\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 117, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/117\hfil Solutions to systems of PDEs] {Solutions to systems of partial differential equations with weighted self-reference and heredity} \author[P. K. Anh, N. T. T. Lan, N. M. Tuan\hfil EJDE-2012/117\hfilneg] {Pham Ky Anh, Nguyen Thi Thanh Lan, Nguyen Minh Tuan} % in alphabetical order \address{Pham Ky Anh \newline Department of Mathematics, School of Science, Vietnam National University, 334 Nguyen Trai Str., Hanoi, Vietnam} \email{anhpk@vnu.edu.vn, anhpk2009@gmail.com} \address{Nguyen Thi Thanh Lan \newline Faculty of Mathematics and Applications, Saigon University, 273 An Duong Vuong str., w. 3, dist. 5, Ho Chi Minh City, Vietnam} \email{nguyenttlan@gmail.com} \address{Nguyen Minh Tuan \newline Department of Mathematics, University of Education, Vietnam National University, G7 Build., 144 Xuan Thuy Rd., Cau Giay Dist., Hanoi, Vietnam} \email{tuannm@hus.edu.vn} \thanks{Submitted June 18, 2012. Published July 14, 2012.} \subjclass[2000]{35F25, 45G15, 47J35, 47N60, 92D15} \keywords{Hereditary; self-referred; non-linear integro-differential equations; \hfill\break\indent recursive scheme} \begin{abstract} This article studies the existence of solutions to systems of nonlinear integro-differential self-referred and heredity equations. We show the existence of a global solution and the uniqueness of a local solution to a system of integro-differential equations with given initial conditions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \newcommand{\ep}{\varepsilon} \newcommand{\eps}[1]{{#1}_{\varepsilon}} \section{Introduction}\label{intro} Self-referred and hereditary phenomena play an important role in applied sciences, especially that in studying evolution processes of biology. Mathematically, these phenomena can be described by the following model: let $A: X\to\mathbb{R}$ and $B: X\to\mathbb{R}$ be two functionals defined on a function space $X$. Consider the equation \begin{equation}\label{I1} Au(x,t)=u(Bu(x,t),t), \end{equation} where $u=u(x,t)$, $(x,t)\in\mathbb{R}\times[0,+\infty)$ is an unknown function satisfying some initial data at $t=0$, $A$ and $B$ are differential or/and integral operators. For example, if \begin{equation}\label{I2} Bu(x,t)=\int_0^tu(x,\tau)d\tau, \end{equation} then $B$ is called a {\it hereditary} operator. As the unknown function $u$ in the right-hand side of the equation \eqref{I1} depends on itself, equation \eqref{I1} may be called a self-reference equation. Some special cases of \eqref{I1} were originally studied by Volterra in the 20 century (see \cite{MMEP2,EP1} and references therein). It is noticeable to say that some authors considered the variable $t$ as the complex one. In the simple case when $B$ is an identity operator, Eder \cite{Eder} obtained the existence, uniqueness, analyticity of solutions, and the analytic dependence of solutions of the real-variable equation \begin{equation*} %\label{I3} u'(t)=u(u(t)). \end{equation*} Si and Cheng \cite{Si1} investigated a more general functional-differential equation \begin{equation}\label{I4} u'(t)=u(at+bu(t)), \end{equation} where $a\neq 1$ and $b\neq 0$ are complex numbers, and $u:\mathbb{C}\to \mathbb{C}$ is the unknown complex-variable function. In particular, by constructing a convergent power series solution $v(t)$ of a companion equation of the form $$ \beta v'(\beta t)=v'(t)[v(\beta^2 t)-a v(\beta t)+a], $$ the authors \cite{Si1} obtained the analytic solution of \eqref{I4} which is of the form \[ \frac{v(\beta v^{-1}(t)) -at}{b}. \] As a development of \eqref{I4}, Cheng, Si, and Wang \cite{Si2} studied the equation $$ \alpha t+\beta u'(t)=u\big(at+bu'(t)\big), $$ where $\alpha$ and $\beta$ are complex numbers. The main results of \cite{Si2} are the existence theorems for the analytic solutions, and an explicit solution via symmetric methods. Equations of the form \eqref{I1} attract attention of many authors. More investigations can be found in \cite{Stanek95,Stanek97,Stanek98,Stanek00,Stanek00.1,Stanek01,Wan-Tong02,Domoshnitsky02,Hartung05,MMEP1,Domoshnitsky06,MMEP2,EP1,pascali_ut,UL,Stanek09,TL,Bernat}, and references therein. In recent years, Pascali and Miranda obtained many results concerning the self-referred functional-differential equations \cite{MMEP1,MMEP2,EP1,pascali_ut}. For instance, the authors in \cite{MMEP2} studied the initial-value problem \begin{equation}\label{I5} \begin{gathered} \frac{\partial}{\partial t}u(x,t)=u(\frac{1}{t}\int_0^tu(x,s)ds,t),\quad x\in\mathbb{R},\; t\in [0,T],\\ u(x,0)=u_0(x),\quad x\in\mathbb{R}. \end{gathered} \end{equation} The authors claimed that under some suitable conditions problem \eqref{I5} has a unique bounded and continuous solution. Observe that the unknown $u$ in the right-hand side of \eqref{I5} contains a weighted hereditary operator $$ (Bu)(t):=\frac{1}{t}\int_0^tu(x,s)ds. $$ Motivated by the long list of works on self-referred functional-differential equations as mentioned above, we study the following system of two partial-differential equations with self-reference and weighted heredity \begin{equation}\label{I6} \begin{gathered} \frac{\partial}{\partial t}u(x,t)=u\Big(f\big(u(x,t)\big) +v\big(\frac{1}{t}\int_0^tu(x,s)ds +\varphi(u(x,t)),t\big),t\Big) \\ \frac{\partial}{\partial t}v(x,t) =v\Big(g\big(v(x,t)\big)+u\big(\frac{1}{t}\int_0^tv(x,s)ds+\psi(v(x,t)),t\big),t\Big), \end{gathered} \end{equation} associated with the initial conditions \begin{equation}\label{I7} u(x, 0)=u_0(x),\quad v(x, 0)=v_0(x), \end{equation} where $f, g, \varphi, \psi, u_0$ and $v_0$ are given functions satisfying some suitable conditions. This work is devoted to the uniqueness of local solution, and the existence of a global solution of this problem. \section{Preliminaries} To study problem \eqref{I6}-\eqref{I7}, we reduce it to the following system of two integral equations \begin{equation} \begin{gathered}\label{II.1} u(x,t)=u_0(x)+\int_0^tu\Big(f\Big(u(x,s)\Big) +v\Big(\frac{1}{s}\int_0^su(x,\tau)d\tau+\varphi(u(x,s)),s\Big),s\Big)ds,\\ v(x,t)=v_0(x)+\int_0^tv\Big(g\Big(v(x,s)\Big) +u\Big(\frac{1}{s}\int_0^sv(x,\tau)d\tau+\psi(v(x,s)),s\Big),s\Big)ds. \end{gathered} \end{equation} \begin{proposition}\label{prop2.1} If problem \eqref{II.1} has a solution $(u, v)$, then the pair of functions $(u, v)$ solves problem \eqref{I6}-\eqref{I7}. \end{proposition} We omit the proof of this proposition, as it is quite simple. Therefore, we shall investigate problem \eqref{II.1} hereafter. We now define the sequences of real functions $\{u_n\}_{n\ge 1},\, \{v_n\}_{n\ge 1}$ as follows: \begin{equation} \label{II.2} \begin{gathered} u_1(x,t)=u_0(x)+\int_0^tu_0\Big(f\Big(u_0(x)\Big)+v_0\Big(u_0(x) +\varphi\big(u_0(x)\big)\Big)\Big)ds, \\ v_1(x,t)=v_0(x)+\int_0^tv_0\Big(g\Big(v_0(x)\Big)+u_0\Big(v_0(x) +\psi\big(v_0(x)\big)\Big)\Big)ds, \\ \begin{aligned} u_{n+1}(x,t) &=u_0(x)+\int_0^tu_n\Big(f\Big(u_n(x,s)\Big) \\ &\quad +v_n\Big(\frac{1}{s}\int_0^su_n(x,\tau)d\tau +\varphi\big(u_n(x,s)\big),s\Big),s\Big)ds, \end{aligned}\\ \begin{aligned} v_{n+1}(x,t)&=v_0(x)+\int_0^tv_n\Big(g\Big(v_n(x,s)\Big) \\ &\quad +u_n\Big(\frac{1}{s}\int_0^sv_n(x,\tau)d\tau+\psi\big(v_n(x,s)\big), s\Big),s\Big)ds, \end{aligned} \end{gathered} \end{equation} for $x\in\mathbb{R}$ and $t> 0$. We should give the following additional conditions on the functions $u_0, v_0$ and $f, g, \varphi, \psi$: \begin{itemize} \item[(A1)] $u_0$ and $v_0$ are bounded and Lipschitz continuous on $\mathbb{R}$. \item[(A2)] $f, g, \varphi$ and $\psi$ are Lipschitz continuous on $\mathbb{R}$. \end{itemize} The functional inequalities in the next lemma are useful for proving the main results. \begin{lemma}\label{lemma1} Assume that the functions $u_0, v_0$ and $ f, g, \varphi$, and $\psi$ satisfy conditions as in {\rm(A1)--(A2)}. For any $n\ge 1$ there exist two continuous, non-negative functions defined on $\mathbb{R}^+$, say $M_n(t)$ and $N_n(t)$, such that the following two inequalities hold: \begin{gather*} |u_{n+1}(x, t)-u_{n+1}(y, t)|\leq M_{n+1}(t)|x-y|,\quad n\in\mathbb{N},\; x, y\in\mathbb{R}\\ |v_{n+1}(x, t)-v_{n+1}(y, t)|\leq N_{n+1}(t)|x-y|, \quad n\in\mathbb{N},\; x, y\in\mathbb{R}. \end{gather*} Moreover, there is a positive constant $T_1$ such that the non-negative function sequences $\{M_n(t)\}_{n\ge 1}$, $\{N_n(t)\}_{n\ge 1}$ are uniformly bounded on the interval $(0, T_1]$; i.e., there exists a constant $G_0>0$ such that $00, N_0>0$. Let $P, Q, \varpi, \sigma>0$ such that \begin{equation}\label{II.3} \begin{gathered} |f(\alpha_1)-f(\alpha_2)|\leq P|\alpha_1-\alpha_2|,\quad \alpha_1, \alpha_2\in\mathbb{R}\\ % |g(\beta_1)-g(\beta_2)|\leq Q|\beta_1-\beta_2|,\quad \beta_1, \beta_2\in\mathbb{R}\\ % |\varphi(\gamma_1)-\varphi(\gamma_2)|\leq \varpi |\gamma_1-\gamma_2|,\quad \gamma_1, \gamma_2\in\mathbb{R}\\ % |\psi(\eta_1)-\psi(\eta_2)|\leq \sigma |\eta_1-\eta_2|, \quad \eta_1, \eta_2\in\mathbb{R}. \end{gathered} \end{equation} For $n=1$ we have $$ |u_1(x,t)-u_1(y,t)|\leq M_1(t)|x-y|, $$ where \[ M_1(t)=M_0+t(M_0^2P+M_0^2N_0+M_0^2N_0\varpi), \] and $$|v_1(x,t)-v_1(y,t)|\leq N_1(t)|x-y|,$$ where \[N_1(t)=N_0+t(N_0^2Q+M_0N_0^2+M_0N_0^2\sigma).\] \noindent For $n=2$, we derive $$ |u_2(x,t)-u_2(y,t)|\leq M_2(t)|x-y|, $$ where \[ M_2(t)=M_0+\int_0^t\Big(M_1^2(s)P+N_1(s)\frac{1}{2s}\frac{d}{ds} \Big(\int_0^sM_1(\tau)d\tau\Big)^2 +M_1^2(s)N_1(s)\varpi\Big)ds, \] and $$ |v_2(x,t)-v_2(y,t)|\leq N_2(t)|x-y|, $$ where \[ N_2(t)=N_0+ \int_0^t\Big(N_1^2(s)Q+M_1(s)\frac{1}{2s}\frac{d}{ds} \Big(\int_0^sN_1(\tau)d\tau\Big)^2 +M_1(s)N_1^2(s)\sigma\Big)ds. \] We can inductively prove that \begin{equation}\label{II.4} |u_{n+1}(x,t)-u_{n+1}(y,t)|\leq M_{n+1}(t)|x-y|, \end{equation} where \[ M_{n+1}(t)=M_0+\int_0^t\Big(M_n^2(s)P+N_n(s)\frac{1}{2s} \frac{d}{ds}\Big(\int_0^sM_n(\tau)d\tau\Big)^2\\ +M_n^2(s)N_n(s)\varpi\Big)ds, \] and \begin{equation}\label{II.5} |v_{n+1}(x,t)-v_{n+1}(y,t)|\leq N_{n+1}(t)|x-y|, \end{equation} where \begin{align*} N_{n+1}(t)=N_0+\int_0^t\Big(N_n^2(s)Q+M_n(s)\frac{1}{2s} \frac{d}{ds}\Big(\int_0^sN_n(\tau)d\tau \Big)^2 +M_n(s)N_n^2(s)\sigma\Big)ds. \end{align*} Clearly, the functions $M_{n+1}(t)$ and $N_{n+1}(t)$ are non-negative and continuous on $\mathbb{R}$. We shall prove that each one of the function sequences $\{M_{n+1}\}_{n\ge 1}(t)$ and $\{N_{n+1}\}_{n\ge 1}(t)$ is uniformly bounded on some $(0, T_1]$. Indeed, by choosing constants $K_0, H_0$, and $I_0>0$ fulfilling the following conditions $$ N_0+K_0\leq H_0 \quad M_0+K_0\leq I_0 \quad G_0=\max\{H_0, I_0\}, $$ there exists a number $T_1>0$ such that \begin{equation}\label{II.6} \begin{gathered} (M_0^2P+M_0^2N_0+M_0^2N_0\varpi)t\leq K_0,\quad \forall t\in(0,T_1]\\ (N_0^2Q+M_0N_0^2+M_0N_0^2\sigma)t\leq K_0,\quad \forall t\in(0,T_1]\\ (G_0^2P+G_0^3+G_0^3\varpi)t\leq K_0,\\ (G_0^2Q+G_0^3+G_0^3\sigma)t\leq K_0. \end{gathered} \end{equation} Then \begin{equation}\label{II.7} \begin{gathered} 0\leq M_1(t)-M_0=(M_0^2P+M_0^2N_0+M_0^2N_0\varpi)t\leq K_0,\\ 0\leq N_1(t)-N_0=(N_0^2Q+M_0N_0^2+M_0N_0^2\sigma)t\leq K_0. \end{gathered} \end{equation} It follows that \begin{equation}\label{II.8} \begin{gathered} 0\leq M_1(t)\leq K_0+M_0\leq I_0\leq G_0,\\ 0\leq N_1(t)\leq K_0+N_0\leq H_0\leq G_0. \end{gathered} \end{equation} Similarly, \begin{equation} \label{II.9} \begin{aligned} 0\leq M_2(t)-M_0 \leq\int_0^t(G_0^2P+G_0^3+G_0^3\varpi)ds=(G_0^2P+G_0^3+G_0^3\varpi)t \leq K_0,\\ 0\leq N_2(t)-N_0 \leq\int_0^t(G_0^2Q+G_0^3+G_0^3\sigma)ds=(G_0^2Q+G_0^3+G_0^3\sigma)t \leq K_0. \end{aligned} \end{equation} From these inequalities, we have \begin{equation}\label{II.10} \begin{gathered} 0\leq M_2(t)\leq M_0+K_0\leq I_0\leq G_0,\\ 0\leq N_2(t)\leq N_0+K_0\leq H_0\leq G_0. \end{gathered} \end{equation} By induction on $n$ we obtain \begin{equation}\label{II.11} \begin{gathered} 0\leq M_{n+1}(t)\leq M_0+K_0\leq G_0,\\ 0\leq N_{n+1}(t)\leq N_0+K_0\leq G_0, \end{gathered} \end{equation} for every $t\in (0,T_1]$, $T_1>0$. The lemma is proved. \end{proof} We can see that Lemma \ref{lemma1} concerns the properties of the functions $\{u_{n}(x,t)\}$ and $\{v_{n}(x,t)\}$, while Lemma \ref{lemma2} concerns the recursive sequences $\{u_{n+1}(x,t)-u_n(x,t)\}$ and $\{v_{n+1}(x,t)-v_n(x,t)\}$. \begin{lemma}\label{lemma2} Assume that the functions $u_0, v_0$ and $ f, g, \varphi$, and $\psi$ satisfy conditions as in {\rm(A1)--(A2)}. For any $n\ge 1$ there exist two nonnegative, continuous functions, say $A_n(t)$ and $B_n(t)$, satisfying the following two inequalities: \begin{gather*} |u_{n+1}(x,t)-u_n(x,t)|\leq A_{n+1}(t),\quad x\in\mathbb{R},\; t\in\mathbb{R}^+,\\ |v_{n+1}(x,t)-v_n(x,t)|\leq B_{n+1}(t),\quad x\in\mathbb{R},\; t\in\mathbb{R}^+. \end{gather*} Moreover, there is a positive constant $T_2$ such that the both series with general terms $A_n(t)$, and $B_n (t)$ are uniformly convergent on $(0, T_2]$. \end{lemma} \begin{proof} We have \begin{gather*} |u_1(x,t)-u_0(x)|\leq t\|u_0\|_{L^\infty}:=A_1(t),\\ |v_1(x,t)-v_0(x)|\leq t\|v_0\|_{L^\infty}:=B_1(t). \end{gather*} Similarly, \begin{align*} |u_2(x,t)-u_1(x,t)| &\leq\int_0^t\Big(A_1(s)\Big(1+M_0P+M_0N_0\varpi\Big)+M_0B_1(s)\\ &\quad +M_0N_0\frac{1}{s}\int_0^sA_1(\tau)d\tau\Big)ds:=A_2(t), \end{align*} and \begin{align*} |v_2(x,t)-v_1(x,t)| &\leq\int_0^t\Big(B_1(s)\Big(1+N_0Q+M_0N_0\sigma\Big)+N_0A_1(s)\\ &\quad +M_0N_0\frac{1}{s}\int_0^sB_1(\tau)d\tau\Big)ds:=B_2(t). \end{align*} By induction on $n$, we conclude that \begin{equation}\label{II.12} |u_{n+1}(x,t)-u_n(x,t)|\leq A_{n+1}(t), \end{equation} where \begin{align*} A_{n+1}(t)&=\int_0^t\Big(A_n(s)\Big(1+M_{n-1}(s)P+M_{n-1}(s)N_{n-1}(s)\varpi\Big) \\ &\quad +B_n(s)M_{n-1}(s) +M_{n-1}(s)N_{n-1}(s)\frac{1}{s}\int_0^sA_n(\tau)d\tau\Big)ds; \end{align*} and \begin{equation}\label{II.13} |v_{n+1}(x,t)-v_n(x,t)|\leq B_{n+1}(t), \end{equation} where \begin{align*} B_{n+1}(t)&=\int_0^t\Big(B_n(s)\Big(1+N_{n-1}(s)Q+M_{n-1}(s)N_{n-1}(s)\sigma\Big) \\ &\quad +A_n(s)N_{n-1}(s) +M_{n-1}(s)N_{n-1}(s)\frac{1}{s}\int_0^sB_n(\tau)d\tau\Big)ds. \end{align*} For a number $h\in (0, 1/2)$, we can choose $T_2>0$ such that the following two inequalities hold for any $t\in(0, T_2]$, \begin{equation}\label{II.14} \begin{gathered} (1+G_0P+G_0+G_0^2\varpi+G_0^2)t\leq h<\frac{1}{2},\\ (1+G_0Q+G_0+G_0^2\sigma+G_0^2)t\leq h<\frac{1}{2}, \end{gathered} \end{equation} By \eqref{II.14} and Lemma \ref{lemma1}, \begin{equation} \label{II.15} \begin{aligned} 0\leq A_{n+1}(t) &\leq(1+G_0P+G_0^2\varpi+G_0^2)t\|A_n\|_{L^\infty}+G_0t\|B_n\|_{L^\infty} \\ &\leq h(\|A_n\|_{L^\infty}+\|B_n\|_{L^\infty}), \end{aligned} \end{equation} and \begin{equation} \label{II.16} \begin{aligned} 0\leq B_{n+1}(t) &\leq(1+G_0Q+G_0^2\sigma+G_0^2)t\|B_n\|_{L^\infty}+G_0t\|A_n\|_{L^\infty} \\ &\leq h(\|A_n\|_{L^\infty}+\|B_n\|_{L^\infty}). \end{aligned} \end{equation} By induction on $n$, we obtain \[ 0\leq A_{n+1}(t), B_{n+1}(t)\leq h^n\Big(\|A_1\|_\infty+\|B_1\|_\infty\Big), \] for $t\in(0, T_2]$. Therefore, the series with general terms $A_n(.)$ and $B_n(.)$ uniformly converge on the interval $(0, T_2]$. Lemma \ref{lemma2} is proved. \end{proof} \begin{remark} \label{rmk2.4} \rm It is easy to prove inductively that \[ |u_{n+1}(x,t)|\le e^t\|u_0\|_{\infty},\quad |v_{n+1}(x,t)|\le e^t\|v_0\|_{\infty}. \] If we consider $T$ such that $0t_1,\\ u_n(x_2,t)\geq u_n(x_1,t),\quad \forall t\in(0,+\infty),\;x_2>x_1,\\ v_n(x,t_2)\geq v_n(x,t_1),\quad \forall x\in\mathbb{R},\; t_2>t_1,\\ v_n(x_2,t)\geq v_n(x_1,t),\quad \forall t\in(0,+\infty),\;x_2>x_1. \end{gathered} \end{equation} Also, we can prove that (see also remark \ref{rmk2.4}) \begin{equation} \label{III.5} \begin{aligned} 0\leq u_n(x,t)\leq u_{n+1}(x,t)\leq e^T\|u_0\|_{L^\infty}, \\ 0\leq v_n(x,t)\leq v_{n+1}(x,t)\leq e^T\|v_0\|_{L^\infty}, \end{aligned} \end{equation} for all $x\in\mathbb{R}$, $t\in(0,T]$ and $n\in \mathbb{N}$. On the other hand, \begin{gather} |u_{n+1}(x,t_1)-u_{n+1}(x,t_2) \leq \big|\int_{t_1}^{t_2}\|u_0\|_{L^\infty}e^Tds \big| \leq \|u_0\|_{L^\infty}e^T|t_2-t_1|,\label{III.6}\\ |v_{n+1}(x,t_1)-v_{n+1}(x,t_2)| \leq \big|\int_{t_1}^{t_2}\|v_0\|_{L^\infty}e^Tds\big| \leq\|v_0\|_{L^\infty}e^T|t_2-t_1|.\label{III.7} \end{gather} Relations \eqref{III.6} and \eqref{III.7} ensure that $u_n$ and $v_n$ satisfy $(C_4)$. Since the sequences $(u_n)$ and $(v_n)$ are non decreasing, above and upper bounded, there exist the limits \begin{equation} \label{III.8} u_\infty(x,t)=\lim_nu_n(x,t), \quad v_\infty(x,t)=\lim_nv_n(x,t). \end{equation} Since $u_0, v_0, f, g, \varphi$ and $\psi$ are lower semi-continuous and non-decreasing, the functions $f(u_0), g(v_0), v_0\big(u_0+\varphi(u_0)\big)$, and $u_0\big(v_0+\psi(v_0)\big)$ are lower semi-continuous and non-decreasing (see \cite[Lemma 3]{TL}). Hence, $u_0\big(f(u_0)+v_0\big(u_0 +\varphi(u_0)\big)\big)$, and $v_0\big(g(v_0)+u_0\big(v_0 +\psi(v_0)\big)\big)$ are lower semi-continuous and non-decreasing, too. Thus, the lower semi-continuity and the decrease of $u_1(x, t)$ and $v_1(x,t)$ are established. By induction on $n$, we can conclude that $u_n(x,t)$ and $v_n(x,t)$ are lower semi-continuous and non-decreasing. Lemma \ref{lemma4} is proved. \end{proof} \section{Main results} \begin{theorem}[Uniqueness of local solutions]\label{theorem1} Assume that the functions $f$, $g$, $\varphi$, $\psi$, $u_0$, and $v_0$ satisfy {\rm (A1)--(A2)}. Then there exists a positive constant $T_\star$ such that \eqref{II.1} has a unique solution on $R \times (0, T_*]$ denoted by $\{u_*, v_*\}$. Moreover, the functions $u_\infty,\, v_\infty$ are Lipschitz continuous and bounded with respect to each of the variables $x\in\mathbb{R}$, and $ t\in(0, T_\star]$. \end{theorem} \begin{theorem}[Existence of global solutions]\label{theorem2} Assume that $f$, $g$, $\varphi$, $\psi$, $u_0$ and $v_0$ satisfy {\rm (B1)--(B2)}. There exist two functions $u_\infty, v_\infty: \mathbb{R}\times (0, +\infty)\to\mathbb{R}$ satisfying \eqref{II.1} for $t\in(0, +\infty)$. Moreover, these solutions have the properties similar to those of $\{u_n(x,t)\}_{n\geq 1}$ and $\{v_n(x,t)\}_{n\geq 1}$ as in Lemma \ref{lemma4}; namely, the functions $u_\infty, v_\infty$ possess the properties {\rm (C1)--(C4)}. \end{theorem} \begin{proof}[Proof of Theorem \ref{theorem1}] Write $T_*:=\min\{T_1, T_2\}$. By Lemmas \ref{lemma1}, and \ref{lemma2}, the limits $u_{\infty}(x, t)$, $v_{\infty}(x, t)$ of the sequences $\{u_n(x,t)\}_{n\ge 1}$, $\{v_n(x, t)\}_{n\ge 1}$ are bounded on $\mathbb{R}\times (0,T_*]$, Lipschitz continuous with respect to each of variables, and satisfy problem \eqref{II.1}. Now, suppose that $(u_\star,v_\star)$ is another solution of \eqref{II.1} on $\mathbb{R}\times (0,T_*]$ with the same given data. We have % \label{III.9} \begin{align*} &\Big|u_\star\Big(f(u_\star(x,t))+v_\star\Big(\frac{1}{t}\int_0^tu_\star(x,s)ds +\varphi(u_\star(x,t)),t\Big),t\Big) \\ &-u_\infty\Big(f(u_\infty(x,t))+v_\infty\Big(\frac{1}{t}\int_0^tu_\infty(x,s)ds +\varphi(u_\infty(x,t)),t\Big),t\Big)\Big| \\ &\leq\|u_\star-u_\infty\|_{L^\infty}+\Big|u_\infty\Big(f(u_\star(x,t)) +v_\star\Big(\frac{1}{t}\int_0^tu_\star(x,s)ds +\varphi(u_\star(x,t)),t\Big),t\Big) \\ &\quad-u_\infty\Big(f(u_\infty(x,t))+v_\infty\Big(\frac{1}{t}\int_0^tu_\infty(x,s)ds +\varphi(u_\infty(x,t)),t\Big),t\Big)\Big| \\&\leq(1+M_\infty(t)P+M_\infty(t)N_\infty(t) +M_\infty(t)N_\infty(t)\varpi)\|u_\star-u_\infty\|_{L^\infty} \\ &\quad+M_\infty(t)\|v_\star-v_\infty\|_{L^\infty}. \end{align*} From the above inequality and Lemma \eqref{lemma1} we obtain \begin{equation} \label{III.10} \begin{split} &|u_\star(x,t)-u_\infty(x,t)| \\ &\leq \Big(1+G_0P+G_0^2+G_0^2\varpi\Big)t\|u_\star -u_\infty\|_{L^\infty}+G_0t\|v_\star-v_\infty\|_{L^\infty}. \end{split} \end{equation} In addition, we have \begin{equation} \label{III.11} \begin{aligned} &\Big|v_\star\Big(g(u_\star,v_\star)+u_\star\Big(\frac{1}{t}\int_0^tv_\star(x,s)ds +\psi(v_\star(x,t)),t\Big),t\Big)\\ &- v_\infty\Big(g(u_\infty,v_\infty)+u_\infty\Big(\frac{1}{t}\int_0^tv_\infty(x,s)ds +\psi(v_\infty(x,t)),t\Big),t\Big)\Big| \\ &\leq\Big(1+N_\infty(t)Q+M_\infty(t)N_\infty(t)\sigma\Big)\|v_\star -v_\infty\|_{L^\infty} +N_\infty(t)\|u_\star-u_\infty\|_{L^\infty}. \end{aligned} \end{equation} By \eqref{III.11} and Lemma \eqref{lemma1}, we find \begin{equation} \label{III.12} \begin{aligned} &|v_\star(x,t)-v_\infty(x,t)| \\ &\leq\Big(1+G_0Q+G_0^2+G_0^2\sigma\Big)t\|v_\star-v_\infty\|_{L^\infty} +G_0t\|u_\star-u_\infty\|_{L^\infty}. \end{aligned} \end{equation} Combining \eqref{III.10} and \eqref{III.12}, we obtain \begin{equation}\label{III.13} \begin{aligned} &|u_\star(x,t)-u_\infty(x,t)|\\ &\leq\Big(1+G_0P+G_0+G_0^2+G_0^2\varpi\Big)t \max\{\|u_\star-u_\infty\|_{L^\infty},\|v_\star-v_\infty\|_{L^\infty}\}, \end{aligned} \end{equation} and \begin{equation} \label{III.14} \begin{aligned} &|v_\star(x,t)-v_\infty(x,t)|\\ &\leq\Big(1+G_0Q+G_0+G_0^2+G_0^2\sigma\Big)t \max\{\|u_\star-u_\infty\|_{L^\infty},\|v_\star-v_\infty\|_{L^\infty}\}. \end{aligned} \end{equation} Taking account of \eqref{II.14}, \eqref{III.13} and \eqref{III.14}, we have \begin{equation} \label{III.15} \begin{gathered} |u_\star(x,t)-u_\infty(x,t)|\leq h\max\{\|u_\star -u_\infty\|_{L^\infty},\|v_\star-v_\infty\|_{L^\infty}\}, \\ |v_\star(x,t)-v_\infty(x,t)|\leq h\max\{\|u_\star-u_\infty\|_{L^\infty},\|v_\star-v_\infty\|_{L^\infty}\}, \end{gathered} \end{equation} for all $t\in(0,T_0]$, $x\in\mathbb{R}$. Finally, we conclude that \begin{equation*} \max\{\|u_\star-u_\infty\|_{L^\infty},\|v_\star-v_\infty\|_{L^\infty}\}\leq h\max\{\|u_\star-u_\infty\|_{L^\infty},\|v_\star-v_\infty\|_{L^\infty}\}. \end{equation*} The last inequality ensures the uniqueness of the solution. Theorem \ref{theorem1} is proved. \end{proof} \begin{proof}[Proof of Theorem \ref{theorem2}] Thanks to \eqref{III.4} and \eqref{III.5}, the following two limits exit: \begin{equation} \label{III.17} u_\infty(x,t)=\sup_nu_n(x,t), \quad v_\infty(x,t)=\sup_nv_n(x,t). \end{equation} We shall prove that $u_\infty(x,t), v_\infty(x,t)$ satisfy \eqref{II.1}. From \eqref{III.17} we have \begin{equation} \label{III.18} \begin{aligned} &u_{n+1}(x,t)-u_0(x) \\ &=\int_0^tu_n\Big(f\Big(u_n(x,s)\Big)+v_n \Big(\frac{1}{s}\int_0^su_n(x,\tau)d\tau+\varphi\big(u_n(x,s)\big),s\Big),s\Big)ds \\ &\leq\int_0^tu_\infty\Big(f\Big(u_\infty(x,s)\Big)+v_\infty\Big(\frac{1}{s}\int_0^su_\infty(x,\tau)d\tau+\varphi\big(u_\infty(x,s)\big),s\Big),s\Big)ds, \end{aligned} \end{equation} and \begin{equation} \label{III.19} \begin{aligned} &v_{n+1}(x,t)-v_0(x) \\ &=\int_0^tv_n\Big(g\Big(v_n(x,s)\Big)+v_n \Big(\frac{1}{s}\int_0^sv_n(x,\tau)d\tau+\psi\big(v_n(x,s)\big),s\Big),s\Big)ds \\ &\leq\int_0^tv_\infty\Big(g\Big(v_\infty(x,s)\Big)+u_\infty\Big(\frac{1}{s}\int_0^sv_\infty(x,\tau)d\tau+\psi\big(v_\infty(x,s)\big),s\Big),s\Big)ds. \end{aligned} \end{equation} As ${u_n(x,t)}$, and ${v_n(x,t)}$ are non-decreasing, we have \begin{align}\label{III.20} &u_{n+p}\Big(f\Big(u_{n+p}(x,t)\Big)+v_{n+p}\Big(\frac{1}{t}\int_0^tu_{n+p}(x,s)ds+\varphi\big(u_{n+p}(x,t)\big),t\Big),t\Big) \\ &\geq u_{n}\Big(f\Big(u_{n+p}(x,t)\Big)+v_{n+p}\Big(\frac{1}{t}\int_0^tu_{n+p}(x,s)ds+\varphi\big(u_{n+p}(x,t)\big),t\Big),t\Big), \end{align} and \begin{equation} \label{III.21} \begin{aligned} &v_{n+p}\Big(g\Big(v_{n+p}(x,t)\Big)+u_{n+p}\Big(\frac{1}{t}\int_0^tv_{n+p}(x,s)ds +\psi\big(v_{n+p}(x,t)\big),t\Big),t\Big) \\ &\geq v_{n}\Big(g\Big(v_{n+p}(x,t)\Big)+u_{n+p}\Big(\frac{1}{t} \int_0^tv_{n+p}(x,s)ds+\psi\big(v_{n+p}(x,t)\big),t\Big),t\Big). \end{aligned} \end{equation} From \eqref{III.20} and \eqref{III.21} we deduce \begin{equation} \label{III.22} \begin{aligned} &\lim_{p\to\infty}\int_0^tu_{n+p}\Big(f\Big(u_{n+p}(x, s)\Big)\\ &\quad +v_{n+p}\Big(\frac{1}{s}\int_0^su_{n+p}(x,\tau)d\tau +\varphi\big(u_{n+p}(x,s)\big),s\Big),s\Big)ds \\ &\geq\int_0^tu_n\Big(f\Big(u_\infty(x,s)\Big) +v_\infty\Big(\frac{1}{s}\int_0^su_\infty(x,\tau)d\tau +\varphi\big(u_\infty(x,s)\big),s\Big),s\Big)ds, \end{aligned} \end{equation} and \begin{equation} \label{III.23} \begin{aligned} &\lim_{p\to\infty}\int_0^tv_{n+p}\Big(g\Big(v_{n+p}(x,s)\Big)\\ &\quad +u_{n+p}\Big(\frac{1}{s}\int_0^sv_{n+p}(x,\tau)d\tau +\psi\big(v_{n+p}(x,s)\big),s\Big),s\Big)ds \\ &\geq\int_0^tv_n\Big(g\Big(v_\infty(x,s)\Big)+u_\infty\Big(\frac{1}{s} \int_0^sv_\infty(x,\tau)d\tau+\psi\big(v_\infty(x,s)\big),s\Big),s\Big)ds. \end{aligned} \end{equation} Hence, \begin{equation} \label{III.24} \begin{aligned} &\lim_p[u_{n+p+1}(x,t)-u_0(x)]\\ &=\lim_p\int_0^tu_{n+p}\Big(f\Big(u_{n+p}(x, s)\Big) \\ &\quad +v_{n+p}\Big(\frac{1}{s}\int_0^su_{n+p}(x,\tau)d\tau +\varphi\big(u_{n+p}(x,s)\big),s\Big),s\Big)ds \\ &\geq\int_0^tu_n\Big(f\Big(u_\infty(x,s)\Big) +v_\infty\Big(\frac{1}{s}\int_0^su_\infty(x,\tau)d\tau +\varphi\big(u_\infty(x,s)\big),s\Big),s\Big)ds, \end{aligned} \end{equation} and \begin{equation} \label{III.25} \begin{aligned} &\lim_p[v_{n+p+1}(x,t)-v_0(x)]\\ &=\lim_p\int_0^tv_{n+p}\Big(g\Big(v_{n+p}(x,s)\Big) \\ &\quad +u_{n+p}\Big(\frac{1}{s}\int_0^sv_{n+p}(x,\tau)d\tau +\psi\big(v_{n+p}(x,s)\big),s\Big),s\Big)ds \\ &\geq\int_0^tv_n\Big(g\Big(v_\infty(x,s)\Big) +u_\infty\Big(\frac{1}{s}\int_0^sv_\infty(x,\tau)d\tau +\psi\big(v_\infty(x,s)\big),s\Big),s\Big)ds. \end{aligned} \end{equation} By \eqref{III.24}-\eqref{III.25} we find that \begin{equation} \label{III.26} \begin{aligned} &u_\infty(x,t)-u_0(x)\geq\int_0^tu_\infty\Big(f\Big(u_\infty(x,s)\Big) \\ &\quad +v_\infty\Big(\frac{1}{s}\int_0^su_\infty(x,\tau)d\tau +\varphi\big(u_\infty(x,s)\big),s\Big),s\Big)ds, \end{aligned} \end{equation} and \begin{equation} \label{III.27} \begin{aligned} &v_\infty(x,t)-v_0(x)\geq\int_0^tv_\infty\Big(g\Big(v_\infty(x,s)\Big) \\ &\quad +u_\infty\Big(\frac{1}{s}\int_0^sv_\infty(x,\tau)d\tau +\psi\big(v_\infty(x,s)\big),s\Big),s\Big)ds. \end{aligned} \end{equation} Combining \eqref{III.18}-\eqref{III.19} and \eqref{III.26}-\eqref{III.27} we obtain \begin{equation} \label{III.28} \begin{aligned} u_\infty(x,t)-u_0(x) &=\int_0^tu_\infty\Big(f\Big(u_\infty(x,s)\Big) \\ &\quad +v_\infty\Big(\frac{1}{s}\int_0^su_\infty(x,\tau)d\tau +\varphi\big(u_\infty(x,s)\big),s\Big),s\Big)ds, \end{aligned} \end{equation} and \begin{equation} \label{III.29} \begin{aligned} v_\infty(x,t)-v_0(x) &=\int_0^tv_\infty\Big(v_\infty(x,s)\Big) \\ &\quad +u_\infty\Big(\frac{1}{s}\int_0^sv_\infty(x,\tau)d\tau +\psi\big(v_\infty(x,s)\big),s\Big),s\Big)ds. \end{aligned} \end{equation} The above equalities imply that $(u_\infty, v_\infty)$ is a solution of \eqref{II.1}. On the other hand, it is easily seen that $u_\infty, v_\infty$ are Lipschitz continuous in $t$ on $(0,+\infty)$. The proof is complete. \end{proof} \section{Illustrative Example} Consider the initial-value problem for a system of integro-differential equations \eqref{I6}-\eqref{I7} with the following data: \begin{gather*} u_0(x) =\begin{cases} 1 - |x| &\text{if } |x|\leq 1 \\ 0 &\text{otherwise} \end{cases}\\ v_0(x)=1 \quad\text{for all } x\in\mathbb{R},\\ f(u)=u,\quad g(v)=v, \quad \varphi(u)=\psi(v)=0. \end{gather*} We compute the successive approximations as follows: \begin{equation}\label{IV.1} \begin{gathered} \begin{aligned} u_1(x,t)&=u_0(x)+\int_0^tu_0\Big(f(u_0(x))+v_0(u_0(x)+\varphi(u_0(x)))\Big)ds\\ &=u_0(x)+\int_0^tu_0(u_0(x)+1)ds =u_0(x)+\int_0^t0ds=u_0(x), \end{aligned} \\ \begin{aligned} v_1(x,t)&=v_0(x)+\int_0^tv_0\Big(g(v_0(x))+u_0(v_0(x)+\psi(v_0(x)))\Big)ds\\ &=1+\int_0^t1ds=1+t. \end{aligned} \end{gathered} \end{equation} Similarly, $u_2(x,t)=u_0(x)$, $v_2(x,t)=1+t+(t^2/2)$. Suppose that \begin{equation}\label{IV.4} u_n(x,t)=u_0(x),\quad v_n(x,t)=\sum_{i=0}^n\frac{t^i}{i!}. \end{equation} We can prove inductively that \[ u_{n+1}(x,t)=u_0(x),\quad v_{n+1}(x,t)=\sum_{i=0}^{n+1}\frac{t^i}{i!}. \] Tending $n$ to infinity we obtain \begin{equation}\label{IV.7} u_\star(x,t)=u_0(x),\quad v_\star(x,t)=e^t. \end{equation} In fact, we can choose $u_0(x)$ as a nonnegative, Lipschitz continuous function having a compact support, and $v_0(x) =c $\ as a constant function. Due to the symmetry of the system, the functions $u_0$ and $v_0$ are interchangeable. \subsection*{Concluding remarks} Mathematically, one can provide acceptable assumptions on equations, and add suitable restrictions on initial data of problems so that the solution exists uniquely. 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