\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 $$\label{eq01} \begin{gathered} x'(t) = -a(t)x(t)+ c(t)x'\big ( g(t) \big ) g'(t) + q \big( t, x(t), x(g(t))\big),\\ x(t + T) = x(t). \end{gathered}$$ 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 $$\label{eq02} x(t)= c(t) x(g(t)) + \eta \int_{t-T}^t \big [ q \big (s,x(s), x(g(s))\big) - r(s) x(g(s)) \big] e^{-\int_s^t a(r) \, dr} \, ds,$$ where $$\label{eq03} r(s)= a(s) c(s) + c'(s).$$ \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 $t$. \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} Now divide both sides of \eqref{eq08} by $e^{\int _0^t a(r) \, dr}$. Since $x \in P_T$, then $$\label{eq09} x(t) \frac{1}{\eta} = \int_{t-T}^t c(s) x'(g(s)) g'(s) e^{-\int _s^t a(r) \, dr} + q \big (s, x(s), x(g(s)) \big ) e^{-\int _s^t a(r) \, dr} \, ds.$$ 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 \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} 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 $$\label{eq07} H\psi(t) = c(t) \psi (g(t)) + \eta \int_{t-T}^t \big[ q \big (s, \psi(s), \psi(g(s)) \big) - r(s) \psi(g(s)) \big] e^{-\int_s^t a(r) \, dr} \, ds,$$ 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 $$\label{eq04} \mathcal{B}\psi(t)= c(t)\psi(g(t))$$ and $$\label{eq05} \mathcal{A}\psi(t) = \eta \int_{t-T}^t \big[ q \big (s, \psi(s), \psi(g(s)) \big) - r(s) \psi(g(s)) \big] e^{-\int_s^t a(r) \, dr} \, ds.$$ 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 $$\label{eq06} \|c \| \leq \zeta <1.$$ 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} \begin{thebibliography}{00} \frenchspacing \bibitem {bst} E. Beretta, F. Solimano, and Y. 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