\documentclass[reqno]{amsart} \usepackage{graphicx} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 12, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2008/12\hfil Existence of positive solutions] {Existence of positive solutions for semi-positone systems} \author[ A. K. Dram\'e, D. G. Costa\hfil EJDE-2008/12\hfilneg] {Abdou K. Dram\'e, David G. Costa} % in alphabetical order \dedicatory{Dedicated to Professor Claude Lobry on the occasion of his 60-th birthday} \address{Abdou K. Dram\'e \newline Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV, USA} \email{abdoukhadry.drame@unlv.edu} \address{David G. Costa \newline Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV, USA} \email{costa@unlv.nevada.edu} \thanks{Submitted July 18, 2007. Published January 24, 2008.} \subjclass[2000]{34B18, 47H10} \keywords{Positive solution; semi-positone system; phase plane analysis; \hfill\break\indent Schauder fixed point theorem} \begin{abstract} This paper studies semi-positone systems of equations by using phase plane analysis and fixed point theory. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{pro}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} This paper concerns the existence of positive solutions for the system $$\label{eP} \begin{gathered} -u''=\lambda f(u,v)\\ -v''=\mu\, g(u,v)\\ u(0)=u(1)=0=v(0)=v(1). \end{gathered}$$ where $\lambda, \mu>0$ are positive parameters and the nonlinear functions $f,\; g: [0, \infty)\to \mathbb{R}$ are continuous and satisfy the assumptions: \begin{itemize} \item[(A1)] For $v$ fixed, \begin{gather*} f(s,v)<0,\quad\text{for } 0< s< s_{0}\quad \text{or}\quad s>s_{1}\\ f(s,v)>0,\quad \text{for } s_{0}< s< s_{1},\\ \int_{0}^{s_1}f(s,v)ds>0\,; \end{gather*} \item[(A2)] For $u$ fixed, $g$ satisfies hypotheses similar to those of $f$ in (A1); \item[(A3)] \begin{itemize} \item[(i)] $f$ (resp. $g$) is monotone increasing in $v$ (resp. in $u$) or $f$ (resp. $g$) is monotone decreasing in $v$ (resp. in $u$) or \item[(ii)]$\int_{s_{0}}^{s_{1}}\min_{0\leq v\leq s_{1}}f(u,v)du>-\int_{0}^{s_{0}}\min_{0\leq v\leq s_{1}}f(u,v)du$ (resp. $g$). \end{itemize} \end{itemize} We point out that assumptions (A1) and (A2) on the nonlinearities $f$ and $g$ are natural and necessary for existence of positive solutions when $f(.,v)$ and $g(u,.)$ are negative at zero, as shown by the results of \cite{Dan87} and \cite{Cle87} in the case of a single equation. Systems of type \eqref{eP} are referred to as semi-positone systems. Such problems arise in many situations, for instance, in steady state problems of population models with nonlinear sources. In section 4 we provide an application to a model leading to a system of type \eqref{eP}. We refer the reader to \cite{Cas88}, where Castro and Shivaji initially called such problems (with $f(0)<0$) \emph{non-positone problems}, in contrast with the terminology \emph{positone problems}, coined by Cohen and Keller in \cite{Coh67}, when the nonlinearity $f$ was \emph{positive} and \emph{monotone}. In fact, P. L. Lions provided in \cite{Lio82} an extensive review on positive solutions of scalar positone equations and already remarked that the case when the nonlinearity is negative near the origin was mathematically challenging and important for applications. Semipositone problems with scalar equations have motivated many researchers in the last fifteen years. We refer the reader to the survey paper by Castro-Maya-Shivaji \cite{Cas002} and references therein for a review. On the other hand, semi-positone systems have been studied in \cite{Anu94}, \cite{Hai00} and \cite{Cas001}, where the authors used the method of sub- and super-solutions. However, in their results the nonlinear functions $f$ and $g$ are usually monotonic in both variables $u$ and $v$. In \cite{Cas001}, the construction of sub- and super-solutions was based on the structure $f=f_{1}(u,v)-h_1(x)$ and $g=g_1(u,v) -h_2(x)$ of the nonlinear functions $f$ and $g$, with the functions $f_1$ and $g_1$ having nonnegative partial derivative in the variables $u$ and $v$. Also, in \cite{Dan87} and \cite{Cle87}, semi-positone problems for a scalar equation have been studied under general assumptions on the nonlinearity $f$. As mentioned earlier, it was shown (among other things) that if $f(0)<0$, then the condition $\int_{0}^{s_{1}}f(s)ds>0$ is a necessary condition for existence of positive solutions (sufficiency was also shown in \cite{Cle87}). More recently, the authors in \cite{Cos06} studied semi-positone problems in the ODE case under very weak conditions on $f$ by using phase plane analysis for conservative systems. In the present paper, we prove existence of positive solutions for semi-positone systems under general conditions on the nonlinearities. We prove that the condition given in \cite{Dan87} and \cite{Cle87} is a sufficient condition in the case of systems, and we extend one of the results of \cite{Cos06} to systems of equations. Our approach combines the theory of conservative systems and tools of functional analysis, like fixed point theory. For the rest of this article, we let $C_{0}[0, 1]$ denote the space of continuous functions $v$ on $[0,1]$ satisfying the boundary conditions $v(0)=v(1)=0$, and endowed with the sup norm. Also, we let $\mathcal{K}$ denote the closed, convex cone of nonnegative functions in $C_{0}[0, 1]$, namely, $\mathcal{K}=\{v\in C_{0}[0, 1: v\geq 0\}$. In addition, for a given $v$ in $\mathcal{K}$, we define the continuous function $$f_{v}(s,u)=f(u,v(s)),\quad \text{for } 0\leq s\leq 1,\;u\geq 0,$$ and consider the auxiliary problem $$\label{ePv} \begin{gathered} -u''=\lambda f_{v}(s,u)\\ u(0)=u(1)=0 \end{gathered}$$ \section{Existence result for \eqref{ePv}} The proof of existence of positive solutions for \eqref{ePv} is based on the method of sub and super-solutions. From the function $f_{v}$ we construct the functions $F_{v}$ and $G_{v}$ below, where we shall omit the subscript $v$ in the latter functions for simplicity of notation: $$F(u)=\min_{0\leq s\leq 1}f_{v}(s,u) \quad \text{and}\quad G(u)=\max_{0\leq s\leq 1}f_{v}(s,u).$$ Since $f$ is continuous in $(u,v)$ and $v$ is in $\mathcal{K}$, the functions $F$ and $G$ are well-defined, continuous, and satisfy the following properties: \begin{itemize} \item[(i)] \begin{gather*} F(s), G(s)<0,\quad \text{for } 0\leq s< s_{0}, \text{ or } s>s_{1} \\ F(s), G(s)>0,\quad \text{for } s_{0}0,\quad \int_{0}^{s_{1}}G(s)ds>0 \end{gather*} \item[(ii)] $F(u)\leq f_{v}(s,u)\leq G(u)$, for $0\leq s\leq 1$, and $u\geq 0$. \end{itemize} Note that solutions of the problems $$\label{ePF} \begin{gathered} -u''=\lambda F(u)\\ u(0)=u(1)=0 \end{gathered}$$ and $$\label{ePG} \begin{gathered} -u''=\lambda G(u)\\ u(0)=u(1)=0 \end{gathered}$$ are, respectively, sub-solutions and super-solutions of \eqref{ePv}. \subsection*{Existence of a sub-solution for \eqref{ePv}} We are interested in the existence of a positive solution for problem \eqref{ePF}. We have the following result. \begin{lemma}\label{lem1} Assume that $f$ satisfies assumptions {\rm (A1)} and {\rm (A3)}. Then, there exists $\Lambda_{1}>0$ such that the problem \eqref{ePF} has a positive solution if and only if $\lambda\geq\Lambda_{1}$. \end{lemma} \begin{proof} We prove this lemma by using phase-plane analysis on the conservative system associated to \eqref{ePF}. Let us introduce the parameter $L=\sqrt{\lambda}$, and the new function $w(Lt)=u(t)$. Then problem \eqref{ePF} becomes \begin{gather*} -w''=F(w)\\ w(0)=w(L)=0\,. \end{gather*} Next, by defining $u_{1}=w$ and $u_{2}=u_{1}'$, the above problem is equivalent to the system $$\label{eP1} \begin{gathered} u_{1}'= u_{2}\\ u_{2}'= -F(u_{1})\\ u_{1}(0)= u_{1}(L)=0\,. \end{gathered}$$ Therefore, we study the solutions of the conservative system \eqref{eP1}. It is known that the energy is constant along the trajectories of \eqref{eP1} in the phase plane $(u_{1}, u_{2})$; i.~e., we have $$\label{energy} E=\frac{u_{2}^{2}}{2}+U(u_{1})=\mathrm{constant}\,,$$ where $U$ is the potential energy function given by $$U(x)=\int_{0}^{x}F(\xi)d\xi\,.$$ We represent in Figure 1 and 2, respectively, the potential energy function $U$ and the phase-portrait of the system \eqref{eP1}. \begin{figure}[ht] \begin{center} \includegraphics[width=0.7\textwidth]{fig1} % epotentialU.eps \caption{Potential energy function $U$ } \end{center} \end{figure} \begin{figure}[ht] \begin{center} \includegraphics[width=0.7\textwidth]{fig2} % {sp-portrait.eps \caption{Phase-portrait of system \eqref{eP1}} \end{center} \end{figure} From equation \eqref{energy}, we define the time map $T:\,[0, k_{0})\to (0, \infty)$, $$T(k) = \int_{0}^{u(k)} \frac{d\xi}{\sqrt{2(E(k)-U(\xi))}}$$ for initial conditions $u_{1}(0)=0$ and $u_{2}(0)=k$, where $u(k)$ is the distance reached by $u_{1}(t)$, from the origin, the first time when $u_{2}(t)=0$ in the $u_{1}u_{2}$-plane. Therefore, problem \eqref{eP1} has a positive solution if and only if there is $k\geq 0$ such that $L=2T(k)$. Let $a$ be the first zero of $U$ in $(0, s_{1})$ (see figure 1), $M=U(s_{1})>0$, $m=U(s_{0})$, and $k_{0}=\sqrt{2M}$. Then we have $a\leq u(k)$ for all $0\leq k0$ such that \eqref{ePG} has a positive solution if and only if $\lambda\geq\Lambda_{2}$. \end{lemma} \begin{proof} The proof is similar to that of Lemma 2.1. We recall the problem \eqref{ePG} is \begin{gather*} % \label{ePG} -u''=\lambda G(u)\\ u(0)=u(1)=0 \end{gather*} Similarly, the potential energy function $V$ and the time map $S$ are defined by $$V(x)=\int_{0}^{x}G(\xi)d\xi,\quad S(k)=\int_{0}^{u(k)}\frac{d\xi}{\sqrt{2(E(k)-V(\xi))}}$$ and the associated conservative system is $$\label{eP2} \begin{gathered} x_{1}'=x_{2}\\ x_{2}'=-G(x_{1})\\ x_{1}(0)=x_{2}(L)=0\,. \end{gathered}$$ The time map $S(k)$ has the same shape of the time map $T(k)$ in the proof of Lemma \ref{lem1}. Also, by using arguments similar to those in Lemma \ref{lem1}, we conclude that \eqref{ePG} has a positive solution if and only if $\lambda\geq \Lambda_{2}$, where $\Lambda_{2}=4(\min_{0\leq k< k_{1}}S(k))^{2}$ and $k_{1}=\sqrt{2 V(s_{1})}$. \end{proof} \subsection*{Existence of a positive solution for \eqref{ePv}} Before giving the main result in this section, we prove, in the next lemma, existence of a particular ordered pair $(w_{1}, w_{2})$ formed by a positive solution of \eqref{ePF} and a positive solution of \eqref{ePG}. \begin{lemma}\label{lem3} Assume that $f$ satisfies assumptions {\rm (A1)} and {\rm (A3)}. Then, there exists $\lambda_{0}>0$ such that for any $\lambda\geq \lambda_{0}$, there exists a positive solution $w_{1}$ of \eqref{ePF} and a positive solution $w_{2}$ of \eqref{ePG} with $$w_{1}\leq w_{2}\quad \text{everywhere in } [0, 1].$$ \end{lemma} \begin{proof} Let $\lambda_{0}=\max(\Lambda_{1}, \;\Lambda_{2})$, where $\Lambda_{1}$ and $\Lambda_{2}$ are given in Lemmas \ref{lem1} and \ref{lem2}, and let us consider the potential energy functions $U$ and $V$ corresponding for the systems \eqref{eP1} and \eqref{eP2}. These can be represented together in the same figure as follows (see figure 4). \begin{figure}[ht] \begin{center} \includegraphics[width=0.8\textwidth]{fig4} % UandV.eps \caption{Potential energy functions for \eqref{eP1} and \eqref{eP2}} \end{center} %\label{qpetit} \end{figure} Let $a_{F}$ and $a_{G}$ be, respectively, the first zero of $U$ and $V$ in $(0, s_{1})$ (cf.~figure 4). Then $$T(0)=\int_{0}^{a_{F}}\frac{d\xi}{\sqrt{-2U(\xi)}},\quad S(0)=\int_{0}^{a_{G}}\frac{d\xi}{\sqrt{-2V(\xi)}}, \quad \text{ and } \quad a_{G}\leq a_{F}0, we define \bar k=\sqrt{2V(a_{F})}>0, and denote by (u_{1}, u_{2}) the solution of \eqref{eP1} corresponding to T(0) and by (x_{1},x_{2}) the solution of \eqref{eP2} corresponding to S(\bar k). Clearly, T(0)=S(\bar k) and u_{1}(T(0))=a_{F}=x_{1}(S(\bar k)). The solutions (u_{1}, u_{2}) and (x_{1}, x_{2}) are represented together in figure 5. \begin{figure}[ht] \begin{center} \includegraphics[width=0.7\textwidth]{fig5} % sol1and2.eps \caption{Special solutions of \eqref{eP1} and \eqref{eP2}} \end{center}%\label{qpetit} \end{figure} Now, we claim that u_{1}(t)\leq x_{1}(t) for all 0\leq t\leq L. The proof of this claim will be done by integrating backward the systems \eqref{eP1} and \eqref{eP2}. Indeed, we first introduce the functions \varphi_{1}(t)=u_{1}(\frac{L}{2}-t) and \varphi_{2}(t)=u_{2}(\frac{L}{2}-t), so that we have $$\label{e1} \begin{gathered} \varphi_{1}'=-\varphi_{2}\\ \varphi_{2}'=F(\varphi_{1})\\ \varphi_{1}(0)=a_{F},\quad \varphi_{2}(0)=0\,. \end{gathered}$$ Similarly, by introducing \psi_{1}(t)=x_{1}(\frac{L}{2}-t) and \psi_{2}(t)=x_{2}(\frac{L}{2}-t), we have $$\label{e2} \begin{gathered} \psi_{1}'=-\psi_{2}\\ \psi_{2}'= F(\psi_{1})\\ \psi_{1}(0)= a_{F},\quad \psi_{2}(0)=0\,. \end{gathered}$$ Now, assume that there exists t_{0}\in (0, \frac{L}{2}) such that \varphi_{1}(t_{0})= \psi_{1}(t_{0}) and \varphi_{2}(t_{0})= \psi_{2}(t_{0}) (equivalently, u_{1}(\frac{L}{2}-t_{0})= x_{1}(\frac{L}{2}-t_{0}) and u_{2}(\frac{L}{2}-t_{0})= x_{2}(\frac{L}{2}-t_{0}), or in other words the solution's curves in the phase plane touch each other). Then, we have$$ \varphi_{2}'(t_{0})=F(\varphi_{1}(t_{0}))\leq G(\varphi_{1}(t_{0}))=G(\varphi_{1}(t_{0}))=\psi_{2}'(t_{0}). $$Therefore, as t increases from t_{0}, \varphi_{1}'(t) will remain greater than \psi_{1}'(t), that is u_{1}'(\frac{L}{2}-t) remains less than x_{1}'(\frac{L}{2}-t) for all t in [t_{0}, t_{0}+\varepsilon] and for small \varepsilon>0. Hence, u_{1}(\frac{L}{2}-t)\leq x_{1}(\frac{L}{2}-t), for all t\in [t_{0}, t_{0}+\varepsilon]. And repeating this reasoning from t_{0}+\varepsilon, we finally get u_{1}(t)\leq x_{1}(t) for all 0\leq t\leq \frac{L}{2}. By symmetry of the periodic solutions (u_{1}, u_{2}) and (x_{1}, x_{2}) about the horizontal axis in the phase plane, we have u_{1}(t)\leq x_{1}(t) for all \frac{L}{2}\leq t\leq L. To complete the proof, we let w_{1}(t)=u_{1}(L t) and w_{2}(t)=x_{1}(L t). \end{proof} Based on the above lemmas, we state and prove the main result of this section. \begin{theorem}\label{th1} Assume that f satisfies assumptions {\rm (A1)} and {\rm (A3)}. Then, for any \lambda\geq \lambda_{0}, problem \eqref{ePv} has a positive solution u satisfying$$ 0\leq w_{1}\leq u\leq w_{2}\leq a_{F} $$where \lambda_{0}, w_{1} and w_{2} are given in Lemma 2.3. Moreover, u is uniquely defined by the conditions u(\frac{1}{2})=a_{F} and u'(\frac{1}{2})=0. We denote the mapping v\mapsto u by \mathcal{F}_{1}. \end{theorem} \begin{proof} The existence of u is guaranteed by the existence of the particular sub- and super-solutions w_{1} and w_{2} given in Lemma \ref{lem3}. Therefore, we have to prove that u is uniquely defined by the conditions stated above. Indeed, since w_{1}\leq u\leq w_{2} and w_{1}(\frac{1}{2})= w_{2}(\frac{1}{2})=a_{F}, it follows that u satisfies the Cauchy Problem $$\label{e3} \begin{gathered} -u''=F(u)\\ u(\frac{1}{2})=a_{F},\quad u'(\frac{1}{2})=0 \end{gathered}$$ Therefore, by integrating equation (\ref{e3}) forward and backward, we see that u is uniquely defined on both intervals [\frac{1}{2}, 1] and [0, \frac{1}{2}]. This complete the proof. \end{proof} \section{Existence of a positive solution for \eqref{eP}} Let us consider the solution u of \eqref{ePv} given in Theorem \ref{th1}. Observing that u belongs to \mathcal{K}, we consider (by symmetry) the problem $$\label{ePu} \begin{gathered} -v''=\mu g_{u}(s,v)\\ v(0)=v(1)=0 \end{gathered}$$ where g_{u}(s,v)=g(u(s),v) for all 0\leq s\leq 1 and v\geq 0. Then, similarly to problem \eqref{ePv}, we obtain the following result on existence of a positive solution for \eqref{ePu}. \begin{corollary}\label{co1} Assume that g satisfies assumptions {\rm (A2)--(A3)}. Then, there exists \mu_{0}>0 such that, for any \mu\geq \mu_{0}, problem \eqref{ePu} has a positive solution v uniquely defined by v(\frac{1}{2})=a_{G} and v'(\frac{1}{2})=0 and such that$$ 0\leq v\leq s_{1}. $$We denote the mapping u\mapsto v by \mathcal{F}_{2}. \end{corollary} We shall use Schauder's Fixed Point Theorem \cite[p 126]{Mar76} to prove existence of a positive solution for \eqref{eP}. For that, we define the mapping$$ \mathcal{F}=\mathcal{F}_{2}\circ\mathcal{F}_{1}: \mathcal{K} \to \mathcal{K} $$where \mathcal{F}_{1} and \mathcal{F}_{2} are given by Theorem \ref{th1} and Corollary \ref{co1}, respectively. Then, we have the following result. \begin{theorem}\label{th2} Assume assumptions {\rm (A1)--(A3)}. Then, there exist \lambda_{0}>0, \,\mu_{0}>0 such that for any pair (\lambda,\mu) with \lambda\geq \lambda_{0}, \mu\geq \mu_{0}, problem \eqref{eP} has a positive solution (u,v). Furthermore, v is a fixed point of the mapping \mathcal{F}. \end{theorem} \begin{proof} Let \mathcal{K}_{0} be the bounded, closed, convex subset of C_0 [0,1] consisting of those functions v\in\mathcal{K} satisfying 0\leq v\leq s_1; i.~e.,$$ \mathcal{K}_{0} = \{ v\in \mathcal{K}: 0\leq v\leq s_1\}\,. $$On the other hand, by Theorem \ref{th1} and Corollary \ref{co1}, we note that \mathcal{F} : \mathcal{K} \to \mathcal{K} is well-defined and continuous (by continuous dependence of solutions of differential equations on parameters) and, furthermore, \mathcal{F} maps \mathcal{K}_{0} into \mathcal{K}_{0}. Also, by standard regularity, we have that \mathcal{F}(v)\in C^2 [0, 1], so that \mathcal{F} is completely continuous. Therefore, by Schauder's Fixed Point Theorem, \mathcal{F} has a fixed point v\in\mathcal{K}_{0}. And, if we denote by u the solution of problem \eqref{ePv} given in Theorem \ref{th1} with v being the fixed point of \mathcal{F}, then u\in\mathcal{K}_{0} and the pair (u,v) is a positive solution of \eqref{eP}. \end{proof} \section{An example} We shall consider the following reaction-diffusion system that describes the dynamics of two competing species in an environment \Omega that we assume to be one-dimensional. Let u(t,x) and v(t,x) represent the concentrations or population densities of the two species. With the standard non-dimensionalization process, we may assume the diffusion coefficient equal to 1, and \Omega= (0, 1 ). The boundary of the environment is considered to be hostile to the species (so, one has zero Dirichlet boundary conditions) and we assume that the reproduction or growth of the species follows the law \rho_{1}(u,v)=\lambda\frac{mu} {K+u+\frac{u^2}{\alpha}}(v+1) for the species u and \rho_{2}(u,v)=\mu\frac{mv} {K+v+\frac{v^2}{\alpha}}(u+1) for the species v. Then, the dynamics of the two species is described by $$\label{expl1} \begin{gathered} \frac{\partial u}{\partial t}= \frac{\partial^2 u}{\partial x^2}+\lambda(\frac{mu}{K+u+\frac{u^2}{\alpha}}-\alpha)(v+1)\\ \frac{\partial v}{\partial t}=\frac{\partial^2 v}{\partial x^2}+\mu(\frac{mv}{K+v+\frac{v^2}{\alpha}}-\alpha)(u+1)\\ u(0) = u(1) = 0,\quad v(0) = v(1) = 0\,. \end{gathered}$$ In the above system the the constants K, \alpha, \lambda and \mu are positive and the terms -\lambda(v+1) and -\mu(u+1) represent a mortality or harvesting of the species occurring in the interior of the environment. The steady state problem associated with \eqref{expl1} is given by $$\label{expl} \begin{gathered} -u''=\lambda f(u,v)\\ -v''=\mu g(u,v)\\ u(0)=u(1)=0=v(0)=v(1)\,, \end{gathered}$$ where \begin{gather*} f(u,v)=\Big(\frac{mu}{K+u+\frac{u^2}{\alpha}} -\alpha\Big)(v+1)\,,\\ g(u,v)=\Big(\frac{mv}{K+v+\frac{v^2}{\alpha}} -\alpha\Big)(u+1) \end{gather*} In what follows, we show that our assumptions (A1)--(A3) are satisfied for this system. We only check the assumptions for the function f since the situation for g is similar. \noindent (A1) is satisfied: Let \alpha_{0}:=m+2K-2\sqrt{K(m+K)}. Then, for any 0<\alpha<\alpha_{0}, we have (m-\alpha)^{2}>4\alpha K and the function f satisfies f(s_{0},v)=f(s_{1},v)=0 for any v\geq 0, where$$ s_{0}=\frac{m-\alpha-\sqrt{(m-\alpha)^{2}-4\alpha K}}{2},\quad s_{1}=\frac{m-\alpha+\sqrt{(m-\alpha)^{2}-4\alpha K}}{2}. In addition, f(s,v)<0 for 0s_{1}, and f(s,v)>0 for s_{0}0 such that \varphi(\alpha)>0 for all \alpha\leq \alpha_{1}, hence \int_{0}^{s_{1}}f(u,v)du>0 for all 0<\alpha\leq \alpha_{1}. This shows that assumption (A1) is satisfied. Similarly, we see that assumption (A2) is satisfied. \noindent (A3) part (ii) is satisfied: Let us consider the integral \begin{align*} I&=\int_{0}^{s_{0}}\min_{0\leq v\leq s_{1}}f(u,v)du+\int_{s_{0}}^{s_{1}}\min_{0\leq v\leq s_{1}}f(u,v)du \\ &=\int_{0}^{s_{0}}\min_{0\leq v\leq s_{1}}\Big(\frac{mu}{K+u+\frac{u^2}{\alpha}}-\alpha\Big)(v+1)du\\ &\quad + \int_{s_{0}}^{s_{1}}\min_{0\leq v\leq s_{1}}\Big( \frac{mu}{K+u+\frac{u^2}{\alpha}}-\alpha\Big)(v+1)du\,. \end{align*} Since \frac{mu}{K+u+\frac{u^2}{\alpha}}-\alpha\leq 0 \quad \text{for all } u \in [0, s_{0}] $$and$$ \frac{mu}{K+u+\frac{u^2}{\alpha}}-\alpha\geq 0 \quad \text{for all } u\in [s_{0}, s_{1}]\,,  we have \begin{align*} I&=\int_{0}^{s_{0}}\Big(\frac{mu} {K+u+\frac{u^2}{\alpha}}-\alpha\Big)(s_{1}+1)du+ \int_{s_{0}}^{s_{1}}\Big(\frac{mu} {K+u+\frac{u^2}{\alpha}}-\alpha\Big)du\\ &= s_{1}\int_{0}^{s_{0}}\Big(\frac{mu} {K+u+\frac{u^2}{\alpha}}-\alpha\Big)du +\int_{0}^{s_{1}}\Big(\frac{mu}{K+u+\frac{u^2} {\alpha}}-\alpha\Big)du\,. \end{align*} Therefore, as in the case of $\varphi (\alpha)$ above, there exists $\alpha_{2}>0$ such that $I>0$ for $0<\alpha<\alpha_{2}$. \subsection*{Conclusion} For any $\alpha$ satisfying $0<\alpha\leq \min(\alpha_{0}, \alpha_{1}, \alpha_{2})$, the functions $f$ and $g$ satisfy the assumptions (A1)--(A3). \begin{remark} \label{rmk1} \rm The existence result given in this paper is valid for a system of $n$ equations (with $n>2$), the proof being similar to the one presented here. 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