\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 176, pp. 1--9.\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-2012/176\hfil Existence of solutions]
{Existence of solutions for a  nonlinear fractional
boundary value problem via a local minimum theorem}

\author[C. Bai \hfil EJDE-2012/176\hfilneg]
{Chuanzhi Bai}  % in alphabetical order

\address{Chuanzhi Bai \newline
Department of Mathematics, Huaiyin Normal University,
Huaian, Jiangsu 223300, China}
\email{czbai8@sohu.com} 

\thanks{Submitted July 30, 2012. Published October 12, 2012.}
\subjclass[2000]{58E05, 34B15, 26A33}
\keywords{Critical points; fractional differential equations;
\hfill\break\indent  boundary-value problem}

\begin{abstract}
 This article concerns the existence of solutions to the  nonlinear
 fractional boundary-value problem
 \begin{gather*}
 \frac{d}{dt} \Big({}_0 D_t^{\alpha-1}({}_0^c D_t^{\alpha} u(t))
 -{}_t D_T^{\alpha-1}({}_t^c D_T^{\alpha} u(t))\Big)
 +\lambda f(u(t)) = 0, \quad\text{a.e. } t \in [0, T], \\
 u(0) = u(T) = 0,
 \end{gather*}
 where $\alpha \in (1/2, 1]$,  and $\lambda$ is a positive real parameter.
 The approach is based on a local minimum theorem established by Bonanno.
\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 been  proved to be  valuable
tools in the  modeling  of many phenomena in various fields of physic, 
chemistry, biology, engineering and economics.  There has been significant development
in fractional differential equations, one can see the monographs of
 Miller and Ross \cite{m2}, Samko et al \cite{s1},
Podlubny \cite{p1}, Hilfer \cite{h1}, Kilbas et al \cite{k1} and the 
papers  \cite{a1,a2,b1,b2,b3,b4,k2,l1,w1,w2,z1} and the
 references therein. 

Critical point theory has been very useful in determining
the existence of solution for integer order differential equations
with some boundary conditions, for example
\cite{c1,l1,l2,m1,r1,t1}.
 But until now, there are few results on the solution to
fractional BVP which were established by the critical point theory,
since it is often very difficult to establish a suitable space and
variational functional for fractional BVP. Recently, Jiao and Zhou
\cite{j1} investigated the fractional boundary-value problem
\begin{gather*}
\frac{d}{dt} \Big(\frac{1}{2} {}_0 D_t^{-\beta}(u'(t)) + \frac{1}{2}
 {} _t D_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{gather*}
 by using the critical point theory, where $_0 D_t^{-\beta}$ and
 $_t D_T^{-\beta}$
are the left and right Riemann-Liouville fractional integrals of order 
$0 \le \beta < 1$ respectively,
$F : [0, T] \times \mathbb{R}^N \to \mathbb{R}$ is a given function and
$\nabla  F(t, x)$ is the gradient of $F$ at $x$.

In this article,  by using a local minimum theorem established by
 Bonanno in \cite{b5},
a new approach is provided to investigate the existence of solutions to the
following  fractional boundary value problems
\begin{equation}
\begin{gathered}
\frac{d}{dt} \Big({}_0 D_t^{\alpha-1}({}_0^c D_t^{\alpha} u(t))
-  {}_t D_T^{\alpha-1}({}_t^c D_T^{\alpha} u(t))\Big)
+ \lambda f(u(t)) = 0, \quad\text{a.e. } t \in [0, T], \\
   u(0) = u(T) = 0,
  \end{gathered}   \label{e1.1}
\end{equation}
  where $\alpha \in (1/2, 1]$, ${}_0 D_t^{\alpha-1}$ and
${}_t D_T^{\alpha-1}$ are the left and
 right  Riemann-Liouville fractional integrals of order $1-\alpha$ respectively,
$_0^c D_t^{\alpha}$ and $_t^c D_T^{\alpha}$ are the left and  right Caputo
 fractional derivatives of order $0< \alpha \leq 1$ respectively, $\lambda$ is
a positive real parameter,  and $f : \mathbb{R} \to \mathbb{R}$ is a
continuous function.


\section{Preliminaries}

 In this section, we  introduce some definitions and properties of the 
fractional calculus which are used in  this article.

\begin{definition}[\cite{k1}] \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 a function $f$  are defined by
\begin{gather*}
{}_a D_t^{-\alpha}f(t) = \frac{1}{\Gamma(\alpha)}
\int_a^t (t-s)^{\alpha-1} f(s) ds, \quad t \in [a, b], \; \alpha > 0,\\
{}_t D_b^{-\alpha}f(t) = \frac{1}{\Gamma(\alpha)}
\int_t^b (s-t)^{\alpha-1} f(s) ds, \quad t \in [a, b], \; \alpha > 0,
\end{gather*}
 provided the right-hand sides are pointwise defined on
$[a, b]$, where $\Gamma(\alpha)$ is the standard gamma function.
\end{definition}

\begin{definition}[\cite{k1}] \label{def2.2}  \rm
 Let $\gamma \ge 0$ and $n \in \mathbb{N}$.

(i) If $\gamma \in (n-1, n)$ and $f \in AC^n([a, b], \mathbb{R}^N)$, then the
left and right Caputo fractional derivatives of order $\gamma$ for
function $f$ denoted by $_a^c D_t^{\gamma}f(t)$ and $_t^c
D_b^{\gamma}f(t)$, respectively, exist almost everywhere on $[a,
b]$, $_a^c D_t^{\gamma}f(t)$ and $_t^c D_b^{\gamma}f(t)$ are
represented by
\begin{gather*}
{}_a^c D_t^{\gamma}f(t) =
\frac{1}{\Gamma(n-\gamma)} \int_a^t (t-s)^{n-\gamma-1} f^{(n)}(s)
ds, \quad t \in [a, b],\\
{}_t^c D_b^{\gamma}f(t) =
\frac{(-1)^n}{\Gamma(n-\gamma)} \int_t^b (s-t)^{n-\gamma-1} f^{(n)}(s)
ds, \quad t \in [a, b], 
\end{gather*}
 respectively.

(ii) If $\gamma = n - 1$ and $f \in AC^{n-1}([a, b], \mathbb{R}^N)$, then
$_a^c D_t^{n-1}f(t)$ and $_t^c D_b^{n-1}f(t)$ are represented by
\[
{}_a^c D_t^{n-1}f(t) = f^{(n-1)}(t), 
\quad\text{and}\quad _t^c D_b^{n-1}f(t) = (-1)^{(n-1)} f^{(n-1)}(t), \quad t
\in [a, b].
\]
\end{definition}

With these definitions, we have the rule for fractional integration
by parts, and the composition of the Riemann-Liouville  fractional
integration operator with the Caputo fractional differentiation
operator, which were proved in \cite{k1,s1}.

\begin{proposition}[\cite{k1,s1}] \label{prop2.1}
We have the following property of fractional integration
\begin{equation}
\int_a^b [_a D_t^{-\gamma}f(t)] g(t) dt = \int_a^b [_t
D_b^{-\gamma}g(t)] f(t) dt, \quad  \gamma > 0,\label{e2.1}
\end{equation}
 provided that $f \in L^p([a, b], \mathbb{R}^N)$,
 $g \in L^q([a, b], \mathbb{R}^N)$ and $p \ge 1$,
$q \ge 1$, $1/p + 1/q \le 1 + \gamma$ or $p
\neq  1$, $q \neq  1$, $1/p + 1/q = 1 + \gamma$.
\end{proposition}


\begin{proposition}[\cite{k1}] \label{prop2.2 }
Let $n \in \mathbb{N}$ and $n-1 < \gamma \le n$.
 If $f \in AC^n([a, b], \mathbb{R}^N)$
or $f \in C^n([a, b], \mathbb{R}^N)$, then
\begin{gather*}
{}_a D_t^{-\gamma}({}_a^c D_t^{\gamma} f(t)) 
= f(t) - \sum _{j=0}^{n-1} \frac{f^{(j)}(a)}{j!}(t-a)^j\,,\\
{}_t D_b^{-\gamma}({}_t^c D_b^{\gamma} f(t)) 
= f(t) - \sum _{j=0}^{n-1} \frac{(-1)^jf^{(j)}(b)}{j!}(b-t)^j,
\end{gather*}
 for $t \in [a, b]$. In particular, if $0 < \gamma \le 1$ and
 $f \in AC([a, b], \mathbb{R}^N)$
or $f \in C^1([a, b], \mathbb{R}^N)$, then
\begin{equation}
{}_a D_t^{-\gamma}({}_a^c D_t^{\gamma} f(t)) = f(t) - f(a), \quad \text{and} \quad
_t D_b^{-\gamma}({}_t^c D_b^{\gamma} f(t)) = f(t) - f(b).   \label{e2.2}
\end{equation}
\end{proposition}


 \begin{remark} \label{rmk2.1} \rm
 In view of \eqref{e2.1} and Definition \ref{def2.2}, it is obvious 
that $u \in AC([0, T])$ is a solution of \eqref{e1.1} if and only if $u$ 
is a solution of the  problem
\begin{equation}
\begin{gathered}
\frac{d}{dt} \left({}_0 D_t^{-\beta}(u'(t)) + {}_t D_T^{-\beta}(u'(t))\right)
+ \lambda f(u(t)) = 0, \quad\text{a.e. } t \in [0, T], \\
   u(0) = u(T) = 0,
  \end{gathered} \label{e2.3}
\end{equation}
  where $\beta = 2(1-\alpha) \in [0, 1)$.
\end{remark}


To establish a variational structure for \eqref{e1.1}, it is necessary
to construct appropriate function spaces.
Denote by $C_0^{\infty}[0, T]$ the set of all functions
 $g \in C^{\infty}[0, T]$ with $g(0) = g(T) = 0$.


\begin{definition}[\cite{j1}] \label{def2.3} \rm
 Let $0 < \alpha \le 1$. The fractional derivative space $E_0^{\alpha}$  is defined
 by the closure of  $C_0^{\infty}[0, T]$ with respect to the norm
\[
\|u\|_{\alpha}
  = \Big(\int_0^T |_0^c D_t^{\alpha} u(t)|^2 dt + \int_0^T  |u(t)|^2 dt   \Big)^{1/2},
   \quad \forall u \in E^{\alpha}.
\]
\end{definition}

  \begin{remark} \label{rmk2.2} \rm
 It is obvious that the fractional derivative space $E_0^{\alpha}$  is the space of
 functions $u \in L^2[0, T]$  having an $\alpha$-order Caputo fractional
 derivative $_0^c D_t^{\alpha} u \in L^2[0, T]$ and $u(0) = u(T) = 0$.
\end{remark}


\begin{proposition}[\cite{j1}] \label{prop2.3 }
Let $0  < \alpha \le 1$. The fractional derivative space $E_0^{\alpha}$ is 
reflexive and separable Banach space.
\end{proposition}


 \begin{lemma}[\cite{j1}] \label{lem2.1} 
 Let $ 0 < \alpha \le 1$. For all $u\in E_0^{\alpha}$, we have
\begin{gather}
 \|u\|_{L^2}  \le
\frac{T^{\alpha}}{\Gamma(\alpha+1)} \|\empty_0^c
D_t^{\alpha}u\|_{L^2}, \label{e2.4} \\ 
 \|u\|_{\infty} \le
 \frac{T^{\alpha - 1/2}}{\Gamma(\alpha)(2(\alpha-1)+1)^{1/2}} \|_0^c D_t^{\alpha} 
u\|_{L^2}.  
\label{e2.5}
\end{gather}
\end{lemma}

 By \eqref{e2.4}, we can consider $E_0^{\alpha}$ with respect to the norm
\begin{equation}
\|u\|_{\alpha} = \Big(\int_0^T |_0^c D_t^{\alpha} u(t)|^2 dt\Big)^{1/2}
 = \|_0^c D_t^{\alpha} u\|_{L^2},    \quad \forall u \in E_0^{\alpha} \label{e2.6}
\end{equation}
in the following analysis.


 \begin{lemma}[\cite{j1}] \label{lem2.2}
 Let $ 1/2 < \alpha \le 1$, then for
all any  $u \in E_0^{\alpha}$, we have
\begin{equation}
|\cos (\pi \alpha)| \|u\|_{\alpha}^2
\le  - \int_0^T {}_0^c D_t^{\alpha} u(t) \cdot {}_t^c
D_T^{\alpha} u(t) dt \le \frac{1}{|\cos (\pi \alpha)|}
\|u\|_{\alpha}^2. \label{e2.7}
\end{equation}
\end{lemma}

Our main tools is the local minimum theorem \cite{b5} which is 
recalled below.
Given a set $X$ and two functionals $\Phi, \Psi : X \to \mathbb{R}$, let
\begin{gather}
 \beta(r_1, r_2) = \inf _{v \in \Phi^{-1}(]r_1, r_2[)}
 \frac{\sup_{u \in \Phi^{-1}(]r_1, r_2[)} \Psi(u)  - \Psi(v)}{r_2
- \Phi(v)}, \label{e2.8}\\
 \rho_2(r_1, r_2) = \sup _{v \in \Phi^{-1}(]r_1, r_2[)} \frac{\Psi(v) 
- \sup_{u \in \Phi^{-1}(]-\infty, r_1])} \Psi(u)}{\Phi(v) - r_1},  
\label{e2.9}
\end{gather}
for all $r_1, r_2 \in R$, with $r_1 < r_2$.

 \begin{theorem}[\cite{b5}] \label{thm2.1} 
 Let $X$ be a reflexive real Banach space; $\Phi : X \to \mathbb{R}$ 
be a sequentially weakly lower semicontinuous, coercive and continuously 
Gateaux differential function whose Gateaux derivative admits a continuous inverse
on $X^*$; $\Psi : X \to \mathbb{R}$ be a continuously Gateaux differentiable 
function whose Gateaux derivative is compact. Put $I_{\lambda}
= \Phi - \lambda \Psi$ and assume that there are $r_1, r_2 \in \mathbb{R}$, 
with $r_1 < r_2$, such that
\[
 \beta(r_1, r_2) <  \rho_2(r_1, r_2),
\]
 where $\beta$ and $\rho_2$ are given by  \eqref{e2.8} and \eqref{e2.9}.
Then, for each $\lambda \in  \big(\frac{1}{\rho_2(r_1, r_2)},
\frac{1}{\beta(r_1, r_2)}\big)$ there is $u_{0, \lambda} \in
\Phi^{-1}(]r_1, r_2[)$ such that $I_{\lambda}(u_{0, \lambda})
 \le I_{\lambda}(u) $ for all $u \in \Phi^{-1}(]r_1, r_2[)$ 
and $I'_{\lambda}(u_{0, \lambda}) = 0$. 
\end{theorem}


\section{Main result}

For given $u \in E_0^{\alpha}$, we define functionals
 $\Phi, \Psi : E^{\alpha} \to \mathbb{R}$ as follows:
\[
\Phi(u) := - \int_0^T {}_0^c D_t^{\alpha} u(t) \cdot {}_t^c D_T^{\alpha} u(t) dt,
\quad
 \Psi(u) := \int_0^T F(u(t))dt, 
\]
 where $F(u) = \int_0^u f(s)ds$. Clearly, $\Phi$ and $\Psi$ are Gateaux 
differentiable functional whose Gateaux derivative at the point 
$u \in E_0^{\alpha}$ are given by
\begin{gather*}
\Phi'(u)v = - \int_0^T({}_0^c D_t^{\alpha} u(t) \cdot
{}_t^c D_T^{\alpha} v(t) + {}_t^c D_T^{\alpha} u(t) \cdot {}_0^c D_t^{\alpha} v(t)) dt,
\\
\Psi'(u)v =  \int_0^T f(u(t)) v(t) dt = - \int_0^T \int_0^t f(u(s))ds \cdot v'(t) dt,
\end{gather*}
for every $v \in E_0^{\alpha}$.  By  Definition \ref{def2.2} and \eqref{e2.2},  
we have
\[
 \Phi'(u)v =
 \int_0^T({}_0 D_t^{\alpha-1}({}_0^c D_t^{\alpha} u(t))
 - {}_t D_T^{\alpha-1}({}_t^c D_T^{\alpha} u(t)))  \cdot v'(t) dt.
\]
Hence, $I_{\lambda} = \Phi - \lambda \Psi \in C^1(E_0^{\alpha}, \mathbb{R})$. 
If $u_* \in E_0^{\alpha}$ is a critical  point of $I_{\lambda}$,
then
\begin{equation}
\begin{aligned}
 0 =  I_{\lambda}'(u_*)v 
&=\int_0^T \Big({}_0 D_t^{\alpha-1}({}_0^c D_t^{\alpha} u_*(t))
 - {}_t D_T^{\alpha-1}({}_t^c D_T^{\alpha} u_*(t)) \\
&\quad + \lambda \int_0^t f(u_*(s))ds\Big)  \cdot v'(t) dt,
\end{aligned}\label{e3.1}
\end{equation}
  for $v \in E_0^{\alpha}$. We can choose $v \in E_0^{\alpha}$ such that
\[
v(t) = \sin \frac{2k\pi t}{T}   \quad  \text{ or} \quad
 v(t) = 1 - \cos \frac{2k\pi t}{T},
 \quad k = 1, 2, \dots .
\]
The theory of Fourier series and \eqref{e3.1} imply
\begin{equation}
 {}_0 D_t^{\alpha-1}({}_0^c D_t^{\alpha} u_*(t))
 - {}_t D_T^{\alpha-1}({}_t^c D_T^{\alpha} u_*(t))
+ \lambda \int_0^t f(u_*(s))ds = C \label{e3.2}
\end{equation}
  a.e. on $[0, T]$ for some $C \in \mathbb{R}$. By \eqref{e3.2}, it is
 easy to show  that $u_* \in E_0^{\alpha}$
  is a solution of \eqref{e1.1}.

  By Lemma \ref{lem2.1},  when $\alpha > 1/2$,  for each
 $u \in E_0^{\alpha}$ we have
\begin{equation}
\|u\|_{\infty} \le   \Omega \Big(\int_0^T |_0^c D_t^{\alpha} u(t)|^2 dt
\Big)^{1/2}
 = \Omega \|u\|_{\alpha},
\label{e3.3}
\end{equation}
 where
\begin{equation}
\Omega =  \frac{T^{\alpha-\frac{1}{2}}}{\Gamma(\alpha) \sqrt{2(\alpha-1) + 1}}.
\label{e3.4}
\end{equation}
 Given two constants $c \ge 0$ and $d \neq 0$,
 with $c \neq  \sqrt{\frac{\omega_{\alpha, d}}{|\cos(\pi \alpha)|}} \cdot \Omega$,
 where $\Omega$ as in \eqref{e3.4}.
Put
\[
\omega_{\alpha, d} := \frac{4\Gamma^2(2-\alpha)}{\Gamma(4-2\alpha)}
T^{1-2\alpha} d^2 (2^{2\alpha-1}-1).
\]



\begin{theorem} \label{thm3.1} 
Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function and 
$\frac{1}{2} < \alpha \le 1$.
Assume that there exist a positive constant $c$ and a constant $d \neq  0$ with
\begin{equation}
 \sqrt{\frac{\omega_{\alpha, d}}{|\cos(\pi \alpha)|}} \, \Omega  < c,  \label{e3.5}
\end{equation}
such that
\begin{equation}
0 <  \frac{ \max _{|\eta| \le c} F(\eta)}{c^2 |\cos(\pi \alpha)|} <
\frac{\frac{1}{\Gamma(2-\alpha)|d|}
\int_0^{\Gamma(2-\alpha)|d|} F(x)dx}{ \omega_{\alpha,d}\Omega^2}.
 \label{e3.6}
\end{equation}
 Then, for each
\[
\lambda \in \Big(\frac{\omega_{\alpha,d} \Gamma(2-\alpha)|d|}
{T \int_0^{\Gamma(2-\alpha)|d|} F(x)dx},
  \frac{c^2 |\cos(\pi \alpha)|}{T\Omega^2 \max _{|\eta| \le c} F(\eta)}\Big),
\]
problem \eqref{e1.1} admits at least one  solution $\bar{u}$ such that
$ \|\bar{u}\|_{\alpha} < c/\Omega$.
\end{theorem}

\begin{proof}
Let $\Phi, \Psi$ be the functionals defined above. It is well known that they
satisfy all regularity assumptions requested in Theorem \ref{thm2.1} and that the
 critical point of the functional $\Phi - \lambda \Psi$ in $E_0^{\alpha}$ is
exactly the solution of \eqref{e1.1}.   Put
\begin{equation} \label{e3.7}
\begin{gathered}
  r = \frac{|\cos(\pi \alpha)|}{\Omega^2} c^2, \\
  u_0(t) = \begin{cases}
\frac{2 \Gamma(2-\alpha) d}{T} t, &  t \in [0, T/2),\\
 \frac{2 \Gamma(2-\alpha) d}{T}(T - t), &  t \in [T/2, T]
\end{cases}
\end{gathered}
\end{equation}
 It is easy to check that  $u_0(0) = u_0(T) = 0$ and $u_0 \in L^2[0, T]$.
The direct calculation shows that
\[
 {}_0^c D_t^{\alpha} u_0(t) = \begin{cases}
\frac{2d}{T} t^{1-\alpha}, &  t \in [0,T/2),\\
\frac{2d}{T}(t^{1-\alpha} - 2 (t-\frac{T}{2})^{1-\alpha}), &   t \in [T/2, T]
 \end{cases}
\]
and
\begin{align*}
\|u_0\|_{\alpha}^2
&= \int_0^T ({}_0^c D_t^{\alpha} u_0(t))^2 dt =
\int_0^{\frac{T}{2}} + \int_{T/2}^T ({}_0^c D_t^{\alpha} u_0(t))^2 dt \\
&= \frac{4d^2}{T^2}\Big[\int_0^T t^{2(1-\alpha)} dt
-  4 \int_{T/2}^T  t^{1-\alpha} (t-\frac{T}{2})^{1-\alpha} dt + 4
  \int_{T/2}^T (t-\frac{T}{2})^{2(1-\alpha)} dt \Big]\\
&= \frac{4(1+2^{2\alpha-1})d^2}{3-2\alpha} T^{1-2\alpha}  -  \frac{16d^2}{T^2}
\int_{T/2}^T  t^{1-\alpha} (t-\frac{T}{2})^{1-\alpha} dt  < \infty.
\end{align*}
That is, ${}_0^c D_t^{\alpha} u_0 \in L^2[0, T]$.  Thus,
 $u_0 \in E_0^{\alpha}$.
 Moreover, the direct calculation shows
\[
 {}_t^c D_T^{\alpha} u_0(t) = \begin{cases}
\frac{2d}{T}((T - t)^{1-\alpha} - 2 (\frac{T}{2} - t)^{1-\alpha}), &  t \in [0,T/2),\\
 \frac{2d}{T}(T - t)^{1-\alpha}, &  t \in [T/2, T]
\end{cases}
\]
and
\begin{align*}
 \Phi(u_0)
&= - \int_0^T {}_0^c D_t^{\alpha} u_0(t) \cdot {}_t^c D_T^{\alpha} u_0(t) dt
\\
&= - (\frac{2d}{T})^2 \Big[\int_0^{\frac{T}{2}} t^{1-\alpha}
\Big((T - t)^{1-\alpha} - 2 \big(\frac{T}{2} - t\big)^{1-\alpha}\Big)dt
 \\
&\quad + \int_{T/2}^T (T - t)^{1-\alpha} \cdot
\Big(t^{1-\alpha} - 2 (t-\frac{T}{2})^{1-\alpha}\Big)dt\Big]
\\
&= - (\frac{2d}{T})^2 \Big[\int_0^T t^{1-\alpha} (T - t)^{1-\alpha} dt - 4
\int_0^{\frac{T}{2}} t^{1-\alpha} \big(\frac{T}{2} - t\big)^{1-\alpha}dt \Big]
\\
&= - (\frac{2d}{T})^2 \Big[\frac{\Gamma^2(2-\alpha)}{\Gamma(4-2\alpha)} T^{3-2\alpha}
 - 4 \frac{\Gamma^2(2-\alpha)}{\Gamma(4-2\alpha)} (\frac{T}{2})^{3-2\alpha}\Big]
\\
&= \frac{4\Gamma^2(2-\alpha)}{\Gamma(4-2\alpha)} T^{1-2\alpha}  (2^{2\alpha-1}-1)d^2
= \omega_{\alpha, d},
\end{align*}
 and
\[
 \Psi(u_0) = \int_0^T F(u_0(t)) dt
= \frac{T}{\Gamma(2-\alpha) |d|} \int_0^{\Gamma(2-\alpha) |d|} F(x) dx.
\]
Hence, from \eqref{e3.5},  one has $ 0  < \omega_{\alpha, d} <
  \frac{|\cos(\pi \alpha)|}{\Omega^2} c^2$; that is,
 $0  < \Phi(u_0) < r$.
Moreover, for all $u \in E_0^{\alpha}$ such that $u \in \Phi^{-1}(]-\infty, r])$,
 by \eqref{e2.7} we have
\[
|\cos (\pi \alpha)| \|u\|_{\alpha}^2 \le \Phi(u)  \le r,
\]
 which implies
\begin{equation}
 \|u\|_{\alpha}^2 \le \frac{1}{|\cos(\pi \alpha)|} r.\label{e3.8}
\end{equation}
 Thus,  by \eqref{e3.3}, \eqref{e3.8} and \eqref{e3.7} we obtain
\[
 |u(t)| < \Omega \|u\|_{\alpha}
\le \Omega \sqrt{\frac{r}{|\cos(\pi \alpha)|}} = c,
\quad \forall t \in [0, T].
\]
 Hence,
\[
 \Psi(u) = \int_0^T F(u(t)) dt \le \int_0^T \max _{|\eta| \le c} F(\eta) dt
= T \max _{|\eta| \le c} F(\eta),
\]
for all $u \in E_0^{\alpha}$ such that $u \in \Phi^{-1}(]-\infty, r])$. Hence,
\[
 \sup _{u \in \Phi^{-1}(]-\infty, r])} \Psi(u) \le
 T \max _{|\eta| \le c} F(\eta).
\]
 Hence, one has
\begin{equation}
\begin{aligned}
 \beta(0, r) 
&\le \frac{\sup_{u \in \Phi^{-1}(]-\infty, r])} \Psi(u)
 - \Psi(u_0)}{r - \Phi(u_0)}
\\
& \le  \Omega^2 T \frac{ \max _{|\eta| \le c} F(\eta)
- \frac{1}{\Gamma(2-\alpha)|d|} \int_0^{\Gamma(2-\alpha) |d|}
F(x)dx}{|\cos(\pi \alpha)| c^2 - \omega_{\alpha,d} \Omega^2}
\\
& < \Omega^2 T   \frac{ \max _{|\eta| \le c} F(\eta)
- \frac{\omega_{\alpha,d} \Omega^2}{|\cos(\pi \alpha)| c^2}
 \max _{|\eta| \le c} F(\eta)}{|\cos(\pi \alpha)| c^2 - \omega_{\alpha,d} \Omega^2}
\\
& = \Omega^2 T  \frac{ \max _{|\eta| \le c} F(\eta)}{c^2 |\cos(\pi \alpha)|},
\end{aligned}\label{e3.9}
\end{equation}
by condition \eqref{e3.6}.  On the other hand, if $u \in \Phi^{-1}(]-\infty, 0])$,
then $\Phi(u) \le 0$. Thus, by \eqref{e2.4} and \eqref{e2.7}
  we have $ \|u\|_{L^2} = 0$; that is, $u(t) = 0$,   a.e. $t \in [0, T]$.  Hence,
\begin{equation}
\begin{aligned}
 \rho_2(0, r)
&\ge \frac{\Psi(u_0) - \sup_{u \in \Phi^{-1}(]-\infty, 0])} \Psi(u)}{\Phi(u_0)}
= \frac{\Psi(u_0)}{\Phi(u_0)} \\
& =  T \frac{ \frac{1}{\Gamma(2-\alpha)|d|}
\int_0^{\Gamma(2-\alpha) |d|}  F(x)dx }{\omega_{\alpha,d}}.
\end{aligned}\label{e3.10}
\end{equation}
Thus, by \eqref{e3.9}, \eqref{e3.10} and \eqref{e3.6}  it follows that
 $\beta(0, r)  < \rho_2(0, r)$.
So, from Theorem \ref{thm2.1} for each
\[
\lambda \in   \Big(\frac{\omega_{\alpha,d} \Gamma(2-\alpha)|d|}{T \int_0^{\Gamma(2-\alpha) |d|} F(x)dx},
  \frac{c^2 |\cos(\pi \alpha)|}{T\Omega^2 \max _{|\eta| \le c} F(\eta)}\Big)
   \subset    \Big(\frac{1}{\rho_2(0, r)},  \ \frac{1}{\beta(0, r)}\Big),
\]
the function  $\Phi - \lambda \Psi$ admits at least one critical point $\bar{u}$
such that  $ 0 < \Phi(\bar{u}) < r$; that is,
 $\|\bar{u}\|_{\alpha} < \frac{c}{\Omega}$,
and the conclusion is achieved.
\end{proof}


We conclude with an example that illustrates the results obtained here.
 Let $\alpha = 0.8$, $T = 1$, and $f(u) = \cos (\pi u/3)$.
 Then \eqref{e1.1} reduces to the  boundary-value problem
\begin{equation}
\begin{gathered}
\frac{d}{dt} \Big({}_0 D_t^{-0.2}({}_0^c D_t^{0.8} u(t))
-  {}_t D_1^{-0.2}({}_t^c D_1^{0.8} u(t))\Big)
+ \lambda  \cos (\frac{\pi}{3}u(t))  = 0, \quad\text{a.e. }t \in [0, 1], \\
   u(0) = u(1) = 0.
  \end{gathered}
\label{e3.11}
\end{equation}
 Owing to Theorem \ref{thm3.1}, for each $\lambda \in  (3.2964, 4.30512)$,
 boundary-value problem \eqref{e3.11} admits at least one  solution.
In fact, put $c = 2.5$ and $d = 1$, it is easy to calculate that
$\Omega = 1.1089$, $\omega_{0.8, 1} = 1.4$ and
\[
 \sqrt{\frac{\omega_{0.8, 1}}{|\cos(0.8 \pi)|}} \, \Omega
= 1.4588  < 2.5 =  c.
\]
Moreover, we have
\begin{equation}
\frac{\frac{1}{\Gamma(2-\alpha)|d|} \int_0^{\Gamma(2-\alpha) |d|} F(x)dx}{ \omega_{\alpha,d} \Omega^2}
 = \frac{\frac{1}{\Gamma(1.2)} \int_0^{\Gamma(1.2)} \frac{3}{\pi}
 \sin (\pi x/3) dx}{\omega_{0.8,1} \cdot \Omega^2} = 0.2467,
 \label{e3.12}
\end{equation}
   and
\begin{equation}
  \frac{ \max _{|\eta| \le c} F(\eta)}{c^2 |\cos(\pi \alpha)|}
 = \frac{3/\pi }{2.5^2 \cdot |\cos(0.8 \pi)|} = 0.1889,
 \label{e3.13}
\end{equation}
which implies that condition \eqref{e3.6} holds. Thus, by
Theorem \ref{thm3.1}, for each  $\lambda \in (3.2964, 4.3051)$,  problem
\eqref{e3.11} admits at least one  solution $\bar{u}$ such that
$\|\bar{u}\|_{0.8} < 2.2545$.

\subsection*{Acknowledgements}
This work is supported by Natural Science Foundation of Jiangsu Province 
(grant BK2011407),
and  by the Natural Science Foundation of China (grant 10771212).

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\end{document}