\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 96, 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 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/96\hfil Positive periodic solutions] {Positive periodic solutions of nonlinear first-order functional difference equations with a parameter} \author[Y. Lu\hfil EJDE-2011/96\hfilneg] {Yanqiong Lu} \address{Yanqiong Lu \newline Department of Mathematics, Northwest Normal University Lanzhou, 730070, China} \email{linmu8610@163.com} \thanks{Submitted June 29, 2011. Published July 28, 2011.} \thanks{Supported by grant 11061030 from the NSFC, the Fundamental Research Funds for the \hfill\break\indent Gansu Universities.} \subjclass[2000]{34G20} \keywords{Positive periodic solutions; existence; nonexistence; \hfill\break\indent difference equations; fixed point} \begin{abstract} We obtain the existence and multiplicity of positive $T$-periodic solutions for the difference equations $$\Delta x(n)=a(n,x(n))-\lambda b(n)f(x(n-\tau(n)))$$ and $$\Delta x(n)+a(n,x(n))=\lambda b(n)f(x(n-\tau(n))),$$ where $f(\cdot)$ may be singular at $x=0$. Using a fixed point theorem in cones, we extend recent results in the literature. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \section{Introduction} In recent years, there has been considerable interest in the existence of periodic solutions of the equation $$x'(t)=\tilde a(t, x(t))-\lambda \tilde b(t) \tilde f(x(t-\tau(t))), \label{e1.1}$$ where $\lambda>0$ is a positive parameter, $\tilde a$ is continuous in $x$ and $T$-periodic in $t$, $\tilde b\in C(\mathbb{R}, [0,\infty))$ and $\tau\in C(\mathbb{R}, \mathbb{R})$ are $T$-periodic functions, $\int^T_0\tilde b(t)dt>0$, $f\in C([0,\infty),[0,\infty))$. \eqref{e1.1} has been proposed as a model for a variety of physiological processes and conditions including production of blood cells, respiration, and cardiac arrhythmias. See, for example, \cite{c1,c2,f1,g2,j1,w1,m3,w2,w3,y1} and the references therein. In this article, we study the existence of positive $T$-periodic solutions of a discrete analogues to \eqref{e1.1} of the form $$\Delta x(n)=a(n,x(n))-\lambda b(n)f(x(n-\tau(n))),\quad n\in \mathbb{Z} \label{e1.2}$$ and $$\Delta x(n)+a(n,x(n))=\lambda b(n)f(x(n-\tau(n))),\quad n\in \mathbb{Z}, \label{e1.3}$$ where $\mathbb{Z}$ is the set of integer numbers, $T\in \mathbb{N}$ is a fixed integer, $a:\mathbb{Z}\times[0, \infty)\to[0, \infty)$ is continuous in $x$ and $T$-periodic in $n$, $b:\mathbb{Z}\to[0, +\infty)$, $\tau:\mathbb{Z}\to\mathbb{Z}$ are $T$-periodic and $\sum_{n=0}^{T-1}b(n)>0$, $f\in C((0,+\infty),(0,+\infty))$ and may have a repulsive singularity near $x=0$, $\lambda>0$ is a parameter. So far, relatively little is known about the existence of positive periodic solutions of \eqref{e1.2} and \eqref{e1.3}. To our best knowledge, Ma \cite{m2} dealt with the special equations of \eqref{e1.2} and \eqref{e1.3} of the form $$\Delta x(n)=a(n)g(x(n))x(n)-\lambda b(n)f(x(n-\tau(n))) \label{e1.4}$$ and $$\Delta x(n)+a(n)g(x(n))x(n)=\lambda b(n)f(x(n-\tau(n))), \label{e1.5}$$ with certain values of $\lambda$, for which there exist positive $T$-periodic solutions of \eqref{e1.4} and \eqref{e1.5}, respectively. If $g(x(n))\equiv1$, this special case see \cite{l1,m1,r1}. All these authors \cite{l1,m1,m2,r1} focus their attention on the fact that the number of positive $T$-periodic solutions can be determined by the behaviors of the quotient of $f(x)/x$ at $\{0,+\infty\}$. However, our main results show the number of positive $T$-periodic solutions can be determined by the behaviors of the quotient of $f(x)/x$ at $[0,\infty]$. It is the purpose of this paper to study more general equations \eqref{e1.2} and \eqref{e1.3} and generalize the main results of Ma \cite{m2}. We also establish some existence and multiplicity for \eqref{e1.2} and \eqref{e1.3}, respectively. The main tool we will use is the fixed point index theory \cite{d1,g1}. Throughout this paper, we denote the product of $x(n)$ from $n=a$ to $n=b$ by $\prod_{n=a}^bx(n)$ with the understanding that $\prod_{n=a}^bx(n)=1$ for all $a>b$. The rest of the paper is arranged as follows: In Section 2, we give some preliminary results. In Section 3 we state and prove some existence results of positive periodic solutions for \eqref{e1.2} and \eqref{e1.3}. Finally, Section 4 is devoted to improving some results of Ma \cite{m2}. For related results on the associated differential equations, see Weng and Sun \cite{w2}. \section{Preliminaries} In this article, we make the following assumptions: \begin{itemize} \item[(H1)] There exist functions $a_1, a_2:\mathbb{Z}\to[0,+\infty)$ are $T-$periodic functions such that $\sum_{n=0}^{T-1}a_1(n)>0$, $\sum_{n=0}^{T-1}a_2(n)>0$ and $a_1(n)x(n)\leq a(n,x(n))\leq a_2(n)x(n)$ for $n\in\mathbb{Z}$ and $x>0$. In addition, $\lim_{x\to0}\frac{a(n,x)}{x}$ exists for $n\in \mathbb{Z}$. \item[(H2)] $a(n, x)$ is continuous in $x$ and $T$-periodic in $n$, $b:\mathbb{Z}\to[0, +\infty),\ \tau:\mathbb{Z}\to\mathbb{Z}$ are $T$-periodic and $B:=\sum_{n=0}^{T-1}b(n)>0$; $f\in C((0,+\infty),(0,+\infty))$ and may have a repulsive singularity near $x=0$. \end{itemize} Denote $$\sigma_i=\prod_{s=0}^{T-1}(1+a_i(s))^{-1},\quad i=1,2,\quad m=\frac{\sigma_2}{1-\sigma_2},\quad M=\frac{1}{1-\sigma_1}.$$ From (H1), it is clear that $0<\frac{m}{M}<1$. Let $$E:=\{x:\mathbb{Z}\to\mathbb{R}:x(n+T)=x(n)\}$$ be the Banach space with the norm $\|x\|=\max_{n\in\mathbb{Z}}|x(n)|$. Define the cone $$P:=\{x\in E : x(n)\geq0,\ x(n)\geq\frac{m}{M}\|x\|\},$$ and the operator $A_{\lambda}:P\to E$ by $$(A_\lambda x)(n)=\lambda\sum_{s=n}^{n+T-1} G_x(n,s)b(s)f(x(s-\tau(s))), \quad n\in\mathbb{Z}, \label{e2.1}$$ where $$G_x(n,s)=\frac{\prod_{k=n}^{s}(1+\frac{a(k,x(k))}{x(k)})^{-1}} {1-\prod_{k=1}^{T}(1+\frac{a(k,x(k))}{x(k)})^{-1}},\quad n\leq s\leq n+T.$$ It follows from (H1) that $$m\leq G_x(n,s)\leq M.$$ If (H1) and (H2) hold and $x\in P$, then $$\lambda m\sum_{s=n}^{n+T-1} b(s)f(x(s-\tau(s))) \leq \|A_{\lambda}x\|\leq\lambda M\sum_{s=n}^{n+T-1}b(s)f(x(s-\tau(s))). \label{e2.2}$$ The construction of $G_x(n,s)$ is due to Ma \cite{m2}. Following the approach in \cite{m2}, we can easily prove the following two Lemmas. Similar arguments have been also employed in \cite{r1}. We remark that the process of proofs are similar and are omitted. \begin{lemma} \label{lem2.1} Assume that {\rm (H1), (H2)} hold. Then $A_\lambda(P)\subset P$ and $A_\lambda:P\to P$ is compact and continuous. \end{lemma} \begin{lemma} \label{lem2.2} Assume that {\rm (H1), (H2)} hold. Then $x\in P$ is a solution of \eqref{e1.2} if and only if $x$ is a fixed point of $A_\lambda$ in $P$. \end{lemma} The following well-known result of the fixed point index is crucial in our arguments. \begin{lemma}[\cite{d1,g1}] \label{lem2.3} Let $E$ be a Banach space and $K$ be a cone in $E$. For $r>0$, define $K_r=\{u\in K : \|u\|r>0$ and $$m^2\min_{x\in[\frac{m}{M}r,r]}\frac{f(x)}{x}>M^2 \max_{x\in[R,\frac{M}{m}R]}\frac{f(x)}{x}.\label{e3.1}$$ Then, for each $\lambda$ satisfying $$\frac{M}{m^2B\min_{x\in[\frac{m}{M}r,r]} \frac{f(x)}{x}}<\lambda \leq\frac{1}{MB\max_{x\in[R,\frac{M}{m}R]} \frac{f(x)}{x}},\label{e3.2}$$ equation \eqref{e1.2} has a positive $T$-periodic solution $x$ satisfying r\frac{M}{\lambda m^2 B},\quad\forall x\in[\frac{m}{M}r,r] \quad \text{and}\quad \frac{f(x)}{x}\leq\frac{1}{\lambda MB} ,\quad x\in[R,\frac{M}{m}R]. $$Define the open sets$$ \Omega_1:=\{x\in E: \|x\|\lambda m\sum_{s=n}^{n+T-1}b(s)\frac{M}{\lambda m^2 B} x(s-\tau(s))) \\ &\geq\frac{M}{mB}\sum_{s=0}^{T-1}b(s)\frac{m}{M}r=r=\|x\|. \end{align*} Hence\|A_\lambda x\|>\|x\|$,$x\in \partial\Omega_1\cap P$. From Lemma \ref{lem2.3}, we have that $$i(A_\lambda,\Omega_1\cap P,P)=0.$$ If$x\in \partial\Omega_2\cap P$, then$\|x\|=\frac{M}{m}R$and$R\leq x\leq\frac{M}{m}R. According to \eqref{e2.2}, it follows that \begin{align*} \|A_\lambda x\| &\leq\lambda M\sum_{s=n}^{n+T-1}b(s)f(x(s-\tau(s)))\\ &\leq\lambda M\sum_{s=n}^{n+T-1}b(s)\frac{1}{\lambda MB}x(s-\tau(s))) \\ &\leq\frac{1}{B}\sum_{s=0}^{T-1}b(s)\frac{M}{m}R=\frac{M}{m}R=\|x\|. \end{align*} Hence\|A_\lambda x\|\leq\|x\|$,$x\in \partial\Omega_2\cap P$. From Lemma \ref{lem2.3}, we have that $$i(A_\lambda,\Omega_2\cap P,P)=1.$$ Thus$i(A_\lambda,\Omega_{2}\backslash\bar{\Omega}_{1},P)=1$and$A_\lambda$has a fixed point in$\Omega_{2}\backslash \bar{\Omega}_{1}$, which is a positive$T$-periodic solution of \eqref{e1.2} and $$rr>0 and $$m^2\min_{x\in[R,\frac{M}{m}R]}f(x)/x>M^2\max_{x\in[\frac{m}{M}r,r]} f(x)/x.\label{e3.3}$$ Then, for each \lambda satisfying $$\frac{M}{m^2B\min_{x\in[R,MR/m]} f(x)/x} \leq\lambda<\frac{1}{MB\max_{x\in[m/rM,r]}f(x)/x},\label{e3.4}$$ equation \eqref{e1.2} has a positive T-periodic solution x satisfying rr>0 and$$ \bar{m}^2\min_{x\in[\frac{\bar{m}}{\bar{M}}r,r]} \frac{f(x)}{x}>\bar{M}^2\max_{x\in[R,\frac{\bar{M}}{\bar{m}}R]} \frac{f(x)}{x}. $$Then, for each \lambda satisfying$$ \frac{\bar{M}}{\bar{m}^2B\min_{x\in[\frac{\bar{m}}{\bar{M}}r,r]} \frac{f(x)}{x}}<\lambda\leq\frac{1}{\bar{M}B\max_{x\in[R, \frac{\bar{M}}{\bar{m}}R]}\frac{f(x)}{x}}, $$equation \eqref{e1.3} has a positive T-periodic solution x satisfying rr>0 and$$ \bar{m}^2\min_{x\in[R,\frac{\bar{M}}{\bar{m}}R]} \frac{f(x)}{x}>\bar{M}^2\max_{x\in[\frac{\bar{m}}{\bar{M}}r,r]} \frac{f(x)}{x}. $$Then, for each \lambda satisfying$$ \frac{\bar{M}}{\bar{m}^2B\min_{x\in[R,\frac{\bar{M}}{\bar{m}}R]} \frac{f(x)}{x}}\leq\lambda<\frac{1}{\bar{M}B\max_{x\in[\frac{\bar{m}} {\bar{M}}r,r]}\frac{f(x)}{x}}, $$equation \eqref{e1.3} has a positive T-periodic solution x satisfying r\frac{M}{m^2B\min_{x\in[mc/M,c]}f(x)/x}. \item[(ii)] If i_\infty=1 or 2, then \eqref{e1.2} has i_\infty positive T-periodic solution(s) for 0<\lambda<\frac{1}{MB\max_{x\in[c,Mc/m]}f(x)/x}. \end{itemize} \end{corollary} \begin{proof} (i) If f_0=0, then there exists small enough r_1 such that c>r_1>0 and$$ m^2\min_{x\in[mc/M,c]}\frac{f(x)}{x} \geq M^2\max_{x\in[\frac{m^2}{M^2}r_1,\frac{m}{M}r_1]} \frac{f(x)}{x}\to0 \quad(\text{as } r_1\to0). $$By applying Theorem \ref{thm3.2} with R=\frac{m}{M}c and r=\frac{m}{M}r_1, Equation \eqref{e1.2} has a positive T-periodic solution x satisfying$$\frac{m}{M}r_1c>0$ and $$m^2\min_{x\in[mc/M,c]}\frac{f(x)}{x}\geq M^2 \max_{x\in[\frac{M}{m}R_1,\frac{M^2}{m^2}R_1]}\frac{f(x)}{x}\to0 \quad (\text{as }R_1\to\infty).$$ Thus, by applying Theorem \ref{thm3.1} with $R=\frac{M}{m}R_1$ and $r=c$, there exists a positive $T$-solution $x$ of Eq.\eqref{e1.2} satisfying $$cr_2>0 and$$ M^2\max_{x\in[c,\frac{M}{m}c]}\frac{f(x)}{x} \leq m^2\min_{x\in[\frac{m^2}{M^2}r_2,\frac{m}{M}r_2]} \frac{f(x)}{x}\to\infty \quad(\text{as } r_2\to0). $$Thus, by applying Theorem \ref{thm3.1} with R=c and r=\frac{m}{M}r_2, Equation \eqref{e1.2} has a positive T-periodic solution x satisfying$$ \frac{m}{M}r_2c>0$such that $$M^2\max_{x\in[c,\frac{M}{m}c]}\frac{f(x)}{x} \leq m^2\min_{x\in[\frac{M}{m}R_2,\frac{M^2}{m^2}R_2]} \frac{f(x)}{x}\to\infty \quad (\text{as } R_2\to\infty ).$$ Thus, by applying Theorem \ref{thm3.2} with$R=\frac{M}{m}R_2$and$r=\frac{M}{m}c$, there exists a positive$T$-solution$x$of \eqref{e1.2} satisfying $$\frac{M}{m}cM^2f_\infty, Equation \eqref{e1.2} has a positive T-periodic solution for$$ \frac{M}{m^2Bf_0}<\lambda<\frac{1}{MBf_{\infty}}. $$\item[(2)] If m^2f_\infty>M^2f_0, Equation \eqref{e1.2} has a positive T-periodic solution for$$ \frac{M}{m^2Bf_\infty}<\lambda<\frac{1}{MBf_0}. $$\end{itemize} \end{corollary} \begin{proof} (1) Since m^2f_0>M^2f_\infty, inequality \eqref{e3.1} is satisfied by taking r small enough and R large enough. According to Theorem \ref{thm3.1}, Equation \eqref{e1.2} has a positive T-periodic solution for$$ \frac{M}{m^2B(f_0+\epsilon)}<\lambda<\frac{1}{MB(f_{\infty}-\epsilon)}, $$where \epsilon>0 is sufficiently small. (2) Since m^2f_\infty>M^2f_0, inequality \eqref{e3.3} is satisfied by taking r small enough and R large enough. As a consequence of Theorem \ref{thm3.2}, Equation \eqref{e1.2} has a positive T-periodic solution for$$ \frac{M}{m^2B(f_\infty+\epsilon)}<\lambda <\frac{1}{MB(f_{0}-\epsilon)}, $$where \epsilon>0 is sufficiently small. \end{proof} \begin{remark} \label{rmk4.1} \rm Corollary \ref{coro4.1} improves the results in Ma \cite[Theorem 4.1]{m2}. Since assertion (b) in \cite[Theorem 4.1]{m2} fails to the case \lim_{x\to0^+}f(x)=+\infty, which is due to the definition of M(r)=\max\{f(x): 0\leq x\leq r\}. However, Corollary \ref{coro4.1} is valid to the case \lim_{x\to0^+}f(x)=+\infty and provides more desirable intervals of \lambda. If a(n, x) of \eqref{e1.2} is replaced with a(n)g(x(n))x(n) of \eqref{e1.4}, then Corollary \ref{coro4.2} is exactly the same as \cite[Theorem 4.3]{m2}. \end{remark} The following results are direct consequences of Theorems \ref{thm3.3} and \ref{thm3.4}. \begin{corollary} \label{coro4.3} Assume that {\rm (H1), (H2)} hold and c\in(0, \infty) is a fixed constant, then \begin{itemize} \item[(i)] If i_0=1 or 2, then \eqref{e1.3} has i_0 positive T-periodic solutions for$$ \lambda>\frac{\bar{M}}{\bar{m}^2B \min_{x\in[\frac{\bar{m}}{\bar{M}}c,c]}f(x)/x}. $$\item[(ii)] If i_\infty=1 or 2, then \eqref{e1.3} has i_\infty positive T-periodic solutions for$$ 0<\lambda<\frac{1}{\bar{M}B\max_{x\in[c,\frac{\bar{M}}{\bar{m}}c]} f(x)/x}. $$\end{itemize} \end{corollary} \begin{corollary} \label{coro4.4} Assume that {\rm (H1), (H2)} hold and i_0=i_\infty=0, then \begin{itemize} \item[(1)] If \bar{m}^2f_0>\bar{M}^2f_\infty, Equation \eqref{e1.3} has a positive T-periodic solution for$$ \frac{\bar{M}}{\bar{m}^2Bf_0}<\lambda<\frac{1}{\bar{M}Bf_{\infty}}. $$\item[(2)] If \bar{m}^2f_\infty>\bar{M}^2f_0, Equation \eqref{e1.3} has a positive T-periodic solution for$$ \frac{\bar{M}}{\bar{m}^2Bf_\infty}<\lambda<\frac{1}{\bar{M}Bf_0}.$$\end{itemize} \end{corollary} \begin{remark} \label{rmk4.2}\rm Corollary \ref{coro4.3} improves the results in \cite[Theorem 4.4]{m2}. Since assertion (b) in \cite[Theorem 4.4]{m2} fails to the case$\lim_{x\to0^+}f(x)=+\infty$, which is due to the definition of$M(r)=\max\{f(x): 0\leq x\leq r\}$. However, Corollary \ref{coro4.3} is valid to the case$\lim_{x\to0^+}f(x)=+\infty$and provides more desirable intervals of$\lambda$. If$a(n, x)$of \eqref{e1.3} is replaced with$a(n)g(x(n))x(n)\$ of \eqref{e1.5}, then Corollary \ref{coro4.4} is exactly the same as \cite[Theorem 4.6]{m2}. \end{remark} \begin{thebibliography}{00} \bibitem{c1} S. Cheng, G. Zhang; \emph{Existence of positive periodic solutions for non-autonomous functional differential equations}, Electron. J. Differential Equations 59 (2001) 1-8. \bibitem{c2} S. N. Chow; \emph{Existence of periodic solutions of autonomous functional differential equations}, J. Differential Equations 15 (1974) 350-378. \bibitem{d1} K. Deimling; \emph{Nonlinear Functional Analysis}, Springer, Berlin, 1985. \bibitem{f1} H. I. Freedman, J. 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