\documentclass[twoside]{article} \usepackage{amsfonts,amsmath} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil An existence theorem \hfil EJDE--2003/04} {EJDE--2003/04\hfil Ferenc Izs\'ak \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2003}(2003), No. 04, pp. 1--9. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % An existence theorem for Volterra integrodifferential equations with infinite delay % \thanks{ {\em Mathematics Subject Classifications:} 45J05, 45K05. \hfil\break\indent {\em Key words:} Volterra integrodifferential equation, Schauder fixed point theorem, \hfil\break\indent competitive systems. \hfil\break\indent \copyright 2003 Southwest Texas State University. \hfil\break\indent Submitted October 21, 2002. Published January 7, 2003.} } \date{} % \author{Ferenc Izs\'ak} \maketitle \begin{abstract} Using Schauder's fixed point theorem, we prove an existence theorem for Volterra integrodifferential equations with infinite delay. As an appplication, we consider an $n$ species Lotka-Volterra competitive system. \end{abstract} \newtheorem{theorem}{Theorem}[section] \numberwithin{equation}{section} \section{Introduction} Vrabie \cite[page 265]{vrabie} studied the partial integrodifferential equation \begin{equation}\label{vra1} \begin{gathered} \dot{u}(t)=-Au(t)+\int_{a}^{t}k(t-s)g(s,u(s))\mathrm{d}s \\ u(a)=u_{0}, \end{gathered} \end{equation} where $u:[a,b]\to X$, $X$ is a Banach space, $A:\mathcal{D}(A)\subset X\to X $ is an $M$-accretive operator; $t\in [a,b]$, $g:[a,b]\times X\to X$, $k:[0,a]\to \mathcal{L}(X)$ are continuous functions. The result, existence of solutions on some interval $[a,c)$ was obtained by using the Schauder's fixed point theorem. Schauder's fixed point theorem is a usual tool for proving existence theorems in infinite delay case. In \cite{teng1}, Teng applied it to prove existence theorems for Kolmogorov systems. Another frequently used method (especially for integrodifferential equations) is the Leray-Schauder alternative, see \cite{nto} and its references. Modifying (\ref{vra1}) we investigate the case when the initial function is given on $(-\infty,0]$, which means infinite delay, moreover in the right-hand side we take a function of the integral. This form allows us proving existence theorems for systems. In this case $g$, $k$ in the right hand side have to be also modified. The spirit of the proof is similar to \cite[pages 265--268]{vrabie} but we need some assumptions on $k$ and $g$ and additional spaces and operators have to be introduced to carry out the proof. In section 3 we apply the result to a system (a competition model arising from population dynamics); existence of global solution will be proved. In the compactness arguments we need the following definition. \paragraph{Definition} A family of functions $H\subset L^{1}([a,b];X)$ is 1-equiintegrable if the following two conditions are satisfied: \begin{itemize} \item For all $\epsilon>0$, there exists $\delta$ such that for all $f\in H$, $\lambda(E)<\delta \to \int_{E} \|f(t)\|\mathrm{d}t<\epsilon)$ \item For all $\epsilon>0$, there exists $h>0$ such that for all $f\in H$ and all $h_{0}0$. Let $u_{0}\in \mathcal{D}(A)$ and $K\subset L^{1}([a,b];X)$ be 1-equiintegrable. Then the set $M(K)=\{ u^{f}: u^{f}\textrm{is the mild solution of (\ref{acp})},f\in K\}$ is relatively compact in $\mathcal{C}([a,b];X)$. \end{theorem} \section{An existence result for a class of Volterra-type integrodifferential equations} \subsection*{A class of Volterra-type integrodifferential equations} Let $U$ be an open subset of $X$, and $U_{A}=U\cap \mathcal{D}(A)$, with $(I-\lambda A)^{-1}$ compact. Let $b>a$ and $g=(g_{1},g_{2},\dots ,g_{n})$ be Lipschitz-continuous functions in the second variable, where $g_{i}:(-\infty,b]\times U_{A}\to X $ are bounded and continuous. Let $k=(k_{1},k_{2},\dots ,k_{n})$ be a function such that $k_{i}\in L_{1}([0,\infty), \mathcal{L}(X))$ and \begin{equation} \label{g1} k(t)g(s,u(s))=(k_{1}(t)g_{1}(s,u(s)),k_{2}(t)g_{2}(s,u(s)), \dots,k_{n}(t)g_{n}(s,u(s))). \end{equation} Let the space $X^{n}$ be equipped with the maximum norm, $\|\mathbf{x}\|=\max_{1\le i\le n}\|x_{i}\|$, where $\mathbf{x}=(x_{1},x_{2},\dots,x_{n})$. Let $F:X^{n}\to X$ be a function such that for some constant $M_{F}\in\mathbb{R}$, \begin{equation}\label{plfelt} \|F(x)\|\le M_{F} \|x\| \quad\mbox{and}\quad M_{F}\int_{-\infty}^{0}\|k(-\tau)\|\mathrm{d}\tau\le 1. \end{equation} Consider the problem \begin{gather}\label{foegy1} \dot{u}(t)=-Au(t)+F\big(\int_{-\infty}^{t}k(t-s)g(s,u(s)) \mathrm{d}s\big)\quad\mathrm{for}\quad t\ge a\\ \label{foegy2} u(t)=u_{0}(t-a) \quad \mathrm{for}\quad t\le a, \end{gather} where $u_{0}\in\mathcal{C}((-\infty,0],X)$ is a given bounded, equiintegrable function which fulfills the matching condition \begin{equation}\label{match} u_{0}(0)=F\Big(\int_{-\infty}^{0}k(-s)g(a+s,u_{0}(s))\mathrm{d}s\Big). \end{equation} \begin{theorem} \label{thm2} Under assumptions \eqref{g1} and \eqref{plfelt}, there is a value $c$ in $(a,b)$ such that \eqref{foegy1}-\eqref{foegy2} has a weak solution on $(-\infty,c]$. \end{theorem} \paragraph{Proof:} Note that $k_{i}\in L_{1}([0,\infty), \mathcal{L}(X))$ implies $k\in L_{1}([0,\infty), \mathcal{L}(X^{n},\mathbb{R}^{n}))$ and (\ref{plfelt}) makes sense. This is only a technical supposition because (\ref{foegy1}) could be rewrite with $k/M$ and $Mg$ (instead of $k$, $g$, resp.; $M\in\mathbb{R}$ is sufficiently big) fulfilled (\ref{foegy1}). Let $$ P:\mathcal{C}((-\infty,b],U)\mapsto \mathcal{C}((-\infty,b],U) $$ defined by \begin{equation} \label{1fix} Pf(t)=\begin{cases} F\Big(\int_{-\infty}^{t} k(t-s)g(s,u^{f}(s))\mathrm{d}s\Big) &\mbox{if } t\ge a\\ f(t) &\mbox{if } t\le a, \end{cases} \end{equation} where $u^{f}$ is the weak solution of (\ref{acp}). Observe that $Pf=f$ holds if and only if $u^{f}$ is the weak solution of the equation (\ref{foegy1})-(\ref{foegy2}). Let us choose $\rho>0$ such that \begin{equation} \label{U} B(u(a),\rho):=\{v\in X:\|v-u(a)\|\le \rho\}\subset U. \end{equation} Since $g$ is bounded there is $M\in\mathbb{R}$ such that \begin{equation} \label{1felt} \|g(s,v)\|\le M \quad \textrm{for} \quad (s,v)\in([-\infty,b]\times [U_{A}\cap B(u_{0},\rho)]). \end{equation} Denote by $S(t)$ the semigroup generated by $-A$ on $\mathcal{D}(A)$. Let us choose further $b\ge c_{0}\ge a$ such that for all $t\in[a,c_{0}]$ \begin{equation} \label{ro} \|S(t-a)u_{0}-u_{0}\|+(c_{0}-a)M\le \rho, \end{equation} and $c\in [a,c_{0}]$ such that \begin{equation}\label{cfelt} (c-a)M_{F}\|k\|_{L_{1}}\le 1. \end{equation} Let us define \begin{equation}\label{chelyett} \mathcal{C}_{u_{0}}((-\infty,b],U)=\{u\in\mathcal{C} ((-\infty,b],U):u(t)=u_{0}(t-a)\quad\mathrm{for}\quad t\le a\}. \end{equation} Let $$ H:\mathcal{C}_{u_{0}}((-\infty,b],U)\mapsto \mathcal{C}([a,b],U) $$ be a natural homeomorphism with $(Hf)(t)=f(t)$ for $t\in[a,b]$ and let \begin{equation} \label{K} K_{u_{0}}^{r}:=\{f\in \mathcal{C}([-\infty,c],X):\|Hf(t)\|_{\infty}\le r \:\&\:f(b)=u_{0}(d-a)\quad\mathrm{for}\quad d\le a\}. \end{equation} Obviously $K^{r}_{u_{0}}$ is nonempty, bounded, closed and convex subset of the space $\mathcal{C}_{u_{0}}([-\infty,c],X)$. Observe that $P=P_{1}\circ P_{2}$, where (using the matching condition (\ref{match})) we define $P_{1}:\mathcal{C}_{u{_0}}((-\infty,b],U)\to\mathcal{C}_{u_{0}}((-\infty,b],U) $ as \begin{equation}\label{18} P_{1}v(t)= \begin{cases} F\Big(\int_{-\infty}^{t}k(t-s)g(s,v(s))\mathrm{d}s\Big) & \mbox{if } t\ge a \\ v(t) &\mbox{if } t\le a \end{cases} \end{equation} and $P_{2}:\mathcal{C}_{u_{0}}((-\infty,b],U) \to\mathcal{C}_{u_{0}}((-\infty,b],U)$ is defined as $P_{2}=H^{-1}P_{2}^{*}H$, where $$ P_{2}^{*}:\mathcal{C}([a,b],U)\to\mathcal{C}([a,b],U) $$ and $P_{2}^{*}g(t)$ is the weak solution of the abstract Cauchy problem \begin{equation} \begin{gathered} \dot{u}(t)+Au(t)=g(t)\quad \textrm{for}\quad t\ge a\\ u(a)=g(a)=u_{0}(0). \end{gathered} \end{equation} For details on this problem, we refer the reader to Barbu \cite[page 124]{barbu} and for some applications of this result to \cite[page 35]{vrabie}. Let $f,h\in L_{1}([a,b],X)$ and let $u$, $v$ be solutions, in the weak sense, of \begin{eqnarray}\label{vra35}\nonumber \dot{u}(t)+Au(t)&=&f(t)\\ \dot{v}(t)+Av(t)&=&h(t) \end{eqnarray} with some initial conditions $u(a), v(a)$. Then for $s,t\in[a,b]$ we have \begin{equation}\label{laks1} \|u(t)-v(t)\|\le \|u(s)-v(s)\|+\int_{s}^{t}\|f(\tau)-h(\tau)\|\mathrm{d}\tau. \end{equation} From this inequality, it follows that \begin{align*} \|P_{2}^{*}h_{1}(t)-P_{2}^{*}h_{2}(t)\| &\le \|h_{1}(a)-h_{2}(a)\|+\int_{a}^{t}\|h_{1}(\tau)-h_{2}(\tau)\| \mathrm{d}\tau \\ &\le \|h_{1}-h_{2}\|_{\infty}(t-a+1), \end{align*} which implies the continuity of $P_{2}^{*}$ on $\mathcal{C}([a,b],U)$ and so $P_{2}$ on $K_{u_{0}}^{r}$. Using (\ref{1felt}), (\ref{ro}) and (\ref{laks1}) for $u\in K_{u_{0}}^{r}$, $t\in[a,c_{0}]$ we get \begin{equation} \label{bar} \begin{aligned} \|P_{2}^{*}u(t)-u(a)\|&\le \|P_{2}u(t)-S(t-a)u(a)\|+\|S(t-a)u(a)-u(a)\|\\ &\le \|S(t-a)u(a)-u(a)\|+\int_{a}^{c_{0}}\|g(t)\|\mathrm{d}t\\ &\le \|S(t-a)u(a)-u(a)\|+(c_{0}-a)M\le\rho. \end{aligned} \end{equation} Then we conclude that $P_{2}^{*}u(t)\in B(u(a),\rho)\cap \mathcal{D}(A)$. Consequently, $P_{2}u(t)\in\mathcal{D}(g)$ for $t\ge a$ . By (\ref{plfelt}), (\ref{1felt}), (\ref{cfelt}) and (\ref{18}), for $t\ge a$ we have \begin{align*} \|Pu(t)\|&=\|P_{1}P_{2}u(t)\|=\|F\big(\int_{-\infty}^{t}k(t-s)g(s,P_{2}u(s)) \mathrm{d}s\big)\|\\ &\le M_{F}\sup_{s\in (-\infty,t]}\|g(s,P_{2}u(s))\|\int_{-\infty}^{0} \|k(-\tau)\|\mathrm{d}\tau \\ &\le M_{F}M\int_{-\infty}^{0}\|k(-\tau)\|\mathrm{d}\tau\le M\,. \end{align*} and (\ref{match}) implies that $$ Pu(t)=u(t)\quad\mbox{for } t\le a\,; $$ i.e., $P$ maps $K_{u_{0}}^{M}$ into itself. Since \begin{equation} \label{p1folyt} \begin{aligned} \|(P_{1}v-P_{1}w)(t)\|\\ &=\|F\big(\int_{-\infty}^{t}k(t-s)[g(s,v(s))-g(s,w(s))] \mathrm{d}s\big)\|\\ &=M_{F} \|\int_{-\infty}^{a}k(t-s)[g(s,v(s))-g(s,w(s))]\mathrm{d}s\\ &\quad +\int_{a}^{t}k(t-s)[g(s,v(s))-g(s,w(s))]\mathrm{d}s\|\\ &\le M_{F}[v(a)-w(a)\\ &\quad +\max_{s\in[a,t]}[g(s,v(s))-g(s,w(s))](t-a) \|k(t-s)\|_{\mathcal{L}_{1}}], \end{aligned} \end{equation} the function $P_{1}$ is continuous from $\mathcal{C}_{u_{0}}((-\infty,b];U)$ into itself. Using the continuity of $P_{2}$ we have that $P:K^{M}_{u_{0}}\to K^{M}_{u_{0}}$ is continuous. Since $$ \int_{E}Pf(t)\mathrm{d}t\le \lambda(E)\max_{t}Pf(t)\le \lambda(E)\|k\|_{L^{1}}M $$ and \begin{align*} &\int_{a}^{b-h_{0}}\|Pf(t+h_{0})-Pf(t)\|\mathrm{d}t\\ &\le\|F\|\:\|a-b\|\big(\int_{-\infty}^{t+h_{0}}k(t-s)g(s,u^{f}(s))\mathrm{d}s -\int_{-\infty}^{t}k(t-s)g(s,u^{f}(s))\mathrm{d}s\big)\\ &\le h_{0}\|F\|\:\|a-b\|\:\|k\|_{L_{1}}M \end{align*} we get that $HP(K_{u_{0}}^{M})$ is $1$-equiintegrable. Let us define $$ K_{u_{0}}:=\mathop{\rm cl}(\mathop{\rm conv}P(K^{M}_{u_{0}})). $$ Easy calculations shows that $H(K_{u_{0}})=\mathop{\rm cl}(\mathop{\rm conv}HP(K^{r}_{u_{0}})\mathrm{)}$ is equiintegrable and Theorem \ref{thm1} implies the relative compactness of $P_{2}^{*}H(K_{u_{0}})=HP_{2}(K_{u_{0}})$. Since $H$ is homeomorphism, $P_{2}(K_{u_{0}})$ and $P(K_{u_{0}})=P_{1}P_{2}(K_{u_{0}})$ are relative compact. Since $P(K_{u_{0}})$ is a subset of the closed, bounded and convex set $K_{u_{0}}$, the Schauder fixed point theorem ensures the existence of a fixed point of $P$. \section{Application to an $n$ species Lotka-Volterra \\ competitive system} We prove local existence of solutions for a system, which is a model of an $n$ species competition arising in the population dynamics. Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain with smooth boundary. Feng \cite{feng} studied the system ($i=1,\dots ,N$) \begin{equation} \label{feng} \begin{gathered} (u_{i})_{t}=D_{i}\big[ \Delta u_{i}+u_{i}(a_{i}-u_{i}-\sum_{j\neq i}^{N}\kappa _{ij}u_{j}^{\tau_{ij}})\big] \quad\mbox{on }(0,\infty)\times\Omega\\ u_{i}=0\quad \mbox{in } (0,\infty)\times\partial\Omega\\ u_{i}(s,x)=\eta_{i}(s,x)\quad \mbox{on }\ [-\tau,0]\times\Omega , \end{gathered} \end{equation} where $u_{i}(t,x)$ denotes the density of the $i$-th species at time $t$ and position $x$ (inside a bounded domain $\Omega$ of $\mathbb{R}^{3}$), $u_{j}^{\tau_{ij}}(t,x)=u_{j}(t-\tau_{ij},x)$, $\tau_{ij}>0$, $\tau=\max\{\tau_{ij}\}$, $D_{i}, a_{i}$ are positive, and $\kappa_{ij}$ are nonnegative real numbers. Supposing the existence of a solution (a sufficient condition for this - using upper and lower semisolutions - is formulated in \cite{pao}) the authors describe the attractors of (\ref{feng}). In \cite{teng1}, Teng studies \begin{equation}\label{teng1} \begin{aligned} \frac{\mathrm{d}x_{i}(t)}{\mathrm{d}t}=&x_{i}(t)[a_{i}(t)-g_{i}(t,x_{i}(t))- \sum_{j=1}^{m}c_{ij}P_{j}(x(t-\tau_{i,j}(t)))\\ &-\sum_{j=1}^{m}\int_{-\sigma_{ij}}^{0}\kappa_{ij}(t,s)Q_{j}(x_{j}(t+s)) \mathrm{d}s],\quad (i=1,\dots ,n) \end{aligned} \end{equation} an $n$-species Lotka-Volterra competitive system with delays as an application of existence result for periodic Kolmogorov systems with delay. Detailed study of the non-autonomous Lotka-Volterra models with delay (focused on existence of positive periodic solutions) can be found in \cite{teng2}. We rewrite (\ref{feng}) taking into account that a bounded attractor $A$ has a bounded neighborhood $U$ and $B\in\mathbb{R}$ such that $u(t,x)\in U$ for $t\le t_{0}$ implies $\|u(t,x)\|t_{0}$. $B$ can be considered as a bound determined by the carrying capacity of the territory. Let $b:\mathbb{R}\to\mathbb{R}$ be a bounded, continuous such that $b(x)=x$ for $|x|0$ there are $k_{i}\in L_{1}([0,\infty),\mathcal{L}(X))$ such that for any bounded $(u_{1},u_{2},\dots ,u_{n})$ and for all $t>t_{0}$, \begin{align*} &\int_{-\infty}^{t}k_{i}(t-s)(u_{1}(s),u_{2}(s),\dots ,u_{n}(s)) \mathrm{d}s\\ &- (\kappa_{i1}b(u_{1}^{\tau_{i1}}(t)),\kappa_{i2} b(u_{2}^{\tau_{i2}}(t)),\dots ,\kappa_{in}b(u_{n}^{\tau_{in}}(t))) <\epsilon_{i}\quad (i=1,\dots ,n) \end{align*} and $$ \int_{-\infty}^{t}k_{n+1}(t-s)(u_{1}(s),\dots ,u_{n}(s))\mathrm{d}s- (b(u_{1}(t)),\dots ,b(u_{n}(t)))<\epsilon _{n+1}. $$ Moreover, the terms on the left-hand side of (\ref{app1}) and (\ref{app2}) lead to a more precise model than the original equation did (\ref{feng}) or (\ref{feng2}) since the new terms keep track the past of the population. Finally let $F=(F_{1},\dots ,F_{n})$ where $$ \int_{\infty}^{t}k(t-s)g(s,\mathbf{u}(s))\mathrm{d}s\in [L^{2}(\Omega)]^{n\times (n+1)} $$ and \begin{equation}\label{F} \begin{gathered} F_{i}:[L^{2}(\Omega)]^{n\times (n+1)}\to L^{2}(\Omega),\\ F_{i}(\textbf{x}_{1},\textbf{x}_{2},\dots ,\textbf{x}_{n},\textbf{x}_{n+1}) = a_{i}(\textbf{x}_{n+1})_{i}-(\textbf{x}_{n+1})_{i}^{2}- \sum_{j=1}^{n}(\textbf{x}_{n+1})_{i}(\textbf{x}_{i})_{j}. \end{gathered} \end{equation} Since $k=(k_{1},k_{2},\dots ,k_{n+1})$ and $g=(g_{1},g_{2},\dots g_{n+1})$ fulfill every requirements listed in Theorem \ref{thm2} we get the following \begin{theorem} \label{thm3} Let $u_{i}(s,x)=\eta_{i}(s,x)$ on $[-\tau,0]\times\Omega$ be an initial condition with a priori bound $B$ of the possible solutions of (\ref{gfeng}). Let further $k$, $g$ and $F$ be as defined by (\ref{appr2}), (\ref{appr1}) and (\ref{F}) satisfying the conditions of Theorem \ref{thm2}. Then (\ref{gfeng}) - a modified version of (\ref{feng}) - has a global solution. \end{theorem} We have to prove only the existence of a global solution. Observe that the condition $b>c_{0}$ (required in (\ref{ro}) and in (\ref{bar})) plays no role here because we have not restricted the domain of $g$. By repeating the method for seeking local solution one can choose a constant $c-a$ in each steps, i.e. we have a local solution on $[a,c]$ and then $[a,2c-a]$, $[a,3c-2a]$ and so on, where every local solution fulfills the conditions of Theorem \ref{thm2} which ensures the existence of a global solution. \paragraph{Acknowledgements:} The author is grateful to Gyula Farkas for his advice and to L\'aszl\'o Simon for reading the original manuscript. \begin{thebibliography}{99} \frenchspacing \bibitem{barbu} V. Barbu: \emph{Nonlinear semigroups and differential equations in Banach spaces}, Noordhoff International Publishing, Leyden, 1976. \bibitem{Engel} K. J. Engel, R. 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Teng: \emph{On the periodic solutions of periodic multi-species competitive systems with delays} Applied Mathematics and Computation, 127, (2002) 235-247. \bibitem{teng2} Z. Teng: \emph{On nonautonomous Lotka Volterra models of periodic multi-species competitive systems with delays}, Journal of Differential Equations, 179 (2), (2002) 538-561. \bibitem{vrabie} I. I. Vrabie: \emph{Compactness Methods for Nonlinear Evolutions} (Pitman Monographs and Surveys in Pure and Applied Mathematics), Longman Scientific \& Technical, Essex -- John Wiley \& Sons, Inc., New York, 1987. \bibitem{wu} J. Wu: \emph{Theory and Applications of Partial Differential Equations}, Springer Verlag, New York, 1998. \end{thebibliography} \noindent\textsc{Ferenc Izs\'ak}\\ Department of Applied Analysis, Lor\'and E\"otv\"os University \\ H-1518 Budapest, PO Box 120, Hungary \\ e-mail: bizsu@cs.elte.hu \\ and\\ Faculty of Mathematical Sciences, Universtity of Twente \\ PO Box 217, 7500 AE Enschede, The Netherlands. \end{document}