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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 88, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/88\hfil
A nonlinear neutral periodic differential equation]
{A nonlinear neutral periodic differential equation}

\author[E. R. Kaufmann\hfil EJDE-2010/88\hfilneg]
{Eric R. Kaufmann}

\address{Eric R. Kaufmann \newline
Department of Mathematics \& Statistics\\
University of Arkansas at Little Rock, Little Rock, AR 72204, USA}
\email{erkaufmann@ualr.edu}

\thanks{Submitted March 22, 2010. Published June 25, 2010.}
\subjclass[2000]{34A37, 34A12, 39A05}
\keywords{Fixed point theory; nonlinear dynamic equation; periodic}

\begin{abstract}
 In this article we consider the existence, uniqueness and
 positivity of a first order non-linear periodic differential
 equation. The main tool employed is the Krasnosel'ski\u{\i}'s
 fixed point theorem for the sum of a completely continuous
 operator and a contraction.
\end{abstract}

\maketitle 
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

Let $T > 0$ be fixed. We consider the existence, uniqueness and
positivity of solutions for the nonlinear neutral periodic
equation
\begin{equation}\label{eq01}
    \begin{gathered}
 x'(t) = -a(t)x(t)+ c(t)x'\big ( g(t) \big ) g'(t) + q \big( t, x(t), x(g(t))\big),\\
 x(t + T) = x(t).
    \end{gathered}
\end{equation}

In recent years, there have been several papers written on the
existence, uniqueness, stability and/or positivity of solutions
for periodic equations of forms similar to equation \eqref{eq01};
see \cite{kkr, kr1, kr2, kr3, mr, ynr1, ynr2, ynr3} and references
therein. Neutral periodic equations such as \eqref{eq01} arise in
blood cell models (see for example \cite{bst}, \cite{wcl} and
\cite{xl}) and food-limited population models (see for example
\cite{fdc1, fdc2, fdcsjl, fw, fw2, lk}). In the above mentioned
papers, the nonlinear term $q$ and the function $a$ are assumed to
be continuous in all arguments. We impose much weaker conditions
on the nonlinear term $q$ and the argument function $a$.

The map $f:[0, T] \times \mathbb{R}^n \to \mathbb{R}$ is said to 
satisfy Carath\'{e}odory conditions with respect to $L^1[0, T]$ if 
the following conditions hold.
\begin{itemize}
\item[(i)] For each $z \in \mathbb{R}^n$, the mapping
    $t \mapsto f(t, z)$ is Lebesgue measurable.
 \item[(ii)] For almost all $t \in [0, T]$, the mapping
    $z \mapsto f(t, z)$ is continuous on $\mathbb{R}^n$.
\item[(iii)] For each $r > 0$, there exists
    $\alpha_r \in L^1([0, T], \mathbb{R})$ such that for almost
    all $t \in [0,  T]$ and for all $z$ such that $|z| < r$,
    we have $|f(t, z)| \leq \alpha_r(t)$.
\end{itemize}

In Section 2 we present some preliminary material that we will
employ to show the existence of a solution of \eqref{eq01}. Also,
we state a fixed point theorem due to Krasnosel'ski\u{\i}. We
present our main results in Section 3.

\section{Preliminaries}

Define the set $P_T = \lbrace \psi \in C(\mathbb{R},\mathbb{R}):
\psi (t+T)=\psi (t) \rbrace$ and the norm
$\| \psi \|=\sup_{t \in [0,T]}|\psi(t)|$.
Then $(P_T, \|\cdot\|)$ is a Banach space. We will assume
that the following conditions hold.
\begin{itemize}
\item[(A)] $a \in L^1(\mathbb{R}, \mathbb{R})$ is bounded, 
satisfies $a(t+T) = a(t)$ for all $t$ and
\[
    \quad 1 - e^{-\int_{t-T}^{t}  a(r) \, dr} \equiv \frac{1}{\eta} \neq 0.
\]

\item[(C)] $c \in C^1(\mathbb{R}, \mathbb{R})$ satisfies
 $c(t+T) = c(t)$ for all $t$.

\item[(G)] $g \in C^1(\mathbb{R}, \mathbb{R})$ satisfies
$g(t+T) = g(t)$ for all $t$.

\item[(Q1)]  $q$ satisfies Carath\'{e}odory conditions with respect
 to $L^1[0, T]$, and \\ $q(t+T, x, y) = q(t, x, y)$.

\end{itemize}

In our first lemma, we state the integral equation equivalent
to the periodic equation \eqref{eq01}.

\begin{lemma}\label{lemma2.1}
    Suppose that conditions ($A$), ($C$), ($G$) and ($Q_1$) hold. 
    Then $x \in P_T$ is a solution of equation \eqref{eq01} if, 
    and only if, $x \in P_T$ satisfies
    \begin{equation}\label{eq02}
   x(t)= c(t) x(g(t)) +  \eta \int_{t-T}^t \big [ q \big (s,x(s), x(g(s))\big)
    - r(s) x(g(s)) \big] e^{-\int_s^t  a(r) \, dr} \, ds,
    \end{equation}
    where
    \begin{equation}\label{eq03}
        r(s)= a(s) c(s) + c'(s).
    \end{equation}
\end{lemma}

\begin{proof}
Let $x \in P_{T}$ be a solution of \eqref{eq01}.
We first rewrite \eqref{eq01} in the form
\[
    x'(t) + a(t)x(t) = c(t)x'(g(t)) g'(t) + q\big(t, x(t), x(g(t))\big ).
\]
Multiply both sides of the above equation by $ e^{\int_0^t  a(r)
\, dr}$ and then integrate the resulting equation from $t-T$ to
$t$.
\begin{equation}\label{eq08}
\begin{aligned}
    &x(t)e^{\int _0^t  a(r) \, dr} - x(t-T) e^{\int _0^{t-T}  a(r) \, dr} \\
    & =\int_{t-T}^t c(s) x'(g(s)) g'(s) e^{\int _0^s  a(r) \, dr} 
    + q \big (s, x(s), x(g(s)) \big ) e^{\int _0^s  a(r) \, dr} \, ds.
\end{aligned}
\end{equation}
Now divide both sides of \eqref{eq08} by $ e^{\int _0^t  a(r) \,
dr}$.  Since $x \in P_T$, then
\begin{equation}\label{eq09}
  x(t) \frac{1}{\eta}
  = \int_{t-T}^t  c(s) x'(g(s)) g'(s) e^{-\int _s^t  a(r) \, dr}
  + q \big (s, x(s), x(g(s)) \big ) e^{-\int _s^t  a(r) \, dr} \, ds.
\end{equation}
Consider the first term on the right hand side of \eqref{eq09}.
\[
    \int_{t-T}^t  c(s) x'(g(s)) g'(s) e^{-\int _s^t  a(r) \, dr} ds.
\]
Integrate this term by parts to get,
\begin{align*}
    &\int_{t-T}^t  c(s) x'(g(s) ) g'(s) e^{-\int _s^t  a(r) \, dr} ds \\
    &  =  c(t) x(g(t)) - e^{-\int _{t-T}^t  a(s) \, ds} c(t-T) x(g(t-T))\\
    &\quad  - \int_{t-T}^t  \frac{d}{ds} \big[ c(s)e^{-\int _s^t  a(r) \, dr} \big] 
    x(g(s)) \, ds.
\end{align*}
Since $c(t) = c(t - T)$, $g(t) = g(t - T)$, and $x \in P_T$, then
\begin{equation}\label{eq10}
\begin{aligned}
    &\int_{t-T}^t  c(s) x'(g(s)) g'(s) e^{-\int _s^t  a(r) \, dr} ds \\
    &  = \frac{1}{\eta} c(t) x(g(t)) - \int_{t-T}^t  \frac{d}{ds}
    \big[ c(s)e^{-\int _s^t  a(r) \, dr}  \big] x(g(s)) \, ds
\end{aligned}
\end{equation}
Finally, we put the right hand side of \eqref{eq10} into
\eqref{eq09} and simplify. We obtain that if $x \in P_{T}$ is a
solution of \eqref{eq01}, then $x$ satisfies
\[
x(t) =  c(t)x(g(t)) + \eta \int_{t-T}^t \big[ q \big (s,x(s),
x(g(s)) \big)  - r(s) x(g(s)) \big] e^{-\int_s^t  a(r) \, dr} \, ds,
\]
where
$ r(s) = a(s) c(s) + c'(s)$.

The converse implication is easily obtained and the proof is complete.
\end{proof}

We end this section by stating the fixed point theorem that
we employ to help us show the existence of solutions to
equation \eqref{eq01}; see \cite{MAK}.

\begin{theorem}[Krasnosel'ski\u{\i}] \label{kras}
    Let $\mathbb{M}$ be a closed convex nonempty subset of a
Banach space $\big ( \mathcal{B}, \| \cdot\| \big )$. Suppose that
    \begin{itemize}
        \item[(i)] the mapping $A: \mathbb{M} \to \mathcal{B}$
is completely continuous,
        \item[(ii)] the mapping $B: \mathbb{M} \to \mathcal{B}$
is a contraction, and
        \item[(iii)] $x,y \in \mathbb{M}$, implies
$Ax + By \in \mathbb{M}$.
    \end{itemize}
    Then the mapping $A+B$ has a fixed point in $\mathbb{M}$.
\end{theorem}

\section{Existence Results}

We present our existence results in this section. To this end,
we first define the operator $H$ by
\begin{equation}\label{eq07}
 H\psi(t) = c(t) \psi (g(t)) +  \eta \int_{t-T}^t
 \big[ q \big (s, \psi(s), \psi(g(s)) \big)
    - r(s) \psi(g(s)) \big] e^{-\int_s^t  a(r) \, dr} \, ds,
\end{equation}
where $r$ is given in equation \eqref{eq03}. From
Lemma \ref{lemma2.1} we see that fixed points of $H$ are
solutions of \eqref{eq01} and vice versa.

In order to employ Theorem \ref{kras} we need to express the
operator $H$ as the sum of two operators, one of which is
completely continuous and the other of which is a contraction.
 Let $H\psi(t) = \mathcal{A}\psi(t) + \mathcal{B}\psi(t)$ where
\begin{equation}\label{eq04}
    \mathcal{B}\psi(t)= c(t)\psi(g(t))
\end{equation}
and
\begin{equation}\label{eq05}
    \mathcal{A}\psi(t) = \eta \int_{t-T}^t \big[ q \big (s, \psi(s), 
    \psi(g(s)) \big) - r(s) \psi(g(s))
    \big] e^{-\int_s^t  a(r) \, dr} \, ds.
\end{equation}
Our first lemma in this section shows that $\mathcal{A}:P_{T}\to P_{T}$
is completely continuous.

\begin{lemma}\label{lemma3.1}
    Suppose that conditions {\rm (A), (C), (G), (Q1)} hold.
Then $\mathcal{A}:P_{T}\to P_{T}$ is completely continuous.
\end{lemma}

\begin{proof}
 From \eqref{eq05} and conditions (A), (C), (G) and (Q1),
it follows trivially that $r(\sigma + T) = r(\sigma)$ and
$e^{-\int_{\sigma + T}^{t+T}  a(r) \, dr} = e^{-\int_{\sigma}^{t}
 a(\rho) \, d\rho}$. Consequently, we have that
\[
    \mathcal{A} \psi (t + T) = \mathcal{A}\psi(t).
\]
That is, if $\psi \in P_{T}$ then $\mathcal{A}\psi$ is periodic
with period $T$.

To see that $\mathcal{A}$ is continuous let $\{\psi_i\} \subset P_T$
be such that $\psi_i \to \psi$. By the Dominated Convergence Theorem,
\begin{align*}
      &\lim_{i \to \infty}
 \big | \mathcal{A}\psi_i(t) - \mathcal{A}\psi(t) \big | \\
        & \leq \lim_{i \to \infty} \eta \int_{t-T}^t
\Big \{ |r(s)| \, \big | \psi_i(g(s)) - \psi(g(s)) \big |\\
        & \quad  + \Big | q\big ( s, \psi_i(s), \psi_i(g(s)) \big )
- q \big (s, \psi(s), \psi(g(s))\big ) \Big | \Big \}
  e^{-\int_s^t a(r) \, dr} \, ds\\
        & = \eta \int_{t-T}^t  \lim_{i \to \infty}
\Big \{ |r(s)| \, \big | \psi_i(g(s)) - \psi(g(s)) \big |\\
        & \quad  + \Big | q\big ( s, \psi_i(s), \psi_i(g(s)) \big )
- q \big (s, \psi(s), \psi(g(s))\big ) \Big | \Big \}
 e^{-\int_s^t a(r) \, dr} \, ds
         \to 0.
    \end{align*}
Hence $\mathcal{A}: P_T \to P_T$.

Finally, we show that $\mathcal{A}$ is completely continuous.
Let $\mathcal{B} \subset P_T$ be a closed bounded subset and
let $C$ be such that $\|\psi\| \leq C$ for all $\psi \in \mathcal{B}$.
Then
\begin{align*}
    |\mathcal{A} \psi(t)|
& \leq  \eta \int_{t - T}^t  \Big \{
\big | q\big (s, \psi(s), \psi(g(s)) \big ) \big |
  +  |r(s)| \big | \psi(g(s)) \big| \Big \} e^{-\int_s^t a(r) \, dr}
     \, ds\\
 & \leq  \eta N \Big\{ \int_{t - T}^t  \alpha_C(s) \, ds
+ C  \int_{t - T}^t  |r(s)| \, ds \Big\} \equiv K,
\end{align*}
where $N = \max_{s \in [t - T, t]} e^{-\int_s^t a(r) \, dr}$. And
so, the family of functions $\mathcal{A}\psi$ is uniformly
bounded.

Again, let $\psi \in \mathcal{B}$. Without loss of generality,
we can pick $\tau < t$ such that $t - \tau < T$. Then
\begin{align*}
    &|\mathcal{A} \psi(t)  -  \mathcal{A} \psi(\tau) |\\
        & =  \eta \Big| \int_{t-T}^t
\Big \{ q \big (s, \psi(s), \psi(g(s)) \big ) - \, r(s)
 \psi(g(s)) \Big \} e^{-\int_s^t a(r) \, dr} \, ds  \\
        &  \quad - \, \eta  \int_{\tau-T}^\tau
\Big \{ q \big (s, \psi(s), \psi(g(s)) \big ) - \, r(s)
\psi(g(s)) \Big \} e^{-\int_s^\tau a(r) \, dr} \, ds \Big|.\\
\end{align*}
We can rewrite the left hand side as the sum of three integrals.

We obtain the following.
\begin{align*}
   &|\mathcal{A} \psi(t)  -  \mathcal{A} \psi(\tau) |\\
   & \leq  \eta \int_{\tau}^t  \left \{ \Big | q \big (s, \psi(s), \psi(g(s)) \big ) \Big | + |r(s)| \big | \psi(g(s)) \big | \right \} e^{-\int_s^t a(r) \, dr} \, ds \\
   & \quad +  \eta \int_{\tau-T}^\tau  \left \{ \Big | q \big (s, \psi(s), \psi(g(s)) \big ) \Big | + |r(s)| \big | \psi(g(s)) \big | \right \} \\
   & \quad \times \left | e^{-\int_s^t a(r) \, dr} - e^{-\int_s^\tau a(r) \, dr} \right | \, ds \\
   &  \quad + \eta \int_{\tau-T}^{t-T}  \left \{ \Big | q \big (s, \psi(s), \psi(g(s)) \big ) \Big | + |r(s)| \big | \psi(g(s)) \big | \right \}  e^{-\int_s^\tau a(r) \, dr} \, ds \\
   & \leq  2 \eta N \left \{ \int_\tau^t  a_C(s) + C  | r(s) | \, ds \right \}\\
   & \quad + \eta \int_{t - T}^\tau  \big [ a_C(s) + C  | r(s) | \big ]
   \left | e^{-\int_s^t a(r) \, dr} - e^{-\int_s^\tau a(r) \, dr} \right | \, ds .\\
\end{align*}
Now $\int_\tau^t  a_C(s) + |r(s)| \, ds \to 0$ as $(t - \tau) \to
0$. Also, since
\begin{align*}
    &\int_{t - T}^\tau \big [ a_c(s) + |r(s)| \big ] \left | e^{-\int_s^t a(r) \, dr} - e^{-\int_s^\tau a(r) \, dr} \right | \, ds \\
    & \leq \int_0^T \big [ a_c(s) + |r(s)| \big ] \left | e^{-\int_s^t a(r) \, dr} - e^{-\int_s^\tau a(r) \, dr} \right | \, ds, \\
\end{align*}
and $| e^{-\int_s^t a(r) \, dr} - e^{-\int_s^\tau a(r) \,
dr}| \to 0$ as $(t - \tau) \to 0$, then by the Dominated
Convergence Theorem,
\[
    \int_{t - T}^\tau \big [ a_c(s) + |r(s)| \big ]
\big| e^{-\int_s^t a(r) \, dr} - e^{-\int_s^\tau a(r) \, dr} \big| \, ds \to 0
\]
as $(t - \tau) \to 0$. Thus
$|\mathcal{A} \psi(t) - \mathcal{A} \psi(\tau) | \to 0$
as $(t - \tau) \to 0$ independently of $\psi \in \mathcal{B}$.
As such, the family of functions $\mathcal{A}\psi$ is equicontinuous
on $\mathcal{B}$.

By the Arzel\`{a}-Ascoli Theorem, $\mathcal{A}$ is completely
continuous and the proof is complete.
\end{proof}

Our next lemma gives a sufficient condition under which
$\mathcal{B}: P_T \to P_T$ is a contraction.

\begin{lemma}\label{lemma3.2}
    Suppose
    \begin{equation}\label{eq06}
        \|c \| \leq \zeta <1.
    \end{equation}
    Then $\mathcal{B}: P_T \to P_T$ is a contraction.
\end{lemma}

The proof of the above lemma is trivial and hence is omitted.
We now define some quantities that will be used in the following
theorem. Let $\delta = \max_{t \in [0, T]} e^{-\int_0^t  a(r) \,
dr}$,
 $R = \sup_{t \in [0, T]} |r(t)|,$ $A = \int_0^T |\alpha(s)|
\, ds$,
 $B = \int_0^T  |\beta(s)| \, ds,$ $\Gamma = \int_0^T
|\gamma(s)| \, ds$. Also, we need the following condition on the
nonlinear term $q$.

\begin{itemize}
\item[(Q2)] There exists periodic functions
$\alpha, \beta, \gamma \in L^1[0, T]$, with period $T$, such that
\[
    |q(t, x, y)| \leq \alpha(t) |x| + \beta(t) |y| + \gamma(t),
\]
for all $x, y \in \mathbb{R}.$
\end{itemize}

\begin{theorem}\label{thm1}
    Suppose that conditions {\rm (A), (C), (G), (Q1), (Q2)} hold.
Let $\zeta > 0$ be such that $\|c\| \leq \zeta < 1$.
Suppose there exists a positive constant $J$ satisfying the inequality
    \[
        \Gamma \delta \eta + \big ( \zeta + \delta\eta(RT + A + B)
\big)J \leq J.
    \]
    Then  \eqref{eq01} has a solution $\psi \in P_T$ such
that $\|\psi\| \leq J$.
\end{theorem}

\begin{proof}
Define $\mathbb{M} = \{ \psi \in P_T: \|\psi\| \leq J\}$.
By Lemma \ref{lemma3.1}, the operator
$\mathcal{A}:\mathbb{M} \to P_{T}$ is completely continuous.
 Since $\|c\| \leq \zeta < 1$, then by Lemma \ref{lemma3.2},
 the operator $\mathcal{B}: \mathbb{M} \to P_{T}$ is a contraction.
Conditions, (i) and (ii) of Theorem \ref{kras} are satisfied.
We need to show that condition (iii) is fulfilled.
To this end, let $\psi, \varphi \in \mathbb{M}$. Then
    \begin{align*}
|\mathcal{A} \psi(t) + \mathcal{B} \varphi(t)|
& \leq  |c(t)| \big | \varphi(g(t)) \big |
   + \eta \int_{t-T}^t |r(s)| \big | \psi(g(s))
 \big | e^{-\int_s^t  a(r) \, dr} \, ds\\
&  \quad + \eta \int_{t-T}^t \big | q \big (s, \psi(s), \psi(g(s)) \big)
   \big | e^{-\int_s^t  a(r) \, dr} \, ds\\
& \leq \zeta J + \eta \big ( R\delta J + \Gamma \delta + A \delta J
  + B \delta J \big)\\
& = \Gamma \delta\eta + \big ( \zeta + \delta \eta (R + A + B) \big )
 J \, \leq \, J.
    \end{align*}
Thus $\|A \psi + B \varphi \| \leq J$ and so
$A\psi + B \varphi \in \mathbb{M}$. All the conditions of
Theorem \ref{kras} are satisfied and consequently the
operator $H$ defined in \eqref{eq07} has a fixed point
in $\mathbb{M}$. By Lemma \ref{lemma2.1} this fixed point is a
solution of \eqref{eq01} and the proof is complete.
\end{proof}

The condition (Q2) is a global condition on the function $q$.
In the next theorem we replace this condition with the following
local condition.
\begin{itemize}
    \item[(Q2*)] There exists periodic functions
$\alpha^*, \beta^*, \gamma^* \in L^1[0, T]$, with period
$T$, such that $|q(t, x, y)| \leq \alpha^*(t) |x| + \beta^*(t) |y|
+ \gamma^*(t),$ for all $x, y$ with $|x| < J$ and $|y| < J$.
\end{itemize}
The constants $A^*, B^*$ and $\Gamma^*$ are defined as before
with the understanding that the functions $\alpha^*, \beta^*$
and $\gamma^*$ are those from condition (Q2*).

\begin{theorem}
    Suppose that conditions {\rm (A), (C), (G), (Q1)} hold.
Suppose there exists a positive constant $J$ such that {\rm (Q2*)}
holds and such that the inequality
    \[
        \Gamma^* \delta \eta + \big ( \zeta + \delta \eta (RT + A^* + B^*) \big )J \leq J
    \]
    is satisfied. Then equation \eqref{eq01} has a solution
$\psi \in P_T$ such that $\|\psi\| \leq J$.
\end{theorem}

The proof of the above theorem parallels that of Theorem \ref{thm1}.
For our next result, we give a condition for which there exists a
unique solution of \eqref{eq01}. We replace condition (Q2)
 with the following condition.
\begin{itemize}
    \item[(Q2$^\dag$)] There exists periodic functions
$\alpha^\dag, \beta^\dag, \in L^1[0, T]$, with period $T$,
such that
\[
|q(t, x_1, y_1) - q(t, x_2, y_2)|
\leq \alpha^\dag(t) |x_1 - x_2| + \beta^\dag(t) |y_1 - y_2|,
\]
 for all $x_1, x_2, y_1, y_2 \in \mathbb{R}$.
\end{itemize}

\begin{theorem}
    Suppose that conditions {\rm (A), (C), (G), (Q1), (Q2$^\dag$)} hold.
If
    \[
        \zeta + \delta \eta (RT + A^\dag + B^\dag) < 1,
    \]
    then  \eqref{eq01} has a unique $T$-periodic solution.
\end{theorem}

\begin{proof}
Let $\varphi, \psi \in P_{T}$. By \eqref{eq07} we have for all $t$,
\begin{align*}
    |H \varphi (t) - H \psi(t)|
& \leq  | c(t) | \, \| \varphi - \psi \| + \delta \eta \int_{t-T}^t  | r(s) | 
\, \| \varphi - \psi \| \, ds\\
&\quad +  \delta \eta \int_{t-T}^t \Big | q \big (s, \varphi(s), \varphi(g(s)) \big)
 - q \big (s, \psi(s), \psi(g(s)) \big) \Big | \, ds\\
& \leq  \zeta \| \varphi - \psi \| + R \delta \eta T \| \varphi - \psi \| + \eta(A^\dag 
+ B^\dag) \delta \| \varphi - \psi \|.
\end{align*}
Hence, $\|H \varphi - H\psi\| \leq \big ( \zeta + \eta \delta (R T + A^\dag 
+ B^\dag) \big ) \| \varphi - \psi \|$. By the contraction mapping principal, 
$H$ has a fixed point in $P_{T}$ and by Lemma \ref{lemma2.1}, this fixed point 
is a solution of \eqref{eq01}. The proof is complete.
\end{proof}

For our last result, we give sufficient conditions under which there
exists positive solutions of equation \eqref{eq01}. We begin by
defining some new quantities. Let
\[
    m \equiv \min_{s \in [t-T, t]} e^{-\int_s^t  a(r) \, dr},\quad
    M \equiv \max_{s \in [t - T, t]} e^{-\int_s^t  a(r) \, dr}.
\]
Given constants $0 < L < K$, define the set
$\mathbb{M}_2 = \{ \psi \in P_T: L \leq \psi(t) \leq K, t \in [0, T] \}$.

Assume the following conditions hold.
\begin{itemize}
\item[(C2)] $c \in C^1(\mathbb{R}, \mathbb{R})$ satisfies
$c(t+T) = c(t)$ for all $t$ and there exists a $c^* > 0$
 such that $c^* < c(t)$ for all $t \in [0, T]$.

\item[(Q3)] There exists constants $0 < L < K$ such that
    \[
        \frac{(1 - c^*)L}{\eta m T} \leq q(s, \rho, \rho) 
	- r(s) \rho \leq \frac{(1 - \zeta)K}{\eta M T}
    \]
    for all $\rho \in \mathbb{M}$ and $s \in [t - T, t]$.
\end{itemize}

\begin{theorem}
Suppose that conditions {\rm (A), (C2), (G), (Q1), (Q3)} hold.
Suppose that there exists $\zeta$ such that $\|c\| \leq \zeta < 1$.
Then there exists a positive solution of  \eqref{eq01}.
\end{theorem}

\begin{proof}
As in the proof of Theorem \ref{thm1}, we just need to show
that condition (iii) of Theorem \ref{kras} is satisfied.
Let $\varphi, \psi \in \mathbb{M}$. Then
\begin{align*}
   & \mathcal{A} \psi(t)  +  \mathcal{B} \varphi(t)\\
& = c(t) \varphi(g(t))
 + \eta \int_{t - T}^t  \Big [ q \big (s, \psi(s), \psi(g(s)) \big )
 - r(s) \psi(g(s)) \Big ] e^{-\int_s^t  a(r) \, dr} \, ds\\
& \geq  c^* L + \eta m T \frac{(1 - c^*)L}{\eta m T} = L.
\end{align*}
Likewise,
\[
    \mathcal{A} \psi(t) + \mathcal{B} \varphi(t) \leq \zeta K
+ \eta M T \frac{(1 - \zeta)K}{\eta M T} = K.
\]
By Theorem \ref{kras}, the operator $H$ has a fixed point
in $\mathbb{M}_2$. This fixed point is a positive solution
of \eqref{eq01} and the proof is complete.
\end{proof}

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\end{document}