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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 09, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/09\hfil Fractional differential equations]
{Existence of solutions to fractional order ordinary and delay
differential equations and applications}

\author[S. Abbas\hfil EJDE-2011/09\hfilneg]
{Syed Abbas}

\address{Syed Abbas  \newline
School of Basic Sciences,
Indian Institute of Technology Mandi,
Mandi, H.P. - 175001, India}
\email{sabbas.iitk@gmail.com, abbas@iitmandi.ac.in}

\thanks{Submitted November 16, 2010. Published Janaury 16, 2011.}
\subjclass[2000]{34K40, 34K14}
\keywords{Fractional differential equation;
 fixed point theorems;\hfill\break\indent
maximum interval of existence}

\begin{abstract}
 In this article, we discuss the existence and uniqueness of
 solution to fractional order ordinary and delay differential
 equations. We apply our results on the single species model of
 Lotka Volterra type. Fixed point theorems are the main tool used
 here to establish the existence and uniqueness results. First we
 use Banach contraction principle and then Krasnoselskii's fixed
 point theorem to show the existence and uniqueness of the solution
 under certain conditions. Moreover, we prove that the solution can
 be extended to maximal interval of existence.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}


\section{Introduction}

  Fractional differential equations is a generalization of ordinary
differential equations
and integration to arbitrary non integer orders. The origin of
fractional calculus goes back to Newton and Leibniz in the
seventieth century.  It is widely and efficiently used to describe
many phenomena arising in engineering, physics, economy, and
science. Recent investigations have shown that many physical
systems can be represented more accurately through fractional
derivative formulation \cite{mainardi}. Fractional differential
equations, therefore find numerous applications in the field of
visco-elasticity, feed back amplifiers, electrical circuits,
electro analytical chemistry, fractional multipoles, neuron
modelling encompassing different branches of physics, chemistry
and biological sciences \cite{pod1}. There have been many
excellent books and monographs available on this field \cite{diet,
kilbas, miller,pod1,ru,samko}. In \cite{kilbas}, the authors gave
the most recent and up-to-date developments on fractional
differential and fractional integro-differential equations with
applications involving many different potentially useful operators
of fractional calculus. In a recent work by Jaimini et.al.
\cite{jaimini} the authors have given the corresponding Leibnitz
rule for fractional calculus. For the history of fractional
calculus, interested reader may see the recent review paper by
Machado et. al. \cite{ten}.

Many physical processes appear to exhibit fractional order
behavior that may vary with time or space. The fractional calculus
has allowed the operations of integration and differentiation to
any fractional order. The order may take on any real or imaginary
value. Recently theory of fractional differential equations
attracted many scientists and mathematicians to work on
\cite{shantanu, hilfer, pod,pod1, pod2, pod3, saba}. For the
existence of solutions for fractional differential equations, one
can see \cite{ahmad1, ahmad2, ahmad3, agarwal, varsha, luigi,
deng,kai, sayed, furati, hadid1, hadid2, ibrahim, laksh,
laksh1,yu} and references therein. The results have been obtained
by using fixed point theorems like Picard's, Schauder fixed-point
theorem and Banach contraction mapping principle. About the
development of existence theorems for fractional functional
differential equations, many contribution exists \cite{abbas,
ahmad3,agarwal1,banas,shantanu,laksh2,ahmad4}. Many applications
of fractional calculus amount to replacing the time derivative in
a given evolution equation by a derivative of fractional order.
The results of several studies clearly stated that the fractional
derivatives seem to arise generally and universally from important
mathematical reasons. Recently, interesting attempts have been
made to give the physical meaning to the initial conditions for
fractional differential equations with Riemann-Liouville
fractional derivatives were proposed in \cite{gian,pod,pod2,pod3}.


Ahmed et. al. \cite{ahmad} considered the fractional order
predator-prey model and the fractional order rabies model. They
have shown the existence and uniqueness of solutions of the model
system and also studied the stability of equilibrium points. The
motivation behind fractional order system are discussed in
\cite{ahmad}. Lakshmikantham and Vatsala in \cite{laksh, laksh1}
and Lakshmikantham in \cite{laksh2} defined and proved existence
of the solution of fractional initial value problems.

In this article our aim is to show the existence of the solutions of
the differential equations
\begin{equation}
\begin{gathered}
\frac{d^{\alpha}x(t)}{dt^{\alpha}}=g(t,x(t)), \quad t\in [0,T] \\
 x(0)=x_0, \quad 0<\alpha<1,
\end{gathered} \label{ofde}
\end{equation}
and
\begin{equation}
\begin{gathered}
\frac{d^{\alpha}x(t)}{dt^{\alpha}}=f(t,x(t),x(t-\tau)), \quad
t\in [0,T]\\
x(t)=\phi(t), \quad t\in [-\tau,0] \quad 0<\alpha<1,
\end{gathered} \label{dfde}
\end{equation}
under suitable conditions on $g, f$ and $\phi$. We assume that
$g$ satisfies Lipschitz condition with Lipschitz constant $L_g$
and $f(t,x,y)$ can be written as $f_1(t,x)+f_2(t,x,y)$, where both
$f_1,f_2$ are Lipschitz continuous with Lipschitz constants
$L_{f_1}$ and $L_{f_2}$ respectively. Moreover, we show the
existence of maximum interval of existence for the problems
\eqref{ofde} and \eqref{dfde}. As far as I know these kind of
results are new for fractional differential equations.

Next we apply our results on the following fractional order Lotka
Volterra model for $0<\alpha<1$,
\begin{equation}
\begin{gathered}
\frac{d^{\alpha}x(t)}{dt^{\alpha}}
=(t)\Big(r(t)-a(t)x(t-\tau)\Big), \quad t\in [0,T], \; \tau \ge 0,  \\
 x(t)=\phi(t), \quad t\in [-\tau,0].
\end{gathered} \label{meq1}
\end{equation}
where $\frac{d^{\alpha}}{dt^{\alpha}}$ denotes Riemann–Liouville
derivative of order $\alpha$,   $0< \alpha < 1$.
The coefficients $r(t)$ and $a(t)$ satisfy
$$
r_* \le r(t) \le r^*, \quad a_* \le a(t) \le a^*
$$
which are biologically feasible. We use fixed point theory to show
the existence of a solution. For the fixed point theory and many
related results, interested reader may consult \cite{sm}.

\section{Preliminaries and Results}

 \begin{definition}  \rm
The fractional integral of order $\alpha>0$ of a function
$f: \mathbb{R}^+ \to \mathbb{R}$
 of order $\alpha \in \mathbb{R}^+$ is defined by
 $$
I^{\alpha}_0 f(t)=\frac{1}{\Gamma (\alpha)}
\int_0^t(t-s)^{\alpha-1}f(s)ds,
$$
provided the right side exists pointwise on $\mathbb{R}^+$.
$\Gamma$ is the gamma function.
\end{definition}

For instance, $I^{\alpha}f$ exists for all $\alpha >0$, when $f\in
C^0(\mathbb{R}^+)\cap L^1_{\rm loc}(\mathbb{R}^+);$ note also that
when $f\in C^0(\mathbb{R}^+_0)$ then $I^{\alpha}f \in
C^0(\mathbb{R}^+_0)$ and moreover $I^{\alpha}f(0) = 0$.

\begin{definition}  \rm
The fractional derivative of order $\alpha>0$ of a
function $f: \mathbb{R}^+ \to \mathbb{R}$ is given by
$$
\frac{d^{\alpha}}{dt^{\alpha}}f(t)=\frac{1}{\Gamma (1-\alpha)}
\frac{d}{dt}\int_0^t(t-s)^{-\alpha}f(s)ds
=\frac{d}{dt}I^{1-\alpha}_0h(t).
$$
\end{definition}

Using  fractional calculus, the equation \eqref{ofde} can be
represented by following integral form
$$
x(t)=x_0+\frac{1}{\Gamma \alpha}\int_0^t(t-s)^{\alpha-1}g(s,x(s))ds.
$$
First we discuss the existence of the solution of the following
ordinary fractional differential equation \eqref{ofde}
\begin{gather*}
\frac{d^{\alpha}x(t)}{dt^{\alpha}}
=g(t,x(t)), \quad t \in [0,T]\\
 x(0)=x_0.
\end{gather*}
Define the operator
$$
Tx(t)=x_0+\frac{1}{\Gamma \alpha}\int_0^t(t-s)^{\alpha-1}g(s,x(s))ds.
$$
Let the function $g:B(a,\beta) \to \mathbb{R}$ be bounded by $M$,
where
$$
B(a,\beta)=\{(t,x):\ |t| \le a, \ |x-x_0| \le \beta\}.
$$
We assume that our function $g$ is Lipschitz continuous with
respect to $x$ with Lipschitz constant $L_g$.
Denote $b=\min\{a, \frac{\beta}{M}\}$. Let $C$ be the set
of all continuous functions from $[-b,b]$ to $B(a,\beta)$.
 Consider
\begin{align*}
|Tx(t)-x_0| & \le \frac{1}{\Gamma \alpha}
 \int_0^t(t-s)^{\alpha-1}|g(s,x(s))|ds \\
&\le \frac{M}{\Gamma \alpha} \int_0^t(t-s)^{\alpha-1}ds
 \le \frac{M}{\Gamma \alpha} \int_0^t s^{\alpha-1}ds\\
&\le \frac{M}{\Gamma (\alpha+1)} t^{\alpha}
 \le \frac{M}{\Gamma (\alpha+1)}T^{\alpha}
\end{align*}
Thus for $x$ bounded, continuous, $Tx$ is also bounded,
continuous. We denote $B(a,\beta)$ by $B$ in short.
For $x,y \in B$, we have
\begin{align*}
|Tx(t)-Ty(t)|
&\le \frac{1}{\Gamma \alpha}\int_0^t
(t-s)^{\alpha-1}|g(s,x(s))-g(s,y(s))|ds  \\
&\le \frac{L_g}{\Gamma \alpha}\int_0^t (t-s)^{\alpha-1}|x(s)-y(s)|ds\\
&\le \frac{L_g}{\Gamma \alpha}\Big(\int_0^t
(t-s)^{\alpha-1}ds\Big)\sup_{x\in [0,T]}|x(s)-y(s)|  \\
&\le \frac{L_g}{\Gamma \alpha}\|x-y\|\int_0^t s^{\alpha-1}ds\\
&\le \frac{L_g}{\Gamma (\alpha+1)}\|x-y\|T^{\alpha}
\end{align*}
Thus for
$$
\frac{L_gT^{\alpha}}{\Gamma (\alpha+1)}<1,
$$
 we have $\|Tx-Ty\| <\|x-y\|$.

By the contraction mapping principle, we therefore know that $T$
has a unique fixed point in $B$. This implies that our problem has
a unique solution in $B$. Hence we summarize our result in the
following theorem.

\begin{theorem} \label{thm2.3}
Problem \eqref{ofde} has a unique solution in $B$ provided that
$$
\frac{L_gT^{\alpha}}{\Gamma (\alpha+1)}<1.
$$
\end{theorem}

Now we prove the existence of maximal interval of existence for
the fractional differential equation \eqref{ofde}. The analysis is
similar to analysis done by \cite{schmitt} for ordinary
differential equation. Let $\Omega$ be the open, connected subset
of $[0,T]\times \mathbb{R}$.

\begin{theorem} \label{thm2.4}
Assume that $g:\Omega \to \mathbb{R}$ is continuous and
let $x$  be a solution of the problem defined on some interval $I$.
Then $x$ may be extended as a solution of \eqref{ofde} to
a maximal interval of existence $(\omega_{-},\omega_{+})$ and $(t, x(t)) \to
\partial{\Omega}$ as $t \to \omega_{\pm}$.
\end{theorem}

\begin{proof}
We  need to show only the existence of a right
maximal interval of existence. For the left maximal interval of
existence a similar argument will work. Combining both argument
together will imply the existence of a  maximal interval of
existence. Let $x$ be a solution of \eqref{ofde} with the given
initial condition $x(0)=x_0$ defined on an interval $I= [0, a_x)$
for $a_x>0$. We say that two solutions $x_1,x_2$ of the problem
\eqref{ofde} satisfy $x_1 \prec x_2$, if and only if
\begin{gather*}
x \equiv x_1 \equiv x_2 \quad \text{on }  [0,a_x],\\
x_1  \text{ is defined on }  I_{x_1}=[0, a_{x_1}), \;a_{x_1}>a_x,\\
x_2  \text{ is defined on }  I_{x_2}=[0, a_{x_2}), \; a_{x_2} > a_x,
\end{gather*}
and $a_{x_2} \ge a_{x_1}$, also $x_1\equiv x_2$ on $I_{x_1}$.

To show $\prec$ is a partial order on the set of all
solutions $S$ of \eqref{ofde} which coincide with $x$ on $I$, we
need to show that it is reflexive, antisymmetric and transitive.
It is easy to see that $x_1 \prec x_1$ always holds. Now if $x_1
\prec x_2$ and $x_2 \prec x_1$ we have $x_1 \equiv x_2$ on $I_x$
by choosing $a_{x_1}=a_{x_2}$. Now, if $x_1 \prec x_2,  x_2 \prec
x_3$, we have
\begin{gather*}
x \equiv x_1 \equiv x_2 \quad \text{on }  [0,a_x],\\
x_1 \text{ is defined on } I_{x_1}=[0, a_{x_1}), \; a_{x_1}>a_x,\\
x_2 \text{ is defined on } I_{x_2}=[0, a_{x_2}), \; a_{x_2} > a_x,
\end{gather*}
and $a_{x_2} \ge a_{x_1}$, also $x_1\equiv x_2$ on $I_{x_1}$;
and
\begin{gather*}
x \equiv x_2 \equiv x_3 \quad \text{on} \ [0,a_x],\\
x_2 \text{ is defined on }  I_{x_2}=[0, a_{x_2}), \;a_{x_2}>a_x,\\
x_3 \text{ is defined on }  I_{x_3}=[0, a_{x_3}), \; a_{x_3} > a_x,
\end{gather*}
and $a_{x_3} \ge a_{x_2}$, also $x_2\equiv x_3$ on $I_{x_2}$.

From these two conditions, we can easily obtain
\begin{gather*}
x \equiv x_1 \equiv x_3  \text{ on } [0,a_x],\\
x_1 \text{ is defined on }  I_{x_1}=[0, a_{x_1}), \;a_{x_1}>a_x,\\
x_3 \text{ is defined on }  I_{x_3}=[0, a_{x_3}), \; a_{x_3} > a_x,
\end{gather*}
and $a_{x_3} \ge a_{x_1}$, also $x_3\equiv x_1$ on $I_{x_3}$. Thus
$x_1 \prec x_3$.

Thus $\prec$ is a partial order. Now we verify that the conditions
of the Hausdroff maximum principle (\cite{rudin}) hold and hence
that $S$ contains a maximal element, say $\bar{x}$. This maximal
element $\bar{x}$ cannot be further extended to the right. Let $x$
be a solution of \eqref{ofde} with right maximal interval of
existence $[0, \omega_{+})$. Now, we must show that
 $(t, x(t)) \to \partial{\Omega}$ as $t \to \omega_{+}$;
 that is, given any compact set $K \subset \Omega$, there exists
$t_K$, such that $(t, x(t)) \not \in K$, for $t> t_K$.

For the case $\omega_{+}= \infty$, the conclusion clearly holds.
For the other case, that is if $\omega_{+} < \infty$, we proceed
indirectly. In the later case there exists a compact set
$K \subset \Omega$, such that for every $n=1, 2,\dots$ there exists
$t_n$, $0 < \omega_{+}-t_n < \frac{1}{n}$, and
$(t_n, x(t_n)) \in K$. Since $K$ is compact, there will be a
subsequence, for the convenience call it again $\{(t_n, x(t_n))\}$
such that $\{(t_n, x(t_n))\}$ converges to $(\omega_{+},x^*)$
which belongs to $K$.
Since $(\omega_{+}, x^*) \in K$, it is an interior point of
$\Omega$. We may therefore choose a constant $a> 0$, such that
$Q= \{(t, x) : |\omega_{+}-t| \le a, \; |x-x^*| \le a\} \subset
\Omega$. Thus for $n$ large $(t_n, x(t_n)) \in Q$.
Let $m = \max_{(t,x) \in Q} |f(t, x)|$, and let $n$ be so large
that
$$
0 < \omega_{+}- t_n \le \frac{a}{2m} , \quad |x(t_n)-x^*|
\le \frac{a}{2}.
$$
Then
$$
|x(t_n)-x(t)| < m(\omega_{+}-t_n) \le \frac{a}{2},
$$
for $t <\omega_{+}$, by an easy argument. Therefore,
$\lim_{t\to \omega_{+}}x(t)=x^*$. Hence we may extend $x$
to the right of $\omega_{+}$ contradicting the maximality of $x$.
Hence the result is proved for the fractional ordinary
differential equation \eqref{ofde}.
\end{proof}

Consider the following function $g(t,x(t))=x(t)(r(t)-a(t)x(t))$,
where we assume that $r(t) \in [r_*,r^*]$ and $a(t) \in
[a_*,a^*]$. The corresponding fractional differential equations
represent the evolution model of a single species without delay.
It is easy to see that the function $f$ is Lipschitz and bounded
for any $x \in B$. Thus from the above analysis we obtain the
existence of the solution which can be extended to the maximal
interval. One can easily observe that the above results can be
easily extended to $\mathbb{R}^{n}$.


Moreover, for the fractional delay differential equation
\eqref{dfde}, it is easy to see that if $t \in [0,\tau]$, our
function $x(t-\tau)=\phi(t-\tau)$. Thus in this interval the delay
fractional differential equations behave like non-delay fractional
differential equations,
\begin{gather*}
\frac{d^{\alpha}x(t)}{dt^{\alpha}}=f(t,x(t),\phi(t-\tau)), \quad
t\in [0,\tau]  \\
 x(t)=\phi(t), \quad t\in [-\tau,0], \; 0<\alpha<1.
\end{gather*}
A similar analysis we described above for problem \eqref{ofde} can
be used to show the local existence and uniqueness of the solution
of fractional delay differential equation \eqref{dfde}.

Let us consider the function
$f(t,x(t),x(t-\tau))=x(t)(r(t)-a(t)x(t-\tau))$, where we assume
that $r(t) \in [r_*,r^*]$ and $a(t) \in [a_*,a^*]$. These kind of
function come from the modelling of interspecific competition in
one species with $\tau$ as a maturity time period. The
corresponding fractional differential equations for this function
$f$ is \eqref{meq1}. Thus for $t \in [0,\tau]$, our function is
$x(t)(r(t)-a(t)\phi(t-\tau))$. It is easy to see that the function
$f$ is Lipschitz and bounded for any $x \in B$. Hence by using
similar analysis as mentioned above, we obtain local existence of the
solution.

Now our next target is to use Krasnoselskii's fixed point theorem
to prove the existence and uniqueness of the solution of
fractional delay differential equations \eqref{dfde}. A similar
analysis yield the existence of solution of the problem
\eqref{ofde}.

By a solution $x(t)$ of  \eqref{dfde} we mean that it
satisfy the  relation
$$
x(t)=\phi(0)+\frac{1}{\Gamma \alpha}\int_0^t
(t-s)^{\alpha-1}f(s,x(s),x(s-\tau))ds
$$
for $t\in [0,T]$ and $x(t)=\phi(t)$ for $t \in [-\tau,0]$.

First we mention statement of Krasnoselskii's fixed point theorem.

\begin{theorem}[Krasnoselskii] \label{lemma21}
 Let $B$ be a nonempty closed convex
subset of a Banach space $(X, \|\cdot\|)$. Suppose that
$\Lambda_1$ and $\Lambda_2$ map $B$ into $X$ such that
\begin{itemize}
\item[(i)] for any $x,y \in B$, $\Lambda_1x+\Lambda_2y \in B$,
\item[(ii)] $\Lambda_1$ is a contraction,
\item[(iii)] $\Lambda_2$ is continuous and $\Lambda_2 (B)$ is contained
in a compact set.
\end{itemize}
Then there exists $z\in B$ such that
$z=\Lambda_1z+\Lambda_2z$.
\end{theorem}


Now we prove existence of the solutions for the delay fractional
differential equations \eqref{dfde} using Krasnoselskii's fixed
point theorem. We begin with the assumption that our function $f$
can be written as the sum of two functions of the following form
$$f(t,x(t),y(t))=f_1(t,x(t))+f_2(t,x(t),y(t)),$$ where $f_i$,
$i=1,2$ are Lipschitz continuous functions with Lipschitz constants
$L_{f_i}$ for $i=1,2$. Define the operators $F_1$ and $F_2$ by
\begin{gather*}
F_1x(t)=\phi(0)+\frac{1}{\Gamma \alpha}\int_0^t
(t-s)^{\alpha-1}f_1(s,x(s))ds, \\
F_2x(t)=\frac{1}{\Gamma \alpha}\int_0^t
(t-s)^{\alpha-1}f_2(s,x(s),x(s-\tau))ds.
\end{gather*}
It is easy to see that
\begin{align*}
|F_1x(t)-F_1y(t)|
&\leq  \frac{1}{\Gamma \alpha}\int_0^t
(t-s)^{\alpha-1}|f_1(s,x(s))-f_1(s,y(s))|ds  \\
&\leq \frac{L_{f_1}}{\Gamma \alpha}\int_0^t
(t-s)^{\alpha-1}|x(s)-y(s)|ds  \\
&\leq \frac{L_{f_1}}{\Gamma \alpha}\|x-y\|\int_0^t s^{\alpha-1}ds \\
&\leq  \frac{L_{f_1}}{\Gamma (\alpha+1)}\|x-y\|
T^{\alpha}.
\end{align*}
We obtain
 $$
\|F_1x-F_1y\| \le \frac{L_{f_1}T^{\alpha}}{\Gamma (\alpha+1)}\|x-y\|.
$$
Thus $F_1$ is a contraction provided
$\frac{L_{f_1}T^{\alpha}}{\Gamma (\alpha+1)}<1$.

Further assume that the functions $f_i$, $i=1,2$ satisfy the
 relations
\begin{gather*}
|f_1(t,x(t))| \le M_1|x(t)|,\\
|f_2(t,x(t),y(t))| \le M_2 |x(t)|\times |y(t)|.
\end{gather*}
Let $BC([-\tau, T],\mathbb{R})$ denote the
collection of all bounded and continuous function from
$[-\tau, T]$ to $\mathbb{R}$. Consider the set
$$
D=\{x \in BC([-\tau, T],\mathbb{R}) : |x| \le r\}
$$
where $r$ satisfies
$$
|\phi(0)|+\frac{M_1r+M_2r^2}{\Gamma(\alpha+1)} T^{\alpha} \le r.
$$
For $x \in D$, calculating the norm of the function $F=F_1+F_2$,
we have
\begin{align*}
&|F_1x(t)+F_2x(t)|  \\
&\leq  |\phi(0)|+\frac{1}{\Gamma \alpha}\int_0^t
(t-s)^{\alpha-1}|f_1(s,x(s))+f_2(s,x(s),x(s-\tau))|ds  \\
&\leq  |\phi(0)|+\frac{M_1\|x\|+M_2\|x\|^2}{\Gamma \alpha}\int_0^t
(t-s)^{\alpha-1}ds  \\
&\leq |\phi(0)|+\frac{M_1r+M_2r^2}{\Gamma(\alpha+1)} T^{\alpha}.
\end{align*}
Thus $F_1x+F_2x \in D$.
Moreover for $x \in D$, we obtain
\begin{align*}
|F_2x(t)|
&\le  \frac{1}{\Gamma \alpha}\int_0^t
(t-s)^{\alpha-1}|f_2(s,x(s),x(s-\tau))|ds  \\
& \le \frac{M_2\|x\|^2}{\Gamma \alpha}\int_0^t (t-s)^{\alpha-1}ds\\
&\le  \frac{M_2r^2}{\Gamma (\alpha+1)} T^{\alpha}
\le r.
\end{align*}

To prove the continuity of $F_2$, let us consider a sequence $x_n$
converging to $x$. Taking the norm of $F_2x_n(t)-F_2x(t)$, we have
\begin{align*}
&|F_2x_n(t)-F_2x(t)|  \\
&\leq   \frac{1}{\Gamma \alpha}\int_0^t
(t-s)^{\alpha-1}|f_2(s,x_n(s),x_n(s-\tau))-f_2(s,x(s),x(s-\tau))|ds\\
&\leq \frac{L_{f_2}}{\Gamma \alpha}\int_0^t
(t-s)^{\alpha-1}(|x_n(s)-x(s)|+|x_n(s-\tau)-x(s-\tau)|)ds
 \\ &\leq  \frac{2L_{f_2}}{\Gamma \alpha}\Big(\int_0^t
s^{\alpha-1}ds\Big)\|x_n-x\| \\
&\leq  \frac{2L_{f_2}}{\Gamma (\alpha+1)}T^{\alpha} \|x_n-x\|.
\end{align*}
 From the above analysis we obtain
$$
\|F_2x_n-F_2x\| \le \frac{2L_{f_2}}{\Gamma (\alpha+1)}T^{\alpha}
\|x_n-x\|
$$
and hence whenever $x_n \to x$, $Fx_n \to Fx$. This proves the
continuity of $F_2$.

Now for $t_1 \le t_2 \le T$, we have
\begin{align*}
&|F_2x(t_2)-F_2x(t_1)|  \\
&\leq   \frac{1}{\Gamma \alpha}\Big|\int_0^{t_2}
(t_2-s)^{\alpha-1}f_2(s,x(s),x(s-\tau)ds \\
&\quad - \int_0^{t_1} (t_1-s)^{\alpha-1}f_2(s,x(s),x(s-\tau)ds\Big|  \\
&\leq  \frac{1}{\Gamma \alpha}\Big|\int_0^{t_1}
(t_2-s)^{\alpha-1}f_2(s,x(s),x(s-\tau)ds  \\
&\quad +\int_{t_1}^{t_2} (t_2-s)^{\alpha-1}f_2(s,x(s),x(s-\tau)ds\\
&\quad -\int_0^{t_1} (t_1-s)^{\alpha-1}f_2(s,x(s),x(s-\tau)ds\Big|\\
&\leq \frac{1}{\Gamma \alpha}\int_0^{t_1}|((t_2-s)^{\alpha-1}
-(t_1-s)^{\alpha-1})| |f_2(s,x(s),x(s-\tau)|ds \\
&\quad  +\frac{1}{\Gamma \alpha}\int_{t_1}^{t_2}
|(t_2-s)^{\alpha-1}|\times|f_2(s,x(s),x(s-\tau)|ds\\
&\leq  \frac{M_1r^2}{\Gamma
\alpha}\int_0^{t_1}|(t_2-s)^{\alpha-1}-(t_1-s)^{\alpha-1}|ds
   +\frac{M_2r^2}{\Gamma \alpha}\int_{t_1}^{t_2}
|(t_2-s)^{\alpha-1}|ds  \\ &\leq   \frac{r^2}{\Gamma
(\alpha+1)}\max\{M_1,M_2\}\Big|-2(t_2-t_1)^{\alpha}+t_2^{\alpha}-t_1^{\alpha}\Big|
 \\
&\leq \frac{r^2}{\Gamma(\alpha+1)}\max\{M_1,M_2\}(t_2-t_1)^{\alpha}
\end{align*}
The right-hand side of above expression does not depends on $x$. Thus
we conclude that $F_2(D)$ is relatively compact and hence $F_2$ is
compact by Arzela-Ascoli theorem. Using Krasnoselskii’s fixed
point theorem, we obtain that there exists $z \in D$ such that
$Fz=F_1z+F_2z=z$, which is a fixed point of $F$. Hence the problem
\eqref{dfde} has at least one solution in $D$. We summarize the
above results in the form of the following theorem.

\begin{theorem} \label{tmh1}
Model \eqref{dfde} has a solution in the set $D$ provided
$\frac{L_{f_1}T^{\alpha}}{\Gamma (\alpha+1)}<1$ and
$$
|\phi(0)|+\frac{M_1r+M_2r^2}{\Gamma(\alpha+1)} T^{\alpha} \le r\,.
$$
\end{theorem}

Consider the  function
$f(t,x(t),x(t-\tau))=x(t)(r(t)-a(t)x(t-\tau))$ and let us denote
$$
f_1(t,x(t))=x(t)r(t), \quad
f_2(t,x(t),x(t-\tau))=-a(t)x(t)x(t-\tau).
$$
It is easy  to see that
\begin{gather*}
|f_1(t,x(t))| \le r^*|x(t)|,\\
|f_2(t,x(t),x(t-\tau))| \le a^*|x(t)||x(t-\tau)|.
\end{gather*}

Using  fractional calculus,  \eqref{meq1} can be
representable as an integral form of the type
\begin{gather*}
x(t)=\phi(0)+\frac{1}{\Gamma(\alpha)}\int_0^t(t-s)^{\alpha-1}x(s)
\Big(r(s)-a(s)x(s-\tau)\Big)ds \\
 x(t)=\phi(t), \quad t\in [-\tau,0].
\end{gather*}
Define a mapping $\Lambda$ by
$$
\Lambda x(t)=\Lambda_1x(t)+\Lambda_2x(t),
$$
where
\begin{gather*}
\Lambda_1x(t)=\frac{1}{\Gamma(\alpha)}
\int_0^t(t-s)^{\alpha-1}x(s)r(s)ds ,  \\
\Lambda_2x(t)=-\frac{1}{\Gamma(\alpha)}\int_0^t(t-s)^{\alpha-1}
x(s)a(s)x(s-\tau)ds.
\end{gather*}
One can easily see that in this case our operator $F_1$ coincide
with $\Lambda_1$ and $F_2$ coincides with $\Lambda_2$. Thus our
model systems \eqref{meq1} have at least one solution. We
summarize the result for problem \eqref{meq1} in the form of the
following theorem.

\begin{theorem} \label{tmh1b}
The model \eqref{meq1} has a solution in the set $D$ provided
$\frac{L_{f_1}T^{\alpha}}{\Gamma (\alpha+1)}<1$ and
$$
|\phi(0)|+\frac{r^*r+a^*r^2}{\Gamma(\alpha+1)} T^{\alpha} \le r\,.
$$
\end{theorem}

\begin{remark} \label{rmk2.8} \rm
The above result can be extended for $n$ species competitive
system of the form
\begin{gather*}
\frac{d^{\alpha}x_i(t)}{dt^{\alpha}}
= x_i(t)\Big(r_i(t)-\sum_{j=1}^na_{ij}(t)x_j(t-\tau_{ij})\Big),
\quad t \in [0,T], \; i=1,2,\dots,n. \label{meq2}  \\
x_i(t)=\phi_i(t), \quad t \in [-\tau,0],
\end{gather*}
where $\alpha$, $0< \alpha < 1$, $r_i(t) \in [r_*,r^*]$, and
$ a_{ij}\in [a_*,a^*]$.
\end{remark}

\subsection*{Acknowledgments}
The author would like to thank the anonymous referee for his/her
valuable comments and suggestions, and Prof. Julio G. Dix for his
kind support which help me to improve the manuscript considerably.

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\end{document}