\def\papertitle{Existence, uniqueness and stability of solutions to a delay hematopoiesis model}
\def\paperauthorA{Ahl\`{e}me Bouakkaz$^{*1}$ and Rabah Khemis$^{2}$}
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\title{\papertitle}
\affiliation{\paperauthorA}	% This command replaces \name{The DAFx Crew} 
{$^{1,2}$Laboratory of Applied Mathematics and History and Didactics\\  of Mathematics (LAMAHIS), University of 20 August 1955,\\ Skikda, Algeria}

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	\maketitle
	
\begin{abstract}
This work aims to investigate a delay hematopoiesis model where the delay
depends on both the time and the current density of mature blood cells. Based
on the Banach contraction principle, the Schauder's fixed point theorem and
some properties of a Green's function, we establish several interesting
existence and uniqueness results of positive periodic solutions for the
proposed model. The derived results are new and generalize some previous
studies.
\end{abstract}

\section{\textbf{Introduction}}

Consider the following delay hematopoiesis model:%
\begin{equation}
x^{\prime}\left(  t\right)  =-a\left(  t\right)  x\left(  t\right)  +p\left(
t\right)  \dfrac{x\left(  t-\tau\right)  }{1+x\left(  t-\tau\right)
}-H\left(  t,x(t),x(t-\tau)\right)  , \label{1.1}%
\end{equation}
where $x\left(  t\right)  $ is the density of mature cells in blood
circulation at time $t,$ $a\left(  t\right)  $ is the rate of lost cells from
the circulation, $p\left(  t\right)  $ is the production rate, $H$ is the
harvesting function and the delay $\tau$ describes the duration of the
proliferative phase.

A lot of investigations have revealed that the time and state dependent delays
are more realistic than other types of lags to describe real phenomena
especially in life sciences . Actually, in the production of blood cells, some
growth factors and hormones that control the division of the hematopoietic
stem cells, depend on time and the density of mature cells.

To incorporate this biological information, we need to revisit the previous
model as follows:%
\begin{equation}
x^{\prime}\left(  t\right)  =-a\left(  t\right)  x\left(  t\right)  +p\left(
t\right)  \dfrac{x\left(  t-\tau\left(  t,x(t)\right)  \right)  }{1+x\left(
	t-\tau\left(  t,x(t)\right)  \right)  }-H\left(  t,x(t),x(t-\tau\left(
t,x(t)\right)  )\right)  . \label{1.2}%
\end{equation}
In this work, we assume that the delay in equation (\ref{1.2})\ takes the form
$\tau\left(  t,x\left(  t\right)  \right)  =t-x\left(  t\right)  .$ Under this
assumption, equation (\ref{1.2}) becomes the following iterative differential equation:%

\begin{equation}
x^{\prime}\left(  t\right)  =-a\left(  t\right)  x\left(  t\right)  +p\left(
t\right)  \frac{x^{\left[  2\right]  }\left(  t\right)  }{1+x^{\left[
		2\right]  }\left(  t\right)  }-H\left(  t,x(t),x^{\left[  2\right]
}(t)\right)  , \label{1.3}%
\end{equation}
where $x^{\left[  2\right]  }\left(  t\right)  =x\left(  x\left(  t\right)
\right)  ,$ $a\in\mathcal{C}\left(  \mathbb{R},\left(  0,\infty\right)
\right)  ,$ $p\in\mathcal{C}\left(  \mathbb{R},\left[  0,\infty\right)
\right)  $ and $H\in\mathcal{C}\left(  \left[  0,\omega\right]  \times
\mathbb{R}^{2},\left(  0,\infty\right)  \right)  .$

\section{\textbf{Preliminaries}}

We set the following assumptions:%
\begin{equation}
a\left(  t+\omega\right)  =a\left(  t\right)  ,\text{ }p\left(  t+\omega
\right)  =p\left(  t\right)  ,\text{ }H\left(  t+\omega,x,y\right)  =H\left(
t,x,y\right)  ,\text{ for all }t,x,y\in\mathbb{R}, \label{2.1}%
\end{equation}
and%
\begin{equation}
\left\vert H\left(  t,x_{1},y_{1}\right)  -H\left(  t,x_{2},y_{2}\right)
\right\vert \leq k_{1}\left\vert x_{1}-x_{2}\right\vert +k_{2}\left\vert
y_{1}-y_{2}\right\vert ,\text{ }k_{1},k_{2}>0. \label{2.2}%
\end{equation}
For proving the existence of at least one positive periodic solution of
(\ref{1.3}), we will convert this equation into an integral one before
employing the Schauder's fixed point theorem.

For $L,M>0,$ let
\[
\mathbb{K}=\left\{  x\in\mathbb{X},\text{ }0\leq x\left(  t\right)  \leq
M,\text{ }\left\vert x(t_{2})-x(t_{1})\right\vert \leq L\left\vert t_{2}%
-t_{1}\right\vert ,\text{ }\forall t_{1},t_{2}\in\left[  0,\mathbb{\omega
}\right]  \right\}  ,
\]
a convex bounded and closed subset of the Banach space $\left(  \mathbb{X}%
,\left\Vert .\right\Vert \right)  $ where%
\[
\mathbb{X}=\left\{  x\in\mathcal{C}(\mathbb{R},\mathbb{R}),\text{ }%
x(t+\omega)=x(t)\right\}  ,
\]
and
\[
\left\Vert x\right\Vert =\underset{t\in\mathbb{R}}{\sup}\left\vert
x(t)\right\vert =\underset{t\in\left[  0,\omega\right]  }{\sup}\left\vert
x(t)\right\vert .
\]
Furthermore, we assume that%
\begin{equation}
\mathcal{\omega\beta}\left(  H_{0}+\left\Vert p\right\Vert +\left(
k_{1}+k_{2}\left(  1+L\right)  \right)  M\right)  \leq M, \label{2.3}%
\end{equation}
where $H_{0}=\sup_{t\in\left[  0,\omega\right]  }H\left(  t,0,0\right)  $ and%
\begin{equation}
\beta\left(  H_{0}+\left\Vert p\right\Vert +\left(  k_{1}+k_{2}\left(
1+L\right)  \right)  M\right)  \left(  2+\omega\left\Vert a\right\Vert
\right)  \leq L. \label{2.4}%
\end{equation}
We begin our study by the following result which ensures the conversion of our
problem into an equivalent integral equation:

\begin{lemma}
	\label{Lem1}If the condition (\ref{2.1}) holds, then $x\in\mathbb{K}$ is a
	solution of (\ref{1.3}) if and only if $x$ is a solution of the following
	integral equation:%
	\begin{equation}
	x\left(  t\right)  =\int_{t}^{t+\mathcal{\omega}}\left[  p\left(  s\right)
	\dfrac{x^{\left[  2\right]  }(s)}{1+x^{\left[  2\right]  }(s)}-H\left(
	s,x(s),x^{\left[  2\right]  }(s)\right)  \right]  G\left(  t,s\right)  ds,
	\label{2.5}%
	\end{equation}
	where%
	\begin{equation}
	G\left(  t,s\right)  =\dfrac{e^{%
			%TCIMACRO{\dint _{t}^{s}}%
			%BeginExpansion
			{\displaystyle\int_{t}^{s}}
			%EndExpansion
			a\left(  v\right)  dv}}{e^{%
			%TCIMACRO{\dint _{0}^{\mathcal{\omega}}}%
			%BeginExpansion
			{\displaystyle\int_{0}^{\mathcal{\omega}}}
			%EndExpansion
			a\left(  v\right)  dv}-1}. \label{2.6}%
	\end{equation}
	
\end{lemma}

\begin{remark}
	Let%
	\[
	\dfrac{e^{-\int_{0}^{\mathcal{\omega}}a\left(  v\right)  dv}}{e^{\int
			_{0}^{\mathcal{\omega}}a\left(  v\right)  dv}-1}=\alpha\text{ and }%
	\dfrac{e^{\int_{0}^{\mathcal{\omega}}a\left(  v\right)  dv}}{e^{\int
			_{0}^{\mathcal{\omega}}a\left(  v\right)  dv}-1}=\beta,
	\]
	then%
	\begin{align}
		&  \left.  G(t+\omega,s+\omega)=G(t,s),\right.  \label{2.7}\\
		&  \left.  0<\alpha\leq G\left(  t,s\right)  \leq\beta,\right.  \label{2.8}%
	\end{align}
	and%
	\begin{equation}
	\int_{t_{1}}^{t_{1}+w}\left\vert G\left(  t_{2},s\right)  -G\left(
	t_{1},s\right)  \right\vert ds\leq\beta w\left\Vert a\right\Vert \left\vert
	t_{2}-t_{1}\right\vert .\label{2.9}%
	\end{equation}
	
\end{remark}


\section{\textbf{Main results}}

\subsection{Existence}

Now, to apply Schauder's fixed point theorem, we need to construct an operator
$\mathcal{A}:\mathbb{K\rightarrow X}$ as follows:%
\begin{equation}
\left(  \mathcal{A\varphi}\right)  \left(  t\right)  =\int_{t}%
^{t+\mathcal{\omega}}\left[  p\left(  s\right)  \dfrac{\mathcal{\varphi
	}^{\left[  2\right]  }(s)}{1+\mathcal{\varphi}^{\left[  2\right]  }%
	(s)}-H\left(  s,\mathcal{\varphi}(s),\mathcal{\varphi}^{\left[  2\right]
}(s)\right)  \right]  G\left(  t,s\right)  ds. \label{2.10}%
\end{equation}
From Lemma \ref{Lem1}, fixed points of operator $\mathcal{A}$ are solutions of
equation (\ref{1.3}) and vice versa.

\begin{remark}
	For all $t\in\left[  0,\mathcal{\omega}\right]  $ and $\mathcal{\varphi\in
	}\mathbb{K}$ we have%
	\[
	\left(  \mathcal{A\varphi}\right)  \left(  t+\mathcal{\omega}\right)  =\left(
	\mathcal{A\varphi}\right)  \left(  t\right)  ,
	\]
	so $\mathcal{A}$ is well defined.
\end{remark}

\begin{lemma}
	\label{Lem2}If condition (\ref{2.2}) holds, then operator $\mathcal{A}$ is continuous.
\end{lemma}

\begin{proof}
	Let $\varphi,\theta\in\mathbb{K}$, we have%
	\begin{align*}
		\left\vert \left(  \mathcal{A\varphi}\right)  \left(  t\right)  -\left(
		\mathcal{A\theta}\right)  \left(  t\right)  \right\vert  &  \leq\int
		_{t}^{t+\mathcal{\omega}}G\left(  t,s\right)  p\left(  s\right)  \left\vert
		\dfrac{\mathcal{\varphi}^{\left[  2\right]  }(s)}{1+\mathcal{\varphi}^{\left[
				2\right]  }(s)}-\dfrac{\mathcal{\theta}^{\left[  2\right]  }(s)}%
		{1+\mathcal{\theta}^{\left[  2\right]  }(s)}\right\vert ds\\
		&  +\int_{t}^{t+\mathcal{\omega}}G\left(  t,s\right)  \left\vert H\left(
		s,\mathcal{\varphi}(s),\mathcal{\varphi}^{\left[  2\right]  }(s)\right)
		-H\left(  s,\mathcal{\theta}(s),\mathcal{\theta}^{\left[  2\right]
		}(s)\right)  \right\vert ds.
	\end{align*}
	By using (\ref{2.2}), (\ref{2.8}) and [Lemma 2.1, $\left[  12\right]  $], we
	obtain
	\[
	\left\vert \left(  \mathcal{A\varphi}\right)  \left(  t\right)  -\left(
	\mathcal{A\theta}\right)  \left(  t\right)  \right\vert \leq\beta
	\mathcal{\omega}\left(  \left(  \left\Vert p\right\Vert +k_{2}\right)  \left(
	1+L\right)  +k_{1}\right)  \left\Vert \mathcal{\varphi}-\mathcal{\theta
	}\right\Vert .
	\]
	Consequently, $\mathcal{A}$ is continuous.
\end{proof}

\begin{lemma}
	\label{Lem3}Let conditions (\ref{2.2})-(\ref{2.4}) hold and suppose that%
	\begin{equation}
	p\left(  t\right)  \dfrac{\mathcal{\varphi}^{\left[  2\right]  }%
		(t)}{1+\mathcal{\varphi}^{\left[  2\right]  }(t)}-H\left(  s,\mathcal{\varphi
	}(t),\mathcal{\varphi}^{\left[  2\right]  }(t)\right)  \geq0,\text{ }\forall
	t\in\left[  0,\mathcal{\omega}\right]  . \label{2.11}%
	\end{equation}
	Then%
	\[
	\left(  \mathcal{A}\right)  \left(  \mathbb{K}\right)  \subset\mathbb{K}.
	\]
	
\end{lemma}

\begin{proof}
	\ \ \newline\underline{\textbf{Step 1:}} If $\varphi\in\mathbb{K}$, then%
	\begin{align*}
		\left\vert \left(  \mathcal{A\varphi}\right)  \left(  t\right)  \right\vert
		&  =\int_{t}^{t+\mathcal{\omega}}\left[  p\left(  s\right)  \dfrac
		{\mathcal{\varphi}^{\left[  2\right]  }(s)}{1+\mathcal{\varphi}^{\left[
				2\right]  }(s)}-H\left(  s,\mathcal{\varphi}(s),\mathcal{\varphi}^{\left[
			2\right]  }(s)\right)  \right]  G\left(  t,s\right)  ds\\
		&  \leq\int_{t}^{t+\mathcal{\omega}}\left[  p\left(  s\right)  \dfrac
		{\mathcal{\varphi}^{\left[  2\right]  }(s)}{1+\mathcal{\varphi}^{\left[
				2\right]  }(s)}\right]  G\left(  t,s\right)  ds\\
		&  +\int_{t}^{t+\mathcal{\omega}}\left[  H\left(  s,\mathcal{\varphi
		}(s),\mathcal{\varphi}^{\left[  2\right]  }(s)\right)  \right]  G\left(
		t,s\right)  ds\\
		&  \leq\mathcal{\omega\beta}\left(  H_{0}+\left\Vert p\right\Vert +\left(
		k_{1}+k_{2}\left(  1+L\right)  \right)  M\right)  .
	\end{align*}
	From (\ref{2.3}), we obtain%
	\[
	\left(  \mathcal{A\varphi}\right)  \left(  t\right)  \leq M,
	\]
	and from (\ref{2.11}) we get%
	\[
	0\leq\left(  \mathcal{A\varphi}\right)  \left(  t\right)  .
	\]
	Then%
	\begin{equation}
	0\leq\left(  \mathcal{A\varphi}\right)  \left(  t\right)  \leq M,\text{
	}\forall t\in\left[  0,\mathcal{\omega}\right]  ,\text{ }\forall
	\mathcal{\varphi\in}\mathbb{K}. \label{2.12}%
	\end{equation}
	\underline{\textbf{Step 2:}} Let $\varphi\in\mathbb{K}$ and $t_{1},t_{2}%
	\in\left[  0,\mathcal{\omega}\right]  $. Taking into account the assumptions
	(\ref{2.2}), (\ref{2.8}), (\ref{2.9}) and [Lemma 2.1, $\left[  12\right]  $],
	we arrive at%
	\begin{align*}
		\left\vert \left(  \mathcal{A\varphi}\right)  \left(  t_{2}\right)  -\left(
		\mathcal{A\varphi}\right)  \left(  t_{1}\right)  \right\vert  &  \leq
		2\beta\left(  H_{0}+\left\Vert p\right\Vert +\left(  k_{1}+k_{2}\left(
		1+L\right)  \right)  M\right)  \left\vert t_{2}-t_{1}\right\vert \\
		&  +\beta\omega\left\Vert a\right\Vert \left(  H_{0}+\left\Vert p\right\Vert
		+\left(  k_{1}+k_{2}\left(  1+L\right)  \right)  M\right)  \left\vert
		t_{2}-t_{1}\right\vert \\
		&  =\beta\left(  H_{0}+\left\Vert p\right\Vert +\left(  k_{1}+k_{2}\left(
		1+L\right)  \right)  M\right)  \left(  2+\omega\left\Vert a\right\Vert
		\right)  \left\vert t_{2}-t_{1}\right\vert .
	\end{align*}
	By virtue of (\ref{2.4}), we get%
	\begin{equation}
	\left\vert \left(  \mathcal{A\varphi}\right)  \left(  t_{2}\right)  -\left(
	\mathcal{A\varphi}\right)  \left(  t_{1}\right)  \right\vert \leq L\left\vert
	t_{2}-t_{1}\right\vert . \label{2.13}%
	\end{equation}
	Thanks to (\ref{2.12}) and (\ref{2.13}), we get%
	\[
	\left(  \mathcal{A}\right)  \left(  \mathbb{K}\right)  \subset\mathbb{K},
	\]
	which completes the proof.
\end{proof}

Now, we can formulate our existence result as follows:

\begin{theorem}
	If the assumptions (\ref{2.1})-(\ref{2.4}) and (\ref{2.11}) are fulfilled,
	then equation (\ref{1.3}) has at least one positive periodic solution in
	$\mathbb{K}.$
\end{theorem}

\begin{proof}
	As a consequence of Lemmas 2 and 3, all conditions of Schauder's fixed point
	theorem are staisfied. So, we conclude that $\mathcal{A}$ has at least one
	fixed point which is a solution of equation (\ref{1.3}).
\end{proof}

\subsection{Uniqueness}

\begin{theorem}
	Besides the assumptions (\ref{2.1})-(\ref{2.4}), we suppose that%
	\[
	\beta\mathcal{\omega}\left(  \left(  \left\Vert p\right\Vert +k_{2}\right)
	\left(  1+L\right)  +k_{1}\right)  <1.
	\]
	\newline Then equation (\ref{1.3}) has one and only one solution
	$x\in\mathbb{K}.$
\end{theorem}

\begin{proof}
	The proof is based on using the Banach contraction principle.
\end{proof}
\subsection{Continuous dependence on parameters}

\begin{theorem}
	Under the hypothesis of Theorem 2, the unique solution of equation (\ref{1.3})
	depends continuously on parameters.
\end{theorem}

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\end{document}
