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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 324, pp. 1--10.\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-2016/324\hfil Implicit fractional differential equations]
{Nonlinear implicit differential equations of fractional order at resonance}

\author[M. Benchohra, S. Bouriah, J. R. Graef \hfil EJDE-2016/324\hfilneg]
{Mouffak Benchohra, Soufyane Bouriah, John R. Graef}

\address{Mouffak Benchohra \newline
Laboratory of Mathematics,
University of Sidi Bel-Abbes, P.O. Box 89,
Sidi Bel-Abbes 22000, Algeria}
\email{benchohra@univ-sba.dz}

\address{Soufyane Bouriah \newline
Laboratory of Mathematics,
University of Sidi Bel-Abbes, P.O. Box 89,
 Sidi Bel-Abbes 22000, Algeria}
\email{bouriahsoufiane@yahoo.fr}

\address{John R. Graef \newline
Department of Mathematics,
University of Tennessee at Chattanooga,
Chattanooga, TN 37403, USA}
\email{john-graef@utc.edu}

\thanks{Submitted July 25, 2016. Published December 21, 2016.}
\subjclass[2010]{34A08, 34B15}
\keywords{Caputo's fractional derivative; fractional integral; existence;
\hfill\break\indent  
implicit fractional differential equations;  periodic solutions;
 coincidence degree theory}

\begin{abstract}
 In this article, we obtain an existence result for periodic
 solutions to nonlinear implicit fractional differential equations with
 Caputo fractional derivatives. Our main tools is coincidence degree theory,
 which was first introduced by Mawhin.
 Also we present two examples  to show the applicability of our results.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks


\section{Introduction}

Fractional calculus is a generalization of ordinary differentiation and
integration to arbitrary orders (non-integer). See, for example, 
the monographs \cite{ABN, ABN1, ANAS, BDST,POD} and the references therein.

In recent years, fractional differential equations arise naturally in
various fields such as rheology, fractals, chaotic dynamics, modeling,
control theory, signal processing, bioengineering and biomedical
applications, etc. Fractional derivatives provide an excellent instrument
for the description of memory and hereditary properties of various materials
and processes. We refer the reader to the recent works 
\cite{BAGM, BML, Herm, Hil,KISR, LA, ORTI, P, SBD, TRSV} 
and the references therein.

In this article, we are concerned with the existence of periodic solutions 
to the nonlinear implicit fractional differential equation (IFDE for short)
\begin{gather}
{}^cD^{\alpha }y(t)=f(t,y(t),{}^cD^{\alpha }y(t)),  \quad t\in J:=[0,T], \; T>0,
\; 0<\alpha \leq 1,  \label{e1} \\
\label{e2}
y(0)=y(T),
\end{gather}
where $^cD^{\alpha }$ is the Caputo fractional derivative, and
$f:J\times\mathbb{R}\times\mathbb{R}\to \mathbb{R}$ is a continuous function.

Recently, by means of different tools such as the Banach contract principle, 
Schauder's fixed point, Schaefer's fixed point, the Leray-Schauder 
nonlinear alternative, Monch's fixed point theorem, and the measure
 of noncompactness, initial and boundary value problems for 
implicit fractional differential equations involving Caputo type fractional 
derivatives have extensively been studied in the books \cite{ABN, ABN1} 
and the papers \cite{BeBo1, BeBo2, BeLa1, BeLa2}.

This article is motivated by the works \cite{AFONSO, GeZo, Tan} where 
coincidence degree theory is used for some classes of boundary value problem 
for fractional differential equations of integer as well as noninteger orders. 
This article is  organized as follows. In Section 2, some notations are
introduced and we recall some preliminary concepts about fractional calculus. 
The proof of our main result is presented in Section 3 by applying the 
coincidence degree theory of Mawhin. In the last section, we give 
two examples to illustrate the applicability of our main results.
This paper initiates the application of coincidence degree to the study 
of implicit fractional differential equations.

\section{Preliminaries}

In this section, we introduce notation, definitions, and preliminary facts
that are used throughout this paper. By $C(J,{\mathbb R})$, we denote the
Banach space of continuous functions from $J$ into ${\mathbb R}$ with the
norm
\[
\| y\| _{\infty }= \sup \{|y(t)|:t\in J\}.
\]
By $L^{1}(J)$, we denote the space of Lebesgue-integrable functions
$y:J\to {\mathbb R}$ with the norm
$$
\|y\|_{L^{1}}=\int_{0}^{T}|y(t)|dt.
$$

\begin{definition}[\cite{POD}] \rm
The fractional integral of order $\alpha \in\mathbb{R} _{+}$ of the function 
$h\in L^{1}([0,T]$, $\mathbb{R}_{+})$ is defined by
\[
{}^cI^{\alpha }h(t)=\frac{1}{\Gamma(\alpha) }\int_{0}^{t}(t-s) ^{\alpha -1}h(s)ds,
\]
where $\Gamma $ is the Euler gamma function defined by 
$\Gamma(\alpha) =\int_{0}^{{+}\infty }t^{\alpha -1}e^{-t}dt$, $\alpha >0$.
\end{definition}

\begin{definition}[\cite{KISR}] \rm 
For a function $h$ given on the interval $[0,T]$, the Caputo
fractional derivative of order $\alpha$ of $h$ is defined by
\[
(^cD^{\alpha }h)(t)=\frac{1}{\Gamma (n-\alpha) }
\int_{0}^{t}(t-s) ^{n-\alpha -1}h^{(n)}(s)ds,
\]
where $n=[\alpha] +1$ and $[\alpha] $ denotes the
integer part of the real number $\alpha $.
\end{definition}

\begin{lemma}[\cite{KISR}]  \label{int}
Let $\alpha >0$ and $n=[\alpha] +1$; then
\[
{}^cI^{\alpha }(^cD^{\alpha }f(t))
=f(t)-\sum_{k=0}^{n-1}\frac{f^{(k)}(0) }{k!}t^{k}.
\]
\end{lemma}

\begin{lemma}[\cite{POD}] \label{ziro}
If $\alpha >0$, the homogeneous differential equation of fractional order
\[
{}^cD^{\alpha }h(t)=0
\]
has a solution
\[
h(t)=c_{0}+c_{1}t+c_{2}t^{2}+\ldots+c_{n-1}t^{n-1},
\]
where $c_{i}$, $i=1,\ldots,n$ are constants and $n=[\alpha] +1$. 
\end{lemma}

The following definitions and basic lemmas from coincidence degree theory 
are fundamental in the proof of our main result (see \cite{GaMa, Mawh}).

\begin{definition}  \label{opdeFrdm} \rm
Let $X$ and $Y$ be normed spaces. A linear operator 
$L: \operatorname{dom} L \subset X\to Y$ is said to be a Fredholm operator 
of index zero provided that
\begin{enumerate}
\item $\operatorname{img} L$ is a closed subset of $Y$;
\item $\dim \ker L= \operatorname{codim}\operatorname{img} L< +\infty$.
\end{enumerate}
\end{definition}

It follows from Definition  ref{opdeFrdm} that there exist continuous projections 
$P:X\to X$ and $Q:Y\to Y$ such that
$$
\operatorname{img} P=\ker L, \quad \ker Q=\operatorname{img} L,\quad
 X=\ker L\oplus \ker P,\quad  Y= \operatorname{img} L\oplus \operatorname{img} Q.
$$
This implies that the restriction of $L$ to $\operatorname{dom} L\cap \ker P$,
 which we will denote by $L_{P}$, is an isomorphism onto its image.

\begin{definition} \rm
Let $L$ be a Fredholm operator of index zero and let $\Omega\subseteq X$ be 
a bounded set with $\operatorname{dom} L\cap\Omega\neq\emptyset$. 
The operator $N:\overline\Omega\to Y$ is $L$-compact in $\overline\Omega$ if
\begin{enumerate}
\item the mapping $QN:\overline\Omega\to Y$ is continuous and 
$QN(\overline\Omega)\subseteq Y$ is bounded, and

\item the mapping $(L_{P})^{-1}(I-Q)N:\overline\Omega\to X$ is completely continuous.
\end{enumerate}
\end{definition}

\begin{lemma}[\cite{Oreg}]  \label{lem d'exis}
Let $X$ and $Y$ be Banach spaces and let $\Omega\subset X$ be a bounded open 
symmetric set with $0\in\Omega$.  Let $L:\operatorname{dom} L\subset X\to Y$ 
be a Fredholm operator of index zero with
$\operatorname{dom} L\cap\overline\Omega\neq\emptyset$ and $N:X\to Y$ be an
 $L$-compact operator on $\overline\Omega$. Assume that
$$
Lx-Nx\neq-\lambda(Lx+N(-x))
$$
for all $x\in \operatorname{dom} L\cap\partial\Omega$ and all 
$\lambda\in(0,1]$, where $\partial\Omega$ is the boundary of $\Omega$ 
with respect to $X$. Then the equation $Lx=Nx$ has at least one solution on 
$\operatorname{dom} L\cap\overline\Omega$.
\end{lemma}

\section{Existence of solutions}

Let $ X=\{y\in C(J,\mathbb{R}): y(t)= {}^cI^{\alpha}u(t) : u\in C(J,\mathbb{R}), t\in J\} $ 
with the norm
$$ 
\|y\|_{X}= \max\{\|y\|_{\infty}, \, \|^cD^{\alpha}y\|_{\infty}\} 
$$ 
and $ Y=C(J,\mathbb{R})$ with the norm 
$$
\|u\|_{Y}= \sup \{|u(t)|:t\in J\}.
$$
Define the linear operator $L: \operatorname{dom} L\subseteq X\to Y$ by
\begin{equation}\label{l'opL}
Ly:= {}^cD^{\alpha}y,
\end{equation}
where
$$
\operatorname{dom} L=\{y\in X: {}^cD^{\alpha}y\in Y \quad \mbox{and} \quad
 y(0)=y(T)\}.  
$$
Define the operator $N:X\to Y$ by
\begin{equation}  \label{new-101}
Ny(t):=f(t,y(t),{}^cD^{\alpha}y(t)), \quad t\in J.
\end{equation}
Then  problem \eqref{e1}--\eqref{e2} can be equivalently rewritten as $Ly=Ny$.

\begin{lemma} \label{lem3.1}
Let $L$ be defined by \eqref{l'opL}. Then
$\ker L= \{c: c \in \mathbb{R}\}$ and
$$
\operatorname{img} L=\big\{y\in Y: \int_{0}^{T}(T-s)^{\alpha-1}y(s)ds=0\big\}.
$$
\end{lemma}

\begin{proof}
By Lemma \ref{ziro}, for $t\in J$, $Ly(t)= {}^cD^{\alpha}y(t)=0 $ has the solution
$y(t)=c$, where $c\in \mathbb{R}$.
Then
$$ 
\ker L= \{y(t) = c: c \in \mathbb{R}\}.
$$
For $u\in \operatorname{img} L$, there exists $y\in dom L $ such that 
$ u= Ly \in Y. $ By Lemma \ref{int}, for each $t\in J$ we have
$$
y(t)=y(0)+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}u(s)ds.
$$
Since $ y\in \operatorname{dom} L$, $u$ satisfies
$$
\frac{1}{\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}u(s)ds=0.
$$
On the other hand, suppose $u\in Y$ satisfies
$$
\int_{0}^{T}(T-s)^{\alpha-1}u(s)ds=0.
$$
Let $y(t)= {}^cI^{\alpha}u(t)$; then $u(t)= {}^cD^{\alpha}y(t)$ and so 
$y\in \operatorname{dom} L$. Hence, $u\in \operatorname{img} L$,
so
$$
\operatorname{img} L=\big\{y\in Y: \int_{0}^{T}(T-s)^{\alpha-1}y(s)ds=0\big\},
$$
which completes the proof.
\end{proof}

\begin{lemma} \label{lem3.2}
Let $L $ be defined by \eqref{l'opL}. Then $L $ is a Fredholm operator of index zero, 
and the linear continuous projector operators $P :X\to X $ and 
$ Q : Y \to Y$ can be defined as
$$
Py=y(0), \quad 
Qu(t)=\frac{\alpha}{T^{\alpha}}\int_{0}^{T}(T-s)^{\alpha-1}u(s)ds.
$$
Furthermore, the operator $ L_{P}^{-1}: \operatorname{img} L\to X\cap \ker P$ 
satisfies
$$
L_{P}^{-1}(u)(t)= {}^cI^{\alpha}u(t).
$$
\end{lemma}

\begin{proof}
Clearly, $\operatorname{img} P=\ker L $ and $P^{2}=P$.
 It follows that for each $y\in X$, $y=(y-Py)+Py$, that is, $X=\ker P+\ker L$. 
A simple calculation shows that $\ker P \cap \ker L={0}$.
Therefore,
$X=\ker P\oplus\ker L$.
A similar argument shows that for each $u\in Y$, $Q^{2}u=Qu$ and $u=(u-Q(u))+Q(u)$, 
where $(u-Q(u))\in\ker Q= \operatorname{img}L$.

It follows from $\operatorname{img} L =\ker Q$ and $Q^{2}=Q$, that 
$\operatorname{img} Q\cap \operatorname{img} L={0}$. Then, we have
$Y= \operatorname{img} L\oplus \operatorname{img} Q$.
   Thus,
$$
\dim \ker L=\dim \operatorname{img} Q = \operatorname{codim} \operatorname{img} L.
$$
This means that $L$ is a Fredholm operator of index zero.

To prove that $L_{P}^{-1}$ is the inverse of 
$L|_{\operatorname{dom} L \cap \ker P}$, let $u\in \operatorname{img} L$.
Then
\begin{equation}  \label{L1}
LL_{P}^{-1}(u)= {}^cD^{\alpha}(^cI^{\alpha}u)=u.
\end{equation}
Moreover, for $y\in \operatorname{dom} L\cap \ker P$, we obtain that
$$
L_{P}^{-1}(L(y(t)))= {}^cI^{\alpha}({}^cD^{\alpha}y(t))=y(t)-y(0).
$$
Since $y\in \operatorname{dom} L\cap \ker P, $ we know that $y(0)=0$. Therefore
\begin{equation} \label{L2}
L_{P}^{-1}(L(y(t)))=y(t).
\end{equation}

Combining \eqref{L1} and \eqref{L2} shows that 
$L_{P}^{-1}=(L|_{\operatorname{dom} L\cap \ker P})^{-1}$. 
This proves the lemma.
\end{proof}

In the sequel we  use of the following assumption.
\begin{itemize}
\item[(H1)] There exist constants $K$, $\overline{K} >0 $ with 
$K + \overline{K} < \min\left\{1,\frac{\Gamma(\alpha+1)}{T^{\alpha}}\right\}$ 
such that
$$ 
|f(t,u,v)-f(t,\bar{u},\bar{v})|\leq K|u-\bar{u}|+\overline{K}|v-\bar{v}| \quad
 \text{for } t\in J  \text{ and }  u, \bar{u}, v, \bar{v} \in \mathbb{R}. 
$$
\end{itemize}

\begin{lemma} \label{lem3.3}
Assume {\rm (H1)} holds. Then the operator $N$ is $L$-compact on any bounded 
open set $\Omega\subset X$.
\end{lemma}

\begin{proof}
Define the bounded open set
$\Omega=\{y\in X: \|y\|_{X}<M\}$, where $M$ is a positive constant. 
The proof will be given in a sequence of claims.
\smallskip

\noindent\textbf{Claim 1:} $QN$ is continuous.
The continuity of $QN$ follows from the conditions on $f$ and the Lebesgue 
dominated convergence theorem.
\smallskip

\noindent\textbf{Claim 2:} $QN(\overline\Omega)$ is bounded.
For each $y\in\overline\Omega$ and $t\in J$, we have
\begin{align*}
|QN(y)(t)| 
&\leq \frac{\alpha}{T^\alpha}\int_{0}^{T}(T-s)^{\alpha-1}|f(s,y(s),
 {}^cD^{\alpha}y(s))|ds\\
&\leq \frac{\alpha}{T^\alpha}\int_{0}^{T}(T-s)^{\alpha-1}|f(s,y(s),
 {}^cD^{\alpha}y(s))-f(s,0,0)|ds\\
&\quad + \frac{\alpha}{T^\alpha}\int_{0}^{T}(T-s)^{\alpha-1}|f(s,0,0)|ds\\
&\leq  f^*+\frac{\alpha}{T^\alpha}\int_{0}^{T}(T-s)^{\alpha-1}(K|y(s)|
 +\overline {K}| {}^cD^{\alpha}y(s)|)ds\\
&\leq f^*+M(K+\overline K),
\end{align*}
where $f^*=\sup_{t\in J}|f(t,0,0)|$. Thus,
\[
\|QN(y)\|_{Y} \leq f^*+M(K+\overline K):=R.
\]
This shows that $QN(\overline\Omega)\subseteq Y $ is bounded.
\smallskip

\noindent\textbf{Claim 3:}  $L_{P}^{-1}(I-Q)N:\overline\Omega\to X $ is 
completely continuous.

In view of the Ascoli-Arzel\`a theorem, we need to prove that 
$ L_{P}^{-1}(I-Q)N(\overline\Omega)\subset X $ is equicontinuous and bounded.
First, for each $y\in \overline\Omega $ and $t\in J, $ we have
\begin{align*}
&L_{P}^{-1}(I-Q)Ny(t) \\
&= L_{P}^{-1}(Ny(t)-QNy(t))\\
&={}^cI^{\alpha}\Big[f(t,y(t),{}^cD^{\alpha}y(t))-\frac{\alpha}{T^\alpha}
 \int_{0}^{T}(T-s)^{\alpha-1}f(s,y(s),{}^cD^{\alpha}y(s))\Big]ds\\
&= \frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}f(s,y(s),{}^cD^{\alpha}y(s))ds\\
&\quad - \frac{t^\alpha}{T^{\alpha}\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}
 f(s,y(s),{}^cD^{\alpha}y(s))ds.
\end{align*}
On one hand, for each $y\in \overline\Omega $ and $t\in J, $ we have
\begin{align*}
&|L_{P}^{-1}(I-Q)Ny(t)| \\
&\leq \frac{2}{\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}|f(s,y(s),
 {}^cD^{\alpha}y(s))-f(t,0,0)|ds \\
&\quad  + \frac{2}{\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}|f(t,0,0)|ds\\
&\leq [f^*+M(K+\overline{K})]\frac{2T^{\alpha}}{\Gamma(\alpha+1)}:=B_{1},
\end{align*}
so
\begin{equation}\label{B1}
\|L_{P}^{-1}(I-Q)Ny\|_{\infty}\leq B_{1}.
\end{equation}
On the other hand, 
\begin{equation}  \label{new-601}
\begin{aligned}
&{}^cD^{\alpha}(L_{P}^{-1}(I-Q)Ny(t)) \\
&=f(t,y(t),{}^cD^{\alpha}y(t))-\frac{\alpha}{T^\alpha}
 \int_{0}^{T}(T-s)^{\alpha-1}f(s,y(s),{}^cD^{\alpha}y(s))ds,
\end{aligned}
\end{equation}
which implies that for each $y\in \overline\Omega $ and $t\in J$,
$$
|{}^cD^{\alpha}(L_{P}^{-1}(I-Q)Ny(t))|\leq 2f^*+2M(K+\overline{K}):=B_{2},
$$
so that
\begin{equation}\label{B2}
\|{}^cD^{\alpha}(L_{P}^{-1}(I-Q)Ny)\|_{\infty}\leq B_{2}.
\end{equation}
From inequalities \eqref{B1} and \eqref{B2}, we have
$$
\|L_{P}^{-1}(I-Q)Ny\|_{X}\leq \max\{B_{1},B_{2}\},
$$
which shows that $L_{P}^{-1}(I-Q)N(\overline\Omega)$ is uniformly bounded in $X$.

To prove that $L_{P}^{-1}(I-Q)N(\overline\Omega)$ is equicontinuous, 
notice that for $0\leq t_{1}\leq t_{2}\leq T$ and $y\in \overline{\Omega}$, we have
\begin{align*}
&|L_{P}^{-1}(I-Q) Ny(t_{2})-L_{P}^{-1}(I-Q)Ny(t_{1})|\\
&\leq \frac{f^*+M(K+\overline{K})}{\Gamma(\alpha)}
 \Big[\int_{t_{1}}^{t_{2}}(t_{2}-s)^{\alpha-1}ds+ \int_{0}^{t_{1}}
 |(t_{2}-s)^{\alpha-1}-(t_{1}-s)^{\alpha-1}|ds\Big]\\
&\quad +\Big[\frac{M(K+\overline{K})+f^*}{\Gamma(\alpha+1)}\Big]
 (t_{2}^{\alpha}-t_{1}^{\alpha}).
\end{align*}
As $t_{1}\to t_{2}$, the right-hand side of the above inequality tends to zero.
Now from \eqref{new-601}, we have
\begin{align*}
&|{}^cD^{\alpha}(L_{P}^{-1}(I-Q)Ny)(t_{2})
 - {}^cD^{\alpha}(L_{P}^{-1}(I-Q)Ny)(t_{1})|\\
&\leq |f(t_2,y(t_2), {}^cD^{\alpha}y(t_2))-f(t_1,y(t_1), {}^cD^{\alpha}y(t_1))|.
\end{align*}
As $ t_{1}\to t_{2} $ the right-hand side of the above inequality also
tends to zero.
Thus, $ L_{P}^{-1}(I-Q)N(\overline\Omega)$ is equicontinuous in $X$. 
By the Ascoli-Arzel\`a theorem, $ L_{P}^{-1}(I-Q)N(\overline\Omega)$ 
is relatively compact. As a consequence of Claims 1 to 3, we can conclude 
that the operator $N $ is $L$-compact in $\overline\Omega$, and this 
completes the proof.
\end{proof}

\begin{lemma}  \label{lem2.6}
If condition {\rm (H1)} holds, then there exists a positive number 
$A$, not depending on $\lambda$, such that, if
\begin{equation}\label{cond2.4}
L(y)-N(y)=-\lambda[L(y)+N(-y)], \quad \lambda \in (0,1],
\end{equation}
then $\|y\|_{X}\leq A$.
\end{lemma}

\begin{proof}
Assume {\rm (H1)} holds and that $y\in X$ satisfies \eqref{cond2.4}. Then
$$
L(y)-N(y)=-\lambda L(y)-\lambda N(-y),
$$
so
\begin{equation}  \label{new-501}
L(y)=\frac{1}{1+\lambda}N(y)-\frac{\lambda}{1+\lambda}N(-y).
\end{equation}
Using the definitions of the operators $L$ and $N$ (see \eqref{l'opL} 
and \eqref{new-101}), for each $t\in J$,
we obtain
\begin{align*}
|Ly(t)|=|{}^cD^{\alpha}y(t)| 
&\leq  \frac{1}{1+\lambda}\left|f(t,y(t),{}^cD^{\alpha}y(t))\right|+\frac{\lambda}{1+\lambda}\left|f(t,-y(t),-{}^cD^{\alpha}y(t))\right|\\
&\leq \frac{1}{1+\lambda}[\left|f(t,y(t),{}^cD^{\alpha}y(t))-f(t,0,0)\right|+f^*]\\
&\quad + \frac{\lambda}{1+\lambda}[\left|f(t,-y(t),-{}^cD^{\alpha}y(t))-f(t,0,0)\right|+f^*]\\
&\leq \Big(\frac{1}{1+\lambda}+\frac{\lambda}{1+\lambda}\Big) 
 f^*+\frac{1}{1+\lambda}[K|y(t)|+\overline{K}|{}^cD^{\alpha}y(t)|]\\
&\quad + \frac{\lambda}{1+\lambda}[K|-y(t)|+\overline{K}|- {}^cD^{\alpha}y(t)|]\\
&= f^*+\Big(\frac{1}{1+\lambda}+\frac{\lambda}{1+\lambda}\Big)
 [K|y(t)|+\overline{K}|{}^cD^{\alpha}y(t)|]\\
&= f^*+K|y(t)|+\overline{K}|{}^cD^{\alpha}y(t)|\\
&\leq f^* +K\|y\|_{\infty}+\overline{K}\|{}^cD^{\alpha}y\|_{\infty},
\end{align*}
which implies 
\begin{equation}\label{derive}
\|{}^cD^{\alpha}y\|_{\infty}\leq f^*+K\|y\|_{\infty}+\overline{K}\|{}^cD^{\alpha}y\|_{\infty}.
\end{equation}

By \eqref{new-501}, for each $t\in J$, we have
$$
y(t)=\frac{1}{1+\lambda}L_{p}^{-1}Ny(t)
-\frac{\lambda}{1+\lambda}L_{p}^{-1}N(-y(t)),
$$
and so
\begin{align*}
|y(t)| 
&\leq \frac{1}{(1+\lambda)\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}\left|f(s,y(s),{}^cD^{\alpha}y(s))-f(s,0,0)\right|ds\\
&\quad + \frac{\lambda}{(1+\lambda)\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}\left|f(s,-y(s),- {}^cD^{\alpha}y(s))-f(s,0,0)\right|ds\\
&\quad +\frac{f^{*}T^{\alpha}}{(1+\lambda)\Gamma(\alpha+1)}+\frac{\lambda f^{*}T^{\alpha}}{(1+\lambda)\Gamma(\alpha+1)}\\
&\leq \Big(\frac{1}{1+\lambda}+\frac{\lambda}{1+\lambda}\Big)
 \frac{T^{\alpha}}{\Gamma(\alpha+1)}(K\|y\|_{\infty}+\overline{K}\|{}^cD^{\alpha}y\|_{\infty})\\
&\quad  +\Big(\frac{1}{1+\lambda}+\frac{\lambda}{1+\lambda}\Big)
 \frac{f^{*}T^{\alpha}}{\Gamma(\alpha+1)}\\
&=\frac{T^{\alpha}}{\Gamma(\alpha+1)}(K\|y\|_{\infty}
 +\overline{K}\|{}^cD^{\alpha}y\|_{\infty})
 +\frac{f^{*}T^{\alpha}}{\Gamma(\alpha+1)}.
\end{align*}
Hence,
\begin{equation}  \label{derive2}
\|y\|_{\infty} \leq [f^*+K\|y\|_{\infty}
+\overline{K}\|{}^cD^{\alpha}y\|_{\infty}]\frac{T^{\alpha}}{\Gamma(\alpha+1)}.
\end{equation}
Using the definition of the norm $\|\cdot\|_{X}$, we see that if
$\|y\|_{X}=\|{}^cD^{\alpha}y\|_{\infty}$, then, by \eqref{derive}, we have
\begin{align*}
\|y\|_{X} &\leq f^*+K\|y\|_{\infty}+\overline{K}\|{}^cD^{\alpha}y\|_{\infty}\\
&\leq f^* +K\|y\|_{X}+\overline{K}\|y\|_{X}\\
&= f^*+(K+\overline{K})\|y\|_{X},
\end{align*}
and thus
$$
\|y\|_{X}\leq\frac{f^*}{1-(K+\overline{K})}:=A_{1}.
$$
On the other hand, if $\|y\|_{X}=\|y\|_{\infty}$, then \eqref{derive2} implies
\begin{align*}
\|y\|_{X} 
&\leq [f^*+K\|y\|_{\infty}+\overline{K}\|{}^cD^{\alpha}y\|_{\infty}]
 \frac{T^{\alpha}}{\Gamma(\alpha+1)}\\
&\leq [f^*+K\|y\|_{X}+\overline{K}\|y\|_{X}]\frac{T^{\alpha}}{\Gamma(\alpha+1)}\\
&= [f^*+(K+\overline{K})\|y\|_{X}]\frac{T^{\alpha}}{\Gamma(\alpha+1)},
\end{align*}
and so
$$
\|y\|_{X}\leq \frac{f^*}{\frac{\Gamma(\alpha+1)}{T^{\alpha}}
-(K+\overline{K})}:=A_{2}.
$$
Therefore
$$
\|y\|_{X}\leq \max\{A_{1},A_{2}\}:=A,
$$
and this completes the proof.
\end{proof}

\begin{lemma} \label{lem2.7}
If condition {\rm (H1)} is satisfied, then there is a bounded open set 
$\Omega\subset X $ such that
\begin{equation}\label{cond2.8}
L(y)-N(y)\neq-\lambda[L(y)+N(-y)],
\end{equation}
for all $y\in \partial\Omega $ and all $\lambda\in (0,1]$.
\end{lemma}

\begin{proof}
By (H1) and Lemma \ref{lem2.6}, there exists a positive constant $A$ that 
does not depend on $\lambda$ such that, if $y$ satisfies
$$  
L(y)-N(y)=-\lambda[L(y)+N(-y)], \ \lambda\in (0,1], 
$$
then $\|y\|_{X}\leq A$. Thus, if
\begin{equation} \label{cond2.9}
\Omega=\{y\in X : \|y\|_{X}<B\}
\end{equation}
where $B>A$, we have 
$$
L(y)-N(y)\neq-\lambda[L(y)-N(-y)]
$$
for every $y\in \partial\Omega = \{y\in X : \|y\|_{X}=B\}$ and $\lambda\in (0,1]$.
\end{proof}

We are now ready to prove the main result in our paper.

\begin{theorem} \label{thm1}
If {\rm (H1)} holds, then  problem \eqref{e1}--\eqref{e2}
 has at least one solution.
\end{theorem}

\begin{proof}
From (H1) it is clear that the set $\Omega $ defined by $\eqref{cond2.9}$ 
is symmetric, $0\in \Omega$, and 
$X\cap\overline\Omega=\overline\Omega\neq\emptyset$.
 Furthermore, it follows from Lemma \ref{lem2.7} that if condition (H1)
 is satisfied, then
$$
L(y)-N(y)\neq-\lambda[L(y)-N(-y)]
$$
for all $y\in X\cap\partial\Omega=\partial\Omega $ and all $\lambda\in (0,1]$. 
This together with Lemma \ref{lem d'exis} imply that 
 problem \eqref{e1}--\eqref{e2} has at least one solution, and this 
completes the proof.
\end{proof}

\section{Examples}

In this section we give two examples to illustrate our theorem.

\begin{example}  \label{Example 1n}\rm
Consider the problem for non-linear implicit fractional differential equations
\begin{gather}\label{exp1}
D^{1/2}y(t)= \frac{e^{-t}}{(11+e^{t})}\Big[\frac{|y(t)|}{1+ |y(t)|} 
 -\frac{|D^{1/2}y(t)|}{1+ |D^{1/2}y(t)|}\Big], \quad t\in [0,1], \\
\label{exp3}
y(0)=y(1).
\end{gather}
Here we have
$$  
f(t,u,v)=\frac{e^{-t}}{(11+e^{t})} \Big(\frac{u}{1+u} - \frac{v}{1+v}\Big),  
\quad t\in [0,1], \; u,  v\in [0, +\infty),   
$$
and clearly the function $f$ is jointly continuous. 
For each $u$, $\bar{u}$, $v$, $\bar{v} \in [0, +\infty)$ and $t\in [0,1]$,
\[
| f(t,u,v)-f(t,\bar{u},\bar{v})|  
\leq \frac{e^{-t}}{(11+e^{t}) }[|u-\bar{u}| +|v-\bar{v}|]
\leq \frac{1}{12}[|u-\bar{u}| +|v-\bar{v}|].
\]
Hence, condition (H1) is satisfied with
$ K = \overline{K} = 1/12$ and
$$
 K + \overline{K} = \frac{1}{6} 
< \min\big\{1,\frac{\Gamma(\alpha+1)}{T^{\alpha}}\big\}
=\min\big\{1,\frac{\sqrt{\pi}}{2}\big\}=\frac{\sqrt{\pi}}{2}.
$$
It follows from Theorem \ref{thm1} that  problem \eqref{exp1}--\eqref{exp3} 
has at least one solution.
\end{example}

\begin{example}  \label{Example 2} \rm Consider the problem
\begin{gather}\label{exp4}
D^{1/2}y(t)=\frac{t}{3}\sin y(t)+ \frac{1}{100}\sin{D^{1/2}y(t)}+\frac{1}{2}, \quad
  t\in [0,1], \\
\label{exp6}
y(0)=y(1).
\end{gather}
Here,
$$  
f(t,u,v)=\frac{t}{3}\sin u+ \frac{1}{100}\sin v+\frac{1}{2},\quad
 t \in [0,1], \;  u, v \in \mathbb{R}. 
 $$
which is jointly continuous. For any $u$, $\bar{u}$, $v$, $\bar{v} \in \mathbb{R}$ 
and $t\in [0,1]$,
\begin{align*}
|f(t,u,v)-f(t,\bar{u},\bar{v})| 
& \leq \frac{|t|}{3}|\sin u-\sin \bar{u}|+ \frac{1}{100}|\sin v-\sin \bar{v}|\\
&\leq \frac{1}{3}|u-\bar{u}|+ \frac{1}{100}|v-\bar{v}|.
\end{align*}
Hence, condition (H1) is satisfied with
$K=1/3$, $\overline{K}=1/100$,
and
$$  
K+\overline{K}=\frac{103}{300}
<\min\big\{1,\frac{\Gamma(\alpha+1)}{T^{\alpha}}\big\}
=\frac{\sqrt{\pi}}{2}.
$$
It follows from Theorem \ref{thm1} that problem \eqref{exp4}--\eqref{exp6} 
has at least one solution on $J$. 
\end{example}


\subsection*{Acknowledgments} 
The authors would like to thank the anonymous referee for the very helpful 
and clearly stated recommendations for changes to this article.

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\end{document}