\def\papertitle{LOWER AND UPPER SOLUTIONS FOR CONFORMABLE FRACTIONAL DIFFERENTIAL EQUATIONS }
\def\paperauthorA{Bouharket Bendouma}
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\title{\papertitle}
\affiliation{\paperauthorA} % This command replaces \name{The DAFx Crew}
{Faculty of Mathematics and Informatics, Ibn Khaldoun, Tiaret University,\\
PoBox 78, 14000 za\^{a}roura, Tiaret, Algeria}

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\begin{abstract}
In this article, we study the existence of solutions to conformable
fractional differential equations with periodic boundary condition.
Existence results are obtained by the method of upper and lower
solutions and Schauder's fixed point theorem.
\end{abstract}

\section{Introduction}
Fractional differential equations plays an important role in
describing many phenomena and processes in various fields of science
such as physics, chemistry, control systems, population dynamics,
etc., see \cite{KiSrTr, Podl, SaKiMa}. Recently, a new fractional
derivative called the conformable fractional derivative was
introduced by Khalil et al. \cite{KhHoSa}.\vspace{0.1cm}

\ In this paper, we are concerned with the existence of a solutions
for the following conformable fractional differential equations with
periodic boundary condition:
\begin{equation}\label{eq1.1}
   \begin{cases}
      x^{(\alpha)}(t)=f(t,x(t)),\quad \ \ \ for\ t\in J=[a,b],\ a>0,\vspace{0.2cm}\\
      x(a)=x(b).
   \end{cases}
\end{equation}
Where $0<\alpha\leq1$, $f:J\times \mathbb{R}\rightarrow \mathbb{R}$
is a continuous function, $x^{(\alpha)}(t)$ denotes the conformable
fractional derivative of $x$ at $t$ of order $\alpha$.\vspace{0.1cm}

\ The paper is organized as follows. In Section~\ref{sec:2}, we
present the main concepts of conformable fractional derivative
calculus and we give some useful preliminary results. In
Section~\ref{Sec:3}, we prove existence of solutions to problem
\eqref{eq1.1} by using the method of upper and lower solutions and
Schauder's fixed point theorem.

\section{Preliminaries}
\label{sec:2}

In this section, we introduce the definition of conformable
fractional calculus and their important properties.
\begin{definition}\cite{KhHoSa}
\label{d:cfd} Given a function $f : [0,\infty)\rightarrow
\mathbb{R}$ and a real constant $\alpha\in(0,1]$. The conformable
fractional derivative of $f$ of order $\alpha$ is defined by,
\begin{equation}f^{(\alpha)}(t):=\lim_{\varepsilon\rightarrow0} \frac{f(t+\varepsilon
t^{1-\alpha})-f(t)}{\varepsilon}
\end{equation}
for all $t>0$.\ If $f^{(\alpha)}(t)$ exists and is finite, we say
that $f$ is $\alpha$-differentiable at $t$.

If $f$ is $\alpha$-differentiable in some interval $(0,a), \ a>0$,
and $\lim_{t\rightarrow 0^+}f^{(\alpha)}(t)$ exists, then the
conformable fractional derivative of $f$ of order $\alpha$ at $t=0$
is defined as
$$f^{(\alpha)}(0)=\lim_{t\rightarrow0^+}f^{(\alpha)}(t).$$
\end{definition}

\begin{theorem}\cite{KhHoSa}\label{th:00}
Let $\alpha\in(0,1]$ and $f:[0,\infty)\rightarrow \mathbb{R}$ a
$\alpha$-differentiable function at $t_{0} > 0$,  then $f$ is
continuous at $t_{0}$.
\end{theorem}

\begin{theorem}\cite{KhHoSa}
\label{th:01} Let $\alpha\in(0,1]$ and assume $f,g$ to be
$\alpha$-differentiable at a point $t > 0$. Then,
\begin{enumerate}
\item[(i)] $(a f+b g)^{(\alpha)}=af^{(\alpha)}+b g^{(\alpha)}$, \ \ for all $a,b\in\mathbb{R}$;
\item[(ii)] $(f g)^{(\alpha)}=f g^{(\alpha)}+g f^{(\alpha)}$;
\item[(iii)] $\left(f/g\right)^{(\alpha)}=\dfrac{g f^{(\alpha)}-f g^{(\alpha)}}{g^{2}}$.
\item[(iv)] If, in addition, $f$ is differentiable at a point $t>0$, then
$$f^{(\alpha)}(t)=t^{1-\alpha}f'(t).$$
\end{enumerate}
\end{theorem}
 Additionaly, conformable fractional derivatives of certain
 functions as follow:
\begin{enumerate}
\item $(t^{p})^{(\alpha)}=p\,t^{p-\alpha}$,  for all $p\in\mathbb{R}$.
\item $(\lambda)^{(\alpha)}=0$, for all  $\lambda \in \mathbb{R}$.
\item $(e^{ct})^{(\alpha)}=c\,t^{1-\alpha}e^{ct}$,   for all $c\in\mathbb{R}$.
\item $(e^{\frac{p}{\alpha}t^{\alpha}})^{(\alpha)}=p\,e^{\frac{p}{\alpha}t^{\alpha}}$,  for all $p\in\mathbb{R}$.
\end{enumerate}

\begin{remark}
If $f$ is differentiable at $t$, then f is $\alpha$-differentiable
at $t$.
\end{remark}
We introduce the following space:
\begin{eqnarray*}
C^{\alpha}(J,\mathbb{R})&=& \{f:J\rightarrow \mathbb{R}, \ \mbox{ is
\ $\alpha$-differentiable on $J$ \ and} \; f^{(\alpha)}\in
C(J,\mathbb{R})\}.
\end{eqnarray*}

\begin{definition}$($Conformable fractional integral \cite{KhHoSa}$)$.\label{def:cfI}
Let $\alpha\in(0,1]$ and $f: [a,\infty)\rightarrow \mathbb{R}$. The
conformable fractional integral of $f$ of order $\alpha$ from $a$ to
$t$, denoted by $I_{\alpha}^{a}(f)(t)$, is defined by
$$
I_{\alpha}^{a}(f)(t)
:=\int_{a}^{t}f(s)d_{\alpha}s:=\int_{a}^{t}f(s)s^{\alpha-1}ds.
$$
The considered integral is the usual improper Riemann one.
\end{definition}

\begin{theorem}\cite{KhHoSa}
\label{th:02} If $f$ is a continuous function in the domain of
$I_{\alpha}^{a}$ then, for all $t\geq a$ we have
$$\left(I_{\alpha}^{a}(f)\right)^{(\alpha)}(t)=f(t).$$
\end{theorem}


\begin{lemma}\cite{KhHoSa} \label{propo}
Let $f:(a,b) \rightarrow \mathbb{R}$ be differentiable and $0<
\alpha \leq 1$. Then, for all $t>a$ we have
\begin {equation}
I_\alpha^{a} (f^{(\alpha)})(t)= f(t)-f(a).
  \end{equation}
\end{lemma}
\begin{proposition}\cite{IyNw}
\label{pr1}Let $0 < a < b$, $f:[a, b]\rightarrow \mathbb{R}$ be
continuous function and $0< \alpha < 1$. Then for all $t\in[a, b]$
we have,
\[\left|I_{\alpha}^{a}(f)(t)\right| \leq I_{\alpha}^{a}|f|(t).\]
\end{proposition}

\section{Main Results}
\label{Sec:3} In this section, we establish an existence result for
the problem (\ref{eq1.1}). A solution of problem (\ref{eq1.1}) will
be a function $ x\in C^{\alpha}(J,\mathbb{R})$ for which
(\ref{eq1.1}) is satisfied. We introduce the notion of lower and
upper solutions for the problem \eqref{eq1.1}.
\begin{definition} \label{def2.1} \rm
Let $v \in C^{\alpha}(J,\mathbb{R})$. We say that $v $ is a lower
solution of the boundary value problem \eqref{eq1.1}, if
\begin{itemize}
    \item[(i)] $v^{(\alpha)}(t)\leq f(t,v(t)), \quad \mbox{for all } t \in J$;
    \item[(ii)] $v(a)\leq v(b).$
 \end{itemize}
Let $w\in C^{\alpha}(J,\mathbb{R})$. We say that $w$ is an upper
solution of the boundary value problem \eqref{eq1.1}, if
\begin{itemize}
    \item[(i)] $w^{(\alpha)}(t)\geq f(t,w(t)), \quad \mbox{for all } t \in J$;
    \item[(ii)] $w(a)\geq w(b).$
 \end{itemize}
\end{definition}
we define the sector
$$[v,w] =\{x \in C(J,\mathbb{R}): v(t)\leq x(t)\leq w(t), \; \mbox{for all} \;t\in J \}.$$
We consider the following problem.
\begin{equation}\label{E3.1}
\begin{cases}
  x^{(\alpha)}(t)+ \alpha x(t)=f(t,\bar{x}(t))+\alpha \bar{x}(t), \ \ \ t\in  J,\vspace{0.25cm}\\
 x(a)=x(b). \end{cases}
\end{equation}
where
\begin{equation}\label{E3.02}
 \bar{x}(t)=\left\{
\begin{array}{ll}
  v(t),& \ \ \ \ \ \ \  \ \ \ \ \ \textrm{if $x(t)<v(t)$},\vspace{0.25cm} \\
  x(t),& \ \ \ \ \ \ \ \ \ \ \ \ \ \textrm{if $v(t)\leq x(t)\leq w(t)$},\vspace{0.25cm} \\
  w(t),& \ \ \ \ \ \ \ \ \ \ \ \ \ \textrm{if $x(t)>w(t)$},
\end{array}\right.
\end{equation}
and $v(t)\leq w(t)$ on $J$.\\
We need the following auxiliary proposition.

\begin{proposition}\label{p2.08} For every $g\in C^{\alpha}(J,\mathbb{R})$, the
problem
\begin{equation}\label{EP.031}
   \begin{cases}
     x^{(\alpha)}(t)+ \alpha x(t)=g(t), & t\in J=[a,b], \quad a>0,\quad \; 0<\alpha<1,\vspace{0.2cm}\\
     x(a)=x(b).
   \end{cases}
\end{equation}
has a unique solution $x \in C^{\alpha}(J,\mathbb{R})$ given by:
\begin{equation}
\label{eq.th02}
x(t):=e^{-t^{\alpha}}\left(\dfrac{1}{e^{b^{\alpha}-a^{\alpha}}-1}
 \int_{a}^{b}s^{\alpha-1}e^{s^{\alpha}}g(s)ds +
 \int_{a}^{t}s^{\alpha-1}e^{s^{\alpha}}g(s)ds\right)
\end{equation}
\end{proposition}

Let us define the operators $ \mathcal{N}:C(J,\mathbb{R})\rightarrow
C(J,\mathbb{R})$ by
\begin{align*} \mathcal{ N}(x)(t)=e^{-t^{\alpha}}\left(\dfrac{1}{e^{b^{\alpha}-a^{\alpha}}-1}
 \int_{a}^{b}s^{\alpha-1}e^{s^{\alpha}}\Big(f(s,\bar{x}(s))+\alpha \bar{x}(s)\Big)ds \right. \vspace{0.2cm}\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.+
\int_{a}^{t}s^{\alpha-1}e^{s^{\alpha}}\Big(f(s,\bar{x}(s))+\alpha
\bar{x}(s)\Big)ds \right).\end{align*}

\begin{proposition}\label{P3.002}
If $ v, w \in C^{\alpha}(J,\mathbb{R})$ are, respectively, lower and
upper solutions to \eqref{eq1.1}, such that $v(t)\leq w(t)$, then
the operator $\mathcal{N}$ is compact.
\end{proposition}


Now, we can obtain the main theorem of this section.

\begin{theorem}\label{T3.2}
If $ v, w \in C^{\alpha}(J,\mathbb{R})$ are, respectively, lower and
upper solutions to (\ref{eq1.1}), such that $v(t)\leq w(t)$, then
the problem (\ref{eq1.1}) has a solution $x\in
C^{\alpha}(J,\mathbb{R})$, such that $v(t)\leq x(t)\leq w(t)$ for
all $t\in J$.
\end{theorem}

\begin{example} \label{x2-t} \rm %(3.9)
Consider the periodic problem:
\begin{equation}\label{ES.03}
 \begin{cases}
      x^{(1/3)}(t)=\dfrac{2t-1-x^{5}(t)}{\sqrt{t}}    \  \ \ t \in J=[1/2,1],  \vspace{0.25cm}\\
      x(1/2)= x(1).
  \end{cases}
\end{equation}
Here $n=1$,
  $\alpha=1/3$, and $f(t,x)=\dfrac{2t-1-x^{5}(t)}{\sqrt{t}}$. It is clear that $f$ is a continuous function. Observe that  $v=-1 $ and $w =1$ are,
respectively, lower and upper solutions of (\ref{ES.03}) follows
from the fact that
\[v^{(\frac{1}{3})}(t)=0\leq f(t,v(t))=2\sqrt{t}, \;t \in J,\ \ \ v(1/2)\leq v(1),\]
and
\[w^{(\frac{1}{3})}(t)=0 \geq f(t,w(t))=\frac{2(t-1)}{\sqrt{t}}, \; t\in J, \ \ \ w(1/2)\geq w(1),\]
Theorem \ref{T3.2}, implies that problem \eqref{ES.03} has a
solution $x\in c^{\frac{1}{3}}([1/2,1])$, such that $-1 \le x(t) \le
1$, for all $t \in [1/2 ,1]$.

\end{example}
\section{Conclusion}
In this paper, we present existence of solutions for the conformable
fractional differential equations with periodic boundary condition.
These results are obtained by using the notion of upper and lower
solutions and Schauder's fixed point theorem.


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