\documentclass[reqno]{amsart} \usepackage{hyperref,amssymb} \usepackage{graphicx} \AtBeginDocument{{\noindent\small Seventh Mississippi State - UAB Conference on Differential Equations and Computational Simulations, {\em Electronic Journal of Differential Equations}, Conf. 17 (2009), pp. 159--170.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \setcounter{page}{159} \title[\hfilneg EJDE-2009/Conf/17\hfil Bounded solutions] {Bounded solutions: Differential vs difference equations} \author[J. Mawhin\hfil EJDE/Conf/17 \hfilneg] {Jean Mawhin} \address{Jean Mawhin \newline D\'epartement de math\'ematique, Universit\'e Catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium} \email{jean.mawhin@uclouvain.be} \thanks{Published April 15, 2009.} \subjclass[2000]{39A11, 39A12} \keywords{Difference equations; bounded solutions; lower-upper solutions; \hfill\break\indent Landesman-Lazer conditions; guiding functions} \begin{abstract} We compare some recent results on bounded solutions (over $\mathbb{Z}$) of nonlinear difference equations and systems to corresponding ones for nonlinear differential equations. Bounded input-bounded output problems, lower and upper solutions, Landesman-Lazer conditions and guiding functions techniques are considered. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \section{Introduction} In this paper, we survey some recent results on bounded solutions (over $\mathbb{Z}$) of nonlinear difference equations or systems, and compare them to the corresponding situations for bounded solutions (over $\mathbb{R}$) of nonlinear differential equations or systems. We first give some maximum and anti-maximum principles for bounded solutions of linear differential equations of the form $u'(t) + \lambda u(t) = f(t)$ and of corresponding linear difference equations of the form $\Delta u_{m} + \lambda u_{m} = f_{m} \quad (m \in \mathbb{Z}).$ Then we compare Landesman-Lazer conditions for bounded solutions of Duffing's differential equations $x'' + cx' + g(x) = p(t),$ with those for bounded solutions of Duffing's difference equations $\Delta^2x_{m-1} + c\Delta x_{m} + g(x_{m}) = p_{m} \quad (m \in \mathbb{Z})$ or $\Delta^2x_{m-1} + c\Delta x_{m-1} + g(x_{m}) = p_{m}\quad (m \in \mathbb{Z}).$ Finally, we compare the method of guiding functions for systems of ordinary differential equations $x' = f(t,x)$ and for systems of difference equations $\Delta x_{m} = f_{m}(x_{m}),$ or corresponding discrete dynamical systems $x_{m+1} = g_{m}(x_{m}).$ \section{Bounded input--bounded output problem for first order linear equations} \subsection{Bounded solutions of linear ordinary differential equations} The \\ {\em bounded input-bounded output (BIBO)} problem for the linear ordinary differential equation $$\label{LODE} u'(t) + \lambda u(t) = f(t)$$ consists in finding conditions upon $\lambda$ under which, for each $f \in L^\infty(\mathbb{R})$, \eqref{LODE} has a unique solution $u \in AC(\mathbb{R}) \cap L^\infty(\mathbb{R})$. We denote the usual norm of $v \in L^\infty(\mathbb{R})$ by $|u|_{\infty}$. Such a solution is simply called a {\em bounded solution} of \eqref{LODE}. The BIBO problem was essentially solved as follows by Perron in 1930 \cite{P}. If $\lambda = 0$, we have no uniqueness for $f \equiv 0$, and no existence for $f(t) \equiv 1$. If $\lambda \neq 0$, the homogeneous problem $$\label{HLODE} u'(t) + \lambda u(t) = 0$$ only has the trivial bounded solution. For $\lambda > 0$, $$\label{ODES1} u(t) = \int_{-\infty}^t e^{-\lambda(t-s)}f(s)\,ds$$ is a bounded solution of \eqref{LODE}, and hence the unique one. For $\lambda < 0$, $$\label{ODES2} u(t) = - \int_{t}^{+\infty} e^{-\lambda(t-s)}f(s)\,ds$$ is a bounded solution of \eqref{LODE}, and hence the unique one. We summarize the results in the following \begin{proposition}\label{BSODE} Equation \eqref{LODE} has a unique solution $u \in AC(\mathbb{R}) \cap L^\infty(\mathbb{R})$ for each $f \in L^\infty(\mathbb{R})$ if and only if $\lambda \in \mathbb{R} \setminus\{0\}$. \end{proposition} \subsection{A maximum principle for bounded solutions of differential equations} The following definition is modelled upon the one given in \cite{CMO} in a different context. \begin{definition}\label{MPODE} \rm Given $\lambda \in \mathbb{R} \setminus \{0\}$, the linear operator $d/dt +\lambda I : AC(\mathbb{R}) \cap L^\infty(\mathbb{R}) \to L^\infty(\mathbb{R})$ satisfies a {\em maximum principle} (MP) if, for each $f \in L^\infty(\mathbb{R})$, \eqref{LODE} has a unique solution $u$ and if $f(t) \geq 0$ $(t \in \mathbb{R})$ implies that $\lambda u(t)\geq 0$ $(t \in \mathbb{R})$. The MP is {\em strong} if, furthermore, $f(t)\geq 0$ $(t \in \mathbb{R})$ and $\int_{\mathbb{R}}f > 0$ imply that $\lambda u(t) > 0$ $(t \in \mathbb{R})$). \end{definition} A direct consideration of formulas (\ref{ODES1}) and (\ref{ODES2}) immediately implies the following \begin{proposition}\label{MPODET} If $f \in L^\infty(\mathbb{R})$, the BIBO problem for \eqref{LODE} has a MP if and only if $\lambda \in \;]-\infty,0[\, \cup \,]0,+\infty[$, and the MP is not strong. \end{proposition} \subsection{Bounded solutions of linear difference equations} Let $l^\infty(\mathbb{Z}) = \{u = (u_{m})_{m \in \mathbb{Z}} : \sup_{m \in \mathbb{Z}}|u_{m}| < \infty\}$. Endowed with the norm $|u|_{\infty} := \sup_{m \in \mathbb{Z}}|u_{m}|$, $l^\infty(\mathbb{Z})$ is a Banach space. We denote by $\Delta u_{m} = u_{m+1}-u_{m}$ $(m \in \mathbb{Z})$ the forward difference operator acting on sequences $(u_{m})_{m \in \mathbb{Z}}$. The bounded input-bounded output (BIBO) problem we address is to find the values of $\lambda$ such that, for each $(f_{m})_{m \in \mathbb{Z}}\in l^\infty(\mathbb{Z})$, the linear difference equation $$\label{LDE} \Delta u_{m} + \lambda u_{m} = f_{m} \quad (m \in \mathbb{Z})$$ has a unique solution $(u_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$. We refer those solutions as {\em bounded solutions}. Easy computations show that, for $\lambda = 0$, existence or uniqueness may fail. Namely, for $f_{m} = 0$ $(m \in \mathbb{Z})$, any constant sequence is a solution in $l^\infty(\mathbb{Z})$, and, for $f_{m} = 1$ $(m \in \mathbb{Z})$, the solutions given by $u_{m} = u_{0} + m$ $(m \in \mathbb{Z})$ are all unbounded. Similarly, for $\lambda = 2$, any alternating sequence $(-1)^m c$ is a solution of \eqref{LDE} with $f_{m} = 0$ $(m \in \mathbb{Z})$, and, for $f_{m} = (-1)^m$ $(m \in \mathbb{Z})$ none of the solutions $u_{m} = (-1)^m u_{0} + m(-1)^{m+1}$ $(m \in \mathbb{Z})$ is bounded. Now, for $\lambda \in \mathbb{R} \setminus\{0,2\}$, it is easy to see that the homogeneous difference equation $$\label{HLDE} \Delta u_{m} + \lambda u_{m} = 0$$ only has the trivial solution in $l^\infty(\mathbb{Z})$. On the other hand, if $\lambda \in \,]0,2[\,$, $$\label{BS1} u_{m} = \sum_{k=-\infty}^{m-1}(1-\lambda)^{m-k-1}f_{k} \quad (m \in \mathbb{Z})$$ is a solution of \eqref{LDE} belonging to $l^\infty(\mathbb{Z})$, and hence the unique one. Similarly, if $\lambda \in \,]-\infty,0[\,\cup \,]2,+\infty[\,$, $$\label{BS2} u_{m} = - \sum_{k=m}^{\infty}(1-\lambda)^{m-k-1}f_{k} \quad (m \in \mathbb{Z})$$ is the unique solution of \eqref{LDE} belonging to $l^\infty(\mathbb{Z})$. We summarize the results in the following proposition. \begin{proposition}\label{BSDE} Equation \eqref{LDE} has a unique solution $(u_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$ for each $(f_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$ if and only if $\lambda \in \mathbb{R} \setminus\{0,2\}$. \end{proposition} \subsection{A maximum principle for bounded solutions of difference equations} The following definition is modelled upon the one given in \cite{CMO} in a different context. \begin{definition}\label{MP} \rm Given $\lambda \in \mathbb{R} \setminus \{0\}$, the linear operator $\Delta +\lambda I : l^\infty(\mathbb{Z}) \to l^\infty(\mathbb{Z})$ satisfies a {\em maximum principle} (MP) if for each $(f_{m})_{m \in \mathbb{Z}}\in l^\infty(\mathbb{Z})$, the equation \eqref{LDE} has a unique solution and if $f_{m}\geq 0$ $(m \in \mathbb{Z})$ implies that $\lambda u_{m}\geq 0$ $(m \in \mathbb{Z})$. The maximum principle is said to be {\em strong} if, in addition, $f_{m}\geq 0$ $(m \in \mathbb{Z})$, and $\sup_{m \in \mathbb{Z}}f_{m} > 0$ imply that $\lambda u_{m} > 0$ $(m \in \mathbb{Z})$). \end{definition} Notice that, in the more classical terminology modelled on the one for second order elliptic operators, the above definition corresponds to a {\em maximum principle} when $\lambda < 0$, and to an {\em anti-maximum principle} in the sense of Cl\'ement-Pelletier \cite{CP} when $\lambda > 0$. The following result can be read directly upon formulas (\ref{BS1}) and (\ref{BS2}). \begin{proposition}\label{MPT} The BIBO problem for \eqref{LDE} has a MP if and only if $\lambda \in \;]-\infty,0[\, \cup \,]0,1]$, and this MP is not strong; \end{proposition} \subsection{BIBO problem: linear differential vs linear difference equations} It follows from Propositions \ref{MPODET} and \ref{MPT} that the ranges of values for which a maximum principle hold are different in the differential and the difference cases. The following simple propositions help to understand the reason of this difference. Given a linear operator $L$ between Banach spaces, let $\sigma(L)$ denotes its (complex) spectrum and $\mathcal{R}(L) = \mathbb{C} \setminus \sigma(L)$ denote its resolvent set. The following propositions are analogous to those proved in \cite{CMO} is a different context. \begin{proposition}\label{MPE} If the BIBO problem for $L + \lambda I$, with $L = \Delta$ or $d/dt$ has a MP for some $\lambda \neq 0$, then $$\label{MPEF} |u|_{\infty} \leq \frac{|f|_{\infty}}{|\lambda|}.$$ \end{proposition} \begin{proof} If $u \in L^\infty(\mathbb{R})$ is the solution of \eqref{LODE} and $v = \frac{|f|_{\infty}}{\lambda} \in L^\infty(\mathbb{R})$ the solution of $Lv + \lambda v = |f|_{\infty},$ then $v-u \in L^\infty(\mathbb{R})$ is the solution of $L(v - u) + \lambda(v-u) = |f|_{\infty} - f$ and the MP implies that $\lambda(v - u) \geq 0$, i.e. that $\lambda u \leq |f|_{\infty}.$ Similarly, we have $L(v+u) + \lambda(v + u) = |f|_{\infty} + f$ and hence, by the MP, $\lambda(v + u) \geq 0$, i.e. $\lambda u \geq - |f|_{\infty}$. \end{proof} In the ordinary differential equation case, the estimate (\ref{MPEF}) can also be obtained directly for any $\lambda \in \mathbb{R} \setminus\{0\}$. Indeed, it follows from (\ref{ODES1}) that if $\lambda > 0$, then $|u(t)| \leq |f|_{\infty}\int_{-\infty}^{t}e^{-\lambda(t-s)}\,ds = \frac{1}{\lambda}|f|_{\infty}.$ Similarly, if $\lambda < 0$, we get $|u(t)| \leq |f|_{\infty}\int_{t}^{+\infty}e^{-\lambda(t-s)}\,ds = -\frac{1}{\lambda} |f|_{\infty}.$ In the DE case, the following estimates can be obtained directly from the formulas (\ref{BS1}) and (\ref{BS2}) \begin{gather*} |u|_{\infty} \leq \frac{|f|_{\infty}}{|\lambda|} \quad \text{if } \lambda < 0, \quad |u|_{\infty} \leq \frac{|f|_{\infty}}{|\lambda|} \quad \text{if } 0 <\lambda \leq 1,\\ |u|_{\infty} \leq \frac{|f|_{\infty}}{2-\lambda} \quad \text{if } 1 < \lambda < 2, \quad |u|_{\infty} \leq \frac{|f|_{\infty}}{\lambda - 2} \quad \text{if } 2 < \lambda. \end{gather*} \begin{proposition}\label{MPR} If the BIBO problem for $L + \lambda I$, with $L = \Delta$ or $d/dt$ has a MP for some $\lambda \neq 0$, then $$\label{resolvent} \mathcal{R}(L) \supset \{\mu \in \mathbb{C} : |\mu - \lambda| < |\lambda|\}.$$ \end{proposition} \begin{proof} We have, for $\mu \in \mathbb{C}$, \begin{align*} Lu + \mu u = f & \Leftrightarrow Lu + \lambda u + (\mu - \lambda)u = f \\ & \Leftrightarrow u + (\mu - \lambda)(L + \lambda)^{-1}u = (L + \lambda)^{-1}f, \end{align*} and, using Proposition \ref{MPE}, $|(\mu - \lambda)(L + \lambda)^{-1}u|_{\infty} \leq |\mu - \lambda|\frac{|u|_{\infty}}{|\lambda|},$ so that, for $\frac{|\mu - \lambda|}{|\lambda|} < 1$, equation $Lu + \mu u = f$ has a unique bounded solution. \end{proof} It is easy to check that, for the BIBO problem in the ordinary differential equation case, the spectrum $\sigma(L)$ of $L : AC(\mathbb{R}) \cap L^\infty(\mathbb{R}) \to L^\infty(\mathbb{R})$ is equal to $i\mathbb{R}$. \begin{figure}[ht] \begin{center} \includegraphics[width=0.45\textwidth]{fig1} % spectreODE1 \end{center} \caption{ODE spectrum} \label{fig1} \end{figure} Therefore, for any $\lambda \in \mathbb{R}$, the set $\{\mu \in \mathbb{C} : |\mu - \lambda| < |\lambda|\}$ is always contained in the resolvent set $\mathcal{R}(L)$. Similarly, for the BIBO problem in the difference equation case, the spectrum $\sigma(L)$ of $L : l^\infty(\mathbb{Z}) \to l^\infty(\mathbb{Z})$ is the circle $\{1 + e^{i\theta} : \theta \in [0,2\pi]\}$. Hence, for any $\lambda < 0$, the set $\{\mu \in \mathbb{C} : |\mu - \lambda | < |\lambda|\}$ is contained in $\mathcal{R}(L)$, but, for $\lambda > 0$, this is only true for $\lambda \in \,]0,1]$. This, together with Proposition \ref {MPR}, sheds some light on the fact that the maximum principle for the BIBO problem in the difference case only holds for $\lambda \in \,]-\infty,0[\, \cup \,]0,1]$. Notice also that the estimate $|u|_{\infty} \leq \frac{|f|_{\infty}}{|\lambda|}$ only holds for those values of $\lambda$. \begin{figure}[ht] \begin{center} \includegraphics[width=0.45\textwidth]{fig2} % spectreDE1 \end{center} \caption{DE spectrum} \label{fig2} \end{figure} \section{Bounded input--bounded output problems for some Duffing's equations} \subsection{Linear equations} It is a standard result that the second order linear ordinary differential equation $$\label{2LODE} x'' + cx' + ax = f(t)$$ has a unique solution $x \in AC^1(\mathbb{R}) \cap L^\infty(\mathbb{R})$ for any $f \in L^\infty(\mathbb{R})$ if and only if $a < 0$. \subsection{Duffing's equations} Duffing's differential equations are nonlinear second order differential equations of the form $$\label{DUODE} x'' + cx' + g(x) = p(t),$$ where $c \in \mathbb{R}$, $g : \mathbb{R} \to \mathbb{R}$ and $p : \mathbb{R} \to \mathbb{R}$ are continuous. Correspondingly, we call {\em Duffing difference equations} the second order nonlinear difference equations of the form $$\label{DUDE1} \Delta^2x_{m-1} + c\Delta x_{m} + g(x_{m}) = p_{m} \quad (m \in \mathbb{Z})$$ or $$\label{DUDE2} \Delta^2x_{m-1} + c\Delta x_{m-1} + g(x_{m}) = p_{m}\quad (m \in \mathbb{Z})$$ where $\Delta^2 x_{m-1} = x_{m+1} - 2x_{m} + x_{m-1} \quad (m \in \mathbb{Z}),$ $g \in C(\mathbb{R},\mathbb{R})$, and $c \in \mathbb{R}.$ The bounded input-bounded output (BIBO) problem for (\ref{DUODE}) consists, for given $g$, in determining the inputs $p \in L^\infty(\mathbb{R})$ for which equation (\ref{DUODE}) has at least one solution $u \in AC^1(\mathbb{R}) \cap L^\infty(\mathbb{R})$. This problem was first considered by Ahmad \cite{A}, and then by Ortega \cite{O}, Ortega-Tineo \cite{OT}, and Mawhin-Ward \cite{MW}. Similarly, the bounded input-bounded output (BIBO) problem for (\ref{DUDE1}) or (\ref{DUDE2}) consists, for given $g$, in determining the inputs $(p_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$ for which (\ref{DUDE1}) or (\ref{DUDE2}) has at least one solution $(x_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$. See \cite{BM,M}. \subsection{Bounded lower and upper solutions} We develop a method of lower and upper solutions for the bounded solutions of (\ref{DUDE1}) and (\ref{DUDE2}). We first need a limiting lemma \cite{M}. \begin{lemma}\label{LLDE} Let $f_{m} \in C(\mathbb{R},\mathbb{R})$ $(m \in \mathbb{Z})$, $c \in \mathbb{R}$ Assume that, for each $n \in \mathbb{N}^*$, there exists $(x^n_{m})_{-n-1\leq m\leq n+1}$ such that $\Delta^2 x^n_{m-1} + c\Delta x^n_{m} + f_{m}(x^n_{m}) = 0 \quad (-n \leq m \leq n)$ and such that $\alpha_{m} \leq x^n_{m} \leq \beta_{m}$ $(|m| \leq n+1)$ for some $(\alpha_{m})_{m \in \mathbb{Z}}\in l^\infty(\mathbb{Z})$, $(\beta_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$. Then there exists $(\widehat x_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$ such that $\Delta^2 \widehat x_{m-1} + c\Delta \widehat x_{m} + f_{m}(\widehat x_{m}) = 0, \; \alpha_{m} \leq \widehat x_{m} \leq \beta_{m} \quad (m \in \mathbb{Z}).$ \end{lemma} The same result for $\Delta^2 \widehat x_{m-1} + c\Delta \widehat x_{m-1} + f_{m}(\widehat x_{m}) = 0 \;(m \in \mathbb{Z}).$ The proof is based upon Borel-Lebesgue lemma and Cantor diagonalization process. \smallskip We now define the concept of bounded lower and upper solutions for second order difference equations \cite{M}. Let $f_{m} \in C(\mathbb{R},\mathbb{R})$ $(m \in \mathbb{Z}), \; c \in \mathbb{R}$. \begin{definition}\label{LUSDE} \rm $(\alpha_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$ (resp. $(\beta_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$) is a {\em bounded lower solution} (resp. {\em upper solution}) for $\Delta^2 x_{m-1} + c\Delta x_{m} + f_{m}(x_{m})=0 \quad (m \in \mathbb{Z})$ if \begin{gather*} \Delta^2 \alpha_{m-1}+c\Delta \alpha_{m} + f_{m}(\alpha_{m}) \geq 0\\ (\text{resp.}\quad \Delta^2 \beta_{m-1}+c\Delta \beta_{m} + f_{m}(\beta_{m})\leq 0) \quad (m \in \mathbb{Z}) \end{gather*} \end{definition} A similar definition holds for $\Delta^2 x_{m-1} + c\Delta x_{m-1} + f_{m}(x_{m})=0 \quad (m \in \mathbb{Z}).$ We have the associated existence theorem. \begin{theorem}\label{LUSTDE} If $c \geq 0$ (resp. $c \leq 0$) and \begin{gather*} \Delta^2 x_{m-1} + c\Delta x_{m} + f_{m}(x_{m}) = 0 \quad (m \in \mathbb{Z})\\ (\text{resp.} \quad \Delta^2 x_{m-1} + c\Delta x_{m-1} + f_{m}(x_{m}) = 0 \quad (m \in \mathbb{Z})) \end{gather*} has a lower solution $(\alpha_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$ and an upper solution $(\beta_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$ such that $\alpha_{m} \leq \beta_{m}$ $(m \in \mathbb{Z}),$ then it has a solution $(x_{m})_{m \in \mathbb{Z}}$ such that $\alpha_{m} \leq x_{m} \leq \beta_{m} (m \in \mathbb{Z})$ \end{theorem} \begin{proof} The proof is based upon the existence theorem for lower and upper solutions for the Dirichlet problem \begin{gather*} \Delta^2 x_{m-1}+c\Delta x_{m} + f_{m}(x_{m}) = 0 \quad (-n \leq m \leq n)\\ x_{-n-1} = \alpha_{-n-1},\quad x_{n+1} = \alpha_{n+1} \end{gather*} for each $n$ and the limiting Lemma \ref{LLDE}. \end{proof} An important special case is that of constant lower and upper solutions. \begin{corollary}\label{CLUS} If $c \geq 0$ and if $\exists \alpha \leq \beta$ such that $f_{m}(\beta) \leq 0 \leq f_{m}(\alpha)$ $(m \in \mathbb{Z})$, then $\Delta^2 x_{m-1} + c\Delta x_{m} + f_{m}(x_{m})=0 \quad (m \in \mathbb{Z})$ has a solution $(x_{m})_{m \in \mathbb{Z}}$ such that $\alpha \leq x_{m} \leq \beta$ $(m \in \mathbb{Z}).$ \end{corollary} \begin{example} \rm If $c \geq 0$ and $a > 0$, then for each $(p_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$ $\Delta^2 x_{m-1} + c\Delta x_{m} -a x_{m}=p_{m} \quad (m \in \mathbb{Z})$ has a unique solution $(u_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$. \end{example} Similar results hold if $c \leq 0$ for the equations \begin{gather*} \Delta^2 x_{m-1} + c\Delta x_{m-1} + f_{m}(x_{m})=0 \quad (m \in \mathbb{Z})\\ \Delta^2 x_{m-1} + c\Delta x_{m-1} -a x_{m}= p_{m} \quad (m \in \mathbb{Z}). \end{gather*} In the ordinary differential equation case, a similar result holds for all $c \in \mathbb{R}$ for the equations \begin{gather*} x'' + cx' + f(t,x) = 0\\ x'' + cx' -ax = p(t) \quad (a > 0, \quad p \in L^\infty(\mathbb{R})) \end{gather*} (see \cite{B,OP}). \subsection{Second order linear equations} The following result can be proved like Proposition \ref{BSDE}. \begin{proposition}\label{BSDE1} If $c \not \in \{-2,0\}$ $\Delta x_{m-1} + cx_{m} = h_{m} \quad (m \in \mathbb{Z})$ has a unique solution $(x_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$ for each $(h_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$. \end{proposition} Before dealing with second order difference equations, we introduce some notions and results for sequences with bounded primitive. The corresponding concepts for functions upon $\mathbb{R}$ were introduced in \cite{O}. \begin{definition}\label{Deltaprim} \rm The $\Delta$-primitive $(H^{\Delta}_{m})_{m \in \mathbb{Z}}$ of $(h_{m})_{m \in \mathbb{Z}}$ is any sequence $(H^{\Delta}_{m})_{m \in \mathbb{Z}}$ such that $\Delta H^{\Delta}_{m} = h_{m}$ $(m \in \mathbb{Z})$. \end{definition} Such a $\Delta$-primitive is for example given by $H^{\Delta}_{m} = \begin{cases} \sum_{k=0}^{m-1}h_{k} &\text{if } m \geq 1\\ 0 & \text{if } m = 0\\ -\sum_{k=m}^{-1}h_{k}& \text{if } m \leq -1 \end{cases} \quad (m \in \mathbb{Z})$ We define the space $BP(\mathbb{Z})$ as the set $\{(h_{m})_{m \in \mathbb{Z}} : (H^{\Delta}_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})\}.$ It is easy to check that $BP(\mathbb{Z}) \subsetneq l^\infty(\mathbb{Z})$. The situation is different in the continuous case, where $BP(\mathbb{R}) \not \subset BC(\mathbb{R})$, and $BC(\mathbb{R}) \not \subset BP(\mathbb{R})$. We have now the following result for the BIBO problem for some linear second order difference equations. \begin{proposition}\label{2NDDE1} If $c \not\in \{-2,0\}$, $\Delta^{2}x_{m-1} + c \Delta x_{m} = h_{m} \quad (m \in \mathbb{Z})$ has a solution $(x_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$ if and only if $h \in BP(\mathbb{Z})$. \end{proposition} \begin{proposition}\label{2NDDE2} If $c \not\in \{0,2\}$, $\Delta^2 x_{m-1} + c \Delta x_{m-1} = h_{m} \quad (m \in \mathbb{Z})$ has a solution $(x_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$ if and only if $h \in BP(\mathbb{Z})$. \end{proposition} The corresponding results for ordinary differential equations were proved by Ortega in \cite{O}. \begin{proposition}\label{2NDODE} If $c \neq 0$, equation $x'' + cx' = h(t)$ has a solution $x \in AC(\mathbb{R}) \cap L^\infty(\mathbb{R})$ if and only if $h \in BP(\mathbb{R})$. \end{proposition} We now introduce concepts of generalized mean values to bounded sequences. \begin{definition}\label{LUMV} \rm The {\em lower (resp. upper) mean value} of $(p_{j})_{j \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$ is the real number defined by \begin{gather*} \widehat p := \lim_{n \to \infty}\inf_{m-k \geq n} \Big(\frac{1}{m-k}\sum_{j=k+1}^m p_{j}\Big)\\ \Big(\text{resp.} \quad \widetilde p := \lim_{n \to \infty}\sup_{m-k \geq n} \Big(\frac{1}{m-k}\sum_{j=k+1}^m p_{j}\Big)\Big) \end{gather*} \end{definition} \begin{lemma} \label{MVL} The following statements are equivalent : \begin{itemize} \item[(i)] $\alpha < \widehat p \leq \widetilde p < \beta.$ \item[(ii)] there exists $(p^*_{m})_{m \in \mathbb{Z}} \in BP(\mathbb{Z})$, $(p^{**}_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$ such that $p_{m} = p^*_{m} + p^{**}_{m}$ $(m \in \mathbb{Z})$ and $\alpha < \inf_{k \in \mathbb{Z}}p^{**}_{k} \leq \sup_{k \in \mathbb{Z}}p^{**}_{k} < \beta$. \end{itemize} \end{lemma} \begin{corollary} If $\widehat p = \widetilde p = 0$, then, for each $\epsilon > 0$ there exists $(p^*_{m})_{m \in \mathbb{Z}} \in BP(\mathbb{Z})$, $(p^{**}_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$ such that $p_{m} = p^*_{m} + p^{**}_{m}$ $(m \in \mathbb{Z})$, $\sup_{k \in \mathbb{Z}}|p_{k}^{**}| < \epsilon$. \end{corollary} In the continuous case those results and concepts are due to Ortega-Tineo \cite{OT}. \subsection{Duffing difference equations} We can now prove the following result for the existence of bounded solutions of Duffing difference equations. \begin{theorem}\label{BSDDE} Assume that the following conditions hold. \begin{enumerate} \item $c > 0$, $g \in C(\mathbb{R},\mathbb{R})$, $(p_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$ \item There exists $r_{0} > 0$ and $\delta_{-} < \delta_{+}$ such that $g(y) \geq \delta_{+} \quad \text{ for } y \leq -r_{0}, \quad g(y) \leq \delta_{-} \quad \text{ for } y \geq r_{0}.$ \item $\delta_{-} < \widehat p \leq \widetilde p < \delta_{+}$. \end{enumerate} Then $\Delta^2 x_{m-1} + c\Delta x_{m} + g(x_{m}) = p_{m} \quad (m \in \mathbb{Z})$ has at least one solution $(x_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$. \end{theorem} \begin{proof} Write $p_{m} = p^*_{m} + p^{**}_{m}$ $(m \in \mathbb{Z})$ with $(p^*_{m})_{m \in \mathbb{Z}} \in BP(\mathbb{Z})$, $(p^{**}_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$ and $\delta_{-} < \inf_{k \in \mathbb{Z}}p^{**}_{k} \leq \sup_{k \in \mathbb{Z}}p^{**}_{k} < \delta_{+}$. By Proposition \ref{2NDDE1}, $\Delta^2 x_{m-1} + c\Delta x_{m} = p^*_{m} \quad (m \in \mathbb{Z})$ has a solution $(u_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$. Letting $x_{m} = u_{m} + z_{m}$ $(m \in \mathbb{Z})$, we obtain the equivalent problem $$\label{EQUIV} \Delta^2 z_{m-1} + c\Delta z_{m} + g(u_{m} + z_{m}) - p^{**}_{m} = 0 \quad (m \in \mathbb{Z}).$$ Then $\alpha = -r_{0} - \sup_{k \in \mathbb{Z}}u_{k}$ is a lower solution and $\beta = r_{0} - \inf_{k \in \mathbb{Z}}u_{k}$ an upper solution for (\ref{EQUIV}), and we conclude using Corollary \ref{CLUS}. \end{proof} \subsection{Landesman-Lazer condition} Theorem \ref{BSDDE} gives existence conditions of the Landesman-Lazer type. \begin{corollary} If $c > 0$, $g \in C(\mathbb{R},\mathbb{R})$, $(p_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$, and $$\label{LL} \overline{\lim}_{y \to +\infty}g(y) < \widehat p \leq \widetilde p < \underline{\lim}_{y \to -\infty}g(y)$$ then $\Delta^2 x_{m-1} + c\Delta x_{m} + g(x_{m}) = p_{m} \quad (m \in \mathbb{Z})$ has a solution $(x_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$. \end{corollary} \begin{remark} \rm If, for all $x \in \mathbb{R}$, $- \infty < \overline{\lim}_{y \to +\infty} g(y) < g(x) < \underline{\lim}_{y \to -\infty}g(y) < + \infty$ then $(p_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$ and (\ref{LL}) is necessary for the existence of a bounded solution. \end{remark} Similar results hold for $\Delta^2 x_{m-1} + c \Delta x_{m-1} + g(x_{m}) = p_{m} \quad (c < 0)\quad (m \in \mathbb{Z})$ In the ordinary differential equation case, similar results hold for $x'' + cx' + g(x) = p(t) \quad (c \neq 0)$ (see \cite{MW}). \begin{example} \rm 1. If $c > 0$, $b > 0$, $\Delta^2 x_{m-1} + c\Delta x_{m} - b\frac{x_{m}}{1 + |x_{m}|} = p_{m} \quad (m \in \mathbb{Z})$ has a solution $(x_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$ if and only if $(p_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$ and $-b < \widehat p \leq \widetilde p < b$. 2. If $c > 0$, $b > 0$, and $0 \leq a < 1$, $\Delta^2 x_{m-1} + c\Delta x_{m} - b\frac{x_{m}}{1 + |x_{m}|^{a}} = p_{m} \quad (m \in \mathbb{Z})$ has a solution $(x_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$ if and only if $(p_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$. \end{example} It remains an open problem to prove or disprove that if $c > 0$ and $b > 0$, $\Delta^2 x_{m-1} + c\Delta x_{m} + \frac{bx_{m}}{1 + |x_{m}|} = p_{m} \quad (m \in \mathbb{Z})$ has a solution $(x_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$ if and only if $(p_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$ and $-b < \widehat p \leq \widetilde p < b$. Similarly it is an open problem to prove or disprove that if $c > 0$, $b > 0$, and $0 \leq a < 1,$ $\Delta^2 x_{m-1} + c\Delta x_{m} + \frac{bx_{m}}{1 + |x_{m}|^{a}} = p_{m} \quad (m \in \mathbb{Z})$ has a solution $(x_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$ if and only if $(p_{m})_{m \in \mathbb{Z}} \in l^\infty(\mathbb{Z})$. The corresponding results are true in the ordinary differential equation case \cite{A,O,OT}. \section{Guiding functions for bounded solutions of systems of difference equations} \subsection{Guiding functions for ordinary differential equations} Consider the system $$\label{ODES} x' = f(t,x)$$ where $f \in C(\mathbb{R} \times \mathbb{R}^n,\mathbb{R}^n)$. \begin{definition}\label{GFODE} \rm A guiding function for (\ref{ODES}) is a function $V \in C^1(\mathbb{R}^n,\mathbb{R})$ such that, for some $\rho_{0} > 0$, $\langle \nabla V(x),f(t,x)\rangle \leq 0$ when $\|x\| \geq \rho_{0}$. \end{definition} The following theorem was first proved by Krasnosel'skii-Perov in 1958 \cite{KP}. A simpler proof has been given by Alonso-Ortega in 1995 \cite{AO}. \begin{theorem}\label{KP} If (\ref{ODES}) has a guiding function $V$ such that $\lim_{\|x\| \to \infty}V(x) = +\infty$, then (\ref{ODES}) has a solution $x$ bounded over $\mathbb{R}$. \end{theorem} A natural question is to know if a corresponding result holds for a difference system $x_{n+1}-x_{n} = f_{n}(x_{n}) \quad (n \in \mathbb{Z})$ or, equivalently for a discrete dynamical system $x_{n+1} = g_{n}(x_{n}) \quad (n \in \mathbb{Z}).$ \subsection{Guiding function for difference equations} Let us consider the system $$\label{DES} x_{m+1} = g_{m}(x_{m}) \quad (m \in \mathbb{Z})$$ where $g_{m} \in C(\mathbb{R}^n,\mathbb{R}^n)$ $(m \in \mathbb{Z})$. \begin{definition}\label{GFDE} \rm A guiding function for \eqref{DES} is a function $V \in C(\mathbb{R}^n,\mathbb{R})$, such that, for some $\rho_{0} > 0$, $V(g_{m}(x)) \leq V(x)$ when $\|x\| \geq \rho_{0}$ ($m \in \mathbb{Z}$). \end{definition} The result corresponding to Theorem \ref{KP} would be : if $x_{m+1} = g_{m}(x_{m})$ $(m \in \mathbb{Z})$ has a guiding function $V$ such that $\lim_{\|x\| \to \infty}V(x) = +\infty$, then it has a bounded solution. The following example, given in \cite{BM}, shows that this result is {\em false}. Consider the maps $g_{m} \in C(\mathbb{R},\mathbb{R})$ defined by $g_{m}(x) = \begin{cases} 1 & \text{if }x \leq - 2, \\ mx + 2m+1 &\text{if } - 2 0 such that, for each k \in \mathbb{N}^* \[ x_{m+1} = g_{m}(x_{m}) \quad (-k \leq m \leq k)$ has a solution $(x^k_{m})_{-k\leq m\leq k+1}$, satisfying $\max_{-k \leq m \leq k+1}\|x^k_{m}\| \leq \rho.$ Then there exists a solution $(\widehat x_{m})_{m \in \mathbb{Z}}$ of \eqref{DES} such that $\sup_{m \in \mathbb{Z}}\|\widehat x_{m}\| \leq \rho$. \end{lemma} \begin{theorem}\label{GFTDE} Let $g_{m}\in C(\mathbb{R}^n,\mathbb{R}^n)$ $(m \in \mathbb{Z})$. If \eqref{DES} has a guiding function $V$ with constant $\rho_{0}$ such that $\lim_{\|x\| \to \infty}V(x) = +\infty$ and such that $$\label{BND} \sup_{m \in \mathbb{Z}}\max_{\|x\| \leq \rho_{0}}\|g_{m}(x)\| < \infty ,$$ then \eqref{DES} has a solution $(x_{m})_{m \in \mathbb{Z}} \in (l^\infty(\mathbb{Z}))^n$. \end{theorem} \begin{proof} Take $\rho_{1} > \max\{\rho_{0}, \sup_{m \in \mathbb{Z}}\max_{\|x\| \leq \rho_{0}}\|g_{m}(x)\|\}$. Define $$V_{1} := \max_{\|x\| \leq \rho_{1}}V(x).$$ Take $\rho_{2} > \rho_{1}$ such that $B_{\rho_{0}} \subset B_{\rho_{1}} \subset S_{1} := \{x \in \mathbb{R}^n : V(x) \leq V_{1}\} \subset B_{\rho_{2}}.$ Then it is easy to show that $S_{1}$ is positively invariant under the flow \eqref{DES}. For $n \in \mathbb{N}$ fixed and $(x^n)_{m \geq -n}$ the solution such that $x^n_{-n} = 0$ is such that $x^n_{m} \in S_{1} \subset B_{\rho_{2}} \quad (m \geq -n, \; n \in \mathbb{N}).$ Finally, use Lemma \ref{LL2} to obtain a solution $(x_{m})_{m \in \mathbb{Z}} \in (l^\infty(\mathbb{Z}))^n$. \end{proof} \begin{remark} \rm Inequality (\ref{BND}) trivially holds if $g_{m} = g$ $(m \in \mathbb{Z})$. \end{remark} \begin{thebibliography}{00} \bibitem{A} S. Ahmad, \emph{A nonstandard resonance problem for ordinary differential equations,} Trans. Amer. Math. Soc. \textbf{323} (1991), 857-875. \bibitem{AO} J.M. Alonso and R. Ortega, \emph{Global asymptotic stability of a forced Newtonian system with dissipation}, J. Math. Anal. Appl. \textbf{196} (1995), 965-986. \bibitem{BM} J.B. Baillon, J. Mawhin, \emph{Bounded solutions of some nonlinear difference equations, } in preparation. \bibitem{B} I. Barbalat, \emph{Applications du principe topologique de T. 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