\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 34, pp. 1--8.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/34\hfil Existence of positive periodic solutions] {Existence of positive periodic solutions for neutral functional differential equations} \author[Z. Li, X. Wang,\hfil EJDE-2006/34\hfilneg] {Zhixiang Li, Xiao Wang} % in alphabetical order \address{Zhixiang Li \newline Department of Mathematics and System Science, Science School, National University of Defense Technology, Changsha, 410073, China} \email{zhxli02@yahoo.com.cn} \address{Xiao Wang \newline Department of Mathematics and System Science, Science School, National University of Defense Technology, Changsha, 410073, China} \email{wxiao\_98@yahoo.com.cn} \date{} \thanks{Submitted October 25, 2005. Published March 17, 2006.} \subjclass[2000]{34C25} \keywords{Positive periodic solution; cone; neutral delay differential equation; \hfill\break\indent fixed-point theorem} \begin{abstract} We find sufficient conditions for the existence of positive periodic solutions of two kinds of neutral differential equations. Using Krasnoselskii's fixed-point theorem in cones, we obtain results that extend and improve previous results. These results are useful mostly when applied to neutral equations with delay in bio-mathematics. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \section{Introduction} In this paper, we investigate the existence of positive periodic solutions of the following two kinds of nonlinear neutral functional differential equations \begin{equation} \frac{d}{dt}(x(t)-cx(t-\tau(t)))=-a(t)x(t)+g(t,x(t-\tau(t))), \label{e1.1} \end{equation} and \begin{equation} \frac{d}{dt} (x(t)-c\int_{-\infty}^0K(r)x(t+r)dr)=-a(t)x(t) +b(t)\int_{-\infty}^0K(r)g(t,x(t+r))dr, \label{e1.2} \end{equation} where $a,\tau \in C(\mathbb{R};\mathbb{R})$, $\int_0^\omega a(t)dt>0$, $b\in C(\mathbb{R};(0,\infty))$, $g\in C(\mathbb{R}\times[0,\infty),[0,\infty))$, and $a(t)$, $b(t)$, $\tau(t)$, $g(t,x)$ are $\omega$-periodic functions. $\omega>0$ and $c\in[0,1)$ are two constants. Moreover, $K \in C((-\infty,0],[0,\infty))$ and $\int_{-\infty}^0K(r)dr=1$. The function $a(t)$ admits negative values in bad conditions, since the environment fluctuates randomly. Our work is motivated by \cite{j1,j2,w1}, where the equations \begin{gather*} \frac{d}{dt}x(t)=-a(t)x(t)+g(t,x(t-\tau(t))),\\ \frac{d}{dt}x(t)=-a(t)x(t)+b(t)\int_{-\infty}^0K(s)g(t,x(t+s))ds, \end{gather*} are considered. Since these equations include many important models in mathematical biology, such as Hematopoiesis models, blood cell production and the Nicholson's blowflies models in \cite{g1,g2,g5,j1,j2,k1,l1,l2,s1,w1,w2}, the sufficient conditions for the existence of positive periodic solutions of these equations in \cite{j1,j2,w1} are interesting. Meanwhile, since a growing population is likely to consume more (or less) food than a matured one, depending on individual species, this leads to the neutral functional differential equations. Moreover, it is well-known that periodic solutions of differential equations describe the important modality of the systems. So it is important to study the existence of periodic solutions to \eqref{e1.1} and \eqref{e1.2}. Equations \eqref{e1.1} and \eqref{e1.2} include many mathematical ecological models and population models (directly or after some transformation). For example, there are many Hematopoiesis models, which are modifications from models in \cite{g1,j1,j2,l1,l2,w1,w2}: \begin{gather} \frac{d}{dt}(x(t)-cx(t-\tau(t)))=-a(t)x(t) +b(t)e^{-\beta(t)x(t-\tau(t))}, \label{e1.3} \\ \frac{d}{dt} (x(t)-c\int_{-\infty}^0K(r)x(t+r)dr) =-a(t)x(t)+b(t)\int_{-\infty}^0K(r)e^{-\beta(t)x(t+r))}dr. \label{e1.4} \end{gather} There are more general models for blood cell production, which are variations of models in \cite{g1,g2,j1,j2,l1,l2,w1,w2}: \begin{gather} \frac{d}{dt}(x(t)-cx(t-\tau(t)))=-a(t)x(t)+b(t)\frac{1}{1+x^n(t-\tau(t))}, n>0, \label{e1.5} \\ \frac{d}{dt} (x(t)-cx(t-\tau(t))dr)=-a(t)x(t)+b(t)\frac{x(t-\tau(t))}{1+x^n(t-\tau(t))}, n>0,\label{e1.6} \\ \begin{aligned} &\frac{d}{dt}(x(t)-c\int_{-\infty}^0K (r)x(t+r)dr)\\ &=-a(t)x(t)+b(t)\int_{-\infty}^0K(r)\frac{1}{1+x^n(t+r)}dr,n>0, \end{aligned}\label{e1.7} \\ \begin{aligned} &\frac{d}{dt} (x(t)-c\int_{-\infty}^0K(r)x(t+r)dr)\\ &=-a(t)x(t)+b(t)\int_{-\infty}^0K(r)\frac{x(t+r)}{1+x^n(t+r)}dr,n>0. \end{aligned}\label{e1.8} \end{gather} Meanwhile, there are more Nicholson's blowflies models, which are modifications from models in \cite{g1,g5,j1,j2,l2,w1}: \begin{gather} \frac{d}{dt}(x(t)-cx(t-\tau(t))) =-a(t)x(t)+b(t)x(t-\tau(t))e^{-\beta(t)x(t-\tau(t))}, \label{e1.9} \\ \begin{aligned} &\frac{d}{dt}(x(t)-c\int_{-\infty}^0K(r)x(t+r)dr) \\ &=-a(t)x(t)+b(t)\int_{-\infty}^0K(r)x(t+r)e^{-\beta(t)x(t+r)}dr. \end{aligned}\label{e1.10} \end{gather} In this paper, we obtain sufficient conditions for the existence of positive periodic solutions for the neutral delay differential equations \eqref{e1.1} and \eqref{e1.2}. Our results improve and generalize the corresponding results of Jiang and Wei \cite{j1,j2} and Wan \cite{w1}, when $c=0$ in \eqref{e1.1} and \eqref{e1.2}. In fact, Theorem \ref{thm2.1} extends and improves the corresponding results in \cite[Theorem 2.1]{w1} and \cite[Theorem 2.1]{j2}. Meanwhile, Theorem \ref{thm2.2} improves the corresponding results in \cite[Theorem 2.1]{j1}. For $a(t)>0$ in \cite{w1} and $g(t,x)$ sub-linear or super-linear in \cite{j2}, the assumptions in Theorem \ref{thm2.1} and Theorem \ref{thm2.2} are weaker than theirs. When $c\neq 0$, our main results are new. Due to $c\neq 0$, the methods used by the authors \cite{j1,j2,w1} can not be directly applied to \eqref{e1.1} and \eqref{e1.2}. The proofs of the main results in our paper are based on an application of Krasnoselskii's fixed point theorem in cones (See \cite{d1,g3,g4}). To make use of fixed point theorem in a cone, firstly, we introduce the definition of a cone in a Banach space. \smallskip \noindent\textbf{Definition.} Let $X$ be a Banach space. $K$ is called a cone if $K$ is a closed nonempty subset of $X$ and satisfies \begin{itemize} \item[(i)] $\alpha x+\beta y\in K$, for all $x,y \in K$ and $\alpha,\beta>0$; \item[(ii)] $x,-x\in K$ implies $x=0$. \end{itemize} The following Lemma is due to Krasnoselskii (See \cite{d1,g3,g4}). \begin{lemma} \label{lem1.1} Let $X$ be a Banach space, and let $K\subset X$ be a cone in $X$. Assume $\Omega_1,\Omega_2$ are open subsets of $X$ with $0\in \Omega_1,\bar{\Omega}_1\subset \Omega_2$, and let $$ \Phi : K\cap(\bar{\Omega}_2\setminus\Omega_1)\rightarrow K $$ be a completely continuous operator that satisfies one of the following conditions: \begin{itemize} \item[(i)] $\|\Phi x\|\geq \|x\|,\forall x\in K\cap\partial\Omega_1 $ and $\|\Phi x\|\leq \|x\|,\forall x\in K\cap\partial\Omega_2 $; \item[(ii)] $\|\Phi x\|\geq \|x\|,\forall x\in K\cap\partial\Omega_2 $ and $\|\Phi x\|\leq \|x\|,\forall x\in K\cap\partial\Omega_1 $. \end{itemize} Then $\Phi$ has a fixed point in $K\cap(\bar{\Omega}_2\setminus\Omega_1)$. \end{lemma} For convenience, we need to introduce a few notations and assumptions. Let \begin{gather*} G(t,s)=\frac{\exp(\int_t^sa(r)dr)}{\exp(\int_0^\omega a(r)dr)-1}, \\ A:=\min\{G(t,s):0\leq t,s\leq \omega\}=G(t,t)>0, \\ B:=\max\{G(t,s):0\leq t,s\leq \omega\}=G(t,t+\omega)>0, \\ 0< \sigma=\frac{A}{B}<1, \\ m(y)=\max_{(t,x)\in[0,\omega]\times[0,y]}{g(t,x)},y\geq0. \end{gather*} For \eqref{e1.1}, we assume that \begin{itemize} \item[(H1)] $\liminf_{x\rightarrow 0}\frac{g(t,x)}{x}= \alpha(t)$ and $\limsup_{x\rightarrow\infty}\frac{g(t,x)}{x}= \beta(t)$, where $\alpha(t),\beta(t)$ are continuous $\omega$-periodic functions on $\mathbb{R}$. \item[(H2)] $\int_0^\omega\alpha(t)dt> c\int_0^\omega a(t)dt+\frac{1}{A\sigma}(1-c\sigma)$ and $\int_0^\omega\beta(t)dt< c\int_0^\omega a(t)dt+\frac{1}{B}(1-c)$. \item[(H3)] $g(t,x)\geq ca(t)x,\forall(t,x)\in \mathbb{R}\times [0,r_2]$. \end{itemize} From (H1), there exist two constants $r_1$ and $n$ with $0n. \end{gather*} Let $r_2>\max\{\frac{Bm}{1-c-B\int_0^\omega[\beta(t)-ca(t)]dt},n\}>r_1$, where $m=\omega(m(n)+cn\|a(t)\|)$. For \eqref{e1.2}, we suppose that (H1) holds and \begin{itemize} \item[(P2)] $\int_0^\omega b(t)\alpha(t)dt> c\int_0^\omega a(t)dt+\frac{1}{A\sigma}(1-c\sigma)$ and $\int_0^\omega b(t)\beta(t)dt< c\int_0^\omega a(t)dt+\frac{1}{B}(1-c)$. \item[(P3)] $g(t,x)\geq \frac{ca(t)}{b(t)}x$, for all $(t,x)\in \mathbb{R}\times [0, R_2]$. \end{itemize} From (H1) there exist two constants $ R_1$ and $N$ with $0< R_1N. \end{gather*} Let $$ R_2>\max\{\frac{BM}{1-c-B\int_0^\omega[\beta(t)-ca(t)]dt},N\} > R_1, $$ where $M=\omega(m(N)+cN\|a(t)\|)$. The rest of this paper is organized as follows. In the second section, we give and prove our main results. As applications, in the final section, we apply our main results to some population models and several new results are obtained. \section{Existence of Positive Periodic Solutions} Now we state our main results. \begin{theorem} \label{thm2.1} Assume that (H1)-(H3) hold, then \eqref{e1.1} has at least one positive $\omega$-periodic solution. \end{theorem} \begin{theorem} \label{thm2.2} Assume that (H1),(P2) and (P3) hold, then \eqref{e1.2} has at least one positive $\omega$-periodic solution. \end{theorem} \begin{remark} \label{rml1.1} When $c=0$, (H3) and (P3) hold obviously. In this case, Theorem \ref{thm2.1} extends and improves the corresponding results in \cite[Theorem 2.1]{w1} and \cite[Theorem 2.1]{j2}, Meanwhile, Theorem \ref{thm2.2} improves the corresponding results in \cite[Theorem 2.1]{j1}. If assumes $a(t)>0$ in \cite{w1} and $g(t,x)$ is sub-linear or super-linear in \cite{j2}, clearly, then the assumptions in Theorem \ref{thm2.1} and Theorem \ref{thm2.2} are weaker than theirs. \end{remark} We remark that when $c\neq 0$, our main results are new. Now, we should construct a Banach space $X$ and a cone $K$. Let $X=\{x(t): x(t)\in C(\mathbb{R},\mathbb{R})$, $x(t)=x(t+\omega)$, for all $t\in \mathbb{R}\}$ and defining $\|x(t)\|=\sup_{t\in [0,\omega]}|x(t)|$, for all $x\in X$. Then $X$ is a Banach space with the norm $\| \cdot\|$. Let $K=\{x\in X:x(t)\geq0 , x(t)\geq\sigma\|x(t)\|\}$, it is not difficult to verify that $K$ is a cone in $X$. First, we consider the integral equation \begin{equation} x(t)=\int_t^{t+\omega}G(t,s)[g(s,x(s-\tau(s))) -ca(s)x(s-\tau(s))]ds+cx(t-\tau(t)). \label{e2.1} \end{equation} It is easy to see that $\varphi(t)$ is an $\omega$-periodic solution of \eqref{e1.1} if and only if $\varphi(t)$ is an $\omega$-periodic solution of \eqref{e2.1}. Define an operator on $X$, $x=\Phi x$, for $x\in X$, where $\Phi$ is given by \begin{equation} (\Phi x)(t)=\int_t^{t+\omega}G(t,s)[g(s,x(s-\tau(s))) -ca(s)x(s-\tau(s))]ds+cx(t-\tau(t)). \label{e2.2} \end{equation} Clearly, $\Phi$ is not a completely continuous operator on $X$, since $cx$ is not a completely continuous operator on $X$. Since $\Omega_1$ and $\Omega_2$ defined in \cite{j1,j2,w1} are not suitable to here, we should construct two different sets $\Omega_1$ and $\Omega_2$. \begin{proof}[Proof of Theorem \ref{thm2.1}] We define \begin{gather*} \Omega_1:= \{x\in X:\|x\|r_1, \end{align*} which implies that $\|\Phi x\|>\|x\|$ for $x\in K\cap\partial\Omega_1$. On the other hand, by using the same type of argument as in above, we will obtain that $\Phi$ is a completely continuous operator on $\Omega_2$. Thus, if $x\in K\cap\partial\Omega_2$, then $\|x\|=r_2,\|x'\|\leq\bar{r}_2$ or $\|x\|\leq r_2,\|x'\|=\bar{r}_2$. It follows from \eqref{e2.2} and $(H3)$, either $\|x\|=r_2$, $\|x'\|\leq\bar{r}_2$ or $\|x\|\leq r_2$, $\|x'\|=\bar{r}_2$. We have \begin{align*} (\Phi x)(t) &\leq B\int_t^{t+\omega}(g(s,x(s-\tau(s)))-ca(s)x(s-\tau(s))]ds +cx(t-\tau(t))\\ &\leq B\int_{x(t-\tau(t))\leq n}[g(t,x(s-\tau(s)))-ca(s)x(s-\tau(s))]ds\\ &\quad + B\int_{x(t-\tau(t))> n}[g(t,x(s-\tau(s))) -ca(s)x(s-\tau(s))]ds+cx(t-\tau(t))\\ &\leq Bm+Br_2\int_0^\omega[\beta(t)-ca(t)]dt+cr_20$, which means $x(t)$ is a positive $\omega$-periodic solution of \eqref{e1.1}. \end{proof} Next, we consider the integral equation \begin{equation} \begin{aligned} x(t)&=\int_t^{t+\omega}G(t,s)[b(s)\int_{-\infty}^0K(r)g(s,x(s+r))dr\\ &\quad -ca(s)\int_{-\infty}^0K(r)x(s+r)dr]ds +c\int_{-\infty}^0K(r)x(t+r)dr. \end{aligned}\label{e2.3} \end{equation} Similarly, we see that $\varphi(t)$ is an $\omega$-periodic solution of \eqref{e1.2} if and only if $\varphi(t)$ is an $\omega$-periodic solution of above equation. Define an operator on $X$ $x=\Psi x, $ for $x\in X$, where $\Psi$ is given by \begin{equation} \begin{aligned} (\Psi x)(t)&=\int_t^{t+\omega}G(t,s)[b(s) \int_{-\infty}^0K(r)g(s,x(s+r))dr\\ &\quad -ca(s)\int_{-\infty}^0K(r)x(s+r)dr]ds + c\int_{-\infty}^0K(r)x(t+r)dr. \end{aligned} \label{e2.4} \end{equation} \begin{proof}[Proof of Theorem \ref{thm2.2}] We define \begin{gather*} \Omega_1:= \{x\in X:\|x\|< R_1,\|x'\|<\bar{R}_1\}, \\ \Omega_2:= \{x\in X:\|x\|< R_2,\|x'\|<\bar{R}_2\}, \end{gather*} where $\bar{R}_1=\frac{\|a(t)\| R_1+m( R_1)}{1-c}$ and $\bar{R}_2=\frac{\|a(t)\| R_2+m( R_2)}{1-c}$, where $ R_1$ and $ R_2$ are given in above. Obviously, $0\in\Omega_1,\bar{\Omega}_1\subset\Omega_2$. Next, by using the same arguments in the proof of Theorem \ref{thm2.1}, one can obtain that the operator $\Psi$ satisfies all the conditions in Lemma \ref{lem1.1}. Therefore, $\Psi$ has a fixed point $x\in K\cap(\bar{\Omega}_2\setminus\Omega_1)$. Furthermore, $ R_1\leq\|x\|\leq R_2$ and $x(t)\geq\sigma R_1>0$, which means $x(t)$ is a positive $\omega$-periodic solution of \eqref{e1.2}. \end{proof} \section{Some Applications} In this section, we apply the results obtained in previous section to the study equations \eqref{e1.3}-\eqref{e1.10}. In view of Theorem \ref{thm2.1} and Theorem \ref{thm2.2}, we obtain the following results. \begin{theorem} \label{thm3.1} Assume that \begin{enumerate} \item $a, \tau \in C(\mathbb{R};\mathbb{R})$, $\beta, b\in C(\mathbb{R};(0,\infty))$, $\int_0^\omega a(t)dt>0$, and $a(t),\beta(t), \tau(t)$ are $\omega$-periodic functions, $\omega>0$ and $c\in [0,1)$ are two constants. \item $b(t)e^{-\beta(t)x}\geq ca(t)x$ for all $(t,x)\in \mathbb{R}\times [0,r_2]$, where the definition of $r_2$ is similar to (H3) in section 1. \end{enumerate} Then \eqref{e1.3} has at least one positive $\omega$-periodic solution. \end{theorem} \begin{theorem} \label{thm3.2} Assume that \begin{enumerate} \item $a, \tau\in C(\mathbb{R};\mathbb{R})$, $b\in C(\mathbb{R};(0,\infty))$, $\int_0^\omega a(t)dt>0$, and $a(t), \tau(t)$ are all $\omega$-periodic functions, $\omega>0$ and $c\in[0,1)$ are two constants. \item $b(t)\frac{1}{1+x^n}\geq ca(t)x$ for all $(t,x)\in \mathbb{R}\times [0,r_2]$, where the definition of $r_2$ is similar to (H3) in section 1. \end{enumerate} Then \eqref{e1.5} has at least one positive $\omega$-periodic solution. \end{theorem} \begin{theorem} \label{thm3.3} Assume (1) in Theorem \ref{thm3.2} holds and \begin{itemize} \item[(2)] $b(t)\frac{1}{1+x^n}\geq ca(t)$ for all $(t,x)\in \mathbb{R}\times [0,r_2]$, where the definition of $r_2$ is similar to (H3) in section 1. \end{itemize} Then \eqref{e1.6} has at least one positive $\omega$-periodic solution. \end{theorem} \begin{theorem} \label{thm3.4} Assume (1) in Theorem \ref{thm3.1} holds and \begin{itemize} \item[(2)] $b(t)e^{-\beta(t)x}\geq ca(t)$ for all $(t,x)\in \mathbb{R}\times [0,r_2]$, where the definition of $r_2$ is similar to (H3) in section 1. \end{itemize} Then \eqref{e1.9} has at least one positive $\omega$-periodic solution. \end{theorem} \begin{theorem} \label{thm3.5} Assume that \begin{enumerate} \item $a\in C(\mathbb{R};\mathbb{R})$, $\int_0^\omega a(t)dt>0$, $b, \beta\in C(\mathbb{R};(0,\infty))$, and $a(t),b(t),\beta(t)$ are all $\omega$-periodic functions, $\omega>0$, $0\leq c<1$ are constants. Moreover, $K(r)\in C((-\infty,0],[0,\infty))$ and $\int_{-\infty}^0K(r)dr=1$. \item $e^{-\beta(t)x}\geq \frac{ca(t)}{b(t)}x$ for all $(t,x)\in \mathbb{R}\times [0, R_2]$, where the definition of $ R_2$ is similar to (P3) in section 1. \end{enumerate} Then \eqref{e1.4} has at least one positive $\omega$-periodic solution. \end{theorem} \begin{theorem} \label{thm3.6} Assume that \begin{enumerate} \item $a\in C(\mathbb{R};\mathbb{R})$, $\int_0^\omega a(t)dt>0,b\in C(\mathbb{R};(0,\infty))$ and $a(t),b(t)$ are all $\omega$-periodic functions, $\omega>0$ and $c[0,1)$ are two constants. Moreover, $K(r)\in C((-\infty,0],[0,\infty))$ and $\int_{-\infty}^0K(r)dr=1$. \item $\frac{1}{1+x^n}\geq \frac{ca(t)}{b(t)}x$ for all $(t,x)\in \mathbb{R}\times [0, R_2]$, where the definition of $ R_2$ is similar to (P3) in section 1. \end{enumerate} Then \eqref{e1.7} has at least one positive $\omega$-periodic solution. \end{theorem} \begin{theorem} \label{thm3.7} Assume (1) in Theorem \ref{thm3.6} holds and \begin{itemize} \item[(2)] $\frac{1}{1+x^n}\geq \frac{ca(t)}{b(t)}$ for all $(t,x)\in \mathbb{R}\times [0, R_2]$, where the definition of $ R_2$ is similar to (P3) in section 1. \end{itemize} Then \eqref{e1.8} has at least one positive $\omega$-periodic solution. \end{theorem} \begin{theorem} \label{thm3.8} Assume (1) in Theorem \ref{thm3.5} holds and \begin{itemize} \item[(2)] $e^{-\beta(t)x}\geq \frac{ca(t)}{b(t)}$ for all $(t,x)\in \mathbb{R}\times [0, R_2]$, where the definition of $ R_2$ is similar to (P3) in section 1. \end{itemize} Then \eqref{e1.10} has at least one positive $\omega$-periodic solution. \end{theorem} We remark that when $c=0$, Theorems \ref{thm3.1}--\ref{thm3.8} improve the results in \cite{j1,j2,w1}. \begin{thebibliography}{00} \bibitem{d1} K. 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