\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 85, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{7mm}} \begin{document} \title[\hfilneg EJDE-2018/85\hfil Fractional boundary value problems?] {Superlinear fractional boundary value problems without the Ambrosetti-Rabinowitz condition} \author[B. Ge, J. F. Lu, T. T. Zhao, K. Zhou \hfil EJDE-2018/85\hfilneg] {Bin Ge, Jian-Fang Lu, Ting-Ting Zhao, Kang Zhou} \address{Bin Ge (corresponding author) \newline Department of Applied Mathematics, Harbin Engineering University, China} \email{gebin791025@hrbeu.edu.cn} \address{Jian-Fang Lu \newline Department of Applied Mathematics, Harbin Engineering University, China} \email{1176678630@qq.com} \address{Ting-Ting Zhao \newline Department of Applied Mathematics, Harbin Engineering University, China} \email{1659944641@qq.com} \address{Kang Zhou \newline Department of Applied Mathematics, Harbin Engineering University, China} \email{992983801@qq.com} \dedicatory{Communicated by Vicentiu D. Radulescu} \thanks{Submitted February 23, 2017. Published April 10, 2018.} \subjclass[2010]{35J20, 35J70, 35R20} \keywords{Fractional differential equations; pseudo-gradient; \hfill\break\indent nontrivial solutions; critical point} \begin{abstract} In this article, by means of a direct variational approach and the theory of the fractional differential space, we prove the existence of a nontrivial solution for superlinear fractional boundary value problems without Ambrosetti and Rabinowitz condition. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} Fractional differential equations have recently been proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, neurons, electrochemistry, control, porous media, electromagnetism, etc., (see \cite{1,3,4,6,2,5}). Recently, the study of various mathematical problems with the existence of solutions of various BVP of fractional differential equations has been received considerable attention, we refer the reader to \cite{7,8,9,10,11,12,17,14,15,16,13} for an overview of and references on this subject. In this article we are concerned with the existence of a nontrivial solution for the following Dirichlet problem of fractional order differential equation \begin{equation} \label{eP} \begin{gathered} \frac{d}{dt}\Big( \frac{1}{2}{}_0D_t^{-\beta}(u'(t)) +\frac{1}{2}{}_0D_T^{-\beta}(u'(t))\Big) +\lambda\nabla F(t,u(t))=0,\quad\text{a.a. } t\in[0,T], \\ u(0)=u(T)=0, \end{gathered} \end{equation} where ${_0}D_t^{-\beta}$ and ${_0}D_T^{-\beta}$ are the left and right Riemann-Liouville fractional integrals of order $0\leq\beta<1$, respectively, $\lambda>0$ is a real number, $F:[0,T]\times {\mathbb{R}^{N}}\to \mathbb{R}$ is a given function and $\nabla F(t,x)$ is the gradient of $F$ at $x$. Such a type of behaviour occurs, for example, when $\lambda=1$, in this case \eqref{eP} becomes \begin{equation} \label{eP*} \begin{gathered} \frac{d}{dt}\Big( \frac{1}{2}{}_0D_t^{-\beta}(u'(t)) +\frac{1}{2}{}_0D_T^{-\beta}(u'(t))\Big)+\nabla F(t,u(t))=0, \quad \text{a.a. }t\in[0,T], \\ u(0)=u(T)=0. \end{gathered} \end{equation} There have been many works about the existence of nontrivial solutions to \eqref{eP*} by using variational methods. Jiao and Zhou \cite{21} obtained the existence of solutions for \eqref{eP*} by the Mountain Pass theorem under the Ambrosetti-Rabinowitz (AR) condition. Chen and Tang \cite{22} studied the existence and multiplicity of solutions for the system \eqref{eP*} when the nonlinearity $F(t,\cdot)$ are superquadratic, asymptotically quadratic, and subquadratic, respectively. For more recent results, we can refer to \cite{23,24,25} and the references therein. It is well known, the (AR) condition is quite important not only to ensure that the Euler-lagrange functional associated to problem \eqref{eP*} has a mountain pass geometry, but also to guarantee that Palais-Smale sequence of the Euler-Lagrange functional is bounded. However, this condition is very restrictive and eliminates many interesting and important nonlinearities. Motivated by the works described above, we try to get the existence of a nontrivial solution for problem \eqref{eP} without (AR) condition. To state our main result, we assume that $F(t, x)$ satisfies the following general conditions: \begin{itemize} \item[(A1)] $F(t,0)=0$, $\lim_{|x|\to 0}\frac{F(t,x)}{|x|^2}=0$ uniformly a.e. $t\in[0,T]$. \item[(A2)] There are positive constant positive constants $a$, $b$ and $p>1$ such that $$|\nabla F(t,x)|\leq a+b|x|^{p},\; {\rm for}\;{\rm a.e.}\;t\in [0,T],\; {\rm all}\; x\in{\mathbb{R}^{N}}. $$ \item[(A3)] $\lim_{|x|\to \infty}\frac{F(t,x)}{|x|^2}=+\infty$ uniformly a.e. $t\in[0,T]$. \item[(A4)] There exists a constant $C_{\ast}>0$ such that $$ H(t,y)\leq H(t,x)+C_{\ast}, $$ for any $t\in[0,T]$, $0<|y|<|x|$ or $0<|x|<|y|$, where $H(t,x)=(x,\nabla F(t,x))-2F(t,x)$. \end{itemize} This article is organized as follows. In Sect.2 we introduce the fractional differential space setting that we adopt throughout the paper. In Sect.3 we give the main result and its proof. \section{Preliminary results} In this section, we introduce some basic definitions and properties of the fractional calculus which are used further in this paper. For the proofs, which are omitted, we refer the reader to \cite{21,23,22,18,19,20} or other texts on basic fractional calculus. \begin{definition}[\cite{18})] \label{def2.1} \rm Let $g(t)$ be a function defined on $[a,b]$ and $\mu>0$. The left and right Riemann-Liouville fractional integrals of order $\mu$ for function $g(t)$ denoted by ${}_aD_t^{-\mu}g(t)$ and ${}_tD_b^{-\mu}g(t)$, respectively, are defined by \begin{gather*} {}_aD_t^{-\mu}g(t)=\frac{1}{\Gamma(\mu)} \int_a^{t}(t-s)^{\mu-1}g(s)ds,\quad t\in[a,b], \\ {}_tD_b^{-\mu}g(t)=\frac{1}{\Gamma(\mu)}\int_t^{b}(t-s)^{\mu-1}g(s)ds,\quad t\in[a,b], \end{gather*} where $\Gamma(\mu)=\int_0^{\infty}t^{\mu-1}e^{-t}dt$. \end{definition} \begin{definition}[\cite{18}] \label{def2.2} \rm Let $g(t)$ be a function defined on $[a,b]$. The left and right Riemann-Liouville fractional derivatives of order $\mu$ for function $g(t)$ denoted by ${}_aD_t^{\mu}g(t)$ and ${}_tD_b^{\mu}g(t)$, respectively, are defined by \begin{gather*} {}_aD_t^{\mu}g(t)=\frac{d^{n}}{dt^{n}}\,_aD_t^{\mu-n}g(t) =\frac{1}{\Gamma(n-\mu)}\frac{d^{n}}{dt^{n}} \Big(\int_a^{t}(t-s)^{n-\mu-1}g(s)ds\Big),\\ {}_tD_b^{\mu}g(t)=(-1)^{n}\frac{d^{n}}{dt^{n}}\,_tD_b^{\mu-n}g(t) =\frac{1}{\Gamma(n-\mu)}\frac{d^{n}}{dt^{n}} \Big(\int_t^{b}(t-s)^{n-\mu-1}g(s)ds\Big), \end{gather*} where $t\in[a,b]$, $ n-1\leq\mu1/p$ and $\frac{1}{p}+\frac{1}{q}=1$, then \begin{equation}\label{2.2} \|u\|_{\infty}\leq\frac{T^{\frac{\alpha-1}{p}}}{\Gamma(\alpha) \Big((\alpha+1)q+1\Big)^{\frac{1}{q}}}\|^c_0D_t^{\alpha}u\|_p. \end{equation} \end{proposition} According to \eqref{2.1}, one can consider $E_0^{\alpha,p}$ with respect to the norm \begin{equation}\label{2.3} \|u\|_{\alpha,p}=\|^c_0D_t^{\alpha}u\|_p =\Big(\int_0^{T}|^c_0D_t^{\alpha}u|^{p}dt\Big)^{1/p}. \end{equation} \begin{proposition}[\cite{22}] \label{pro2.3} Define $0<\alpha\leq1$ and $1\frac{1}{p}$, and the sequence $u_{k}$ converges weakly to $u\in E_0^{\alpha,p}$, that is, $u_{k}\rightharpoonup u$. Then $u_{k}\to u$ in $C([0,T],\mathbb{R}^N)$, that is, $\|u_{k}-u\|_{\infty}\to 0$, as $k\to\infty$. \end{proposition} By Definition \ref{def2.3}, for any $u \in AC([0,T],\mathbb{R}^N)$, problem \eqref{eP} is equivalent to the problem \begin{equation} \label{eP1} \begin{gathered} \frac{d}{dt}\Big( \frac{1}{2}{}_0D_t^{\alpha-1}(_0^c D_t^\alpha u(t)) -\frac{1}{2}\,{_t}D_T^{\alpha-1}(_t^c D_T^\alpha u(t))\Big) +\lambda\nabla F(t,u(t))=0,\\\ \text{ a.e. } t\in[0,T], \\ u(0)=u(T)=0, \end{gathered} \end{equation} where $\alpha=1-\frac{\beta}{2}\in(\frac{1}{2},1]$. In the following, we will treat problem \eqref{eP1} in the Hilbert space $E^{\alpha}=E_0^{\alpha,2}$ and corresponding norm $\|u\|_{\alpha}=\|u\|_{\alpha,2}$. It follows from that the functional $\phi:E^{\alpha}\to \mathbb{R}$ given by \begin{equation}\label{2.4} \phi_{\lambda}(u)=-\frac{1}{2}\int_0^{T}(^c_0D_t^{\alpha}u(t) ,^c_tD_{T}^{\alpha}u(t))dt-\lambda\int_0^{T}F(t,u(t))dt,\quad u\in E^{\alpha}, \end{equation} is continuously differentiable on $E^{\alpha}$. Moreover, for $u,v\in E^{\alpha}$, we have \begin{equation}\label{2.5} \begin{aligned} \langle\phi'_{\lambda}(u),v\rangle =&-\frac{1}{2}\int_0^{T}[(^c_0D_t^{\alpha}u(t),^c_tD_{T}^{\alpha}v(t)) +(^c_tD_{T}^{\alpha}u(t),^c_0D_t^{\alpha}v(t))]dt\\ &-\lambda\int_0^{T}(\nabla F (t,u(t)),v(t))dt. \end{aligned}\end{equation} \begin{proposition}[\cite{22}] \label{pro2.4} A function $u\in A C([0,T],\mathbb{R}^N)$ is a solution of \eqref{eP1} if \begin{itemize} \item[(i)] $D^{\alpha}(u(t))$ is derivative for almost every $t\in [0,T]$, \item[(ii)] $u$ satisfies \eqref{eP1}, where $D^{\alpha}(u(t))=\frac{1}{2}_0D_t^{\alpha-1} (^c_0D_t^{\alpha}u(t))-\frac{1}{2}_tD_{T}^{\alpha-1}(^c_tD_{T}^{\alpha}u(t))$. \end{itemize} \end{proposition} \begin{proposition}[\cite{22}] \label{pro2.5} If $\frac{1}{2}<\alpha\leq1$ then for any $u\in E^{\alpha}$, one has \begin{equation}\label{2.6} |\cos(\pi\alpha)|\|u\|_{\alpha}^2 \leq-\int_0^{T}(^c_0D_t^{\alpha}u(t),^c_tD_{T}^{\alpha}u(t))dt \leq \frac{1}{|\cos(\pi\alpha)|}\|u\|_{\alpha}^2. \end{equation} \end{proposition} \begin{proposition}[\cite{22}] \label{pro2.6} Let $\frac{1}{2}<\alpha\leq1$ and $\phi_{\lambda}$ be defined by \eqref{2.4}. If $u\in E^{\alpha}$ is a solution of $\phi'_{\lambda}(u)=0$, then $u$ is a solution of problem \eqref{eP1} which satisfies the problem \eqref{eP}. \end{proposition} \section{Main result and its proof} \begin{lemma} Suppose that {\rm (A1)--(A4)} holds. Then we have the following assertions: \begin{itemize} \item[(a)] $\phi_{\lambda}$ is unbounded from below on $E^{\alpha}$; \item[(b)] $u=0$ is a strict local minimum for the functional $\phi_{\lambda}$. \end{itemize} \end{lemma} \begin{proof} By (A3), for any $M>0$, there exists $K_0>0$, such that \begin{equation}\label{3.1} F(t,x)\geq M |x|^2,\quad \text{for all } |x|\geq K_0,\text{ a.e. }t\in[0,T]. \end{equation} On the other hand, by the mean value theorem and (A2), we obtain \begin{equation}\label{3.2} \begin{aligned} |F(t,x)|=&|(\nabla F(t,\lambda_0x),x)|\\ \leq &|\nabla F(t,\lambda_0x)|\cdot |x|\\ \leq & (a+b\lambda_0^{p}|x|^{p})\cdot|x|\\ \leq &(a+b|x|^{p})\cdot|x|\\ \leq & a K_0+b K_0^{p+1} =:C_{M}, \end{aligned}\end{equation} for for some $\lambda_0\in(0,1), |x|\leq K_0$ and a.e. $t\in[0,T]$. Hence, for any $M>0$, there exists $C_{M}>0$, such that \begin{equation}\label{3.3} F(t,x)\geq M_0 |x|^2-C_{M_0},\quad \text{a.e. } t\in[0,T], \text{ all } x\in\mathbb{R}^N, \end{equation} where $M_0=M+\frac{C_{M}}{K_0^2}$. Choosing $u_0=(0,\dots ,0,\frac{T}{\pi}\sin(\frac{\pi t}{T}))\in E^{\alpha}$, then \begin{equation}\label{3.4} \|u_0\|^2_{2}=\int_0^{T}|u_0|^2dt=\frac{T^{3}}{2\pi^2}\quad\text{and}\quad \|u_0\|_{\alpha}^2\leq \frac{T^{3-2\alpha}}{\Gamma^2(2-\alpha)(3-2\alpha)}. \end{equation} For $\eta>0$, and noting that \eqref{3.3} and \eqref{3.4}, we have \begin{equation}\label{3.5} \begin{aligned} \phi_{\lambda}(\eta u_0 ) =&-\frac{1}{2}\int_0^{T}(^c_0D_t^{\alpha}\eta u_0(t),^c_tD_{T}^{\alpha} \eta u_0(t))dt-\lambda\int_0^{T}F(t,\eta u_0(t))dt\\ \leq& \frac{\eta^2}{2|\cos(\pi\alpha)|}\|u_0\|_{\alpha}^2 -\lambda \int_0^{T}(M_0 \eta ^2|u_0|^2-C_{M_0})dt\\ = &\frac{\eta^2}{2|\cos(\pi\alpha)|}\|u_0\|_{\alpha}^2 -\lambda M_0 \eta ^2\int_0^{T}|u_0|^2dt+\lambda C_{M_0}T\\ \leq&\frac{\eta^2}{2|\cos(\pi\alpha)|}\cdot \frac{T^{3-2\alpha}}{\Gamma^2(2-\alpha)(3-2\alpha)} -\lambda M_0 \eta ^2\cdot \frac{T^{3}}{2\pi^2}+\lambda C_{M_0}T\\ =&\Big(\frac{1}{2|\cos(\pi\alpha)|}\cdot \frac{T^{3-2\alpha}}{\Gamma^2(2-\alpha)(3-2\alpha)} -\lambda M_0\frac{T^{3}}{2\pi^2}\Big)\eta^2+\lambda C_{M_0}T. \end{aligned} \end{equation} If $M_0$ is large enough so that \begin{equation} \label{3.6} \frac{1}{2|\cos(\pi\alpha)|} \frac{T^{3-2\alpha}}{\Gamma^2(2-\alpha)(3-2\alpha)}-\lambda M_0\frac{T^{3}}{2\pi^2} <0, \end{equation} then \begin{equation}\label{3.7} \phi_{\lambda}(\eta u_0 )\to -\infty,\quad\text{as } \eta\to+\infty. \end{equation} This proves (a). By (A1), for any $\varepsilon >0$, there exists $\delta(\varepsilon)>0$, such that \begin{equation}\label{3.8} |F(t,x)|<|x|^2\varepsilon,\quad |x|<\delta. \end{equation} Analogously, by the mean value theorem and (A2), we have \begin{equation}\label{3.9} \begin{aligned} |F(t,x)|\leq&(a+b|x|^{p})|x|\\ =&|x|^{p+1}(a\cdot \frac{1}{|x|^{p}}+b)\\ \leq& |x|^{p+1}( \frac{a}{\delta^{p}}+b)\\ =&C_{\varepsilon}|x|^{p+1}, \end{aligned} \end{equation} where $|x|\geq \delta, t\in[0,T],10$ such that \begin{equation*} \|u\|_{p+1}\leq c\|u\|_{\alpha},\quad \forall u\in E^{\alpha}, \end{equation*} which implies \begin{equation}\label{3.14} \begin{aligned} \phi_{\lambda}(u) \geq & \frac{|\cos(\pi\alpha)|}{2}\|u\|_{\alpha}^2 -\frac{\lambda\varepsilon T^{2\alpha}}{\Gamma^2(\alpha+1)} \|u\|_{\alpha}^2-c_0\|u\|_{\alpha}^{p+1}\\ =&\Big(\frac{|\cos(\pi\alpha)|}{2} -\frac{\lambda\varepsilon T^{2\alpha}}{\Gamma^2(\alpha+1)} \Big)\|u\|_{\alpha}^2-c_0\|u\|_{\alpha}^{p+1}. \end{aligned} \end{equation} For a given $\lambda$, choose that $\varepsilon=\varepsilon(\lambda)$ satisfies $\varepsilon<|\cos(\pi\alpha)|\Gamma^2(\alpha+1)/(2\lambda T^{2\alpha})$, then we have \begin{equation}\label{3.15} \phi_{\lambda}(u)>0,\quad \|u\|_\alpha<\rho \end{equation} for some $\rho\in(0,1)$. So $u=0$ is a strict local minimum for $\phi_{\lambda}$. \end{proof} \begin{lemma} Assume {\rm (A1)--(A3)} hold and $0<\lambda_0<\mu_0, \lambda_0\leq \lambda \leq \mu_0$, $c_{\lambda}=\inf_{\gamma\in P}\max_{z\in[0,1]}\phi_{\lambda}(\gamma(z))$, then $c_{\lambda}/\lambda$ is monotone decreasing and is left semi-con\-tinuous. \end{lemma} \begin{proof} By choosing $\varepsilon>0$, such that $$ \Big(\frac{|\cos(\pi\alpha)|}{2}-\frac{\lambda\varepsilon T^{2\alpha}}{\Gamma^2(\alpha+1)}\Big)\geq \frac{|\cos(\pi\alpha)|}{4}. $$ By \eqref{3.14}, we have \begin{equation}\label{3.16} \phi_{\lambda}(u)\geq\frac{|\cos(\pi\alpha)|}{4}\|u\|_{\alpha}^2 -c_0\|u\|_{\alpha}^{p+1},\quad \forall u\in E^{\alpha},\; 0<\lambda_0<\mu_0, \; c_0>0. \end{equation} That is, there exist $\rho>0$ and $R>0$, such that \begin{equation}\label{3.17} \phi_{\lambda}(u)\geq R, \quad \|u\|_{\alpha}=\rho,\quad \forall \lambda\leq \mu_0. \end{equation} By choosing $e\in E^{\alpha}$ such that $\phi_{\lambda_0}(e)<0$, we infer that \begin{equation*} \frac{\phi_{\lambda}(e)}{\lambda}<\frac{\phi_{\lambda_0}(e)}{\lambda_0}<0,\quad \forall \lambda\in [\lambda_0,\mu_0]. \end{equation*} The same can be done to obtain \begin{equation}\label{3.18} \frac{\phi_{\lambda}(u)}{\lambda}\leq \frac{\phi_{\mu}(u)}{\mu} ,\quad \forall u\in E^{\alpha},\; \mu < \lambda. \end{equation} Define \begin{equation*} P=\big\{\gamma:[0,1]\to E^{\alpha}: \gamma\text{ is continuous and } \gamma(0)=0,\; \gamma(1)=e\big\}. \end{equation*} According to the definition of $c_\lambda$, we have $c_{\mu_0}>0$. Thus map $c: [\lambda_0,\mu_0]\to \mathbb{R}_{+}$ is defined as $c(\lambda)=c_{\lambda}$. In fact, the formula \eqref{3.18} contains $\frac{c_{\lambda}}{\lambda}$ is monotone decreasing, the formula \eqref{3.17} contains $c_{\lambda}\geq R >0$ and $c_{\lambda}$ is bounded from below. Now we prove left semi-continuity of $c_{\lambda}/\lambda$. Fix $\mu\in [\lambda_0,\mu_0]$ and $\varepsilon >0$. Then exists $\gamma\in P$, such that \begin{equation}\label{3.19} c_{\mu} \leq \max_{z\in[0,1]}\phi_{\mu}(\gamma(z))\leq c_{\mu}+\frac{\varepsilon\mu}{8}. \end{equation} Let $r_0=\max_{z\in[0,1]}\big|\int_0^{T}F(t,\gamma(z))dt\big|$. Then, for $\mu < 2\lambda$ and $\frac{1}{\lambda}<\frac{1}{\mu}+\frac{\varepsilon}{2c_{\mu}}$, we can obtain \begin{equation}\label{3.20} \begin{aligned} \phi_{\lambda}(\gamma(z)) =&\phi_{\mu}(\gamma(z))+(\phi_{\lambda}(\gamma(z))-\phi_{\mu}(\gamma(z)))\\ =&\phi_{\mu}(\gamma(z))+(\mu-\lambda)\int_0^{T}F(t,\gamma(z))dt\\ \leq &c_{\mu}+\frac{\varepsilon\mu}{8}+r_0|\mu-\lambda|. \end{aligned} \end{equation} If $|\mu-\lambda|<\frac{\varepsilon\mu}{8r_0}$, then $c_{\lambda}\leq c_{\mu}+\frac{\varepsilon\mu}{4}$. Hence, if $\lambda<\mu$, then \begin{equation}\label{3.21} \begin{aligned} \frac{c_{\mu}}{\mu}-\varepsilon &<\frac{c_{\mu}}{\mu}<\frac{c_{\lambda}}{\lambda} \leq\frac{c_{\mu}+\frac{\varepsilon\mu}{4}}{\lambda} =\frac{c_{\mu}}{\lambda}+\frac{\varepsilon\mu}{4\lambda}\\ &\leq \frac{c_{\mu}}{\lambda}+\frac{\varepsilon\mu}{4}\frac{2}{\mu} \leq c_{\mu}(\frac{1}{\mu}+\frac{\varepsilon}{2c_{\mu}}) +\frac{\varepsilon}{2} =\frac{c_{\mu}}{\mu}+\varepsilon. \end{aligned} \end{equation} Hence, $c_{\lambda}/\lambda$ and $c_{\lambda}$ are left semi-continuous. \end{proof} \begin{remark} \label{rem3.1} \rm We recall that the map $b:[\lambda_0,\mu_0]\to \mathbb{R}_{+} $, given by $b(\lambda)=\frac{c_{\lambda}}{\lambda}$, is monotone decreasing. Thus, $b_{\lambda}$ and $c_{\lambda}$ are differentiable at almost all values $\lambda\in(\lambda_0,\mu_0)$. \end{remark} \begin{lemma}\label{lem3.3} There exists $C>0$, such that \begin{equation}\label{3.22} \|\phi_{\mu}'(u)-\phi_{\lambda}'(u)\|_{(E^{\alpha})^{\ast}} \leq C(1+\|u\|_{\alpha}^{p})|\mu-\lambda|,\quad \forall \lambda,\;\mu>0. \end{equation} \end{lemma} \begin{proof} By (A2), we have \begin{equation}\label{3.23} |\nabla F(t,u)|\leq a+b|u|^{p}. \end{equation} For all $v\in E^{\alpha}$ with $\|v\|_{\alpha}\leq1$, we have \begin{equation}\label{3.24} \begin{aligned} |\langle \phi_{\mu}'(u)-\phi_{\lambda}'(u) ,v\rangle| =&|\lambda-\mu| |\int_0^{T}(\nabla F(t,u(t)),v)dt|\\ \leq&|\lambda-\mu| |\int_0^{T}|\nabla F(t,u(t))||v|dt\\ \leq&|\lambda-\mu| |\int_0^{T}|(a+b|u|^{p})|v|dtr\\ \leq&|\lambda-\mu| |\int_0^{T}|(a+b|u|^{p})\max_{t\in[o,T]}|v|dt. \end{aligned} \end{equation} Since the embedding $E^{\alpha}\hookrightarrow C[0,T]$ is continuous, there exists a constant $c_1>0$ such that \begin{equation*} \|v\|_{\infty}\leq c_1\|v\|_{\alpha},\; \|v\|_p\leq c_1\|v\|_{\alpha},\;\; \forall\;v\in E^{\alpha}, \end{equation*} which implies that \begin{equation}\label{3.25} \begin{aligned} |\langle \phi_{\mu}'(u)-\phi_{\lambda}'(u) ,v\rangle| \leq&|\lambda-\mu|\|v\|_{\infty}(aT+b\|u\|_p^{p})\\ \leq&|\lambda-\mu| c_1\|v\|_{\alpha}(aT+b c_1\|u\|_{\alpha}^{p})\\ \leq&|\lambda-\mu| c_1\|v\|_{\alpha}(aT+b c_1\|u\|_{\alpha}^{p}). \end{aligned} \end{equation} So that there exists $C>0$, such that \begin{equation*} \|\phi_{\mu}'(u)-\phi_{\lambda}'(u)\|_{(E^{\alpha})^{\ast}}\leq C(1+\|u\|_{\alpha}^{p})|\mu-\lambda|,\quad \forall \lambda,\;\mu>0. \end{equation*} \end{proof} \begin{lemma}\label{lem3.4} Assume that map $c:[\lambda_0,\mu_0]\to \mathbb{R}_{+}$, satisfies $c(\lambda)=c_{\lambda}$ and $c(\lambda)$ is differentiable at point $\mu$, then there exists a sequence $\{u_{n}\}\in E^{\alpha}$, such that \begin{equation*} \phi_{\mu}(u_{n})\to c_{\mu}, \quad \phi_{\mu}'(u_{n})\to 0, \quad -\int_0^{T}(^c_0D_t^{\alpha}u(t),^c_tD_{T}^{\alpha}u(t))dt\leq C_{2}, \end{equation*} as $n\to \infty$, where $C_{2}=2c_{\mu}+2\mu(2-c'(\mu))+1$. \end{lemma} \begin{proof} Assume, by contradiction, that the lemma is false. Then there exists $\delta>0$, such that \begin{equation*} \|\phi'_{\mu}(u)\|\geq 2\delta,\;\;\forall\;u\in \mathbb{N}_{\delta}, \end{equation*} where $N_{\delta}=\{u\in E^{\alpha}:-\int_0^{T}(^c_0D_t^{\alpha}u(t),^c_tD_{T}^{\alpha}u(t))dt\leq C_{2}, |\phi_{\mu}(u)-c_{\mu}|<\delta\}.$ There exists constant $C_{3}$, such that \begin{equation}\label{3.26} \begin{aligned} \big|\int_0^{T}F(t,u)dt\big| &=\frac{1}{2\mu}|2\phi_{\mu}(u)-\int_0^{T} -(^c_0D_t^{\alpha}u(t),^c_tD_{T}^{\alpha}u(t))dt|\\ &\leq C_{3},\;\;\forall\;u\in \mathbb{N}_{\delta}. \end{aligned} \end{equation} Let $V:N_{\delta}\to E^{\alpha}$ be the pseudo-gradient vector field for $\phi_{\mu}$ in $N_{\delta}$, that is, $V$ is locally Lipschitz, $\|V\|\leq 1$ and \begin{equation}\label{3.27} \phi'_{\mu}(u)\cdot (V(u))\leq -\delta,\quad \forall u\in \mathbb{N}_{\delta} \end{equation} (see \cite{26}). Now, fix a sequence $\{\lambda_{n}\} \subseteq (\lambda_0,\mu_0)$ such that $\mu\leq \lambda_{n+1}<\lambda_{n}$, $\lambda_{n}\to \mu$, $|\lambda_{n}-\mu|\leq \frac{\delta}{4}$, $|c_{\mu}-c_{\lambda_{n}}|\leq \frac{\delta}{4}$, for each $n$, let $\gamma_{n}\in P$ be such that \begin{equation}\label{3.28} \max_{z\in[0,1]}\phi_{\mu}(\gamma_{n}(z))\leq c_{\mu}+(\lambda_{n}-\mu). \end{equation} Consider the set \begin{equation*} A_{n}=\{z\in[0,1]:\phi_{\lambda_{n}}(\gamma_{n}(z)) >c_{\lambda_{n}}-(\lambda_{n}-\mu)\}. \end{equation*} By definition of $c_{\lambda_{n}}$, $A_{n}$ is nonempty. If $v\in \gamma_{n}(A_{n})$, we have \begin{equation}\label{3.29} \begin{aligned} \int_0^{T}F(t,v)dt =&\frac{\phi_{\mu}(v)-\phi_{\lambda_{n}}(v)}{\lambda_{n}-\mu}\\ \leq&\frac{c_{\mu}+(\lambda_{n}-\mu)-c_{\lambda_{n}} +(\lambda_{n}-\mu)}{\lambda_{n}-\mu}\\ =&\frac{c_{\mu}-c_{\lambda_{n}}}{\lambda_{n}-\mu}+2. \end{aligned} \end{equation} Since $c(\mu)$ is differentiable, we have \begin{equation}\label{3.30} \begin{aligned} c_{\lambda_{n}} =&c_{\mu}+c'(\mu)(\lambda_{n}-\mu)+o((\lambda_{n}-\mu)^2)\\ =&c_{\mu}+c'(\mu)(\lambda_{n}-\mu)+o_{n}(1)(\lambda_{n}-\mu); \end{aligned} \end{equation} that is, $c_{\mu}-c_{\lambda_{n}}=[c'_{\mu}+o_{n}(1)](\mu-\lambda_{n})$. So $\int_0^{T}F(t,v)dt=-c'_{\mu}+2+o_{n}(1)$. Since $\int_0^{T}-(^c_0D_t^{\alpha}v(t),^c_tD_{T}^{\alpha}v(t))dt=2\phi_{\mu}(v) +2\mu\int_0^{T}F(t,v(t))dt$, we have \begin{equation}\label{3.31} \begin{aligned} &\int_0^{T}-(^c_0D_t^{\alpha}v(t),^c_tD_{T}^{\alpha}v(t))dt\\ &\leq 2c_{\mu}+2(\lambda_{n}-\mu)+2\mu(-c'(\mu)+2+o_{n}(1)) \leq C_{2}, \end{aligned} \end{equation} for $n$ large. It is easy to see that inequality \eqref{3.26} is satisfied for $v\in \gamma_{n}(A_{n})$. Thus $\gamma_{n}(A_{n})\subset N_{\delta}$, since \begin{equation}\label{3.32} \begin{gathered} c_{\lambda_{n}}-(\lambda_{n}-\mu)\leq\phi_{\lambda_{n}}(v),\quad \phi_{\mu}(v)\leq c_{\mu}+(\lambda_{n}-\mu), \\ |\phi_{\lambda_{n}}(v)-\phi_{\mu}(v)|=|\lambda_{n}-\mu||\int_0^{T}F(t,v)dt|\leq c_{3}|\lambda_{n}-\mu|. \end{gathered} \end{equation} So $v\in \mathbb{N}_{\delta}$, that is $\gamma_{n}(A_{n})\subset N_{\delta}$. Using \eqref{3.27} we have $\phi'_{\lambda_{n}}\cdot(V(u))<-\frac{\delta}{2}$, for all $u\in \mathbb{N}_{\delta}$. Now consider a Lipschitz continuous cut-off function $\eta$ such that $0\leq\eta\leq1$, $\eta(u)=0$ in $u \not\in \mathbb{N}_{\delta}$, and $\eta(u)=1$ for $u\in \mathbb{N}_{\frac{\delta}{2}}$. Let $\phi$ be the flow generated by $\eta V$, that is, \begin{equation}\label{3.33} \begin{gathered} \frac{\partial\phi}{\partial r}(u,r)=\eta(\phi(u,r))V(\phi(u,r)), \quad r\geq 0,\\ \phi(u,0)=u. \end{gathered} \end{equation} Integrating the both sides of the equation, we have \begin{equation*} \int_0^{r}\frac{\partial\phi(u,t)}{\partial t}dt =\int_0^{r}\eta(\phi(u,t))V(\phi(u,t))dt, \end{equation*} That is $\phi(u,t)=u+\int_0^{r}\eta(\phi(u,t))V(\phi(u,t))dt$ is the solution of \eqref{3.33}. Applying the ODE uniqueness we have: If $u \not\in \mathbb{N}_{\delta}$, then $\phi(u,r)=u $ for all $r\geq 0$. If $u\in \mathbb{N}_{\delta}$, then $\phi(u,r)\in \mathbb{N}_{\delta}$ for all $r\geq0$, If $u\in E^{\alpha}$, then $\frac{\partial\phi}{\partial r}(u,r)=V(\phi(u,r))$ and $$ \phi_{\lambda_{n}}(\phi(u,r))\frac{\partial\phi}{\partial r}(u,r)<-\frac{\delta }{2}\leq 0,\;\forall r\geq0, $$ If $\phi(u,r)\in \mathbb{N}_{\frac{\delta}{2}}$, for all $r\in[0,r_0]$, then $\phi_{\lambda_{n}}(\phi(u,r))\leq \phi_{\lambda_{n}}(u)-\frac{\delta r_0}{2}$. It is easy to see that if $u\in \mathbb{N}_{\frac{\delta}{2}}$, then $\phi_{\lambda_{n}}(\phi(u,1))\leq \phi_{\lambda_{n}}(u)-\frac{\delta}{2}$. Since $e\not\in \mathbb{N}_{\delta}$, we have $\phi(e,r)=e$ and $\phi(0,r)=0$, for all $r\geq0$, and then $\phi(\gamma,r)\in P$, for all $r\in \mathbb{R}$ and $\gamma\in P$. This implies that $h_{n}(z)=\phi(\gamma_{n}(z),1)$ is continuous path in $P$ such that \begin{equation*} \phi_{\lambda_{n}}(h_{n}(z))\leq \phi_{\lambda_{n}}(\gamma_{n}(z)), \end{equation*} and then for its maximum point $s_{n}\in[0,1]$, we have $s_{n}\in A_{n}$ and \begin{equation*} c_{\mu}-o_{n}(1)=c_{\lambda_{n}}\leq \max_{z\in[0,1]}\phi_{\lambda_{n}}(h_{n}(z))=\phi_{\lambda_{n}}(h_{n}(s_{n}))\leq \phi_{\lambda_{n}}(\gamma_{n}(s_{n}))-\frac{\delta}{2}. \end{equation*} That is, \begin{equation}\label{3.34} \phi_{\lambda_{n}}(\gamma_{n}(s_{n}))\geq c_{\mu}-o_{n}(1)+\frac{\delta}{2}. \end{equation} On the other hand, by \eqref{3.28} and \eqref{3.32}, we obtain \begin{equation}\label{3.35} \phi_{\lambda_{n}}(\gamma_{n}(s_{n}))\leq \phi_{\mu}(\gamma_{n}(s_{n}))+C_{3}|\lambda_{n}-\mu|\leq c_{\mu}+(1+C_{3})|\lambda_{n}-\mu|. \end{equation} According to \eqref{3.34} and \eqref{3.35}, we have \begin{equation*} c_{\mu}+\frac{\delta}{2}\leq \lim_{n\to\infty}\phi_{\lambda_{n}}(\gamma_{n}(s_{n}))\leq c_{\mu}, \end{equation*} which is a contradiction. So the original conclusion is true. \end{proof} From Lemma \ref{lem3.4} we can obtain the following statement. \begin{lemma}\label{lem3.5} For almost all $\lambda>0$, $c_{\lambda}$ is a critical value for $\phi_{\lambda}$. \end{lemma} \begin{theorem} Suppose that {\rm (A1)--(A4)} holds, then for any $\lambda>0$, problem \eqref{eP} has a nontrivial solution. \end{theorem} \begin{proof} For $c_{\lambda}$ is left semi-continuity, applying Lemma \ref{lem3.5}, for any $\mu>0$, there exists a sequences $\{u_{n}\}\subseteq E^{\alpha}$ and $\lambda_{n}\subseteq \mathbb{R}$, such that \begin{equation}\label{3.36} \begin{gathered} \lambda_{n}\to \mu,\quad c_{\lambda_{n}}\to c_{\mu},\quad n\to\infty.\\ \phi_{\lambda_{n}}(u_{n})=c_{\lambda_{n}},\quad \phi'_{\lambda_{n}}(u_{n})=0. \end{gathered} \end{equation} We only need to show that $\{u_{n}\}$ is bounded in $E^{\alpha}$. If $\{u_{n}\}$ is unbounded, we may assume, without loss of generality, that $\|u_{n}\|_{\alpha}\to \infty$ as $n\to \infty$. Let $\omega_{n}=\frac{u_{n}}{\|u_{n}\|_{\alpha} }$, then $\omega_{n}\in E^{\alpha}$ with $\|\omega_{n}\|_{\alpha}=1$. Then there are $\omega\in E^{\alpha} $ and $h\in L^{p+1}([0,T])$ such that $\omega_{n}\rightharpoonup \omega$ in $E^{\alpha}$, $\omega_{n}\to \omega$ in $C([0,T],\mathbb{R}_{+})$ and $L^{p+1}([0,T])$, $\omega_{n}(t)\to \omega(t)$, a.e. $t\in[0,T]$, $n\to\infty$, $|\omega_{n}(t)|\leq h(t)$, a.e. $t\in[0,T]$, for all $n\in\mathbb{N}$. Let $\Omega_0 =\{t\in [0,T]: \omega(t)\not=0\}$. If $t\in \Omega_0$, then by (A3), \begin{equation}\label{3.37} \lim_{n\to+\infty}\frac{F(t,u_{n}(t))}{u_{n}^2(t)} \omega_{n}^2(t) =+\infty \end{equation} and \begin{equation}\label{3.38} \begin{aligned} &\lim_{n\to+\infty}\frac{\int_0^{T}F(t,u_{n}(t))dt}{\int_0^{T} -(^c_0D_t^{\alpha}u_{n}(t),^c_tD_{T}^{\alpha}u_{n}(t))dt}\\ &=\lim_{n\to+\infty}\Big(\frac{1}{2\lambda_{n}} -\frac{\phi_{\lambda_{n}(u_{n})}}{\lambda_{n} \int_0^{T}-(^c_0D_t^{\alpha}u_{n}(t),^c_tD_{T}^{\alpha}u_{n}(t))dt}\Big) =\frac{1}{2\mu}. \end{aligned} \end{equation} By Proposition \ref{pro2.5} and Fatou's lemma \begin{equation}\label{3.39} \begin{aligned} +\infty =& \int_0^{T}\lim_{n\to\infty}|\cos(\pi\alpha)| \frac{F(t,u_{n}(t))}{u_{n}^2(t)}\cdot \omega_{n}^2(t)dt\\ \leq&\lim_{n\to\infty}\int_0^{T}|\cos(\pi\alpha)| \frac{F(t,u_{n}(t))}{u_{n}^2(t)}\cdot \omega_{n}^2(t)dt\\ \leq&\lim_{n\to\infty}\int_0^{T} \frac{F(t,u_{n}(t))}{\frac{1}{|\cos(\pi\alpha)|}\|u_{n}(t)\|_{\alpha}^2}dt\\ \leq&\lim_{n\to+\infty}\frac{\int_0^{T}F(t,u_{n}(t))dt}{\int_0^{T} -(^c_0D_t^{\alpha}u_{n}(t),^c_tD_{T}^{\alpha}u_{n}(t))dt} =\frac{1}{2\mu}. \end{aligned} \end{equation} This is a contradiction. This shows that $\Omega_0$ has zero measure. Hence $\omega=0$ a.e. $t\in[0,T]$. Let $z_{n}\in[0,1]$, such that $\phi_{\lambda_{n}}(z_{n}u_{n})=\max_{z\in[0,1]}\phi_{\lambda_{n}}(z u_{n})$. By \eqref{2.4} and \eqref{2.5}, we have \begin{equation}\label{3.40} 2\phi_{\lambda_{n}}(z_{n}u_{n}) =-\int_0^{T}(^c_0D_t^{\alpha}z_{n}u_{n},^c_tD_{T}^{\alpha}z_{n}u_{n})dt -2\lambda_{n}\int_0^{T}F(t,z_{n}u_{n})dt \end{equation} and \begin{equation}\label{3.41} \begin{aligned} \phi'_{\lambda_{n}}(z_{n}u_{n})(z_{n}u_{n}) =&-\frac{1}{2}\int_0^{T}(^c_0D_t^{\alpha}z_{n}u_{n}(t), ^c_tD_{T}^{\alpha}z_{n}u_{n}(t))dt\\ &-\frac{1}{2}\int_0^{T}(^c_tD_{T}^{\alpha}z_{n}u_{n}(t), ^c_0D_t^{\alpha}z_{n}u_{n}(t))dt\\ &-\lambda_{n}\int_0^{T}(\nabla F (t,z_{n}u_{n}(t)), z_{n}u_{n}(t))dt. \end{aligned} \end{equation} From $\phi'_{\lambda_{n}}(z_{n}u_{n})(z_{n}u_{n})=0$, we have \begin{equation}\label{3.42} \begin{aligned} 2\phi_{\lambda_{n}}(z u_{n}) \leq& 2\phi_{\lambda_{n}}(z_{n}u_{n})-\phi'_{\lambda_{n}}(z_{n}u_{n})(z_{n}u_{n})\\ =&\lambda_{n}\int_0^{T}(\nabla F (t,z_{n}u_{n}(t)), z_{n}u_{n}(t))-2F(t,z_{n}u_{n}))dt. \end{aligned} \end{equation} By assumption (A4), it follows that \begin{equation}\label{3.43} H(t,z_{n}u_{n}) =(z_{n}u_{n}, \nabla F(t,z_{n}u_{n}))-2F(t,z_{n}u_{n}), \end{equation} and $|z_{n}u_{n}|\leq |u_{n}|$. Therefore, \begin{equation}\label{3.44} H(t,z_{n}u_{n})\leq (u_{n}, \nabla F(t,u_{n}))-2F(t,u_{n})+C_{\ast}. \end{equation} By \eqref{3.42} and \eqref{3.44}, we obtain \begin{equation}\label{3.45} 2\phi_{\lambda_{n}}(z u_{n})\leq\lambda_{n}\int_0^{T}[(u_{n}, \nabla F(t,u_{n}))-2F(t,u_{n})+C_{\ast}]dt. \end{equation} Since $\phi_{\lambda_{n}}(u_{n})=c_{\lambda_{n}},\;\phi'_{\lambda_{n}}(u_{n})=0$, it follows that \begin{gather*} 2c_{\lambda_{n}}=-\int_0^{T}(^c_0D_t^{\alpha}u_{n}(t),^c_tD_{T}^{\alpha}u_{n}(t))dt -2\lambda_{n}\int_0^{T}F(t,u_{n}(t))dt, \\ \phi'_{\lambda_{n}}(u_{n})(u_{n}) =-\int_0^{T}(^c_0D_t^{\alpha}u_{n}(t),^c_tD_{T}^{\alpha}u_{n}(t))dt -\lambda_{n}\int_0^{T}(\nabla F (t,u_{n}(t)),u_{n}(t))dt. \end{gather*} So $2\phi_{\lambda_{n}}(z u_{n})\leq\lambda_{n}C_{\ast}T+2c_{\lambda_{n}}$,\;{\rm for}\;{\rm all}\;$z\in[0,1]$. On the other hand, for all $r_0>0$, \begin{equation}\label{3.46} \begin{aligned} 2\phi_{\lambda_{n}}(r_0\omega_{n}) =&-\int_0^{T}(^c_0D_t^{\alpha}r_0\omega_{n},^c_tD_{T}^{\alpha}r_0\omega_{n})dt -2\lambda_{n}\int_0^{T}F(t,r_0\omega_{n})dt,\\ =&|\cos(\pi\alpha)|r_0^2-O_{n}(1), \end{aligned} \end{equation} which contradicts $2\phi_{\lambda_{n}}(z u_{n})\leq\lambda_{n}C_{\ast}T+2c_{\lambda_{n}}$. This contradiction shows $\{u_{n}\}$ is unbounded in $E^{\alpha}$. The proof is complete. \end{proof} \subsection*{Acknowledgements} This work was supported by the National Natural Science Foundation of China (No. 11201095), by the Youth Scholar Backbone Supporting Plan Project of Harbin Engineering University (No. 307201411008), by the Fundamental Research Funds for the Central Universities (No. 2018), by the Postdoctoral Research Startup Foundation of Heilongjiang (No. LBH-Q14044), and by the Science Research Funds for Overseas Returned Chinese Scholars of Heilongjiang Province (No. LC201502). \begin{thebibliography}{00} \bibitem{7} R. P. Agarwal, D. O'Regan, S. Stanek; \emph{Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations.} J. Math. Anal. Appl., \textbf{ 371} (2010), 57--68. \bibitem{8} Z. Bai and H. Lu; \emph{Positive solutions for boundary value problem of nonlinear fractional differential equations.} J. Math. Anal. Appl., \textbf{311} (2005), 495--505. \bibitem{21} J. Chen, X.H. Tang; \emph{Existence and multiplicity of solutions for some fractional boundary value problem via critical point theory.} Abstr. Appl. Anal., ID: 648635, 2012, 1-21. \bibitem{1} K. Diethelm, A.D. Freed; \emph{On the solution of nonlinear fractional order differential equations used in the modeling of viscoelasticity}, in: F. Keil, W. Mackens, H. Voss, J. Werther (Eds.), Scientific Computing in Chemical Engineering IIComputational Fluid Dynamics, Reaction Engineering and Molecular Properties, Springer-Verlag, Heidelberg, 1999, pp. 217-224. \bibitem{9} M. El-Shahed, J.J. Nieto; \emph{Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order.} Comput. Math. Appl., \textbf{59} (2010), 3438--3443. \bibitem{10} C. S. Goodrich; \emph{Existence of a positive solution to a class of fractional differential equations.} Appl. Math. Lett., \textbf{23} (2010), 1050--1055. \bibitem{23} B. Ge; \emph{Multiple solutions for a class of fractional boundary value problems.} Abstr. Appl. Anal., 2012, Article ID: 468980. \bibitem{11} J. R. Graef, L. Kong, Q. Kong; \emph{Application of the mixed monotone operator method to fractional boundary value problems.} Fract. Differ. Calc., \textbf{2} (2012), 87--98. \bibitem{12} J. R. Graef, L. Kong, B. Yang; \emph{Positive solutions for a semipositone fractional boundary value problem with a forcing term.} Fract. Calc. Appl. Anal., \textbf{15} (2012), 8--24. \bibitem{3} W. G. Glockle, T. F. Nonnenmacher; \emph{A fractional calculus approach of self- similar protein dynamics.} Biophys. J., \textbf{68} (1995), 46--53. \bibitem{4} R. Hilfer; \emph{Applications of Fractional Calculus in Physics}, World Scientific, Singapore, 2000. \bibitem{17} W. H. Jiang; \emph{The existence of solutions for boundary value problems of fractional differential equations at resonance.} Nonlinear Anal: TMA., \textbf{74} (2011), 1987--1994. \bibitem{22} F. Jiao, Y. Zhou; \emph{Existence of solutions for a class of fractional boundary value problems via critical point theory.} Comput. Math. Appl., \textbf{62} (2011), 1181--1199. \bibitem{24} L. J. Kong; \emph{Existence of solutions to boundary value problems arising from the fractional advection dispersion equation, Electron.} J. Differential Equations, \textbf{106} 2013, 1--15. \bibitem{6} J. W. Kirchner, X. Feng, C. Neal; \emph{Fractal streamchemistry and its implications for contaminant transport in catchments.} Nature, \textbf{403} (2000), 524--526. \bibitem{18} A. A. Kilbas, H. M. Srivastava, J. J. Trujillo; \emph{Theory and Applications of Fractional Differential Equations}, Elsevier, Amsterdam, 2006. \bibitem{2} B. N. Lundstrom, M. H. Higgs, W. J. Spain, A. L. Fairhall; \emph{Fractional differentiation by neocortical pyramidal neurons.} Nat. Neurosci., \textbf{11} (2008), 1335--1342. \bibitem{25} Y. N. Li, H. R. Sun, Q. G. Zhang; \emph{Existence of solutions to fractional boundary-value problems with a parameter.} Electron. J. Differential Equations, \textbf{141} 2013, 1--12. \bibitem{14} S. Liang, J. H. Zhang; \emph{Positive solutions for boundary value problems of nonlinear fractional differential equation.} Nonlinear Anal: TMA., \textbf{71} (2009), 5545--5550. \bibitem{5} F. Mainardi; \emph{Fractional calculus: some basic problems in continuum and statistical mechanics}, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien, 1997, pp. 291-348. \bibitem{19} R. L. Magin; \emph{Fractional Calculus in Bioengineering}, Begell House, 2006. \bibitem{20} I. Podlubny; \emph{Fractional Differential Equations}, Academic Press, New York, 1999. \bibitem{26} R. S. Palais; \emph{Critical point theory and the minimax principle.} in: Global Analysis, in: Proc. Sympos. Pure Math., vol. 15, 1968, pp. 185--212. \bibitem{15} X. Zhang, L. Liu, Y. H. Wu; \emph{Multiple positive solutions of a singular fractional differential equation with negatively perturbed term.} Math. Comput. Modelling., \textbf{55} (2012), 1263--1274. \bibitem{16} X. Zhang, L. Liu, Y. H. Wu; \emph{The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives.} Appl. Math. Comput., \textbf{218} (2012), 8526-¨C8536. \bibitem{13} Y. Zhao, S. Sun, Z. Han, M. Zhang; \emph{Positive solutions for boundary value problems of nonlinear fractional differential equations.} Appl. Math. Comput., \textbf{ 217} (2011), 6950--6958. \end{thebibliography} \end{document}