$ given on the interval $[a,b]$, the $\alpha th$ Riemann-Liouville fractional-order derivative of $h$, is defined by $(D^{\alpha}_{a+}h)(t)=\frac{1}{\Gamma(n-\alpha)}\big(\frac{d}{dt}\big)^{n}\int_ a^t(t-s)^{n-\alpha-1}h(s)ds.$ Here $n=[\alpha]+1$ and $[\alpha]$ denotes the integer part of $\alpha$. \end{definition} \begin{definition}[\cite{KiMa}] \rm For a function $h$ given on the interval $[a,b]$, the Caputo fractional-order derivative of order $\alpha$ of $h$, is defined by $$(^{c}D_{a+}^{\alpha}h)(t)=\frac{1}{\Gamma(n-\alpha)}\int_ a^t(t-s)^{n-\alpha-1}h^{(n)}(s)ds,$$ where $n=[\alpha]+1$. \end{definition} \section{Existence of Solutions} Consider the set of functions \begin{align*} PC(J,\mathbb{R})&=\{y: J\to \mathbb{R}: y\in C((t_k,t_{k+1}],\mathbb{R}), \; k=0,\dots,m \text{ and there exist }\\ &\quad y(t^{-}_{k}) \text{ and } y(t^{+}_{k}), \; k=1,\dots,m \text{ with } y(t^{-}_{k})=y(t_{k})\}. \end{align*} This set is a Banach space with the norm $$\|y\|_{PC}=\sup_{t\in J}|y(t)|.$$ Set $J':=[0,T]\backslash\{t_{1},\dots,t_{m}\}$. \begin{definition} \rm A function $y\in PC(J,\mathbb{R})$ whose $\alpha$-derivative exists on $J'$ is said to be a solution of \eqref{e1}--\eqref{e3} if $y$ satisfies the equation $^{c}D^{\alpha}y(t)=f(t,y(t))$ on $J'$, and satisfy the conditions \begin{gather*} \Delta y|_{t=t_{k}}= I_{k}(y(t_{k}^{-})), \ k=1,\dots,m,\\ y(0)= y_{0} \end{gather*} \end{definition} To prove the existence of solutions to \eqref{e1}--\eqref{e3}, we need the following auxiliary lemmas. \begin{lemma}[\cite{Zha}] \label{l1} Let $\alpha > 0$, then the differential equation $$^{c}D^{\alpha}h(t)=0$$ has solutions $h(t)=c_{0}+c_{1}t+c_{2}t^{2}+\dots +c_{n-1}t^{n-1} , c_{i}\in \mathbb{R}$, $i=0,1,2,\dots,n-1$, $n=[\alpha]+1$. \end{lemma} \begin{lemma}[\cite{Zha}] \label{l2} Let $\alpha > 0$, then $${I^{\alpha}}^{c}D^{\alpha}h(t)=h(t)+ c_{0}+c_{1}t+c_{2}t^{2}+\dots+c_{n-1}t^{n-1}$$ for some $c_{i}\in \mathbb{R}$, $i=0,1,2,\dots,n-1$, $n=[\alpha]+1$. \end{lemma} As a consequence of Lemma \ref{l1} and Lemma \ref{l2} we have the following result which is useful in what follows. \begin{lemma}\label{l3} Let $0< \alpha\leq 1$ and let $h: J \to\mathbb{R}$ be continuous. A function $y$ is a solution of the fractional integral equation $$\label{e4} y(t)=\begin{cases} y_{0}+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}h(s)ds & \text{if } t\in [0,t_{1}], \\[3pt] y_{0}+\frac{1}{\Gamma(\alpha)}\sum_{i=1}^{k}\int_{t_{i-1}}^{t_{i}} (t_{i}-s)^{\alpha-1}h(s)ds\\ + \frac{1}{\Gamma(\alpha)} \int_{t_{k}}^{t}(t-s)^{\alpha-1}h(s)ds +\sum_{i=1}^{k}I_{i}(y(t_{i}^{-})), & \text{if } t\in (t_{k},t_{k+1}], \end{cases}$$ where $k=1,\dots,m$, if and only if $y$ is a solution of the fractional IVP \begin{gather}\label{e5} ^{c}D^{\alpha}y(t)= h(t), \quad t\in J', \\ \label{e6} \Delta y|_{t=t_{k}}= I_{k}(y(t_{k}^{-})), \quad k=1,\dots,m, \\ \label{e7} y(0)= y_{0}. \end{gather} \end{lemma} \begin{proof} Assume $y$ satisfies \eqref{e5}-\eqref{e7}. If $t\in [0,t_{1}]$ then $$^{c}D^{\alpha}y(t)= h(t).$$ Lemma \ref{l2} implies $$y(t)=y_{0}+\frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1}h(s)ds.$$ If $t\in (t_{1},t_{2}]$ then Lemma \ref{l2} implies \begin{align*} y(t)&=y(t_{1}^{+})+\frac{1}{\Gamma(\alpha)} \int_{t_{1}}^{t}(t-s)^{\alpha-1}h(s)ds\\ &= \Delta y|_{t=t_{1}}+y(t_{1}^{-})+\frac{1}{\Gamma(\alpha)} \int_{t_{1}}^{t}(t-s)^{\alpha-1}h(s)ds\\ &= I_{1}(y(t_{1}^{-}))+y_{0}+\frac{1}{\Gamma(\alpha)} \int_{0}^{t_{1}}(t_{1}-s)^{\alpha-1}h(s)ds +\frac{1}{\Gamma(\alpha)} \int_{t_{1}}^{t}(t-s)^{\alpha-1}h(s)ds. \end{align*} If $t\in (t_{2},t_{3}]$ then from Lemma \ref{l2} we get \begin{align*} y(t)&= y(t_{2}^{+})+\frac{1}{\Gamma(\alpha)} \int_{t_{2}}^{t}(t-s)^{\alpha-1}h(s)ds\\ &= \Delta y|_{t=t_{2}}+y(t_{2}^{-})+\frac{1}{\Gamma(\alpha)} \int_{t_{2}}^{t}(t-s)^{\alpha-1}h(s)ds\\ &= I_{2}(y(t_{2}^{-}))+I_{1}(y(t_{1}^{-}))+y_{0}+\frac{1}{\Gamma(\alpha)} \int_{0}^{t_{1}}(t_{1}-s)^{\alpha-1}h(s)ds\\ &\quad +\frac{1}{\Gamma(\alpha)} \int_{t_{1}}^{t_{2}}(t_{2}-s)^{\alpha-1}h(s)ds+\frac{1}{\Gamma(\alpha)} \int_{t_{2}}^{t}(t-s)^{\alpha-1}h(s)ds. \end{align*} If $t\in (t_{k},t_{k+1}]$ then again from Lemma \ref{l2} we get (\ref{e4}). Conversely, assume that $y$ satisfies the impulsive fractional integral equation (\ref{e4}). If $t\in [0,t_{1}]$ then $y(0)=y_{0}$ and using the fact that $^{c}D^{\alpha}$ is the left inverse of $I^{\alpha}$ we get $$^{c}D^{\alpha}y(t)= h(t), \quad \text{for each } t\in [0,t_{1}].$$ If $t\in [t_{k},t_{k+1})$, $k=1,\dots,m$ and using the fact that $^{c}D^{\alpha}C=0$, where $C$ is a constant, we get $$^{c}D^{\alpha}y(t)= h(t), \text{for each } t\in [t_{k},t_{k+1}).$$ Also, we can easily show that $$\Delta y|_{t=t_{k}}= I_{k}(y(t_{k}^{-})), \quad k=1,\dots,m.$$ \end{proof} Our first result is based on Banach fixed point theorem. \begin{theorem}\label{thm1} Assume that \begin{itemize} \item[(H1)] There exists a constant $l>0$ such that $|f(t,u)-f(t,\overline u)|\leq l |u-\overline u|$, for each $t\in J$, and each $u, \overline u \in \mathbb{R}$. \item[(H2)] There exists a constant $l^*>0$ such that $|I_{k}(u)-I_{k}(\overline u)|\leq l^*|u-\overline u|$, for each $u, \overline u \in \mathbb{R}$ and $k=1,\dots,m$. \end{itemize} If $$\label{eq1} \big[\frac{T^{\alpha}l(m+1)} {\Gamma(\alpha+1)}+ml^*\big] < 1,$$ then \eqref{e1}-\eqref{e3} has a unique solution on $J$. \end{theorem} \begin{proof} We transform the problem \eqref{e1}--\eqref{e3} into a fixed point problem. Consider the operator $F:PC(J,\mathbb{R})\to PC(J,\mathbb{R})$ defined by \begin{align*} F(y)(t)&=y_{0}+\frac{1}{\Gamma(\alpha)}\sum_{00$such that$ |f(t,u)|\leq M$for each$ t\in J$and each$u\in \mathbb{R}$. \item[(H5)] The functions$I_{k}:\mathbb{R}\to\mathbb{R}$are continuous and there exists a constant$ M^* >0$such that$|I_{k}(u)|\leq M^*$for each$u\in \mathbb{R}$,$k=1,\dots,m$. \end{itemize} Then \eqref{e1}-\eqref{e3} has at least one solution on$J$. \end{theorem} \begin{proof} We shall use Schaefer's fixed point theorem to prove that$F$has a fixed point. The proof will be given in several steps. \textbf{Step 1:}$F$is continuous. Let$\{y_{n}\}$be a sequence such that$y_{n}\to y$in$PC(J,\mathbb{R})$. Then for each$t\in J\begin{align*} |F(y_{n})(t)-F(y)(t)| &\leq \frac{1}{\Gamma(\alpha)}\sum_{00, there exists a positive constant $\ell$ such that for each $y\in B_{\eta^*}=\{y\in PC(J,\mathbb{R}): \|y\|_{\infty}\leq \eta^* \}$, we have $\|F(y)\|_{\infty}\leq \ell$. By (H4) and (H5) we have for each $t\in J$, \begin{align*} |F(y)(t)|&\leq |y_{0}|+\frac{1}{\Gamma(\alpha)}\sum_{00$such that $$\frac{\overline M}{|y_{0}|+\psi(\overline M) \frac{mT^{\alpha}\phi_{f}^{0}}{\Gamma(\alpha+1)} + \psi(\overline M)\frac{T^{\alpha}\phi_{f}^{0}}{\Gamma(\alpha+1)} + m\psi^*(\overline M)}>1,$$ where$\phi_{f}^{0}=\sup\{\phi_{f}(t): \ t\in J\}$. \end{itemize} Then \eqref{e1}-\eqref{e3} has at least one solution on$J$. \end{theorem} \begin{proof} Consider the operator$F$defined in Theorems \ref{thm1} and \ref{thm2}. It can be easily shown that$F$is continuous and completely continuous. For$\lambda\in [0,1]$, let$y$be such that for each$t\in J$we have$y(t)=\lambda (Fy)(t)$. Then from (H6)-(H7) we have for each$t\in J, \begin{align*} |y(t)|&\leq |y_{0}|+\frac{1}{\Gamma(\alpha)}\sum_{00 such that $|g(u)|\leq M^{**}$ for each $u\in PC(J,\mathbb{R})$. \item[(H10)] There exists a constant $k>0$ such that $|g(u)-g(\overline u)|\leq l^{**} |u-\overline u|$ for each $u, \overline u \in PC(J,\mathbb{R})$. \item[(H11)] There exists $\psi^{**}:[0,\infty)\to (0,\infty)$ continuous and nondecreasing such that $|g(u)|\leq \psi^{**}(|u|)$ for each $u\in PC(J,\mathbb{R})$. \item[(H12)] There exists an number $\overline M^{*}>0$ such that $$\frac{\overline M^*}{|y_{0}|+\psi^{**}(\overline M^{*})+\psi(\overline M^{*})\frac{mT^{\alpha}\phi_{f}^{0}}{\Gamma(\alpha+1)} + \psi(\overline M^{*})\frac{T^{\alpha}\phi_{f}^{0}}{\Gamma(\alpha+1)} +m\psi^*(\overline M^{*})}>1,$$ \end{itemize} \begin{theorem}\label{nt1} Assume that {\rm (H1), (H2), (H10)} hold. If $$\label{neq1} \big[\frac{T^{\alpha}l(m+1)} {\Gamma(\alpha+1)}+ml^*+l^{**}\big] < 1,$$ then the nonlocal problem \eqref{ne1}-\eqref{ne3} has a unique solution on $J$. \end{theorem} \begin{proof} We transform the problem (\ref{ne1})--(\ref{ne3}) into a fixed point problem. Consider the operator $\tilde F:PC(J,\mathbb{R})\to PC(J,\mathbb{R})$ defined by \begin{align*} \tilde F(y)(t) &= y_{0}-g(y)+\frac{1}{\Gamma(\alpha)}\sum_{0\frac{3}{10}, which is satisfied for some $\alpha\in (0,1]$. Then by Theorem \ref{thm1} the problem \eqref{ex1}-\eqref{ex3} has a unique solution on $[0,1]$ for values of $\alpha$ satisfying (\ref{ex4}). \begin{thebibliography}{99} \bibitem{ABH} R. P. Agarwal, M. Benchohra and S. Hamani; Boundary value problems for fractional differential equations, \emph{Adv. Stud. Contemp. Math.} \textbf{16} (2) (2008), 181-196. \bibitem{BaDa} A. Babakhani and V. Daftardar-Gejji; Existence of positive solutions for multi-term non-autonomous fractional differential equations with polynomial coefficients. \emph{Electron. J. Differential Equations} \textbf{2006}(2006), No. 129, 12 pp. \bibitem{BaDa1} A. Babakhani and V. Daftardar-Gejji; Existence of positive solutions for $N$-term non-autonomous fractional differential equations. \emph{Positivity} \textbf{9} (2) (2005), 193--206. \bibitem{BaSi} D. D. Bainov, P. S. Simeonov; \emph{Systems with Impulsive effect,} Horwood, Chichister, 1989. \bibitem{BeBe} M. Belmekki and M. Benchohra; Existence results for Fractional order semilinear functional differential equations, \emph{Proc. A. Razmadze Math. Inst.} \textbf{146} (2008), 9-20. \bibitem{BeBeGo} M. Belmekki, M. Benchohra and L. Gorniewicz; Semilinear functional differential equations with fractional order and infinite delay, \emph{Fixed Point Th.} \textbf{9} (2) (2008), 423-439. \bibitem{BeGrHa} M. Benchohra, J. R. Graef and S. Hamani; Existence results for boundary value problems with nonlinear fractional differential equations, \emph{Appl. Anal.} \textbf{87} (7) (2008), 851-863. \bibitem{BeHaNt} M. Benchohra, S. Hamani and S. K. Ntouyas; boundary value problems for differential equations with fractional order, \emph{Surv. Math. Appl.} \textbf{3} (2008), 1-12. \bibitem{BeHeNt1} M. Benchohra, J. Henderson and S. K. Ntouyas; \emph{Impulsive Differential Equations and Inclusions}, Hindawi Publishing Corporation, Vol. 2, New York, 2006. \bibitem{BeHeNtOu1} M. Benchohra, J. Henderson, S. K. Ntouyas and A. Ouahab; Existence results for fractional order functional differential equations with infinite delay, \emph{J. Math. Anal. Appl.} \textbf{338} (2008), 1340-1350. \bibitem{By1} L. Byszewski; Theorems about existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, \emph{J. Math. Anal. Appl.} \textbf{162} (1991), 494-505. \bibitem{By2} L. Byszewski; Existence and uniqueness of mild and classical solutions of semilinear functional-differential evolution nonlocal Cauchy problem. \emph{Selected problems of mathematics}, 25--33, 50th Anniv. Cracow Univ. Technol. Anniv. Issue, 6, Cracow Univ. Technol., Krakow, 1995 \bibitem{ByLa} L. Byszewski and V. Lakshmikantham; Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, \emph{Appl. Anal.} \textbf{40} (1991), 11-19. \bibitem{DaJa1} V. Daftardar-Gejji and H. Jafari; Boundary value problems for fractional diffusion-wave equation. \emph{Aust. J. Math. Anal. Appl.} \textbf{3} (1) (2006), Art. 16, 8 pp. (electronic) \bibitem{DeRo} D. Delbosco and L. Rodino; Existence and uniqueness for a nonlinear fractional differential equation, \emph{J. Math. Anal. Appl.} \textbf{204} (1996), 609-625. \bibitem{DiFr} K. Diethelm and A.D. Freed; On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, in Scientifice Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties'' (F. Keil, W. Mackens, H. Voss, and J. Werther, Eds), pp 217-224, Springer-Verlag, Heidelberg, 1999. \bibitem{DiFo} K. Diethelm and N. J. Ford; Analysis of fractional differential equations, \emph{J. Math. Anal. Appl.} \textbf{265} (2002), 229-248. \bibitem{DiWa} K. Diethelm and G. Walz; Numerical solution of fractional order differential equations by extrapolation, \emph{Numer. Algorithms} \textbf{16} (1997), 231-253. \bibitem{El} A. M. A. El-Sayed; Fractional order evolution equations, \emph{J. Fract. Calc.} \textbf{7} (1995), 89-100. \bibitem{El1} A. M. A. El-Sayed; Fractional order diffusion-wave equations, \emph{Intern. J. Theo. Physics} \textbf{35} (1996), 311-322. \bibitem{El2} A. M. A. El-Sayed; Nonlinear functional differential equations of arbitrary orders, \emph{Nonlinear Anal.} \textbf{33} (1998), 181-186. \bibitem{FuTa} K. M. Furati and N.-e. Tatar; Behavior of solutions for a weighted Cauchy-type fractional differential problem. \emph{J. Fract. Calc.} \textbf{28} (2005), 23--42. \bibitem{FuTa1} K. M. Furati and N.-e. Tatar; An existence result for a nonlocal fractional differential problem. \emph{J. Fract. Calc.} \textbf{26} (2004), 43--51. \bibitem{GaKlKe} L. Gaul, P. Klein and S. Kempfle; Damping description involving fractional operators, \emph{Mech. Systems Signal Processing} \textbf{5} (1991), 81-88. \bibitem{GlNo} W. G. Glockle and T. F. Nonnenmacher; A fractional calculus approach of self-similar protein dynamics, \emph{Biophys. J.} \textbf{68} (1995), 46-53. \bibitem{GrDu} A. Granas and J. Dugundji; \emph{Fixed Point Theory}, Springer-Verlag, New York, 2003. \bibitem{HePo} N. Heymans and I. Podlubny; Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. \emph{Rheologica Acta} \textbf{45} (5) (2006), 765–-772. \bibitem{Hil} R. Hilfer; \emph{Applications of Fractional Calculus in Physics}, World Scientific, Singapore, 2000. \bibitem{KM} E. R. Kaufmann and E. Mboumi; Positive solutions of a boundary value problem for a nonlinear fractional differential equation, \emph{Electron. J. Qual. Theory Differ. Equ.} 2007, No. 3, 11 pp. \bibitem{KiMa} A. A. Kilbas and S. A. Marzan; Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, \emph{Differential Equations} \textbf{41} (2005), 84-89. \bibitem{KST} A. A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo; \emph{Theory and Applications of Fractional Differential Equations}. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006. \bibitem{LBS} V. Lakshmikantham, D. D. Bainov and P. S. Simeonov; \emph{Theory of Impulsive Differntial Equations}, Worlds Scientific, Singapore, 1989. \bibitem{Mai} F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, in Fractals and Fractional Calculus in Continuum Mechanics'' (A. Carpinteri and F. Mainardi, Eds), pp. 291-348, Springer-Verlag, Wien, 1997. \bibitem{MeScKiNo} F. Metzler, W. Schick, H. G. Kilian and T. F. Nonnenmacher, Relaxation in filled polymers: A fractional calculus approach, \emph{J. Chem. Phys.} \textbf{103} (1995), 7180-7186. \bibitem{MiRo} K. S. Miller and B. Ross, \emph{An Introduction to the Fractional Calculus and Differential Equations}, John Wiley, New York, 1993. \bibitem{MoHa} S. M. Momani and S. B. Hadid, Some comparison results for integro-fractional differential inequalities. \emph{J. Fract. Calc.} \textbf{24} (2003), 37-44. \bibitem{MoHaAl} S. M. Momani, S. B. Hadid and Z. M. Alawenh; Some analytical properties of solutions of differential equations of noninteger order, \emph{Int. J. Math. Math. Sci.} \textbf{ 2004} (2004), 697-701. \bibitem{OlPs} K. B. Oldham and J. Spanier; \emph{The Fractional Calculus}, Academic Press, New York, London, 1974. \bibitem{Pod} I. Podlubny, \emph{Fractional Differential Equation}, Academic Press, San Diego, 1999. \bibitem{Pod1} I. Podlubny; Geometric and physical interpretation of fractional integration and fractional differentiation, \emph{ Fract. Calculus Appl. Anal.} \textbf{5} (2002), 367-386. \bibitem{PoPeViLeDo} I. Podlubny, I. Petra\v s, B. M. Vinagre, P. O'Leary and L. Dor\v cak; Analogue realizations of fractional-order controllers. Fractional order calculus and its applications, \emph{Nonlinear Dynam.} \textbf{29} (2002), 281-296. \bibitem{SaKiMa} S. G. Samko, A. A. Kilbas and O. I. Marichev, \emph{Fractional Integrals and Derivatives. Theory and Applications}, Gordon and Breach, Yverdon, 1993. \bibitem{SaPe} A. M. Samoilenko, N. A. Perestyuk; \emph{Impulsive Differential Equations} World Scientific, Singapore, 1995. \bibitem{YuGa} C. Yu and G. Gao; Existence of fractional differential equations, \emph{J. Math. Anal. Appl.} \textbf{310} (2005), 26-29. \bibitem{Zha} S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional diffrential equations, \emph{Electron. J. Differential Equations} vol. 2006 (2006), No. 36, pp, 1-12. \end{thebibliography} \end{document}