\documentclass[twoside]{article} \usepackage{amssymb, amsmath} \pagestyle{myheadings} \markboth{\hfil Positive periodic solutions \hfil EJDE--2002/55} {EJDE--2002/55\hfil Youssef N. Raffoul \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2002}(2002), No. 55, pp. 1--8. \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} \\ % Positive periodic solutions of nonlinear functional difference equations % \thanks{ {\em Mathematics Subject Classifications:} 39A10, 39A12. \hfil\break\indent {\em Key words:} Cone theory, positive, periodic, functional difference equations. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Submitted December 22, 2001. Published June 13, 2002.} } \date{} % \author{Youssef N. Raffoul} \maketitle \begin{abstract} In this paper, we apply a cone theoretic fixed point theorem to obtain sufficient conditions for the existence of multiple positive periodic solutions to the nonlinear functional difference equations $$x(n+1)=a(n)x(n)\pm \lambda h(n) f(x(n-\tau(n))).$$ \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \numberwithin{equation}{section} \section{Introduction} Let $\mathbb{R}$ denote the real numbers, $\mathbb{Z}$ the integers and $\mathbb{R}^{+}$ the positive real numbers. Given $ab$, \section{Positive periodic solutions} We now state Krasnosel'skii fixed point theorem \cite{k}. \begin{theorem}[Krasnosel'skii] \label{thm2.1} Let $\mathcal{B}$ be a Banach space, and let $\mathcal{P}$ be a cone in $\mathcal{B}$. Suppose $\Omega_{1}$ and $\Omega_2$ are open subsets of $\mathcal{B}$ such that $0 \in \Omega_1 \subset \overline{\Omega}_1 \subset \Omega_2$ and suppose that $$T:\mathcal{P} \cap ( \overline{\Omega}_2 \backslash \Omega_1) \to \mathcal{P}$$ is a completely continuous operator such that \begin{itemize} \item [(i)] $\|Tu \|\leq \|u \|$, $u \in \mathcal{P} \cap \partial \Omega_1$, and $\|Tu \|\geq \|u \|$, $u\in {\mathcal{P}} \cap \partial\Omega_2$; or \item [(ii)] $\|Tu \|\geq \|u \|$, $u\in {\mathcal{P}} \cap \partial \Omega_1$, and $\|Tu \|\leq \|u \|$, $u\in {\mathcal{P}} \cap \partial \Omega_2$. \end{itemize} Then $T$ has a fixed point in ${\mathcal{P}} \cap (\overline{\Omega}_2 \backslash \Omega_1)$. \end{theorem} Let $\mathcal{X}$ be the set of all real $T$-periodic sequences. This set endowed with the maximum norm $\|x\|= \max_{n\in[0,T-1]}|x(n)|$, $\mathcal{X}$ is a Banach space. The next Lemma is essential in obtaining our results. \begin{lemma} \label{lm2.2} $x(n)\in \mathcal{X}$ is a solution of equation \eqref{e1.1} if and only if \vskip .1cm $$\label{e2.1} x(n)=\lambda \sum^{n+T-1}_{u=n}G(n,u)h(u)f(x(u-\tau(u)))$$ where $$\label{e2.2} G(n,u)=\frac{\prod^{n+T-1}_{s=u+1}a(s)} {1-\prod^{n+T-1}_{s=n}a(s)}, \quad u \in [n, n+T-1].$$ \end{lemma} Note that the denominator in $G(n,u)$ is not zero since $0 < a(n)<1$ for $n \in [0,T-1]$. The proof of Lemma 2.1 is easily obtained by noting that \eqref{e1.1} is equivalent to $$\triangle \Big(\prod^{n-1}_{s=-\infty}a^{-1}(s)x(n)\Big) =\lambda h(n)f(x(n-\tau(n))\prod^{n}_{s=-\infty}a^{-1}(s).$$ By summing the above equation from $u=n$ to $u=n+T-1$ we obtain \eqref{e2.1}. Note that since $0< a(n)< 1$ for all $n\in [0,T-1]$, we have $$N \equiv G(n,n)\leq G(n,u)\leq G(n,n+T-1)=G(0,T-1)\equiv M$$ for $n\leq u \leq n+T-1$ and $$1\geq \frac{G(n,u)}{G(n,n+T-1)}\geq \frac{G(n,n)}{G(n,n+T-1)}=\frac{N}{M}>0.$$ For each $x\in X$, define a cone by $$\mathcal{P} = \big\{ y \in \mathcal{X}: y(n)\geq 0, n\in \mathbb{Z} \quad\mbox{and}\quad y(n) \geq \eta \|y \|\big\},$$ where $\eta = N/M$. Clearly, $\eta \in(0,1)$. Define a mapping $T:\mathcal{X}\to \mathcal{X}$ by $$(Tx)(n)=\lambda \sum^{n+T-1}_{u=n}G(n,u)h(u)f(x(u-\tau(u))$$ where $G(n,u)$ is given by \eqref{e2.2}. By the nonnegativity of $\lambda$, $f$, $a$, $h$, and $G$, $Tx(n) \geq 0$ on $[0,T-1]$. It is clear that $(Tx)(n+T)=(Tx)(n)$ and $T$ is completely continuous on bounded subset of $\mathcal{P}$. Also, for any $x \in \mathcal{P}$ we have \begin{eqnarray*} (Tx)(n)&=& \lambda \sum^{n+T-1}_{u=n}G(n,u)h(u)f(x(u-\tau(u)))\\ &\leq &\lambda \sum^{T-1}_{u=0}G(0,T-1)h(u)f(x(u-\tau(u)). \end{eqnarray*} Thus, $$\|Tx \|= \max_{n \in [0, T-1]} |Tx(n)| \leq \lambda \sum_{u=0}^{T-1} G(0,T-1) h(u)f(x(u-\tau(u))).$$ Therefore, \begin{eqnarray*} Tx(n)&=& \lambda \sum_{u=n}^{n+T-1} G(n,u) h(u) f(x(u-\tau(u))) \\ &\geq& \lambda N \sum_{u=0}^{T-1} h(u)f(x(u-\tau(u))) \\ &=& \lambda N \sum_{u=0}^{T-1} \frac{G(0,T-1)}{M}h(u)f(x(u-\tau(u))) \\&\geq& \eta \|Tx \|. \end{eqnarray*} That is, $T\mathcal{P}$ is contained in $\mathcal{P}$. \hfill$\diamondsuit$ \smallskip In this paper we shall make the following assumptions. \begin{itemize} \item[(A1)] the function $f: \mathbb{R^{+}}\to \mathbb{R^{+}}$ is continuous \item [(A2)] $h(n)> 0$ for $n\in \mathbb{Z}$ \item [(L1)] $\lim_{x\to 0}\frac{f(x)}{x}=\infty$ \item [(L2)] $\lim_{x\to \infty}\frac{f(x)}{x}=\infty$ \item [(L3)] $\lim_{x\to 0}\frac{f(x)}{x}=0$ \item [(L4)] $\lim_{x\to \infty}\frac{f(x)}{x}=0$ \item [(L5)] $\lim_{x\to 0}\frac{f(x)}{x}=l$ with $0 0$ be such that $$\frac{1}{\eta B(L-\epsilon)}\leq\lambda \leq\frac{1}{A(l+\epsilon)}.$$ By condition (L5), there exists $H_1>0$ such that $f(y)\leq (l+\epsilon)y$ for $00$. \end{theorem} \paragraph{Proof:} Apply (L1) and choose $H_{1}>0$ such that if $00$, set $$H_{2}=\max\{2H_{1}, \lambda QA\}.$$ Then if $x\in \mathcal{P}$ and $\|x\|=H_{2}$, \begin{eqnarray*} Tx(n)&\leq& \lambda N \sum^{T-1}_{u=0}G(n,u)h(u)\\&\leq& \lambda Q A \leq H_{2}. \end{eqnarray*} Now assume $f$ is unbounded. Apply condition $(L4)$ and set $\epsilon_1>0$ such that if $x>\epsilon_1$, then $$f(y)<\frac{y}{\lambda A}.$$ Set $H_{2}=\max\{2H_1, \epsilon_1\}$ and define $\Omega_{2}=\{x\in \mathcal{P}:\|x\| 1$ for all $n \in [0,T-1]$. In view of \eqref{e2.7} we have that $$\label{e2.8} x(n)=\lambda \sum^{n+T-1}_{u=n}K(n,u)h(u)f(x(u-\tau(u)))$$ where $$\label{e2.9} K(n,u)=\frac{\prod^{n+T-1}_{s=u+1}a(s)} {\prod^{n+T-1}_{s=n}a(s)-1}, \quad \quad u \in [n, n+T-1].$$ Note that the denominator in $G(n,u)$ is not zero since $a(n)>1$ for $n \in [0,T-1]$. Also, it is easily seen that since $a(n)> 1$ for all $n\in [0,T-1]$, we have $$M \equiv K(n,n)\geq K(n,u)\geq K(n,n+T-1)=K(0,T-1)\equiv N$$ for $n\leq u \leq n+T-1$ and $$1\geq \frac{K(n,u)}{K(n,n)}\geq \frac{K(n,n+T-1)}{K(n,n)}=\frac{N}{M}>0.$$ Finally, by defining $$A^{1}=\underset{0\leq n \leq T-1}{\max}\sum^{T-1}_{u=0}K(n,u)h(u)$$ and $$B^{1}=\underset{0\leq n \leq T-1}{\min}\sum^{T-1}_{u=0}K(n,u)h(u)$$ similar theorems and corollaries can be easily stated and proven regarding equation \eqref{e2.7}. We conclude this paper with the following open problems. Assume that (A1) and (A2) hold. In view of this paper, what can be said about equations \eqref{e1.1} and \eqref{e2.7} when: \begin{enumerate} \item The conditions $(L1)$ and $(L2)$ hold? \item The conditions $(L3)$ and $(L4)$ hold? \item $0 1$ in \eqref{e1.1} for all $n\in[0, T-1]$? \end{enumerate} \begin{thebibliography}{00} \frenchspacing \bibitem {dh} A. Datta and J. Henderson, Differences and smoothness of solutions for functional difference equations, {\em Proceedings Difference Equations} {\bf 1} (1995), 133-142. \bibitem {sc} S. Cheng and G. Zhang, Existence of positive periodic solutions for non-autonomous functional differential equations, {\em Electronis Journal of Differential Equations} {\bf 59} (2001), 1-8. \bibitem {hp} J. Henderson and A. Peterson, Properties of delay variation in solutions of delay difference equations, {\em Journal of Differential Equations} {\bf 1} (1995), 29-38. \bibitem {aw2} R.P. Agarwal and P.J.Y. Wong, On the existence of positive solutions of higher order difference equations, {\em Topological Methods in Nonlinear Analysis} {\bf 10} (1997) 2, 339-351. \bibitem {er} P.W. Eloe, Y. Raffoul, D. Reid and K. Yin, Positive solutions of nonlinear Functional Difference Equations, {\em Computers and Mathematics With applications} {\bf 42} (2001) , 639-646. \bibitem {hl} J. Henderson and S. Lauer, Existence of a positive solution for an nth order boundary value problem for nonlinear difference equations, {\em Applied and Abstract Analysis}, in press. \bibitem {hh} J. Henderson and W. N. Hudson, Eigenvalue problems for nonlinear differential equations, {\em Communications on Applied Nonlinear Analysis} {\bf 3} (1996), 51-58. \bibitem {k} M. A. Krasnosel'skii, Positive solutions of operator Equations {\em Noordhoff, Groningen}, (1964). \bibitem {m} F. Merdivenci, Two positive solutions of a boundary value problem for difference equations, {\em Journal of Difference Equations and Application} {\bf 1} (1995), 263-270. \bibitem {r} Y. Raffoul, Periodic solutions for scalar and vector nonlinear difference equations, {\em Pan-American Journal of Mathematics} {\bf 9} (1999), 97-111. \bibitem {y} W. Yin, Eigenvalue problems for functional differential equations, {\em Journal of Nonlinear Differential Equations} {\bf 3} (1997), 74-82. \end{thebibliography} \noindent\textsc{Youssef N. Raffoul }\\ Department of Mathematics, University of Dayton \\ Dayton, OH 45469-2316 USA \\ e-mail:youssef.raffoul@notes.udayton.edu \end{document}