\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{cite} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 100, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2015 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2015/100\hfil Periodic solutions] {Periodic solutions for nonlinear neutral delay integro-differential equations} \author[A. Bellour, E. Ait Dads \hfil EJDE-2015/100\hfilneg] {Azzeddine Bellour, El Hadi Ait Dads} \address{Azzeddine Bellour \newline Department of Mathematics, Ecole Normale Superieure de Constantine, Constantine, Algeria} \email{bellourazze123@yahoo.com} \address{El Hadi Ait Dads \newline University Cadi Ayyad, Department of Mathematics, Faculty of Sciences Semlalia B.P. 2390, Marrakech, Morocco.\newline UMMISCO UMI 209, UPMC, IRD Bondy France. Unit\'{e} Associ\'{e}e au CNRST URAC 02} \email{aitdads@uca.ma} \thanks{Submitted December 14, 2014. Published April 14, 2015.} \makeatletter \@namedef{subjclassname@2010}{\textup{2010} Mathematics Subject Classification} \makeatother \subjclass[2010]{45D05, 45G10, 47H30} \keywords{Neutral delay integro-differential equation; periodic solution; \hfill\break\indent Perov's fixed point theorem} \begin{abstract} In this article, we consider a model for the spread of certain infectious disease governed by a delay integro-differential equation. We obtain the existence and the uniqueness of a positive periodic solution, by using Perov's fixed point theorem in generalized metric spaces. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} %\newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} The existence of positive solutions to the integral equation with non constant delay \begin{equation} x(t)=\int_{t-\sigma (t)}^{t}f(s,x(s))ds, \label{A1} \end{equation} was considered in \cite{Ait1, Ezzinbi1, Das1, Torre1}. This equation is a mathematical model for the spread of certain infectious diseases with a contact rate that varies seasonally. Here $x(t)$ is the proportion of infectious in population at time $t$, $\sigma (t)$ is the length of time in which an individual remains infectious, $f(t,x(t))$ is the proportion of new infectious per unit of time (see, for example, \cite{24, Bran, 10}). Ait Dads and Ezzinbi \cite{Ait1} and Ding et al \cite{Ding1} studied the existence of a positive pseudo almost periodic solution. Ezzinbi and Hachimi \cite{Ezzinbi1}, Torrej\'{o}n \cite{Torre1}, Xu and Yuan \cite{Xu} showed the existence of a positive almost periodic solution. The existence of a positive almost automorphic solution was studied in \cite{Ding2, LK, Das1}. Bica and Mure\c{s}an \cite{Bica11, Bica1} studied the existence and uniqueness of a positive periodic solution of \eqref{A1} by using Perov's fixed point theorem in the case of $f$ depends also on $x'(t)$ and $\sigma(t)=\sigma$ is constant. Ait Dads and Ezzinbi \cite{Ait2} considered the existence of a positive almost periodic solution, Ding et al \cite{Ding} studied the existence of positive almost automorphic solutions for the neutral nonlinear delay integral equation \begin{equation} x(t)=\gamma x(t-\tau)+(1-\gamma)\int_{t-\tau}^{t}f(s,x(s))ds, \label{AA1} \end{equation} where $0\leq \gamma <1$. We refer to \cite{Lhachimi, AitDads} for the meaning of \eqref{AA1} in the context of epidemics. In this paper, we consider the more general equation \begin{equation} x(t)=\gamma x(t-\sigma (t))+(1-\gamma )\int_{t-\sigma (t)}^{t}f(s,x(s),x'(s))ds, \label{1} \end{equation} Equation \eqref{1} includes many important integral and functional equations that arise in biomathematics (see for example \cite{AEA,Bica3,Bica1,Bran,Azam,Das,Das1,Sar,10}). We would to use Perov's fixed point theorem to obtain conditions for the existence and uniqueness of a positive periodic solution to \eqref{1}. This work is motivated by the work of Wei Long and Hui-Sheng Ding \cite{Das1}. Moreover, the results obtained in this paper generalize several ones obtained in \cite{Bica3, Bica1,Sar,10}, and the main goal in this work is to study the existence and uniqueness of solutions when $\sigma (t)$ is not constant in \eqref{1}. \section{Preliminaries and generalized metric spaces} In this section, we recall the following notation and results in generalized metric spaces. \begin{definition}[\cite{Perov1}] \rm Let $X$ be a nonempty set and $d:X\times X\to \mathbb{R}^n$ be a mapping such that for all $x,y,z\in X$, one has: \begin{itemize} \item[(i)] $d(x,y)\geq 0_{\mathbb{R}^n}$ and $d(x,y)=0_{\mathbb{R}^n} \Leftrightarrow x=y$, \item[(ii)] $d(x,y)=d(y,x)$, \item[(iii)] $d(x,y)\leq d(x,z)+d(z,y)$,\\ where for $x=(x_{1},x_{2},\dots,x_{n})$ and $y=(y_{1},y_{2},\dots,y_{n})$ from $\mathbb{R}^n$, we have $x\leq y\Leftrightarrow x_{i}\leq y_{i}$, for $i=\overline{1,n}$. \end{itemize} Then $d$ is called a generalized metric and $(X,d)$ is a generalized metric space. \end{definition} \begin{definition}[\cite{AS}] \rm Let $( E,\| \cdot\| ) $ be a generalized Banach space, the norm \newline $\|\cdot\| :E\to \mathbb{R}^n$ has the following properties: \begin{itemize} \item[(i)] $\| x\| \geq 0$ for all $x$ $\in E$ and $\|x\| =0$ if and only if $x=0$. \item[(ii)] $\| \lambda x\| =| \lambda |\,\| x\| $ for all $\lambda \in K=\mathbb{R}$ or $\mathbb{C}$ and for all $x\in E$. \item[(iii)] $\| x+y\| \leq \| x\| +\|y\| $ for all $x,y\in E$ (the inequalities are defined by components in $\mathbb{R}^n)$. \end{itemize} \end{definition} \begin{remark} \rm A generalized Banach space is a generalized complete metric space. \end{remark} \begin{definition}[\cite{Perov1}] \rm If $(E,d)$ is a generalized complete metric space and $T:E\to E$ which satisfies the inequality \begin{equation*} d(Tx,Ty)\leq Ad(x,y)\quad \text{for all }x,y\in E, \end{equation*} where $A$ is a matrix convergent to zero (the norms of it its eigenvalues are in the interval $[0, 1)$). We say that $T$ is a Picard operator or generalized contraction. \end{definition} We recall the following Perov's fixed point theorem. \begin{theorem}[\cite{Perov1}] \label{th0} Let $(E,d)$ be a complete generalized metric space. If $T:E\to E$ is a map for which there exists a matrix $A\in M_{n}(\mathbb{R})$ such that \begin{equation*} d(Tx,Ty)\leq Ad(x,y),\quad \forall x,y\in E \end{equation*} and the norms of the eigenvalues of $A$ are in the interval $[0, 1)$, then $T$ has a unique fixed point $x^{\ast }\in E$ and the sequence of successive approximations $x_{m}=T^{m}(x_{0})$ converges to $x^{\ast }$ for any $x_{0}$ $\in E$. Moreover, the following estimation holds \begin{equation*} d(x_{m},x^{\ast })\leq A^{m}(I_{n}-A)^{-1}d(x_{0},x_{1}),\quad \forall m\in \mathbb{N}^{\ast }. \end{equation*} \end{theorem} \section{Main result} In this section, we study the existence and uniqueness of a positive and periodic solution for the equation \eqref{1}. We consider the following functional spaces \begin{gather*} P(\omega )=\{ x\in C(\mathbb{R)}:x(t+\omega )=x(t),\; \forall t\in \mathbb{R}\} \\ P^{1}(\omega )=\{ x\in C^{1}(\mathbb{R)}:x(t+\omega )=x(t),\; \forall t\in \mathbb{R}\} \\ K^{+}(\omega )=\{ x\in P^{1}(\omega ): x(t)\geq 0,\; \forall t\in \mathbb{R}\} \end{gather*} and denote by $E$ the product space $E=K^{+}(\omega )\times P(\omega)$ which is a generalized metric space with the generalized metric $d_{C}:E\times E\to \mathbb{R}^{2}$, defined by \begin{equation*} d_{C}((x_{1},y_{1}),(x_{2},y_{2})) =(\|x_{1}-x_{2}\|+\|x'_{1}-x'_{2}\| ,\| y_{1}-y_{2}\| ) \end{equation*} where $\| u\| =\max \{ |u(t)| :t\in [ 0,\omega ] \} $ for any $u\in P(\omega )$. Before stating the main result, we need the following lemma. \begin{lemma} $(E, d_{C})$ is a complete generalized metric space. \end{lemma} \begin{proof} Let $(x^n)=(x_{n}, y_{n})$ be a Cauchy sequence, then for any $\epsilon=(\epsilon_1, \epsilon_2)>0$, there exists $n_{0}\in \mathbb{N}$ such that for all $n, m \geq n_0$, we have $d_{C}((x_{m},y_{m}),(x_{n},y_{n}))\leq \epsilon$. Hence, for all $n, m \geq n_0$, $\| x_{m}-x_{n}\|+\| x'_{m}-x'_{n}\| \leq \epsilon_1$ and $\| y_{m}-y_{n}\|\leq \epsilon_2$. Then, $(x_{n})$, $(x_{n}')$ and $(y_{n})$ are Cauchy sequences in $P(\omega)$. It is clear that $(P(\omega), \|\cdot\|)$ is a Banach space, hence there exists $y\in P(\omega)$ such that \begin{equation}\label{l1} \lim_{n\to +\infty}\| y_n-y \|=0 \end{equation} and there exist $x, w\in P(\omega)$ such that $\lim_{n\to +\infty}\| x_n-x \|=\lim_{n\to +\infty}\| x_n'-w\|=0$. Now, since for all $n \geq n_0$, and all $t \in \mathbb{R}^{+}$, \begin{equation*} x_n(t)=x_n(0)+\int_{0}^{t}x_n'(s)ds. \end{equation*} Then, by Lebesgue's Dominated Convergence Theorem, $x'(t)=w(t)$ for all $t \in \mathbb{R}^{+}$. Therefore, for all $n \in \mathbb{N}$ and all $t \in \mathbb{R}$, $x_n(t)\geq 0$. Then for all $t \in \mathbb{R}$, $x(t)\geq 0$. As a consequence, $x\in K^{+}(\omega)$ and \begin{equation}\label{l2} \lim_{n\to +\infty}(\| x_n-x \|+\| x_n'-x' \|)=0\,. \end{equation} Finally, we deduce, by \eqref{l1} and \eqref{l2}, that $(x^n)$ converges to $(x, y)\in E$ and $(E, d_{C})$ is a complete generalized metric space. \end{proof} Equation \eqref{1} will be studied under the following assumptions: \begin{itemize} \item[(H1)] $f\in C(\mathbb{R}\times \mathbb{R}^{+}\times \mathbb{R}, (0, +\infty))$ and there exists $\omega >0$ such that \begin{equation*} f(t+\omega ,x,y)=f(t,x,y)\text{ }, \quad \forall (t,x,y)\in \mathbb{R}\times \mathbb{R}^{+}\times \mathbb{R}. \end{equation*} \item[(H2)] There exist $\alpha $, $\beta >0$ such that \begin{equation*} | f(t,u,v)-f(t,u',v')| \leq \alpha | u-u'| +\beta | v-v'|, \end{equation*} for all $t\in \mathbb{R}$ and all $u,u'\in \mathbb{R}^{+}$, for all $v,v'\in \mathbb{R}$. \item[(H3)] $\sigma \in P^{1}(\omega )$ and $\inf_{t\in [ 0,\omega ]} \sigma (t)=\sigma _{0}>0$. Let $\sigma _{1}=\sup_{t\in [0,\omega ] }\sigma (t)$, $\sigma _{2}=\sup_{t\in [ 0,\omega ]}| \sigma '(t)|$ and assume that $\gamma(1+\sigma_2)< 1$. \end{itemize} Under the hypothesis (H1)--(H3), we will use Perov's fixed point theorem to prove the following main result. \begin{theorem}\label{th1} %\label{th5} If the hypotheses {\rm (H1)--(H3)} hold, and if \begin{equation*} \gamma (1+\sigma _{2})+(1-\gamma )\beta ( 2+\sigma _{2})+L +\sqrt{\zeta }<2, \end{equation*} where, $L=\max(\gamma(1+\sigma_2), \gamma+(1-\gamma)\alpha(2+\sigma_1+\sigma_2))$ and \begin{align*} \zeta &=\gamma ^{2}\sigma _{2}(2+\sigma _{2})+(1-\gamma)^{2} \beta (2+\sigma_{2})( \beta (2+\sigma _{2})+4\alpha(2+\sigma _{1}+\sigma _{2}))+(L-\gamma)^{2} \\ &\quad +2\gamma \beta (1-\gamma )(1+\sigma _{2})(2+\sigma _{2})-2L(\beta(1-\gamma) (2+\sigma _{2})+\gamma\sigma _{2}). \end{align*} Then, \eqref{1} has a unique solution in $K^{+}(\omega)$. \end{theorem} \begin{proof} If we differentiate \eqref{1} with respect to $t$ and denoting $x'(t)=y(t)$, for all $t \in \mathbb{R}$, we obtain \begin{align*} y(t)&=\gamma (1-\sigma '(t))y(t-\sigma (t)) + ( 1-\gamma ) \Big[ f(t,x(t),y(t))\\ &\quad -(1-\sigma '(t))f(t-\sigma (t), x(t-\sigma (t)),y(t-\sigma (t)))\Big], \end{align*} which leads to \begin{align*} x(t)&= \gamma x(t-\sigma (t))+(1-\gamma )\int_{t-\sigma (t)}^{t}f(s,x(s),y(s))ds, \\ y(t)&= \gamma (1-\sigma '(t))y(t-\sigma (t)) + ( 1-\gamma ) \Big[ f(t,x(t),y(t))\\ &\quad -(1-\sigma '(t))f(t-\sigma (t),x(t-\sigma (t)),y(t-\sigma (t)))\Big]. \end{align*} Let $T:E\to C(\mathbb{R})\times C(\mathbb{R})$ be the map defined by \begin{equation*} T(x,y)(t)=\begin{pmatrix} T_{1}(x,y)(t) \\ T_{2}(x,y)(t) \end{pmatrix}, \end{equation*} where \begin{equation*} T_{1}(x,y)(t)=\gamma x(t-\sigma (t))+(1-\gamma )\int_{t-\sigma (t)}^{t}f(s,x(s),y(s))ds, \end{equation*} and \begin{equation} \begin{split} T_{2}(x,y)(t)&=\gamma(1-\sigma '(t))y(t-\sigma (t)) + ( 1-\gamma )\Big[ f(t,x(t),y(t))\\ &\quad -(1-\sigma '(t))f(t-\sigma (t),x(t-\sigma (t)),y(t-\sigma (t)))\Big]. \end{split}\label{aaaaa} \end{equation} From Conditions (H1) and (H3), one has that $T_{1}(E)\subset C^{1}(\mathbb{R})$. Hence, from the condition that $f$ is $\omega $-periodic with respect to $t$, it follows that $T_{1}(E)\subset K^{+}(\omega )$. indeed \begin{align*} T_{1}(x,y)(t+\omega) & =\gamma x(t+\omega -\sigma(t+\omega))+(1-\gamma )\int_{t+\omega -\sigma(t+\omega)}^{t+\omega }f(s,x(s),y(s))ds \\ & =\gamma x(t-\sigma(t))+(1-\gamma)\int_{t-\sigma(t)}^{t}f(s-\omega ,x(s-\omega ),y(s-\omega ))ds \\ & =T_{1}(x,y)(t),\quad \forall t\in \mathbb{R},\;\forall (x,y)\in E. \end{align*} In addition in the same way, one has $T_{2}(x,y)(t+\omega )=T_{2}(x,y)(t)$. Consequently, $T(E)\subset E$. Moreover, from Conditions (H2), \begin{align*} &| T_{1}(x_{1},y_{1})(t)-T_{1}(x_2,y_2)(t)| +| T'_{1}(x_{1},y_{1})(t)-T'_{1}(x_2,y_2)(t)| \\ &\leq \gamma | x_{1}(t-\sigma(t))-x_2(t-\sigma(t))|\\ &\quad +(1-\gamma)\int_{t-\sigma(t)}^{t}[ \alpha | x_{1}(s)-x_2(s)| +\beta | y_{1}(s)-y_2(s)| ] ds \\ &\quad +\gamma (1-\sigma'(t))| x'_{1}(t-\sigma(t))-x'_2(t-\sigma(t))|\\ &\quad +(1-\gamma)(\alpha| x_{1}(t)-x_2(t)|+\beta | y_{1}(t)-y_2(t)|)\\ &\quad +(1-\gamma)|1-\sigma'(t)|\alpha| x_{1}(t-\sigma(t))-x_2(t-\sigma(t))|\\ &\quad + (1-\gamma)|1-\sigma'(t)|\beta | y_{1}(t-\sigma(t))-y_2(t-\sigma(t))|\\ &\leq L (\Vert x_{1}-x_2\Vert+\Vert x'_{1}-x'_2\Vert) +(1-\gamma) \beta (1+\sigma_{1}+\sigma_{2})\Vert y_{1}-y_2\Vert \end{align*} where \[ L=\max(\underbrace{\gamma(1+\sigma_2)}_{r_{1}}, \underbrace{\gamma+(1-\gamma)\alpha(2+\sigma_1+\sigma_2)}_{r_{2}}). \] Similarly, one has \begin{align*} &| T_{2}(x_{1},y_{1})(t)-T_{2}(x_2,y_2)(t)| \\ &\leq (2+\sigma_2)(1-\gamma )\alpha \Vert x_{1}-x_2\Vert +[ \gamma(1+\sigma_2)+(1-\gamma)\beta(2+\sigma_2)] \Vert y_{1}-y_2\Vert \\ & \leq (2+\sigma_2)(1-\gamma )\alpha (\| x_{1}-x_2\|+\| x'_{1}-x'_2\|) \\ &\quad +[ \gamma(1+\sigma_2)+(1-\gamma)\beta(2+\sigma_2)] \| y_{1}-y_2\|. \end{align*} So \begin{equation*} d_{C}(T(x_{1},y_{1}) , T(x_{2},y_{2}) ) \leq A\begin{pmatrix} \Vert x_{1}-x_{2}\Vert + \Vert x'_{1}-x'_{2}\Vert \\ \Vert y_{1}-y_{2}\Vert \end{pmatrix} , \end{equation*} where \[ A=\begin{pmatrix} L & (1-\gamma )\beta (2+\sigma _{1}+\sigma _{2}) \\ & \\ (2+\sigma _{2})(1-\gamma )\alpha & \gamma (1+\sigma _{2})+(1-\gamma )\beta (2+\sigma _{2}) \end{pmatrix}. \] The eigenvalues of this matrix are \begin{gather*} \lambda _{1}= \frac{1}{2}[\gamma (1+\sigma _{2})+(1-\gamma )\beta ( 2+\sigma _{2})+L +\sqrt{\zeta }], \\ \lambda _{2}= \frac{1}{2}[\gamma (1+\sigma _{2})+(1-\gamma )\beta ( 2+\sigma _{2})+L - \sqrt{\zeta }] \end{gather*} where \begin{align*} \zeta &=\gamma ^{2}\sigma _{2}(2+\sigma _{2})+(1-\gamma)^{2} \beta (2+\sigma_{2})( \beta (2+\sigma_{2}) +4\alpha(2+\sigma _{1}+\sigma _{2}))+(L-\gamma)^{2} \\ &\quad +2\gamma \beta (1-\gamma )(1+\sigma _{2})(2+\sigma _{2})-2L(\beta(1-\gamma) (2+\sigma _{2})+\gamma\sigma _{2}). \end{align*} In what follows, we show that the eigenvalues of the matrix $A$ are nonnegative real numbers ($\lambda _{1},\lambda _{2}\in \mathbb{R}^{+})$. \smallskip \noindent\textbf{Step 1:} We show that $\lambda _{1},\lambda _{2}$ are real numbers. It suffices to prove that $\zeta \geq 0.$\newline We have two cases: \smallskip \noindent\textbf{Case 1:} If $L=r_1$, then \begin{align*} \zeta &=\gamma ^{2}\sigma _{2}(2+\sigma _{2})+(1-\gamma)^{2}\beta (2+\sigma_{2})( \beta (2+\sigma _{2})+4\alpha(2+\sigma _{1}+\sigma _{2}))+\gamma^{2}\sigma_{2}^{2} \\ &\quad+2\gamma \beta (1-\gamma )(1+\sigma _{2})(2+\sigma _{2}) -2\gamma(1+\sigma_{2})(\beta(1-\gamma)(2+\sigma _{2})+\gamma\sigma _{2})\\ &=\gamma ^{2}\sigma _{2}(2+\sigma _{2})+(1-\gamma)^{2}\beta (2+\sigma_{2}) ( \beta (2+\sigma _{2})+4\alpha(2+\sigma _{1}+\sigma _{2})) +\gamma^{2}\sigma_{2}^{2}\\ &\quad -2\gamma^{2}\sigma _{2}(1+\sigma _{2})\\ &=(1-\gamma)^{2}\beta (2+\sigma_{2})( \beta (2+\sigma _{2})+4\alpha(2+\sigma _{1}+\sigma _{2}))\geq 0. \end{align*} \noindent\textbf{Case 2:} If $L=r_2$, then \begin{align*} \zeta &=\gamma ^{2}\sigma _{2}(2+\sigma _{2})+(1-\gamma)^{2} \beta (2+\sigma_{2})( \beta (2+\sigma _{2})+4\alpha(2+\sigma _{1}+\sigma _{2}))\\ &\quad +(1-\gamma)^{2}\alpha^{2}(2+\sigma_{1}+\sigma_{2})^{2} +2\gamma \beta (1-\gamma )(1+\sigma _{2})(2+\sigma _{2})\\ &\quad -2(\gamma+(1-\gamma)\alpha(2+\sigma_1+\sigma_2))(\beta(1-\gamma) (2+\sigma _{2})+\gamma\sigma _{2})\\ &=\gamma ^{2}\sigma _{2}(2+\sigma _{2})+(1-\gamma)^{2}\beta (2+\sigma_{2}) ( \beta (2+\sigma_{2})+2\alpha(2+\sigma _{1}+\sigma _{2}))\\ &\quad +(1-\gamma)^{2}\alpha^{2}(2+\sigma_{1}+\sigma_{2})^{2} +2\gamma \beta (1-\gamma )\sigma _{2}(2+\sigma _{2})\\ &\quad -2\gamma\sigma _{2}(\gamma+(1-\gamma)\alpha(2+\sigma_1+\sigma_2)) \\ &=\gamma ^{2}\sigma _{2} ^{2}+(1-\gamma)^{2}\beta (2+\sigma_{2})( \beta (2+\sigma _{2})+2\alpha(2+\sigma _{1}+\sigma _{2}))\\ &\quad +(1-\gamma)^{2}\alpha^{2}(2+\sigma_{1}+\sigma_{2})^{2} +2\gamma \beta (1-\gamma )\sigma _{2}(2+\sigma _{2})\\ &\quad -2\gamma\sigma _{2}(1-\gamma)\alpha(2+\sigma_1+\sigma_2) \\ &=(1-\gamma)^{2}\beta (2+\sigma_{2})( \beta (2+\sigma _{2})+2\alpha(2+\sigma _{1}+\sigma _{2})) +2\gamma \beta (1-\gamma )\sigma _{2}(2+\sigma _{2})\\ &\quad +((1-\gamma)\alpha(2+\sigma_1+\sigma_2)-\gamma\sigma _{2})^{2}\geq 0. \end{align*} \smallskip \noindent\textbf{Step 2:} We show that $\lambda _{2}$ is nonnegative. We have two cases: \smallskip \noindent\textbf{Case 1:} If $L=r_1$, then \begin{align*} &[ \gamma (1+\sigma _{2})+(1-\gamma )\beta ( 2+\sigma _{2})+L ] ^{2}-\zeta \\ &=4\gamma^{2}(1+\sigma _{2})^{2}+(1-\gamma )^{2}\beta^{2} (2+\sigma _{2})^{2}+4\gamma(1+\sigma _{2})(1-\gamma )\beta(1+\sigma _{2})-\zeta\\ &= 4\gamma^{2}(1+\sigma _{2})^{2}+4(1-\gamma )\beta(2+\sigma _{2})(r_1-r_2+\gamma) \geq 0. \end{align*} \smallskip \noindent\textbf{Case 2:} If $L=r_2$, then \begin{align*} &[ \gamma (1+\sigma _{2})+(1-\gamma )\beta ( 2+\sigma _{2})+L ] ^{2}-\zeta \\ &=[\gamma (2+\sigma _{2})+(1-\gamma )\beta ( 2+\sigma _{2})+(1-\gamma )\alpha(2+\sigma _{1}+\sigma _{2})] ^{2}-\zeta \\ &=\gamma^{2} (2+\sigma _{2})^{2}+(1-\gamma)^{2}\beta (2+\sigma_{2}) ( \beta (2+\sigma_{2})+2\alpha(2+\sigma _{1}+\sigma _{2}))\\ &\quad +2\gamma \beta (1-\gamma )(2+\sigma _{2})^{2} +(1-\gamma)^{2}\alpha^{2}(2+\sigma_{1}+\sigma_{2})^{2}\\ &\quad +2\gamma(1-\gamma)(2+\sigma _{2})\alpha(2+\sigma_{1}+\sigma_{2})-\zeta\\ &= 4\gamma^{2}(1+\sigma _{2})+4\gamma \beta (1-\gamma )(2+\sigma _{2}) +4\gamma (1-\gamma )\alpha (2+\sigma _{1}+\sigma _{2})(1+\sigma _{2}) \geq 0. \end{align*} Which implies that $\lambda _{2}\geq 0$. We remark that $\lambda _{1}>\lambda _{2}$, this implies that $\lambda _{1}$ and $\lambda _{2}$ belong to the open unit disc of $\mathbb{R}^{2}$ if and only if $\lambda _{1}<1$, which is equivalent to \begin{equation*} \gamma (1+\sigma _{2})+(1-\gamma )\beta ( 2+\sigma _{2})+L +\sqrt{\zeta }<2. \end{equation*} Then, by Perov's fixed point theorem, the operator $T$ has a unique solution $x^{\ast}=(x_{\ast}, y_{\ast})\in K^{+}(\omega)\times P(\omega)$, which implies that $x_{\ast}\in C^{1}(\mathbb{R})$, and for all $t \in \mathbb{R}$, \begin{align*} (x_{\ast})'(t) &=\gamma(1-\sigma '(t))(x_{\ast})'(t-\sigma (t)) + ( 1-\gamma ) \Big[ f(t,x_{\ast}(t),y_{\ast}(t))\\ &\quad -(1-\sigma '(t))f(t-\sigma (t),x_{\ast}(t-\sigma (t)),y_{\ast}(t-\sigma (t))) \Big]. \end{align*} Hence, by using \eqref{aaaaa}, for all $t \in \mathbb{R}$, \[ ((x_{\ast})'-y_{\ast})(t)=\gamma(1-\sigma '(t))((x_{\ast})'-y_{\ast})(t-\sigma (t)). \] Then, $\|(x_{\ast})'-y_{\ast}\|\leq \gamma(1+\sigma_{2})\|(x_{\ast})'-y_{\ast}\|$. We deduce, by Condition (H3), that $(x_{\ast})'=y_{\ast}$ and $x_{\ast}$ is the unique solution of \eqref{1}. \end{proof} To illustrate this result, we have the following example. \begin{example} \rm Consider \eqref{1} where $f$ is $\omega$-periodic with respect to $t$ and $\sigma$ is $\omega$-periodic, $\gamma =\sigma _{1}=\sigma _{2}=\frac{1}{4}$, $\alpha =\frac{1}{6}$, $\beta =\frac{1}{5}$, then $\gamma(1+\sigma _{2})=\frac{5}{16}<1$, $L=\max(\frac{5}{16}, \frac{9}{16})=\frac{9}{16}$ and \begin{equation*} \gamma (1+\sigma _{2})+(1-\gamma )\beta ( 2+\sigma _{2})+L +\sqrt{\zeta }=\frac{67+\sqrt{2749}}{80}\cong 1.86<2. \end{equation*} Thus, by Theorem \ref{th1}, Equation \eqref{1} has a unique positive $\omega$-periodic solution. \end{example} The following proposition gives an estimation of the error between the exact solution and the approximate solution of \eqref{1}. \begin{proposition} Under the assumptions of Theorem \ref{th1}, the solution of \eqref{1}, which is obtained by the successive approximations method starting from any $x^{0}=(x_{0},y_{0})\in E$, satisfies the estimate \[ d_{C}(x^{m},x^{\ast })\leq \frac{1}{\mu(\lambda_{1}-\lambda_{2})} \begin{pmatrix} e_{1}\lambda_{1}^{m}+e_{2}\lambda_{2}^{m}& e_{3}\lambda_{1}^{m}+e_{4}\lambda_{2}^{m}\\ e_{5}\lambda_{1}^{m}+e_{6}\lambda_{2}^{m}& e_{7}\lambda_{1}^{m}+e_{8}\lambda_{2}^{m} \end{pmatrix} \times d_{C}(x^{1},x^{0}), \] where $\mu=(1-L)(1-\gamma (1+\sigma _{2})-\beta (2+\sigma _{2}) (1-\gamma))-(1-\gamma)^{2}\alpha \beta(2+\sigma _{2})(2+\sigma _{1} +\sigma _{2})$, $x^{m}=T(x^{m-1})$, $x^{m}=(x_{m},y_{m})$, for all $m\in \mathbb{N}^{\ast }$ and \begin{equation} \begin{aligned} e_{1}&= (a(L-\lambda_{2})-c^{2}), e_{2}=(a(\lambda_{1}-L)+c^{2}) \\ e_{3}&= (b(L-\lambda_{2})-c(1-L)), e_{4}=(b(\lambda_{1}-L)+c(1-L)) \\ e_{5}&= (L-\lambda_{1})(\frac{a(L-\lambda_{2})}{c}-c), e_{6} =(L-\lambda_{2})(c-\frac{a(L-\lambda_{1})}{c}) \\ e_{7}&= (L-\lambda_{1})(\frac{b(L-\lambda_{2})}{c}+L-1), e_{8} =(L-\lambda_{2})(1-L-\frac{b(L-\lambda_{1})}{c}) \end{aligned}\label{e1.1} \end{equation} such that \begin{align*} a&= 1-\gamma (1+\sigma _{2})-\beta (2+\sigma _{2})(1-\gamma) \\ b&= (1-\gamma )\beta (2+\sigma _{1}+\sigma _{2})\\ c&= (2+\sigma _{2})(1-\gamma)\alpha. \end{align*} \end{proposition} \begin{proof} From Theorem \ref{th0}, by the conditions of Theorem \ref{th1}, one has \begin{equation*} d_{C}(x^{m},x^{\ast })\leq A^{m}(I-A)^{-1}d_{C}(x^{1},x^{0}),\quad \forall m\in \mathbb{N}^{\ast }. \end{equation*} We have \begin{equation*} A^{m}=\frac{1}{\lambda_{1}-\lambda_{2}} \begin{pmatrix} (L-\lambda_{2})\lambda_{1}^{m}+(\lambda_{1}-L)\lambda_{2}^{m} & (1-\gamma)\alpha(2+\sigma_{2})(\lambda_{2}^{m}-\lambda_{1}^{m}) \\ \frac{(L-\lambda_{1})(L-\lambda_{2})(\lambda_{1}^{m} -\lambda_{2}^{m})}{(1-\gamma)\alpha(2+\sigma_{2})} & (\lambda_{1}-L)\lambda_{1}^{m}+(L-\lambda_{2})\lambda_{2}^{m} \end{pmatrix}, \end{equation*} and \begin{equation*} (I-A)^{-1}=\frac{1}{\mu} \begin{pmatrix} \underbrace{1-\gamma (1+\sigma _{2})-\beta (2+\sigma _{2})(1-\gamma)}_{a} & \underbrace{(1-\gamma )\beta (2+\sigma _{1}+\sigma _{2})}_{b}\\ \underbrace{(2+\sigma _{2})(1-\gamma)\alpha}_{c} & 1-L \end{pmatrix}, \end{equation*} where $\mu=(1-L)(1-\gamma (1+\sigma _{2})-\beta (2+\sigma _{2})(1-\gamma)) -(1-\gamma)^{2}\alpha \beta(2+\sigma _{2})(2+\sigma _{1}+\sigma _{2})$. Which implies \begin{equation*} A^{m}(I-A)^{-1}=\frac{1}{\mu(\lambda_{1}-\lambda_{2})} \begin{pmatrix} e_{1}\lambda_{1}^{m}+e_{2}\lambda_{2}^{m} & e_{3}\lambda_{1}^{m}+e_{4}\lambda_{2}^{m}\\ e_{5}\lambda_{1}^{m}+e_{6}\lambda_{2}^{m} & e_{7}\lambda_{1}^{m}+e_{8}\lambda_{2}^{m} \end{pmatrix}, \end{equation*} where $e_{i}$, $i=1,\dots,8$ are given by \eqref{e1.1}. \end{proof} \begin{thebibliography}{99} \bibitem{AEA} E. Ait Dads, O. Arino, K. Ezzinbi; \emph{Existence de solution p\'{e}riodique d'une \'{e}quation int\'{e}grale non lin\'{e}aire \`{a} retard d\'{e}pendant du temps}, Journal Facta Universitatis, Ser. Math, and Inf. (1996), 79-92. \bibitem{Lhachimi} E. Ait Dads, P. Cieutat, L. Lhachimi; \emph{Existence of positive almost periodic or ergodic solutions for some neutral nonlinear integral equations}, Journal of Differential and integral equations. 22 (2009), 1075-1096. \bibitem{Ait2} E. Ait Dads, K. Ezzinbi; \emph{Almost periodic solution for some neutral nonlinear integral equation}, Nonlinear Anal. 28 (1997), 1479-1489. \bibitem{Ait1} E. Ait Dads, K. Ezzinbi; \emph{Existence of positive pseudo almost periodic solution for a class of functional equations arising in epidemic problems}, Cybernetics and Systems Analysis. 30 (1994), 133-144. \bibitem{24} J. B\'{e}lair; \emph{Population models with state-dependent delays}, in Mathematical Population Dynamics (O. Arino, D. E. Axelrod and M. Kimmel, eds.), Marcel Dekker, New York. (1991), 165-176. \bibitem{Bica3} A. M. Bica, S. Mure\c{s}an; \emph{Periodic solutions for a delay integro-differential equations in Biomathematics}, RGMIA Res. Report Coll. 6 (2003), 755-761. \bibitem{Bica11} A. M. Bica, S. Mure\c{s}an; \emph{Parameter dependence of the solution of a delay integro-differential equation arising in infectious diseases}, Fixed Point Theory. 6 (2005), 79-89. \bibitem{Bica1} A. M. Bica, S. Mure\c{s}an; \emph{Smooth dependence by LAG of the solution of a delay integro-differential equation from Biomathematics}, Communications in Mathematical Analysis. 1 (2006), 64-74. \bibitem{Bran} K. L. Cooke, J. L. Kaplan; \emph{A periodicity thseshold theorem for epidemics and population growth}, Math. Biosciences. 31 (1976), 87-104. \bibitem{Ding1} H. S. Ding, Y. Y. Chen, G. M. N'Gu\'{e}r\'{e}kata; \emph{Existence of positive pseudo almost periodic solutions to a class of neutral integral equations}, Nonlinear Analysis TMA. 74 (2011), 7356-7364. \bibitem{Ding} H. S. Ding, J. Liang, G. M. N'Gu\'{e}r\'{e}kata, T. J. Xiao; \emph{Existence of positive almost automorphic solutions to neutral nonlinear integral equations}, Nonlinear Analysis TMA. 69 (2008), 1188-1199. \bibitem{Ding2} H. S. Ding, J. Liang, T. J. Xiao; \emph{Positive almost automorphic solutions for a class of non-linear delay integral equations}, Applicable Analysis. 88 (2009), 231-242. \bibitem{Ezzinbi1} K. Ezzinbi, M. A. Hachimi; \emph{Existence of positive almost periodic solutions of functional equations via Hilbert’s projective metric}, Nonlinear Anal. 26 (1996) 1169-1176. \bibitem{Azam} C. A. Iancu; \emph{Numerical method for approximating the solution of an integral equation from biomathematics}, Studia Univ. Babes-Bolyai, Mathematica. 43 (1988), 37-45. \bibitem{LK} L. Kikina, K. Kikina; \emph{Positive almost automorphic solutions for some nonlinear integral equations}, Int.\ journal of Math. Analysis. 5 (2011), 1459-1467. \bibitem{Das} D. Guo, V. Lakshmikantham; \emph{Positive solutions of nonlinear integral equations arising in infections diseases}, J. Math. Anal. Appl. 134 (1988), 1-8. \bibitem{AitDads} L. Lhachimi; \emph{Contribution \`{a} l'\'{e}tude qualitative et quantitative de certaines \'{e}quations fonctionnelles: Etude de cas d'\'{e}quations int\'{e}grales \`{a} retard et de type neutre, \'{e}quations diff\'{e}rentielles et \'{e}quations aux diff\'{e}rences.}, Th\`{e}se de doctorat, Universit\'{e} Cadi Ayyad Marrakech, 2010. \bibitem{Das1} W. Long, H. S. Ding; \emph{Positive almost automorphic solutions for some nonlinear delay integral equations}, Electronic Journal of Differential Equations. 2008, \textbf{57} (2008), 1-8. \bibitem{Sar} S. Mure\c{s}an, A. Bica; \emph{Parameter dependence of the solution of a delay integro-differential equation arising in infectious diseases}, Fixed Point Theory. 6 (2005), 79-89. \bibitem{Perov1} A. I. Perov, A. V. Kibenko; \emph{On a general method to study the boundary value problems}, Iz. Akod. Nank. 30 (1966), 249-264. \bibitem{10} H. L. Smith; \emph{On periodic solutions of a delay integral equation modelling epidemics}, J. Math. Biology. 4 (1977), 69-80. \bibitem{AS} A. Szilard; \emph{A note on Perov's fixed point theorem}, Fixed point theory. 4 (2003), 105-108. \bibitem{Torre1} R. Torrej\'{o}n; \emph{Positive almost periodic solutions of a nonlinear integral equation from the theory of epidemics}, J. Math. Analysis Applic. 156 (1991), 510-534. \bibitem{Xu} B. Xu, R. Yuan; \emph{The existence of positive almost periodic type solutions for some neutral nonlinear integral equation}, Nonlinear Analysis. 60 (2005), 669-684. \end{thebibliography} \end{document}