\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 80, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/80\hfil Positive solutions] {Positive solutions for fractional differential equations with variable coefficients} \author[Y. Chen\hfil EJDE-2012/80\hfilneg] {Yi Chen, Zhanmei Lv} \address{Yi Chen \newline School of Mathematical Sciences and Computing Technology\\ Central South University\\ Changsha, Hunan 410083, China} \email{mathcyt@163.com} \address{Zhanmei Lv \newline School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, China} \email{cy2008csu@163.com} \thanks{Submitted April 3, 2012. Published May 18, 2012.} \thanks{Supported by grant CX2011B079 from the Hunan Provincial Innovation Foundation \hfill\break\indent for Postgraduate.} \subjclass[2000]{26A33, 34A08, 34A12} \keywords{Fractional differential equations; fixed point theorem; \hfill\break\indent positive solution; multiplicity solution} \begin{abstract} In this article, we study the existence of the positive solutions for a class of differential equations of fractional order with variable coefficients. The equation of this type plays an important role in the description and modeling of control systems, such as $PD^{\mu}$-controller. The differential operator is taken in the Riemann-Liouville sense. Our analysis relies on the Leggett-Williams fixed point theorem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} Fractional calculus is a generalization of the ordinary differentiation and integration. It plays an important role in science, engineering, economy, and other fields, see \cite{d1,h1, k1,l1,m1,m2,o1,p1}. For example, the book \cite{m2} details the use of fractional calculus in the description and modeling of systems, and in a range of control design and practical applications. And today there are many papers dealing with the fractional differential equations due to its various applications, see \cite{a1,a2,b1,b2,b3,l3,n1,s1,s2,s3,z1,z2}. In \cite{m2}, the authors considered the dynamic model of an immersed plate, which is modeled by $$\label{e1} \begin{gathered} A_BD^{2}_{0+}y(t)+B_BD^{1.5}_{0+}y(t)+C_By(t)=f(t),\nonumber\\ y(0)=y'(0)=0. \end{gathered}$$ As indicated in \cite{p1}, a fractional order $PD^{\mu}$-controller can be more suitable for the control of "reality" than integer order. For example, the fractional-order $PD^{\mu}$-controller can be characterized by (see \cite[equation (9.33)]{p1}) $$\label{e2} a_2D^{\beta}_{0+}y(t)+T_dD^{\mu}_{0+}y(t)+a_1D^{\alpha}_{0+}y(t)+(a_0+K)y(t) =Kw(t)+T_dD^{\mu}_{0+}w(t),$$ where $\alpha<\mu<\beta$. And, \eqref{e1} and \eqref{e2} are the particular case of the equation of type \eqref{maineq} in our paper. And for the system of this type, we can find its many other real applications in \cite{o1,p1,r1} and in \cite[Chapter 14-18]{m2}. Problems of this type, with constant coefficients, have provoked some interest in recent literature, such as \cite{b1,s1,s2,s3} and references therein. In \cite{s1}, the author indicated that: Some of the earlier results of this type contains errors in the proof of equivalence of the initial value problems and the corresponding Volterra integral equations (see survey paper by Kilbas and Trujillo \cite{k2})". Motivated by these papers, in this paper, we consider the following initial value problems of fractional differential equations with variable coefficients $$\label{maineq} \begin{gathered} D^{\alpha_n}_{0+}u(t)-\sum_{j=1}^{n-1}a_{j}(t)D^{\alpha_{j}}_{0+}u(t) =f(t,u(t)),\quad 0 \leq t \leq 1,\\ u(0)=u'(0)=0, \end{gathered}$$ where $0< \alpha_1 < \alpha_2 < \dots < \alpha_{n-1}< \alpha_n-1< 1 < \alpha_{n} < 2$, $n \geq 2$, $n \in \mathbb{Z}$, $a_n \in \mathbb{R}$, $f:[0,1]\times [0,+\infty)\to [0,+\infty)$ is continuous and $a_j:[0,1]\to (0,+\infty)$ ($j=1,2,\dots,n-1$) are continuously differentiable. We will study the problem \eqref{maineq} in the Banach space $C[0,1]$ equipped with the maximum norm $\|\cdot\|$. To the best of our knowledge, the results on the existence of solutions for the fractional differential equations with variable coefficients are relatively scare. The variable coefficients cause the problem more complex. The main difficulty in dealing with such issues is that the classical integration by parts formula is no longer applicable for the fractional integration. And how to get the equivalent integral equation of the problem \eqref{maineq} differs from the equations with constant coefficients. In the paper we solve these problems. This article is organized as follows. In Section 2, we present some results of fractional calculus theory and auxiliary technical lemmas, which are used in the next section. Section 3, applying the results of Section 2, we obtain the existence and multiplicity results of the positive solutions for the problem \eqref{maineq} by the Leggett-Williams fixed point theorem in a cone. Then an example is given in Section 4 to demonstrate the application of our results. \section{Preliminaries} First of all, we present the necessary definitions and fundamental facts on the fractional calculus theory. These can be found in \cite{k1,m1,p1}. \begin{definition}[\cite{k1,o1,p1}] \label{def2.1} \rm The Riemann-Liouville fractional integral of order $\nu>0$ of a function $h:(0, \infty)\to \mathbb{R}$ is given by $$I^{\nu}_{0+}h(t)=D^{-\nu}_{0+}h(t)=\frac{1}{\Gamma(\nu)}\int^t_0(t-s)^{\nu-1}h(s)ds$$ provided that the right-hand side is pointwise defined on $(0, \infty)$. \end{definition} \begin{definition}[\cite{k1,o1,p1}]\label{def2.2} \rm The Riemann-Liouville fractional derivative of order $\nu>0$ of a continuous function $h:(0, \infty)\to \mathbb{R}$ is given by $$D^{\nu}_{0+}h(t)=\frac{1}{\Gamma(n-\nu)}\big(\frac{d}{dt}\big)^n \int^t_0(t-s)^{n-\nu-1}h(s)ds,$$ where $n=[\nu]+1$, provided that the right-hand side is pointwise defined on $(0, \infty)$. \end{definition} \begin{lemma}[\cite{b3}]\label{lem2.1} Assume that $h(t) \in C(0, 1)\cap L(0, 1)$ with a fractional derivative of order $\nu>0$ that belongs to $C(0, 1)\cap L(0, 1)$. Then $$I^{\nu}_{0+}D^{\nu}_{0+}h(t)=h(t)+C_1t^{\nu-1}+C_2t^{\nu-2}+ \dots +C_Nt^{\nu-N},$$ for some $C_i \in \mathbb{R}$, $i=1,2,\ldots,N$, where $N$ is the smallest integer such that $N \geq \nu$. \end{lemma} \begin{lemma}[\cite{k1,o1,p1}]\label{lem2.2} If $\nu_1,\nu_2,\nu>0$, $t \in [0,1]$ and $h(t) \in L[0,1]$, then $$I^{\nu_1}_{0+}I^{\nu_2}_{0+}h(t)=I^{\nu_1+\nu_2}_{0+}h(t),\quad D^{\nu}_{0+}I^{\nu}_{0+}h(t)=h(t).$$ \end{lemma} \begin{lemma}[\cite{l3,p1}]\label{lem2.3} If $h(t) \in C[0,1]$ and $\nu>0$, then we have $$\left[I^{\nu}_{0+}h(t) \right]_{t=0}=0,\quad\text{or}\quad \lim_{t\to 0}\frac{1}{\Gamma(\nu)}\int^t_0(t-s)^{\nu-1}h(s)ds=0.$$ \end{lemma} Let \begin{gather}\label{gi} g_j(t,s)=(\alpha_n-1)a_j(s)-(t-s)a_j'(s),\quad (t,s) \in [0,1]\times[0,1],\\ \label{hj} h_j(t,\tau)=\int_0^1{\xi}^{-\alpha_{j}}(1-\xi)^{\alpha_{n}-2}g_j(t,\tau+\xi(t-\tau)) d\xi,\quad (t,\tau) \in [0,1]\times[0,1], \end{gather} where $j=1,2,\dots,n-1$. It is obvious that $g_j(t,s),h_j(t,\tau)$ are continuous and that for $00$, for $j=1,2,\dots,n-1$. \end{lemma} \begin{proof} In view of \eqref{bj}, we have \begin{equation*} a_j(t)=e^{b_j(t)},\ \ {a_j}'(t)={b_j}'(t)e^{b_j(t)},\ \ j=1,2,\dots,n-1. \end{equation*} \par Then, by $(H_1)$, we deduce that \begin{align*} g_j(t,s)&= (\alpha_n-1)a_j(s)-(t-s)a_j'(s)\\ &= (\alpha_n-1)e^{b_j(t)}-(t-s)b_j'(t)e^{b_j(t)}\\ &=e^{b_j(t)} \big((\alpha_n-1)-(t-s)b_j'(t)\big)>0. \end{align*} The proof is complete. \end{proof} For convenience, denote \begin{gather*} M_j=\max_{0\leq t \leq 1,\, 0\leq s \leq 1}g_j(t,s),\quad m_j=\min_{0\leq t \leq 1,\, 0\leq s \leq 1}g_j(t,s),\quad j=1,2,\dots,n-1; \\ P_1=\sum_{j=1}^{n-1}\frac{M_j B(1-\alpha_j,\alpha_n-1)}{(\alpha_n-\alpha_j) \Gamma{(\alpha_n)}\Gamma{(1-\alpha_j)}} =\sum_{j=1}^{n-1}\frac{M_j}{(\alpha_n-1)\Gamma{(\alpha_n-\alpha_j+1)}}; \\ P_2=\sum_{j=1}^{n-1}\frac{m_j B(1-\alpha_j,\alpha_n-1)}{(\alpha_n-\alpha_j) \Gamma{(\alpha_n)}\Gamma{(1-\alpha_j)}} =\sum_{j=1}^{n-1}\frac{m_j}{(\alpha_n-1)\Gamma{(\alpha_n-\alpha_j+1)}}; \end{gather*} We can easily show that $M_j \geq m_j >0$ and $P_1\geq P_2 >0$. Then we have the following lemma. \begin{lemma}\label{lem2.5} Let $f:[0,1]\times [0,+\infty)\to [0,+\infty)$ is continuous and $a_j:[0,1]\to (0,+\infty)\ (j=1,2,\dots,n-1)$ are continuously differentiable, then $$\label{est_h} m_j B(1-\alpha_j,\alpha_n-1)\leq h_j(t,\tau)\leq M_j B(1-\alpha_j,\alpha_n-1),\quad j=1,2,\dots,n-1,$$ where $B(\cdot,\cdot)$ is the Beta function. \end{lemma} \begin{proof} According to \eqref{hj}, for each $j=1,2,\dots,n-1$, we have $h_j(t,\tau) \leq M_j \int_0^1{\xi}^{-\alpha_{j}}(1-\xi)^{\alpha_{n}-2}d\xi = M_j B(1-\alpha_j,\alpha_n-1),\quad (t,\tau) \in [0,1]\times[0,1].$ Analogously, \begin{align*} h_j(t,\tau) \geq m_j B(1-\alpha_j,\alpha_n-1),\ (t,\tau) \in [0,1]\times[0,1]. \end{align*} Then we obtain \eqref{est_h}. \end{proof} \begin{definition} \rm $u(t) \in C[0,1]$ is called a solution of the problem \eqref{maineq} if $u'(t)$ exists in $[0,1]$ and $u(t)$ satisfied the equation and the initial conditions in \eqref{maineq}. \end{definition} \begin{lemma}\label{lem1} Let $f:[0,1]\times [0,+\infty)\to [0,+\infty)$ is continuous and $a_j:[0,1]\to (0,+\infty)$ $(j=1,2,\dots,n-1)$ are continuously differentiable, then $u(t)$ is a solution of the equation \eqref{maineq} if and only if $u(t)\in C[0,1]$ is the solution of the integral equation $$\label{equivlentequation} \begin{split} u(t)&=\sum_{j=1}^{n-1}\frac{1}{\Gamma({\alpha_{n}})\Gamma({1-\alpha_{j}})} \int_0^t(t-\tau)^{\alpha_{n}-\alpha_{j}-1}u(\tau)h_j(t,\tau) d\tau\\ &\quad + \frac{1}{\Gamma({\alpha_{n}})} \int_0^t(t-\tau)^{\alpha_{n}-1}f(\tau,u(\tau))d\tau \end{split}$$ \end{lemma} \begin{proof} Necessity''. Applying Lemma \ref{lem2.1} and the initial conditions, we have $$\label{prooflem1} u(t)= \lambda_1\sum_{j=1}^{n-1}I_{0+}^{\alpha_n} \left(a_j(t)D_{0+}^{\alpha_j}u(t)\right)+\lambda_2I_{0+}^{\alpha_n}f(t,u(t)).$$ Combining Definition \ref{def2.2} and Lemma \ref{lem2.3}, we have \begin{align*} &I_{0+}^{\alpha_n}\big(a_j(t)D_{0+}^{\alpha_j}u(t)\big)\\ &=\frac{1}{\Gamma{(\alpha_n)}}\int_0^t(t-s)^{\alpha_n-1}a_j(s) \Big(\frac{d}{ds}\frac{1}{\Gamma{(1-\alpha_j)}}\int_0^ s(s-\tau)^{-\alpha_j}u(\tau)d\tau\Big)ds\\ & =\frac{1}{\Gamma{(\alpha_n)}\Gamma{(1-\alpha_j)}} \Big((t-s)^{\alpha_n-1}a_j(s)\int_0^s(s-\tau)^{-\alpha_j}u(\tau)d\tau\Big) \Big|_{s=0}^{s=t}\\ &\quad -\frac{1}{\Gamma{(\alpha_n)}\Gamma{(1-\alpha_j)}} \int_0^t\int_0^s\frac{d}{ds}\Big((t-s)^{\alpha_n-1}a_j(s)\Big) (s-\tau)^{-\alpha_j}u(\tau)d\tau ds\\ & =\frac{1}{\Gamma{(\alpha_n)}\Gamma{(1-\alpha_j)}} \int_0^t\int_0^s(t-s)^{\alpha_n-2}g_j(t,s)(s-\tau)^{-\alpha_j}u(\tau)d\tau ds\\ & =\frac{1}{\Gamma{(\alpha_n)}\Gamma{(1-\alpha_j)}} \int_0^t\int_{\tau}^t(t-s)^{\alpha_n-2}g_j(t,s)(s-\tau)^{-\alpha_j}u(\tau) ds\,d\tau\\ &=\frac{1}{\Gamma{(\alpha_n)}\Gamma{(1-\alpha_j)}} \int_0^t(t-\tau)^{\alpha_{n}-\alpha_{j}-1}u(\tau) \int_0^1{\xi}^{-\alpha_{j}}(1-\xi)^{\alpha_{n}-2}\\ &\quad\times g_j(t,\tau+\xi(t-\tau))d\xi d\tau\\ &=\frac{1}{\Gamma({\alpha_{n}})\Gamma({1-\alpha_{j}})} \int_0^t(t-\tau)^{\alpha_{n}-\alpha_{j}-1}u(\tau)h_j(t,\tau)d\tau, \end{align*} where $j=1,2,\dots,n-1$. Thus, in view of \eqref{prooflem1}, we derive \eqref{equivlentequation}. Sufficiency''. Suppose that $u(t)\in C[0,1]$ is the solution of \eqref{equivlentequation}. Then we canshow that $u'(t)$ exists in $[0,1]$ by \eqref{equivlentequation}. Also by exploiting Lemma \ref{lem2.2} and Lemma \ref{lem2.3}, we can deduce the equation \eqref{maineq} easily and prove that the initial conditions are satisfied. This completes the proof. \end{proof} \begin{theorem}[\cite{l2}]\label{thm3} Let $(E,\|\cdot\|)$ be a Banach space, $P\subset E$ be a cone of $E$ and $c>0$ be a constant. Suppose that there exists a concave nonnegative continuous functional $\omega$ on $P$ with $\omega(x) \leq \|x\|$ for all $x \in \overline{P}_c$. Let $B: \overline{P}_c \to \overline{P}_c$ be a completely continuous operator. Assume there are numbers $a, b$ and $d$ with $0a\} \neq \emptyset$ and $\omega(Bx)>a$ for all $x\in P(\omega,a,b)$; \item[(2)] $\|Bx\|a$ for all $x\in P(\omega, a, c)$ with $\|Bx\|>b$. \end{itemize} Then $B$ has at least three fixed points $x_1,\ x_2$ and $\ x_3$ in $\overline{P}_c$. Furthermore, $x_1 \in P_d$, $\ x_2 \in \{x \in P(\omega,a,c): \omega(x)>a\};$ $x_3 \in \overline{P}_c\backslash (P(\omega,a,c)\cup\overline{P}_d)$. \end{theorem} \section{Existence and multiplicity of positive solutions} Let $$K=\big\{x \in C[0,1]: x(t)\geq0,\, t \in [0,1], \min_{t \in [0,l_1]}x(t)\geq L \|x\|\big\},$$ where $0< l_1 <1$ and $00$, there exists a $\delta_j>0\ (\delta_j<1)$ such that $$|h_j(t_1,\tau_1)-h_j(t_2,\tau_2)|\leq \frac{\varepsilon}{3\gamma_2},$$ for all $(t_1,\tau_1),(t_2,\tau_2)\in[0,1]\times[0,1]$ with $|t_1-t_2|\leq\delta_j$ and $|\tau_1-\tau_2|\leq\delta_j$, $j=1,2,\dots,n-1$. Next prove that $A(U)$ is equicontinuous. For the given $\varepsilon>0$, there exists $\rho_j>0(1\leq j\leq n-1)$ such that $|{t_2}^{\alpha_n-\alpha_j}-{t_1}^{\alpha_n-\alpha_j}|<\varepsilon/(3\gamma_1)$, where $|t_2-t_1|<\rho_j$. Let $$\delta=\min\big\{\delta_1,\ \delta_2,\ \dots,\ \delta_{n-1},\ \rho_1,\ \rho_{2}, \dots, \rho_{n-1},(\frac{\varepsilon}{3})^{1/{(\alpha_n-\alpha_{n-1})}}\big\}.$$ For each $u \in U$, $t_1,t_2 \in [0,1]$ with $|t_1-t_2|\leq \delta (t_1 \frac{\Gamma{(\alpha_n+1)}(1-C_1P_2L)}{{l_1^{\alpha_n}}}; \end{gather*} \item[(H4)]$P_1<1$and$f(t,u)< \Gamma{(\alpha_n+1)}(1-P_1)d$, for all$(t,u)\in[0,1]\times[0,d]$. \end{itemize} Then problem \eqref{maineq} has at least three positive solutions$u_1, u_2$and$ u_3$in$\overline{K}_b$. Furthermore,$u_1 \in K_d$;$u_2 \in \{u \in K(\omega,a,b): \omega(u)>a\}$;$u_3 \in \overline{K}_b\backslash (K(\omega,a,b)\cup\overline{K}_d)$. \end{theorem} \begin{proof} From Section 2, we know that$K_d=\{u \in K: \|u\|a\} \neq \emptyset$and$\omega(Tu)>a$for all$u\in K(\omega,a,b)$. Take$u_0(t)=(a+b)/2$, then$\omega(u_0)=\min_{t \in [l_1,\,l_2]}u_0(t)=(a+b)/2>a$and$\|u_0\|=(a+b)/2a\} \neq \emptyset$. For each$u\in K(\omega,a,b)$, applying condition$(H_3)$, the definition of$Kand Lemma \ref{lem2.5}, we can show \begin{align*} \omega(T u)&= \min_{t \in [l_1,\,l_2]} \Big(\sum_{j=1}^{n-1}\frac{1}{\Gamma({\alpha_{n}}) \Gamma({1-\alpha_{j}})}\int_0^t(t-\tau)^{\alpha_{n}-\alpha_{j}-1} u(\tau)h_j(t,\tau)d\tau \\ &\quad + \frac{1}{\Gamma({\alpha_{n}})}\int_0^t(t-\tau)^{\alpha_{n}-1} f(\tau,u(\tau))d\tau\Big)\\ &\geq \min_{t \in [l_1,\,l_2]} \Big(\sum_{j=1}^{n-1}\frac{1}{\Gamma({\alpha_{n}})\Gamma({1-\alpha_{j}})} \int_0^t(t-\tau)^{\alpha_{n}-\alpha_{j}-1}u(\tau)h_j(t,\tau)d\tau\Big)\\ &\quad +\min_{t \in [l_1,\,l_2]} \Big(\frac{1}{\Gamma({\alpha_{n}})} \int_0^t(t-\tau)^{\alpha_{n}-1}f(\tau,u(\tau))d\tau\Big)\\ &\geq \min_{t \in [l_1,\,l_2]}\Big(\sum_{j=1}^{n-1} \frac{m_j B(1-\alpha_j,\alpha_n-1)}{\Gamma({\alpha_{n}}) \Gamma({1-\alpha_{j}})}\int_0^{l_1}(t-\tau)^{\alpha_{n}-\alpha_{j}-1} u(\tau)d\tau\Big)\\ &\quad + \frac{1}{\Gamma({\alpha_{n}})}\int_0^{l_1}(l_1-\tau)^{\alpha_{n}-1} f(\tau,u(\tau))d\tau\\ &\geq \min_{t \in [l_1,\,l_2]}\Big(\sum_{j=1}^{n-1} \frac{m_j B(1-\alpha_j,\alpha_n-1)}{(\alpha_{n}-\alpha_{j}) \Gamma({\alpha_{n}})\Gamma({1-\alpha_{j}})}La [t^{\alpha_n-\alpha_j}-(t-l_1)^{\alpha_n-\alpha_j}]\Big)\\ &\quad + \frac{1}{\Gamma({\alpha_{n}})}\int_0^{l_1}(l_1-\tau)^{\alpha_{n}-1} C_2ad\tau\\ &\geq C_1P_2La+\frac{C_2a}{\Gamma({\alpha_{n}+1})}l_1^{\alpha_{n}}\\ &>C_1P_2La +{(1-C_1P_2L)a}=a, \end{align*} which implies\omega(T u)>a$. Step 3:$\|Tx\|0$, such that$f(t,u) \leq \sigma u$, for all$u \geq r_1$. Let$\beta=\max_{0\leq t \leq r_1}u(t)$, then $$0\leq f(t,u) \leq \sigma u+\beta,\quad 0\leq u<+\infty.$$ Let$b>\max\{{\beta}/\big(\Gamma{(\alpha_n+1)}(1-P_1)-\sigma\big),\ a\}$. Combining this with condition (H3'), we obtain the conditions (H2) and (H3). Therefore, the conclusion of Theorem \ref{thm4.1} holds. \end{proof} \begin{corollary}\label{cor4.2} If the condition {\rm (H4)} in Theorem \ref{thm4.1} is replaced by \begin{itemize} \item[(H4')]$P_1<1$and$ \limsup_{u \to 0^+}\max_{t\in[0,1]}\frac{f(t,u)}{u}<\Gamma{(\alpha_n+1)}(1-P_1)$. \end{itemize} Then the conclusion of Theorem \ref{thm4.1} holds. \end{corollary} \section{Examples} To illustrate our main results, we present an example: $$\label{e4.1} \begin{gathered} D_{0+}^{1.5}u(t)-a_3(t)D_{0+}^{0.3}u(t)-a_2(t) D_{0+}^{0.2}u(t)-a_1(t)D_{0+}^{0.1}u(t) =f(t,u(t)),\\ u(0)=u'(0)=0, \quad 0\leq t\leq1, \end{gathered}$$ where \begin{gather*} a_1(t)=\ln (t^2+{4}),\quad a_2(t)=\frac{1}{8}(\sin t+1), \quad a_3(t)=\frac{1}{12}\big(\frac{t^2}{t^2+1}+1\big),\\ f(t,u)=(t^2+1)u+\beta, \quad \beta >0. \end{gather*} Note that $$0\leq {b_1}'(t) <\frac{1}{2\ln{4}}<0.5,\quad 0\leq {b_2}'(t) <0.5,\quad 0\leq {b_3}'(t)\leq \frac{1}{6}<0.5.$$ A simple computation shows that$P_1\approx 0.5146$. Let$l_1=0.98$,$L=0.95$,$d=2\beta$,$a=20\beta/9$,$b=4\beta$,$C_1<1$, and$C_2=1.4\$. It is easy to check that all the hypotheses in Theorem \ref{thm4.1} are satisfied. Thus, we conclude that problem \eqref{e4.1} has at least three positive solutions. \begin{thebibliography}{00} \bibitem{a1} Ravi P. Agarwal, Donal O'Regan, Svatoslav Stan\v{e}k; Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations, \emph{J. Math. Anal. Appl.} \textbf{371} (2010) 57-68. \bibitem{a2} Bashir Ahmad, Juan J. Nieto; Existence of solutions for nonlocal boundary value problems of higher-Order nonlinear fractional differential equations, \emph{Bound. Value Probl.} \textbf{2009} (2009), p.\,11 Article ID 708576. \bibitem{b1} A. Babakhani and V. Daftardar-Gejji; Existence of positive solutions of nonlinear fractional differential equations, \emph{J. Math. Anal. Appl.} \textbf{278} (2003) 434-442. \bibitem{b2} Zhanbing Bai; On positive solutions of a nonlocal fractional boundary value problem, \emph{Nonlinear Anal.} \textbf{72} (2010) 916-924. \bibitem{b3} Z. Bai, H. L\"{u}l; Positive solutions for boundary value problem of nonlinear fractional differential equation, \emph{J. Math. Anal. Appl.} \textbf{311} (2005) 495-505. \bibitem{d1} Kai Diethelm; The Analysis of Fractional Differential Equations, Springer, Berlin, 2010. \bibitem{h1} R. Hilfer; Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, 2000. \bibitem{k1} A. A. Kilbas, H. M. Srivastava, J. J Trujillo; Theory and Applications of Fractional Differential Equations, Elsevier B. V., Amsterdam, 2006. \bibitem{k2} A. A. Kilbas, J. J. Trujillo; Differential equations of fractional order: methods, results and problems. \emph{I. Appl. Anal.} \textbf{78} (2001) 153-192. \bibitem{l1} V. Lakshmikantham, S. Leela, J. Vasundhara Devi; Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, 2009. \bibitem{l2} R. W. Leggett, L. R. Williams; Multiple positive fixed-points of nonlinear operators on ordered Banach spaces, \emph{Indiana Univ. Math. J. } \textbf{28} (1979) 673-688. \bibitem{l3} C. P. Li, W. H. Deng; Remarks on fractional derivatives, \emph{Appl. Math. Comput.} \textbf{187} (2007) 777-784. \bibitem{m1} K. S. Miller, B. Ross; An introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. \bibitem{m2} Concepci\'on A. Monje, Yangquan Chen, Blas M. Vinagre, Dingy\"u Xue, Vicente Feliu; Fractional-order Systems and Controls: Fundamentals and Applications (Advances in Industrial Control), Springer-Verlag, London, 2010. \bibitem{n1} Juan J. Nieto; Maximum principles for fractional differential equations derived from Mittag-Leffler functions, \emph{Appl. Math. Lett.} \textbf{23} (2010) 1248-1251. \bibitem{o1} Manuel Duarte Origueira, Fractional Calculus for Scientists and Engineers, Springer, Berlin, 2011. \bibitem{p1} I. Podlubny; Fractional Differential Equations, Academic Press, San Diego, 1999. \bibitem{r1} Yu. Rossikhin, M. V. Shitikova; Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, \emph{Appl.Mech. Rev.} \textbf{50} (1997) 15-67. \bibitem{s1} Hussein A. H. Salem; Global monotonic solutions of multi term fractional differential equations, \emph{Appl. Math. Comput. } {\bf217} (2011) 6597-6603. \bibitem{s2} Mirjana Stojanovi\'c; Existence-uniqueness result for a nonlinear n-terms fractional equation, \emph{J. Math. Anal. Appl.} \textbf{353} (2009) 244-255. \bibitem{s3} Jiangping Sun, Yahong Zhao; Multiplicity of positive solutions of a class of nonlinear fractional differential equations, \emph{Comput. Math. Appl.} \textbf{49} (2005) 73-80. \bibitem{z1} Shuqin Zhang; Monotone iterative method for initial value problem involving Riemann-Liouville fractional derivatives, \emph{Nonlinear Anal.} \textbf{71} (2009) 2083-2097. \bibitem{z2} S. Zhang; Positive solutions for boundary-value problems of nonlinear fractional differential equations, \emph{Electron. J. Differential Equations} \textbf{36} (2006) 1-12. \end{thebibliography} \end{document}