\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 116, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2016 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2016/116\hfil Existence of solutions] {Existence of solutions for fractional differential equations with Dirichlet boundary conditions} \author[K. Ben Ali, A. Ghanmi, K. Kefi \hfil EJDE-2016/116\hfilneg] {Khaled Ben Ali, Abdeljabbar Ghanmi, Khaled Kefi} \address{Khaled Ben Ali \newline D\'epartement de Math\'ematiques, Facult\'e des Sciences de Tunis\\ Campus Universitaire, 2092 Tunis, Tunisia} \email{benali.khaled@yahoo.fr} \address{Abdeljabbar Ghanmi \newline Department of Mathematics, Faculty of Sciences and Arts Khulais, University of Jeddah, Saudi Arabia. \newline Departement of Mathematics, Faculty of Sciences Tunis El Manar, 1060 Tunis, Tunisia} \email{Abdeljabbar.ghanmi@lamsin.rnu.tn} \address{Khaled Kefi \newline D\'epartement de Math\'ematiques, Facult\'e des Sciences de Tunis\\ Campus Universitaire, 2092 Tunis, Tunisia} \email{khaled\_kefi@yahoo.fr} \thanks{Submitted January 28, 2016. Published May 10, 2016.} \subjclass[2010]{26A33, 58E05, 35J60} \keywords{Fractional differential equation; left and right fractional derivatives; \hfill\break\indent boundary value problem; Nehari manifold} \begin{abstract} In this article, we apply the Nehari manifold to prove the existence of a solution of the fractional differential equation \begin{gather*} \frac{d}{dt} \Big(\frac12 {\,}_0D_t^{-\beta}(u'(t)) +\frac12 {\,}_tD_T^{-\beta}(u'(t)))= f(t,u(t)) + \lambda h(t)|u(t)|^{r-2}u(t), \\ \text{a.e } t\in [0,T],\\ u(0)=u(T)=0, \end{gather*} where $ _0D_t^{-\beta},\; _tD_T^{-\beta}$ are the left and right Riemann-Liouville fractional integrals, respectively, of order $0< \beta < 1$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} Recently, there has been surge in the interest for fractional differential equations in fields such as: from physics, chemistry, aerodynamics, electrical circuits, diffusion, electro dynamics of complex medium, and applied mathematics. Among the researchers studying such equations, we can quote for example the authors in \cite{R1,R9,R11,R15}. Researchers have examined some problems related to these types of equations by using methods such as fixed point theorem, coincidence degree theory, and critical point theory; see \cite{R2,R3,R4,R5,R6,R7,R8,R10,R12,R13,R14,R16,R17,R18,R19,R20,R21, R22,R25,R26,R27,R28,R29}. As an example Jiao and Zhou \cite{R7} studied the boundary-value problem \begin{equation} \label{e1} \begin{gathered} \frac{d}{d t}\Big(\frac{1}{2} {\,}_0D_t^{-\beta}(u'(t)) +\frac{1}{2} {\,}_tD_T^{-\beta}(u'(t))\Big) +\nabla F(t,u(t))=0,\quad\text{a.e. } t \in [0,T],\\ u(0)=u(T)=0, \end{gathered} \end{equation} where $0< \beta < 1$, and $_0D_t^{-\beta}$ and $_tD_T^{-\beta}t$ are the left and right Riemann-Liouville fractional integrals of order $\beta$, respectively, $F : [0,T]\times \mathbb{R}^N\to \mathbb{R}$, and $\nabla F(t,x)$ is the gradient of $F$ with respect to $x$. By using the mountain pass theorem, they showed the existence of a solution. Bai \cite{R4} and other researchers considered the problem \begin{equation} \label{e2} \begin{gathered} \frac{d}{d t}\Big(\frac{1}{2} {\,}_0D_t^{\alpha-1}(_0^{C}D_t^{\alpha}(u(t))) +\frac{1}{2} {\,}_tD_T^{\alpha-1}( _t^{C}D_T^{\alpha}u(t))\Big) +\lambda a(t) f(u(t))=0,\\ \text{a.e. }t \in [0,T],\\ u(0)=u(T)=0 \end{gathered} \end{equation} where $\alpha \in (1/2,1]$, and $_0D_t^{\alpha-1}$ and $_tD_T^{\alpha-1}$ are the left and right Riemann-Liouville fractional integrals of order $1-\alpha$, where $_0^{C}D_t^{\alpha}(u(t))$ and $_t^{C}D_T^{\alpha}(u(t))$ are the left and right Caputo fractional derivatives of order $\alpha$. By using a critical-point theorem established by Bonanno, he proved the existence of a solution to this problem. We mention also the works \cite{R5,R6,R7,R8}, where by using critical point theory, the existence and multiplicity of solutions have been established for the related problems. In this article, we attempt to highlight the use of the Nehari method to prove the existence of solutions to the problem \begin{equation} \label{e3} \begin{gathered} \frac{d}{d t}\Big(\frac{1}{2} {\,}_0D_t^{-\beta}(u'(t)) +\frac{1}{2} {\,}_tD_T^{-\beta}(u'(t))\Big) =f(t,u(t))+ \lambda h(t)|u(t)|^{r-2}u(t),\\ \text{a.e. }t \in [0,T],\\ u(0)=u(T)=0, \end{gathered} \end{equation} where $\lambda $ is a positive parameter, $10$ and $20$, $M>0$ such that \begin{equation} \label{e1.2} 0< \mu F(t,s)\leq sf(t,s) \end{equation} for all $t\in \mathbb{R}$ and $|s|\geq M$, where \begin{equation} \label{e1.3} F(t,s)=\int_0^s f(t,x)dx. \end{equation} \item[(H4)] The map $t\to t^{-1}sf(x,ts)$ is increasing on $(0,+\infty)$, for every $x\in \mathbb{R}$ and $s\in \mathbb{R}$. \item[(H5)] $h$ is a nonnegative continuous function on $\Omega$. \end{itemize} Our main result is the following. \begin{theorem} \label{thm1.1} Assuming {\rm (H1)--(H5)}, boundary value problem \eqref{e3} has at least one weak solution. \end{theorem} This article is organized as follows. In Section 2, some preliminaries on the fractional calculus are presented. In Section 3, we set up the variational framework of problem \eqref{e3} and give some necessary lemmas. Section 4 presents the proof of the main result. An example is given in Section 5 to illustrate our main result. \section{Preliminaries results and fractional calculus} In this section, we introduce some notation, definitions, and preliminary facts on fractional calculus which are used throughout this paper. \begin{definition} \label{def2.1} \rm Let $f$ be a function defined on $[a,b]$. The left and right Riemann-Liouville fractional integrals of order $\alpha$ for function $f$ are defined, respectively, by \begin{gather*} _{a}D_t^{-\alpha} f(t)=\frac{1}{\Gamma(\alpha)} \int_0^t(t-s)^{\alpha -1}f(s)ds,\quad t\in [a,b],\;\alpha >0,\\ {}_tD_b^{-\alpha} f(t)=\frac{1}{\Gamma(\alpha)} \int_t^b(t-s)^{\alpha -1}f(s)ds,\quad t\in [a,b],\;\alpha >0, \end{gather*} provided that the right-hand side integral is pointwise defined on $[a,b]$. \end{definition} \begin{definition} \label{def2.2} \rm Let $f$ be a function defined on $[a,b]$. The left and right Riemann-Liouville fractional derivatives of order $\alpha$ for function $f$ are defined, respectively, by \begin{equation} \label{e2.1} \begin{aligned} _{a}D_t^{\alpha} f(t) &= \frac{d^{n}}{d{t^n}}_{a}D_t^{\alpha-n} f(t) \\ &= \frac{1}{\Gamma(\alpha)}\frac{d^{n}}{d{t^n}} \int_0^t(t-s)^{n-\alpha -1}f(s)ds, \quad t\in [a,b],\;\alpha >0,\\ _tD_b^{\alpha} f(t) &= (-1)^n\frac{d^{n}}{d{t^n}}_tD_b^{\alpha-n} f(t) \\ &=\frac{(-1)^n}{\Gamma(\alpha)}\frac{d^{n}}{d{t^n}} \int_{b}^t(s-t)^{n-\alpha -1}f(s)ds, \quad t\in [a,b],\;\alpha >0, \end{aligned} \end{equation} provided that the right-hand side integral is pointwise defined on $[a,b]$. \end{definition} \begin{definition} \label{def2.3} \rm If $\alpha \in (n-1,n)$ and $f\in AC^n([a,b],\mathbb{R})$, then the left and right Caputo fractional derivatives of order $\alpha$ for function $f$ are defined, respectively, by \begin{equation} \begin{aligned} _{a}^{C}D_t^{\alpha} f(t) &= _{a}D_t^{\alpha-n}\frac{d^{n}}{d{t^n}} f(t) \\ &= \frac{1}{\Gamma(\alpha)}\int_0^t(t-s)^{n-\alpha -1}f^{n}(s)ds, \quad t\in [a,b],\;\alpha >0,\\ _t^{C}D_b^{\alpha} f(t) &= (-1)^n _tD_b^{\alpha-n}\frac{d^{n}}{d{t^n}}f(t) \\ & = \frac{(-1)^n}{\Gamma(\alpha)}\int_t^b(s-t)^{n-\alpha -1}f^n(s)ds, \quad t\in [a,b],\;\alpha >0, \end{aligned} \end{equation} where $t\in[a,b]$. \end{definition} \begin{lemma}[\cite{R7}] \label{L1} The left and right Riemann-Liouville fractional integral operators that is have the property of a semigroup; that is, \begin{equation} \label{e2.3} \int [ _{a}D_t^{-\alpha}f(t)]g(t)dt =\int [_tD_b^{-\alpha}g(t)]f(t)dt,\quad \alpha >0, \end{equation} provided $f\in L^p([a,b],\mathbb{R}),g\in L^q([a,b],\mathbb{R})$ and $p\geq q$, $q\geq 1$, $1/p+1/q\leq 1+\alpha$ or $p\neq1$, $q\neq1$, $1/p+1/q=1+\alpha$. \end{lemma} \begin{lemma}[\cite{R7}]\label{L2} Assume that $n-1 <\alpha 1/p$ and $1/p+1/q=1$, then \begin{equation} \|u\|_{\infty}\leq \frac {T^{\alpha-1/p}}{\Gamma (\alpha )[(\alpha -1)q+1]^{1/q}} \| _0^{C}D_t^{\alpha}u\|_{L^p}. \end{equation} \end{lemma} According to \eqref{Nlp}, we can consider $E^\alpha$ equivalent norm with respect to the \begin{equation} \label{Neq} \|u\|_{\alpha,p}= \| _0^{C}D_t^{\alpha}u\|_{L^p},\|u\| = \| _0^{C}D_t^{\alpha}u\|_{L^2}. \end{equation} \begin{lemma}[\cite{R6}] \label{L5} Let $0<\alpha \leq 1$ and $11/p$ and the sequence ${u_k}$ converge weakly to $u$ in $E_0^{\alpha,p}$; that is, $u_k\rightharpoonup u$. Then $u_k\to u$ in $C([0,T],R)$; that is, $\|u-u_k\|-\infty\to 0$ as $k\to \infty$. \end{lemma} Similar to the proof of \cite[Proposition 4.1]{R14}, we have the following property. \begin{lemma}\label{L6} If $1/2<\alpha \leq 1$, for any $u \in E^{\alpha}$, one has \begin{equation} \label{e3.6} |\cos(\pi \alpha)|\|u\|^2 \leq - \int_0^T (_0^{C}D_t^{\alpha}u(t), _t^{C}D_T^{\alpha}u(t))dt \leq \frac{1}{ |\cos(\pi \alpha)|}\|u\|^2. \end{equation} \end{lemma} To obtain a weak solution of boundary-value problem \eqref{e3}, we assume that $u$ is a sufficiently smooth solution of \eqref{e3}. Multiplying \eqref{e3} by an arbitrary $v\in C_0^\infty (0,T)$, we have \begin{equation}\label{SLF} \begin{aligned} &-\int_0^T\Big( \frac{d}{dt}\big(\frac{1}{2} _0D_t^{-\beta}(u'(t)) +\frac{1}{2} _tD_T^{-\beta}(u'(t))\big),v(t)\Big)dt\\ &= \int_0^T (f(t,u(t),v(t))dt+\lambda \int_0^T (h(t)| u(t)|^{r-2}u(t),v(t))dt. \end{aligned} \end{equation} Observe that \begin{equation} \begin{aligned} &-\frac{1}{2}\int_0^T \Big(\frac{d}{dt}\big( _0D_t^{-\beta}u'(t)+ _tD_T^{-\beta}u'(t)\big),v(t)\Big)dt\\ &= \frac{1}{2}\int_0^T\Big(( _0D_t^{-\beta}u'(t),v'(t)) +( _tD_T^{-\beta}u'(t),v'(t))\Big)dt\\ &= \frac{1}{2}\int_0^T\Big(( _0D_t^{-\beta/2}u'(t),_tD_T^{-\beta/2}v'(t)) +( _tD_T^{-\beta/2}u'(t),_0D_t^{-\beta/2}v'(t))\Big)dt. \end{aligned} \end{equation} As $u(0)=u(T)=v(0)=v(T)=0$, we have \begin{equation} \begin{gathered} _0D_t^{-\beta/2}u'(t)= _0D_t^{1-\beta/2}u(t),\\ _tD_T^{-\beta/2}u'(t)= -_tD_T^{1-\beta/2}u(t),\\ _0D_t^{-\beta/2}v'(t)= _0D_t^{1-\beta/2}v(t),\\ _tD_T^{-\beta/2}v'(t)= -_tD_T^{1-\beta/2}v(t). \end{gathered} \end{equation} Then \eqref{SLF} is equivalent to \begin{equation} \label{SLFeq} \begin{aligned} &\int_0^T -\frac{1}{2}[( _0D_t^{\alpha}u(t),_tD_T^{\alpha}v(t))+(_tD_T^{\alpha}u(t),_0D_t^{\alpha}v(t))]dt\\ &= \int_0^T (f(t,u(t),v(t))dt+ \lambda\int_0^T (h(t)| u(t)|^{r-2}u(t),v(t))dt. \end{aligned} \end{equation} Since \eqref{SLFeq} is well defined for $u,v\in E^\alpha$, we define weak solution of \eqref{e3} as follows. \begin{definition} \rm $u$ is a weak solution of \eqref{e3} if \begin{equation} \label{DefSLF} \begin{aligned} &\int_0^T -\frac{1}{2}[( _0D_t^{\alpha}u(t),_tD_T^{\alpha}v(t))+(_tD_T^{\alpha}u(t), _0D_t^{\alpha}v(t))]dt\\ &= \int_0^T (f(t,u(t),v(t))dt+ \lambda\int_0^T ((t)| u(t)|^{r-2}u(t),v(t))dt. \end{aligned} \end{equation} for every $v\in E^\alpha$. \end{definition} We consider the functional $I:E^\alpha \to \mathbb{R}$, defined by \begin{equation} \label{FunEng} I(u)=\int_0^T\big[-\frac{1}{2}( _0D_t^{\alpha}u(t),_tD_T^{\alpha}u(t))-F(t,u(t)) -\frac{ \lambda}{r}h(t)|u(t)|^{r}\big]dt, \end{equation} where $ F(t,u)=\int_0^u f(t,s)ds$. From \cite[Theorem 4.1]{R6}, we can get that if $1/2<\alpha \leq 1$, then the functional $I$ is continuously differentiable on $E^\alpha$. Since $I$ is continuously differentiable on $E^\alpha$, we have \begin{equation} \label{DiffI} \begin{aligned} \langle I'(u),v\rangle &=-\int_0^T \frac{1}{2}[( _0D_t^{\alpha}u(t),_tD_T^{\alpha}v(t))+(_tD_T^{\alpha}u(t), _0D_t^{\alpha}v(t))]dt\\ &\quad - \int_0^T (f(t,u(t),v(t))dt- \lambda\int_0^T (h(t)| u(t)|^{r-2}u(t),v(t))dt\,, \end{aligned} \end{equation} for $u,v\in E^\alpha$. Hence, a critical point of $I$ is a weak solution of \eqref{e3}. To study the solvability of \eqref{e3}, we use the so-called Nehari method. There is one-to-one correspondence between the critical points of $I$ and weak solutions of \eqref{e3}. Now, we define \begin{equation} \label{N2} \mathcal{N}=\{u\in E^\alpha \backslash \{0\}: \langle I'(u),u\rangle=0\}. \end{equation} Then we know that any nonzero critical point of $I$ must be in $\mathcal{N}$. Define \begin{equation} \label{N3} \begin{aligned} \phi(u) &=\langle I'(u),u\rangle\\ &=-\int_0^T( _0D_t^{\alpha}u(t),_tD_T^{\alpha}u(t))dt - \int_0^T (f(t,u(t),u(t))dt \\ &\quad - \lambda\int_0^T (h(t)| u(t)|^{r-2}u(t),u(t))dt. \end{aligned} \end{equation} \begin{lemma}\label{L7} Assume {\rm (H1)--(H5)} are satisfied. If $u\in \mathcal{N}$ is critical point of $I|_{\mathcal{N}}$, then $I'(u)=0$. \end{lemma} \begin{proof} For $u\in \mathcal{N}$, together with (H4) \begin{equation} \label{N4} \begin{aligned} &\langle \phi'(u),u\rangle \\ &=-\int_0^T 2( _0D_t^{\alpha}u(t),_tD_T^{\alpha}u(t))dt\\ &\quad - \int_0^T (\frac{\partial}{\partial u}f(t,u(t)) u^2(t)+f(t,u(t)) u(t))dt - \lambda r \int_0^T h(t)| u(t)|^{r}dt\\ &= \int_0^T 2(f(t,u(t),u(t))dt - \int_0^T (\frac{\partial}{\partial u}f(t,u(t))u^2(t)+f(t,u(t)) u(t))dt\\ &\quad + 2\lambda\int_0^T h(t)| u(t)|^{r}dt - \lambda r \int_0^T h(t)| u(t)|^{r}dt\\ &= \int_0^T (f(t,u(t))u(t)-\frac{\partial}{\partial u}f(t,u(t)).u^2(t))dt - \lambda (2-r) \int_0^T h(t)| u(t)|^{r}dt\\ &<0. \end{aligned} \end{equation} If $u \in \mathcal{N}$ is a critical point of $I|_{\mathcal{N}}$, there exists a Lagrange multiplier $\lambda \in \mathbb{R}$, such that $I'(u)=\lambda \phi'(u)$. Then we have \begin{equation} \label{N5} \langle I'(u),u\rangle =\lambda \langle \phi'(u),u\rangle =0. \end{equation} From \eqref{N4} we obtain $\lambda = 0$. Consequently $I'(u)=0$. The proof is complete. \end{proof} \section{Proof of main result} The proof is done in two steps. \smallskip \noindent\textbf{Step 1:} For any $u\in E^\alpha\setminus\{0\}$, there is a unique $y=y(u)$ such that $y(u)u \in \mathcal{N}$ and one has $ I(yu)=\max_{z}I(zu)>0$. Indeed, we claim that there exist constants $\delta>0,\rho>0$ such that $I(u)>0$ for all $ u\in B_\rho(0)\setminus \{0\}$ and $I(u)\geq \delta$ for all $u\in \partial B_\rho(0)$. That is, 0 is a strict local minimizer of $I$. In fact, by $(H_3)$ we obtain that for all $\epsilon >0$ there exists $C_\epsilon >0$ such that \begin{equation} \label{e4.1} | F(t,u)| \leq \frac{\epsilon}{2}| u|^2+C_\epsilon |u|^p. \end{equation} Then from Lemmas \ref{L4} and \ref{L6}, we have \begin{align} I(u) &= - \frac{1}{2}\int_0^T ( _0^CD_t^{\alpha}u(t),_t^CD_T^{\alpha}u(t))dt - \int_0^T F(t,u(t))dt - \frac{\lambda}{r}\int_0^T h(t)|u(t)|^{r}dt \nonumber \\ &\geq - \frac{1}{2}\int_0^T ( _0^CD_t^{\alpha}u(t),_t^CD_T^{\alpha}u(t))dt - \frac{\epsilon}{2}\int_0^T| u|^2dt - C_\epsilon \int_0^T|u|^p dt \nonumber \\ &\quad - \frac{\lambda}{r}T\| h\|_\infty \| u\|^r_\infty \label{e4}\\ &\geq \frac{1}{2} |\cos(\pi \alpha)| \| u\|^2 - \frac{\epsilon}{2}\int_0^T| u|^2dt - C_\epsilon \int_0^T|u|^p dt - C_\lambda \| u\|^r \label{e5} \\ &\geq \Big(\frac{1}{2} |\cos(\pi \alpha)| - \frac{\epsilon}{2}\frac{T^{2\alpha}}{\Gamma^2(\alpha +1)} \Big)\| u\|^2 - C_\epsilon \Big( \frac{T^{p+\alpha -1/2}}{\Gamma(\alpha)[(\alpha -1)2+1]^{1/2}} \Big)^p\| u\|^p \nonumber \\ &\quad - C_\lambda \Big( \frac{T^{r+\alpha -1/2}}{\Gamma(\alpha) [(\alpha -1)2+1]^{1/2}}\Big)^r\| u\|^r.\label{e6} \end{align} Choose $\epsilon$ such that $ \epsilon/2 (T^{2\alpha}/\Gamma^2(\alpha +1))=(1/4)|\cos(\pi \alpha)|$; then \begin{equation} \label{e7} \begin{aligned} I(u) &\geq \frac14 |\cos(\pi \alpha)|\| u\|^2 - \Big( C_\epsilon \Big( \frac{T^{p+\alpha -1/2}}{\Gamma(\alpha) [(\alpha -1)2+1]^{1/2}}\Big)^p \\ &\quad + C_\lambda \Big( \frac{T^{r+\alpha -1/2}} {\Gamma(\alpha)[(\alpha -1)2+1]^{1/2}}\Big)^r\Big) \| u\|^r \\ &=\| u\|^2\Big((1/4)|\cos(\pi \alpha)| - \Big( C_\epsilon \Big( \frac{T^{p+\alpha -1/2}} {\Gamma(\alpha)[(\alpha -1)2+1]^{1/2}}\Big)^p \\ &\quad + C_\lambda \Big( \frac{T^{r+\alpha -1/2}} {\Gamma(\alpha)[(\alpha -1)2+1]^{1/2}}\Big)^{r}\Big)\| u\|^{r-2}\Big) \end{aligned} \end{equation} Choose $\rho >0$, such that \begin{align*} &\Big(C_\epsilon \Big(\frac{T^{p+\alpha -1/2}}{\Gamma(\alpha)} [(\alpha -1)2+1]^{1/2}\Big)^p +C_\lambda\Big(\frac{T^{r+\alpha -1/2}}{\Gamma(\alpha)} [(\alpha -1)2+1]^{1/2}\Big)^r\Big)\rho^{p-2}\\ &=\frac18|\cos(\pi \alpha)\| u\|^2. \end{align*} Then we have $I(u)\geq (1/8)|\cos(\pi \alpha)\| u\|^2$. Let $\delta=(1/8)|\cos(\pi \alpha)\| u\|^2$; then we have get that there exist constants $\delta >0,\rho>0$ such that $I(u)>0$ for all $u\in B_\rho(0)\backslash\{0\}$ and $I(u)\geq \delta $ for all $u\in \partial B_\rho(0)$. Next, we claim that $I(yu)\to -\infty$, as $y\to \infty$. In fact, by (H4), there exists a constant $A>0$ such that $F(t,u)\geq A|u|^\mu$ for $|u|\geq M$. On the other hand, we can easily get that there exists a constant $B$ such that $F(t,u)\geq B$ for $| u| \leq M$. Then together with Lemma \ref{L6}, we have \[ %4.4 I(yu)\leq \frac{y^2}{2|\cos(\pi \alpha)|}\|u\|^2- Ay^\mu \int_0^T| u|^\mu dt -B- \frac{\lambda }{r}y^r\int_0^T h(t)|u(t)|^{r}dt. \] Then, we can get that $I(yu)\to -\infty$, as $y\to \infty$. Let $g(y):=I(yu)$ for $y>0$. From what we have proved, there hat at least one $ y_u=y(u)>0$ such that \begin{equation} \label{e4.5} g(y_u)=\max_{z\geq 0}g(z)=\max_{z\geq 0}I(zu)=I(y_uu). \end{equation} We prove next that $g(y)$ has a unique critical point for $y>0$. Consider a critical point \begin{equation} \label{e4.6} \begin{aligned} g'(y)&=\langle I'(yu),u\rangle\\ &=- \int_0^T y( _0D_t^{\alpha}u,_tD_T^{\alpha}u)dt-\int_0^T f(t,yu)u\,dt - \lambda y^{r}\int_0^Th(t)|u|^{r}dt\\ =0 \end{aligned} \end{equation} Then,from (H5), we have \begin{align*} %4.7 g''(y) &=- \int_0^T ( _0D_t^{\alpha}u,_tD_T^{\alpha}u) - \int_0^T \frac{\partial f(t,yu)}{\partial (yu)}u^2dt - \lambda ry^{r-1}\int_0^Th(t)|u|^{r}dt\\ &= \int_0^T \frac{f(t,yu)u}{y}dt - \int_0^T \frac{\partial f(t,yu)}{\partial (yu)}u^2dt - \lambda r y^{r-1}\int_0^Th(t)|u|^{r}dt <0. \end{align*} So we know that if $y$ is a critical point of $g$, then it must be a strict local maximum. This implies the uniqueness. Finally, from \begin{equation} \label{e4.8} g'(y)=\langle I'(yu),u\rangle= \frac{1}{y}\langle I'(yu),yu\rangle , \end{equation} we see $y$ is critical point if $yu \in \mathcal{N}$. Define $m=\inf_{\mathcal{N}}I$. Then we can get that $m\geq \inf_{\partial B_{\rho}(0)}I\geq \delta>0$. \smallskip \noindent\textbf{Step 2:} There exists $u\in \mathcal{N}$ such that $I(u)=m$. We claim that both $I$ and $\phi$ are weakly lower semicontinuous. In fact, according to Lemma \ref{L5}, if $u_k\rightharpoonup u $ in $E^\alpha$, then $u_k\to u$ in $C([0,T],\mathbb{R})$. Therefore, $F(t,u_k(t))\to F(t,u(t))$ a.e. $t\in [0,T]$. By the Lebesgue dominated convergence theorem, we have \[ \int_0^T F(t,u_k(t))dt\to \int_0^T F(t,u(t))dt \] which means that the functional $ u\to \int_0^TF(t,u(t))dt$ is weakly continuous on $E^\alpha$. Similarly $u\to \int_0^T f(t,u(t))u(t)dt$ is weakly continuous on $E^\alpha$. Furthermore, $\int_0^T h(t)|u_k(t)|^{r}dt\to \int_0^T h(t)|u(t)|^{r}dt$. Since $E^\alpha$ is Hilbert space, from \eqref{Neq} and Lemma \ref{L6}, we can easily obtain that $- \int_0^T ( _0^CD_t^{\alpha}u(t),_t^CD_T^{\alpha}u(t))dt$ is weakly lower semicontinuous on $E^\alpha$. Then both $I$ and $\phi$ are weakly lower semicontinuous. Since $\mu F(t,u)-uf(t,u)$ is continuous for $t\in [0,T]$ and $| x| \leq M$, there exists $B> 0$, such that \begin{equation} \label{e4.9} F(t,u)\leq \frac{1}{\mu}f(t,u)+B,\quad t\in [0,T],\; | x| \leq M. \end{equation} From (H4) we obtain \begin{equation} \label{e4.10} F(t,u)\leq \frac{1}{\mu}f(t,u)+B,\quad t\in [0,T],\; x \in \mathbb{R}. \end{equation} Let $\{u_k\}\in \mathcal{N}$ be a minimizing sequence; that is, $I(u_k)\to m$, $I'(u_k)\to 0$ as $k\to\infty$. Then \begin{align*} %4.11 & m+o(1)\\ &=I(u_k) \\ &= - \frac{1}{2}\int_0^T ( _0^CD_t^{\alpha}u_k(t),_t^CD_T^{\alpha}u_k(t))dt - \int_0^T F(t,u_k(t))dt- \frac{\lambda}{r}\int_0^T h(t)|u_k(t)|^{r}dt, \\ &\geq - \frac{1}{2}\int_0^T ( _0^CD_t^{\alpha}u_k(t),_t^CD_T^{\alpha}u_k(t))dt -\frac{1}{\mu}\int_0^T u_kf(t,u_k)dt-BT- C_\lambda\| u_k\|^r, \\ &= \big(\frac{1}{\mu}-\frac{1}{2}\big) \int_0^T ( _0^CD_t^{\alpha}u_k(t),_t^CD_T^{\alpha}u_k(t))dt + \frac{1}{\mu}\langle I'(u_k),u_k\rangle -BT- C_\lambda\| u_k\|^r,\\ &\geq \big(\frac{1}{2}-\frac{1}{\mu}\big)|\cos(\pi \alpha)|\| u_k\|^2 - \frac{1}{\mu}\| I'(u_k)\|\| u_k\|-BT- C_\lambda\| u_k\|^r. \end{align*} By $\mu >2>r$ and $I'(u_k)\to 0$, we obtain that $u_k$ is bounded in $E^\alpha$. Since $E^\alpha$ is a reflexive space, going to a subsequence if necessary, we may assume that $u_k\to u$ in $C([0,T],\mathbb{R})$. Since $\phi$ is weakly lower semicontinuous and ${u_k} \in \mathcal{N}$, we first have \begin{equation} \label{e4.12} \phi(u)\leq \liminf_{k\to\infty}\phi(u_k)=0. \end{equation} Then we have $u\neq 0$. In fact, if $u=0$, then $u_k\to u$ in $C([0,T],\mathbb{R})$. By $\phi(u_k)=0$, we obtain $\| u_k\|\to 0$. This is a contradiction with ${u_k} \in \mathcal{N}$. Then from Step 1, there exists a unique $y>0$ such that $yu \in \mathcal{N}$. From this and $I$ being weakly lower semicontinuous, we have \[ %\label{e4.13} m \leq I(yu) \leq \liminf_{k\to\infty}I(yu_k) \leq \lim_{k\to\infty}I(yu_k) \leq \lim_{k\to\infty}I(u_k)=m \] Then we obtain that $m$ is achieved at $yu\in \mathcal{N}$. Finally, from Step 1 and Step2, we obtain $u\in \mathcal{N}$ such that $I(u)=m=\inf_{\mathcal{N}}I$ which is a critical point of $I|_\mathcal{N}$. On the other hand from Lemma \ref{L7} we have $I'(u)=0$. Consequently \eqref{e3} has a weak solution such that $I(u)=m$. The proof is complete. \section{An example} In this section, we give an example to illustrate our results. Let $g$ and $h$ be two nonnegative continuous functions on $[0,T]$, we consider the problem \begin{gather*} \frac{d}{d t}\Big(\frac{1}{2}_0D_t^{-\frac{1}{2}}(u'(t)) +\frac{1}{2}_tD_T^{-\frac{1}{2}}(u'(t))\Big) =g(t)|u(t)|^{p-2}u(t)+ \lambda h(t)|u(t)|^{r-2}u(t), \\ \text{a.e. } t \in [0,T],\\ u(0)=u(T)=0, \end{gather*} where $1