\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{amssymb,graphicx}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 78, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2014/78\hfil Maximum and minimum principles]
{Maximum and minimum principles for nonlinear transport equations
on discrete-space domains}

\author[J. Volek \hfil EJDE-2014/78\hfilneg]
{Jon\'a\v{s} Volek}  % in alphabetical order

\address{Jon\'a\v{s} Volek \newline
Department of Mathematics and NTIS \\
New Technologies for the Information Society - European Centre of Excellence\\
Faculty of Applied Sciences, University of West Bohemia in Pilsen
Univerzitn\'i 8, 306 14 Pilsen, Czech Republic}
\email{volek1@kma.zcu.cz}

\thanks{Submitted  December 30, 2013. Published March 19, 2014.}
\subjclass[2000]{82C70, 34A33, 35B50}
\keywords{Nonlinear transport equation; discrete-space domain; 
\hfill\break\indent maximum principles; existence;
uniqueness; sign preservation; uniform stability;
\hfill\break\indent nonlinear comparison theorem}

\begin{abstract}
 We consider nonlinear scalar transport equations on the domain with discrete space
 and continuous time. As a motivation we derive a conservation law on these domains.
 In the main part of the paper we prove maximum and minimum principles that are
 later applied to obtain an a priori bound which is applied in the proof of
 existence of solution and its uniqueness. Further, we study several consequences
 of these principles such as boundedness of solutions, sign preservation, uniform
 stability and comparison theorem which deals with lower and upper solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction} \label{sec:Introduction}

The transport equation is one of the simplest nonlinear partial differential 
equations. Its importance follows from the fact that it describes traveling 
waves and that it forms the basis for study of hyperbolic equations of second order. 
The reader can see, e.g., \cite{Lo} for details about transport PDE.

We study transport equations on the domain with discrete space and continuous time. 
This is a combination of difference and differential equations. 
As an application of these models we can mention semidiscrete numerical methods 
of Rothe or Galerkin (see \cite{LVe,Ro}). We consider nonlinear equations that 
arise from conservation laws. Linear equations that combine continuous, 
discrete and time-scale variables are studied in \cite{StVo}. 
In that paper authors present some interesting relations between equations 
of this type and stochastic processes of Poisson--Bernoulli type.

In recent years so called dynamical systems on lattices have been studied extensively.
In \cite{ChPa,Chow,MP} authors deal with these related problems and focus 
on PDEs of reaction--diffusion type on finite space lattices. 
Their results can be helpful, e.g., in the modelling of binary alloys 
(see \cite{Chow}).

Moreover, in the last few years the analysis of equations on infinite lattices 
has attracted some researchers. We can refer to \cite{Che,CheGu_1,CheGu_2,Zi} 
for the introduction to these problems. These papers are concerned mainly with 
existence of traveling waves in discrete reaction--diffusion equations and 
their properties. The reader is invited to see \cite{Zi} where the main ideas 
and principles of this field are presented. Problems in 
\cite{Che,CheGu_1,CheGu_2,Zi} are solved often by topological methods using 
fixed point theorems, degree theory, comparison principles and lower and upper 
solutions.

Our analysis can contribute to this mathematical area. Our problem can be 
understood as an equation on infinite lattice. With the help of maximum and 
minimum principles we derive new comparison theorem that deals with ordering 
of lower and upper solutions.

In general, we study simpler problems than reaction-diffusion equations but 
on the other hand, our work can be interesting for another reason as well. 
It can be useful just from the point of view of maximum and minimum principles. 
These principles are strong tools in the theory of differential equations. 
They have many applications and important consequences. We can mention, e.g., 
a priori bounds that can be applied in proofs of existence and uniqueness 
of solution, oscillation results. For the review about these topics in ODEs 
and PDEs see \cite{PrWe} or more recent book \cite{PuSe}. 
In discrete problems these principles have rich behavior. The reader is invited 
to see papers \cite{MaThTo,St,StTh_1,StTh_2} or survey book about partial 
difference equations \cite{Ch} for further details. Consequently, we want 
to explore if the transport equation where we combine continuous and discrete 
approach has some fruitful properties as in these works.

The structure of our paper is as follows. First, we motivate our study, derive 
a conservation law in discrete space and formulate our main problem in 
Section \ref{sec:Conservation_law}. In Section \ref{sec:Max_min_principles} 
we prove maximum and minimum principles for the nonlinear equation by
 the so-called \textit{stairs method}. Then we deal with existence and 
uniqueness of solution in Section \ref{sec:Existence_and_uniqueness} and with 
other consequences in Section \ref{sec:Consequences}. 
In Section \ref{sec:Similar_problem}, we study a related nonlinear problem. 
At the end of the paper, in Section \ref{sec:Conclusion}, we present some open 
problems and directions of future research.

We denote the intervals $ [0, +\infty) $ and $ (0, +\infty) $ by 
$ \mathbb{R}_{0}^{+} $ and $ \mathbb{R}^{+} $ respectively. 
Partial derivative of $ u(x,t) $ w.r.t. $t $ is denoted by $ u_{t}(x,t) $
 and partial difference w.r.t. $ x $ by
\[
 \nabla_{x} u(x,t) = u(x,t) - u(x-1,t).
\]

\section{Conservation law and nonlinear transport equation} 
\label{sec:Conservation_law}

As a motivation we derive the conservation law in discrete space. 
It leads to partial equations on discrete-space domain. Corresponding continuous 
conservation laws are presented, e.g., in \cite{Lo}.

We consider one dimensional discrete space. We simulate it by integers. 
Further, we suppose the density $ u = u(x,t) $ which changes continuously 
in time and which is distributed in discrete space. The magnitude $ u $ 
can express, e.g., the concentration of mass or population, energy etc.

We denote by $ \varphi $ the flux of $ u $. The flux $ \varphi(i,t) $, 
$ i \in \mathbb{Z} $, $ t \in \mathbb{R}_{0}^{+} $, quantifies the amount 
of $ u $ that passes between positions $ x = i $ and $ x = i+1 $ in time $ t $. 
Further, $ f = f(x,t) $ is the source function.

Therefore, consider an arbitrary space segment between $ x = i $ and $ x = j $ 
when $ i < j $. The time change of total amount in that space segment between
 $ x=i $ and $ x = j $ is given by
\begin{equation} \label{eq:cons:global}
  \frac{\mathrm{d}}{\mathrm{d}t} \sum_{x=i}^{j} u(x,t)
= \varphi(i-1,t) - \varphi(j,t) + \sum_{x=i}^{j} f(x,t).
\end{equation}
We call \eqref{eq:cons:global} the conservation law in global form.
 Let us modify \eqref{eq:cons:global} as follows
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}t} \sum_{x=i}^{j} u(x,t) 
& =  - \big[ \varphi(j,t)  - \varphi(j-1,t) + \varphi(j-1,t)
- \dots - \varphi(i,t) + \varphi(i,t) \\
&\quad  - \varphi(i-1,t) \big] + \sum_{x=i}^{j} f(x,t),
\end{align*}
 and finally, we obtain
\[
\sum_{x=i}^{j} \big[  u_{t}(x,t) + \nabla_{x} \varphi(x,t) - f(x,t) \big] = 0.
\]
 The space segment is arbitrary and thus, the following conservation law 
in local form has to hold necessarily
\begin{equation} \label{eq:cons:local}
  u_{t}(x,t) + \nabla_{x} \varphi(x,t) =  f(x,t).
\end{equation}

We study the case of
\[
 \varphi(x,t) = F(x,t,u(x,t)) \quad \text{when} \quad
F: \mathbb{Z} \times \mathbb{R}_{0}^{+} \times \mathbb{R} \to \mathbb{R}.
\]
This leads to the nonlinear transport equation with discrete space.
Therefore, we deal with the following initial-boundary value problem (I-BVP):
\begin{equation} \label{eq:pr_f:problem_DF(u)}
 \begin{gathered}
 u_{t}(x,t) + \nabla_{x} F(x,t,u(x,t)) = f(x,t), \quad
 x \in \mathbb{Z}, \; x > a \in \mathbb{Z}, \; t \in \mathbb{R}^{+},\\
 u(x,0) = \phi(x), \quad \phi: \mathbb{Z} \to \mathbb{R},\\
 u(a,t) = \xi(t), \quad \xi \in C(\mathbb{R}_{0}^{+})
\cap C^{1}(\mathbb{R}^{+}),
 \end{gathered}
\end{equation}
where $ F: \mathbb{Z} \times \mathbb{R}_{0}^{+} \times \mathbb{R} \to \mathbb{R} $
 and $ f: \mathbb{Z} \times \mathbb{R}_{0}^{+} \to \mathbb{R} $.
We prove maximum and minimum principles for lower and upper solutions.

\begin{definition} \label{df:pr_f:lower_upper_DF(u)} \rm
 The function $ v(x,t) $ is called a
\emph{lower solution} of \eqref{eq:pr_f:problem_DF(u)} if
\begin{gather*}
 v_{t}(x,t) + \nabla_{x} F(x,t,v(x,t)) \leq f(x,t), \quad x \in \mathbb{Z}, \;
 x > a \in \mathbb{Z},    \; t \in \mathbb{R}^{+},\\
 v(x,0) \leq \phi(x), \quad x \in \mathbb{Z}, \; x > a \in \mathbb{Z},\\
 v(a,t) \leq \xi(t), \quad t \in \mathbb{R}_{0}^{+}.
\end{gather*}
The function $ w(x,t) $ is an \emph{upper solution} of \eqref{eq:pr_f:problem_DF(u)} 
if
 \begin{gather*}
 w_{t}(x,t) + \nabla_{x} F(x,t,w(x,t)) \geq f(x,t), \quad x \in \mathbb{Z}, \;
 x > a \in \mathbb{Z},    \; t \in \mathbb{R}^{+},\\
 w(x,0) \geq \phi(x), \quad x \in \mathbb{Z}, \; x > a \in \mathbb{Z},\\
 w(a,t) \geq \xi(t), \quad t \in \mathbb{R}_{0}^{+}.
 \end{gather*}
\end{definition}

\section{Maximum and minimum principles} \label{sec:Max_min_principles}

In this section we derive main tools of our study, the maximum and minimum 
principles. Let us mention that if we consider problem \eqref{eq:pr_f:problem_DF(u)} 
with more general difference
\[
 \nabla_{x}^{(\mu)} u(x,t) = \frac{u(x,t) - u(x-\mu,t)}{\mu}
\]
with arbitrary step $ \mu > 0 $ we can prove following results in the similar way. 
Hence, for the sake of simplicity we suppose only difference with unitary step 
$ \nabla_{x} u(x,t)$.
Next technical lemma helps us in the proof of maximum principle.
\medskip

\begin{lemma} \label{lemma:max:increasing}
 Let $ F: \mathbb{Z} \times \mathbb{R}_{0}^{+} \times \mathbb{R} \to \mathbb{R} $
satisfy
 \begin{itemize}
 \item[(A1)] $ F(\chi , \tau , \omega) $ is increasing in 
$ \chi $, i.e., for all $ \chi_{1} < \chi_{2}  $ there is
 $$ F(\chi_{1},\tau, \omega) \leq F(\chi_{2}, \tau , \omega),$$

 \item[(A2)] $ F(\chi , \tau , \omega) $ is strictly increasing in $ \omega $, 
i.e., for all $   \omega_{1} < \omega_{2} $ there is 
$$ F(\chi, \tau , \omega_{1}) < F(\chi, \tau, \omega_{2}).$$
\end{itemize}
Then the following holds:
\begin{gather}
 \text{if $F(\chi_{1},\tau, \omega_{1}) \leq F(\chi_{2}, \tau, \omega_{2})$
then $\chi_{1} \leq \chi_{2}$ or $\omega_{1} \leq \omega_{2}$}, \label{eq:max:lemma_strict_NO}
 \\
 \text{if $F(\chi_{1},\tau, \omega_{1}) < F(\chi_{2}, \tau, \omega_{2})$
then $\chi_{1} < \chi_{2}$ or $\omega_{1} < \omega_{2}$}. \label{eq:max:lemma_strict}
\end{gather}
\end{lemma}

\begin{proof}
 We show only \eqref{eq:max:lemma_strict_NO}. The proof of 
\eqref{eq:max:lemma_strict} is similar. Let us suppose by  contradiction 
that $ \chi_{1} > \chi_{2} $ and $ \omega_{1} > \omega_{2} $. Then we have
 \[
 F(\chi_{2}, \tau , \omega_{2}) \stackrel{\text{\upshape{(A1)}}}{\leq} 
F(\chi_{1}, \tau, \omega_{2})   \stackrel{\text{\upshape{(A2)}}}{<} F(\chi_{1},
 \tau , \omega_{1}),
 \]
a contradiction with the assumption of 
$ F(\chi_{1}, \tau, \omega_{1}) \leq F(\chi_{2}, \tau,   \omega_{2}) $.
\end{proof}

\begin{theorem}[Maximum principle] \label{thrm:max:max_DF(u)}
 Assume that $ F(\chi,\tau,\omega) $ satisfies {\rm (A1)} and \\
{\rm (A2)} and 
$ f(\chi,\tau) \leq 0 $ for all $ \chi \in \mathbb{Z}, \chi > a $, 
$ \tau \in \mathbb{R}^{+} $. Let $ u(x,t) $ be a lower solution 
of \eqref{eq:pr_f:problem_DF(u)}. Then
 \[
 u(x,t) \quad \leq  \sup_{ 
  \begin{array}{c}
 x \in \mathbb{Z},x \geq a\\
 t \in \mathbb{R}_{0}^{+} 
 \end{array}
 } \lbrace \phi(x), \xi(t) \rbrace
 \]
holds for all $ x \in \mathbb{Z} $, $ x \geq a $, and for all 
$ t \in \mathbb{R}_{0}^{+} $.
\end{theorem}

\begin{proof}
 We prove the statement by the so-called \textit{stairs method}. 
The idea of our proof is shown on \mbox{Figure  \ref{fig:stairs}}. 
First, we denote
\[ %\label{eq:NTE_DS_max_11}
  M :=  \sup_{
  \begin{array}{c}
 x \in \mathbb{Z},x \geq a\\
 t \in \mathbb{R}_{0}^{+} \\
 \end{array}
 } \lbrace \phi(x), \xi(t) \rbrace.
 \]
 Assume by contradiction that there exist $ x_{0} \in \mathbb{Z} $, $ x_{0} > a $, 
and $ t_{0} \in  \mathbb{R}^{+} $ such that
 \begin{equation} \label{eq:max:contr_ass}
  u(x_{0},t_{0}) > M.
 \end{equation}
Now from assumptions (A2), \eqref{eq:max:contr_ass} and from the fact that 
$ u(x,t) $ is a lower solution we  obtain
 \begin{gather}
 u_{t}(x_{0},t_{0}) \leq F(x_{0}-1,t_{0},u(x_{0}-1,t_{0})) 
- F(x_{0},t_{0},u(x_{0},t_{0})),
 \label{eq:max:eq_with_fullwrited_difference}\\
 u_{t}(x_{0},t_{0}) < F(x_{0}-1,t_{0},u(x_{0}-1,t_{0})) 
- F(x_{0},t_{0},M). \label{eq:max:eq_with_M}
 \end{gather}
 Now there are two possibilities.

(1) If $ F(x_{0}-1,t_{0},u(x_{0}-1,t_{0})) > F(x_{0},t_{0},M) $ 
then from \eqref{eq:max:lemma_strict} in   Lemma \ref{lemma:max:increasing} 
we get $ u(x_{0}-1,t_{0}) > M $. Hence, in this case we define
\[
 x_{1} = x_{0}-1 \quad \text{and} \quad t_{1} = t_{0}.
 \]

(2) The second possibility is that 
$ F(x_{0}-1,t_{0},u(x_{0}-1,t_{0})) \leq F(x_{0},t_{0},M) $ holds. 
From  \eqref{eq:max:eq_with_M} there is $ u_{t}(x_{0},t_{0}) < 0 $. 
Therefore, the function $ u(x_{0},t) $ is   strictly decreasing in 
$ t = t_{0} $ and we can define
\[
 \overline{t_{0}} = \inf \lbrace \tau = [0,t_{0}]: 
 u(x_{0},t) \text{ is strictly decreasing on  the interval } 
(\tau, t_{0}) \rbrace.
 \]
 If $ \overline{t_{0}} = 0 $ then we have a contradiction with the definition 
of $ M $ via the initial  condition $ \phi(x) $. If $ \overline{t_{0}} > 0 $ 
then there is necessarily $ u_{t}(x_{0}, \overline{t_{0}}) =  0 $ 
and from \eqref{eq:max:eq_with_fullwrited_difference} we obtain
 \[
 F(x_{0},\overline{t_{0}},u(x_{0},\overline{t_{0}})) 
\leq F(x_{0}-1,\overline{t_{0}}, u(x_{0}-1,   \overline{t_{0}})).
 \]
Then \eqref{eq:max:lemma_strict_NO} in Lemma \ref{lemma:max:increasing} 
implies $ u(x_{0},  \overline{t_{0}}) \leq u(x_{0}-1,\overline{t_{0}}) $ 
which gives
 \[
 M < u(x_{0},t_{0}) < u(x_{0},\overline{t_{0}}) \leq u(x_{0}-1,\overline{t_{0}}).
 \]
Consequently, in this case we define
 \[
 x_{1} = x_{0}-1 \quad \text{and} \quad t_{1} = \overline{t_{0}}.
 \]


 Finally, we have $ u(x_{1},t_{1}) > M $. If we continue iteratively 
then after at most $ x_{0} - a $  steps we get a contradiction with definition 
of $ M $. 
\end{proof}

 \begin{figure}[ht]
 \begin{center}
\includegraphics[width=0.5\textwidth]{fig1}  % stairs_method.png
\end{center}
\caption{The idea of the stairs method.
The dotted line shows the situation when only possibility 
(1) occurs which yields a contradiction via the boundary condition $\xi(t)$. 
The bold line shows the combination of possibilities (1) and (2) and a 
contradiction via the boundary condition $\xi(t)$ again. 
The dashed line shows the  situation when we get a contradiction via 
the initial condition $\phi(x)$  in  possibility (2).} \label{fig:stairs}
 \end{figure}

Next we have the minimum principle which can be proved  by a stairs method
similarly to the one in Theorem \ref{thrm:max:max_DF(u)}.

\begin{theorem}[Minimum principle] \label{thrm:max:min_DF(u)}
 Assume that $ F(\chi,\tau,\omega) $ satisfies {\rm (A2)} and
\begin{itemize}
 \item[(A3)] $ F(\chi,\tau,\omega) $ is decreasing in $ \chi $,
 \end{itemize}
 and $ f(\chi,\tau) \geq 0 $ for all $ \chi \in \mathbb{Z}, \chi > a $, 
$ \tau \in \mathbb{R}^{+} $. Let $ u(x,t) $ be an upper solution 
of \eqref{eq:pr_f:problem_DF(u)}. Then
 \[
 \inf_{ \begin{array}{c}
 x \in \mathbb{Z},\, x \geq a\\
 t \in \mathbb{R}_{0}^{+} 
 \end{array}
 } \lbrace \phi(x), \xi(t) \rbrace \quad \leq \quad u(x,t)
 \]
holds for all $ x \in \mathbb{Z} $, $ x \geq a $, and for all 
$ t \in \mathbb{R}_{0}^{+} $.
\end{theorem}

\section{Existence and uniqueness of solution} \label{sec:Existence_and_uniqueness}

In this section we use maximum and minimum principles as a priori bounds 
to prove the existence and uniqueness of solution of \eqref{eq:pr_f:problem_DF(u)}. 
The proof is based on induction and further, we use the following lemma about
 global solution of IVP for ordinary differential equation.

\begin{lemma}[{\cite[Corollary 8.64]{KePe}} ] \label{lemma:exist:extend}
 Consider the following IVP for ordinary differential equation
\begin{equation} \label{eq:exist:ODE}
 \begin{gathered}
  u'(t) = g(t,u(t)), \quad g: I \times \mathbb{R}^{n} \to \mathbb{R}^{n} , \\
  u(t_{0}) = u_{0}, \quad u_{0} \in \mathbb{R}^{n},
 \end{gathered}
 \end{equation}
 when $ I \subset \mathbb{R} $ is an interval. Assume that 
$ h: \mathbb{R}_{0}^{+} \to \mathbb{R} ^{+} $ is continuous and there is a 
$ v_{0} \in \mathbb{R}_{0}^{+} $ such that
\[
 \int_{v_{0}}^{+\infty} \frac{\mathrm{d}s}{h(s)} = +\infty .
\]
Let the function $ g: [t_{0},+\infty ) \times \mathbb{R}^{n} \to \mathbb{R}^{n}  $ 
be continuous  and let
\[
 \| g(\tau,\omega) \|  \leq h ( \| \omega \|)
 \]
hold for all $ (\tau,\omega) \in [t_{0},+\infty ) \times \mathbb{R}^{n} $. 
Then for all $ u_{0} \in  \mathbb{R}^{n} $ with $ \| u_{0} \| \,\, \leq v_{0} $ 
all solutions of \eqref{eq:exist:ODE} exist on $  [t_{0},+\infty ) $.
\end{lemma}

\begin{theorem}[Existence and uniqueness] \label{thrm:exist:exist_DF(u)}
 Suppose that:
\begin{itemize}
 \item[(A4)] $ \phi(x) $, $ \xi(t) $ are bounded;
i.e., there exist $ K > 0 $ such that for all $ x \in \mathbb{Z}, x \geq a $, 
and for all $ t \in \mathbb{R}_{0}^{+} $ $ \vert \phi(x) \vert \leq K $ and 
$ \vert \xi(t) \vert \leq K $ hold,

 \item[(A5)] $ f(\chi, \tau) = 0 $ identically,
 \end{itemize}
the function $ F = F(\tau,\omega) $ is independent of $ \chi $, 
satisfies {\rm (A2)} and
\begin{itemize}
 \item[(A6)] $ F(\tau,\omega) $ is continuous w.r.t. $ \tau $ on 
$ \mathbb{R}_{0}^{+} $,
 \item[(A7)] $ F(\tau,\omega) $ is locally Lipschitz continuous w.r.t. 
$ \omega $ on $ \mathbb{R}_{0}^{+}   \times \mathbb{R} $,
 i.e., for all $ \tau_{0} \in \mathbb{R}_{0}^{+} $ and for all 
$ \omega_{0} \in \mathbb{R} $ there exists a   rectangle
  \[
  \mathcal{R}(\tau_{0}, \omega_{0}) = \left\lbrace (\tau, \omega) \in \mathbb{R}_{0}^{+} \times \mathbb{R}:    \quad 0 \leq \tau - \tau_{0} \leq a , \vert \omega - \omega_{0} \vert \leq b \right\rbrace
  \]
  \noindent and $ L = L(\tau_{0}, \omega_{0}) > 0 $ such that for all 
$ (\tau, \omega_{1}), (\tau, \omega_{2})   \in \mathcal{R}(\tau_{0}, \omega_{0}) $ 
there is
  \[
  \vert F(\tau, \omega_{1}) - F(\tau, \omega_{2}) \vert 
\leq L \vert \omega_{1} - \omega_{2} \vert,
  \]

 \item[\upshape{(A8)}] $ F(\tau,\omega) $ is sublinear w.r.t. $ \omega $,
 i.e., there exist $ A, B > 0 $ such that for all $ \tau \in \mathbb{R}_{0}^{+} $ 
and for all $ \omega \in \mathbb{R} $ there is
 \[
  \vert F(\tau, \omega) \vert \leq A \vert \omega \vert + B .
 \]
 \end{itemize}
Then \eqref{eq:pr_f:problem_DF(u)} possesses a unique solution $ u(x,t) $ 
which is defined for all $ x \in  \mathbb{Z} $, $ x \geq a $, and 
$ t \in \mathbb{R}_{0}^{+} $.
\end{theorem}

\begin{proof}
 We prove the statement by induction on $ x \in \mathbb{Z}$, $x \geq a $.

(1) For $ x = a $ we put $ u(a,t) = \xi(t) $.

(2) Let us have a solution $ u(\overline{x},t) $ which is unique and defined 
for all $ \overline{x} \in \mathbb{Z} $, $ a \leq \overline{x} < x $, 
on $ \mathbb{R}_{0}^{+} $. Then for fixed $ x $ we get from 
\eqref{eq:pr_f:problem_DF(u)} the following IVP for ordinary differential equation
\begin{equation} \label{eq:exist:ODE_proof}
  \begin{gathered}
  u_{t}(x,t) = F(x-1,t,u(x-1,t)) - F(x,t,u(x,t)),\\
  u(x,0) = \phi(x), \quad \phi(x) \in \mathbb{R},
  \end{gathered}
 \end{equation}
where $ F(x-1,t,u(x-1,t)) $ is a given function of $ t $ from the induction
hypothesis.

$\bullet$   Assumptions (A6), (A7) and Picard-Lindel\"of's theorem 
(see \cite[Theorem 8.13]{KePe}) imply
  the existence and uniqueness of a local solution $ u(x,t) $ 
of \eqref{eq:exist:ODE_proof} on some    small interval $ [0, \delta] $, 
$ \delta > 0 $.

$\bullet$ We can make the  estimate
  \begin{align*}
 &\vert F(x-1,t,u(x-1,t)) - F(x,t,u(x,t)) \vert \\
 &\leq | F(x-1,t,u(x-1,t)) | + | F(x,t,u(x,t)) | \\
&\stackrel{\text{\upshape{(A8)}}}{\leq} A | u(x-1,t) | + A | u(x,t) | + 2B \\
&\stackrel{\text{Th. \ref{thrm:max:max_DF(u)}+Th. \ref{thrm:max:min_DF(u)}+(A4)}}
{\leq} A | u(x,t) | + AK + 2B. 
  \end{align*}
If we define $ g(t,u) = F(x-1,t,u(x-1,t)) - F(x,t,u) $, $ h(s) = As + AK + 2B $  
   and $ v_{0} = \vert \phi(x) \vert $ then assumptions of Lemma 
\ref{lemma:exist:extend} are satisfied.    
Therefore, the local solution $ u(x,t) $ can be extended to the whole
 $ \mathbb{R}_{0}^{+} $.

$\bullet$ Finally, we have to check if there is no other solution from some 
time $ t_{0} > 0 $ which disjoins from $ u(x,t) $ in $ t_{0} $. 
Hence, suppose by contradiction that there is a $ t_{0} > 0 $ such that there 
exist two solutions $ u_{1}(x,t) $ and $ u_{2}(x,t) $ of \eqref{eq:exist:ODE_proof}
 with $ u_{1}(x,t) = u_{2}(x,t) $ on $ [0, t_{0}] $ and 
$ u_{1}(x,t) \neq u_{2}(x,t) $ on $ (t_{0}, t_{0} + \epsilon) $, 
$ \epsilon > 0  $. Let us denote $ u_{t_{0}} = u_{1}(x,t_{0}) $ and 
investigate the solvability of the IVP
 \begin{equation} \label{eq:exist:ODE_proof_2}
  \begin{gathered}
  u_{t}(x,t) = F(x-1,t,u(x-1,t)) - F(x,t,u(x,t)), \quad t > t_{0},\\
  u(x,t_{0}) = u_{t_{0}}.
  \end{gathered}
 \end{equation}
The right-hand side of equation in \eqref{eq:exist:ODE_proof_2} is unique 
by induction hypotheses. Functions $ u_{1}(x,t) $, $ u_{2}(x,t) $ 
solve \eqref{eq:exist:ODE_proof_2} on $ [t_{0}, t_{0} + \epsilon) $. 
But assumptions of Picard--Lindelöf's theorem are also satisfied for
 \eqref{eq:exist:ODE_proof_2} thanks to (A6), (A7) and consequently, 
there cannot be two distinct solutions. This is a contradiction which 
finishes the proof. 
\end{proof}

\begin{remark} \label{rem:exist:only_exist}\rm
 If we omit the assumption (A7) of local Lipschitz continuity of 
$ F(\tau, \omega) $ in Theorem    \ref{thrm:exist:exist_DF(u)} then 
the uniqueness is not guaranteed and we get only the existence result by the same  
procedure with the help of Cauchy--Peano's theorem (see \cite[Theorem 8.27]{KePe}) 
instead of Picard--Lindel\"of's theorem.
\end{remark}

We present the following example for an illustration what functions 
$ F(\chi, \tau, \omega) $ can be considered in Theorem \ref{thrm:exist:exist_DF(u)}.

\begin{example} \label{eq:exist:ex} \rm
 Assumptions of Theorem \ref{thrm:exist:exist_DF(u)} are satisfied, e.g., 
for following functions $ F(\tau, \omega) $:
\begin{itemize}
 \item $ F(\tau, \omega) = k(\tau) \omega $ when $k(\tau) > 0 $ (linear equation),
 \item $ F(\tau, \omega) = k(\tau) \arctan{\omega}$ when $k(\tau) > 0 $.
 \end{itemize}
 For the following function $ F $ we have only existence guaranteed 
(cf. Remark  \ref{rem:exist:only_exist}):
\begin{itemize}
 \item
 $F(\tau, \omega) =  \begin{cases}
  - \sqrt[3]{-\omega}, & \text{for } \omega < 0,\\
  \sqrt[3]{\omega}, & \text{for } \omega \geq 0.
 \end{cases}$
\end{itemize}
\end{example}


\section{Consequences of maximum and minimum principles} \label{sec:Consequences}

In this section we study well-known consequences of maximum and minimum principles. 
Corresponding results for classical differential equations can be found in 
\cite{PrWe}.
The next two corollaries follow immediately from Theorems \ref{thrm:max:max_DF(u)} 
and \ref{thrm:max:min_DF(u)}.

\begin{corollary}[Boundedness of solutions]
\label{cor:cons:boundedness}
 Let $ F = F(\tau, \omega) $ satisfy Assumption {\rm (A2)}, 
$f(\chi, \tau) = 0 $ identically, $\phi(x) $ and $ \xi(t) $ be  bounded and 
$ u(x,t) $ be a solution of \eqref{eq:pr_f:problem_DF(u)}. 
Then $ u(x,t) $ is  bounded.
\end{corollary}

\begin{corollary}[Sign preservation] \label{cor:cons:sign}
 Let $ F = F(\chi, \tau, \omega) $ satisfy {\rm (A2)} and {\rm (A3)}, \\
$ f(\chi,\tau) $ be nonnegative, $ \phi(x) $ and $ \xi(t) $ be nonnegative and 
$ u(x,t) $ be a  solution of \eqref{eq:pr_f:problem_DF(u)}. Then $ u(x,t) $ 
is nonnegative.
\end{corollary}

Last application of maximum and minimum principles from Theorems 
\ref{thrm:max:max_DF(u)} and \ref{thrm:max:min_DF(u)} is the uniform 
stability of solutions of the linear problem and its consequences. 
Thus, let us consider the linear problem
\begin{equation} \label{eq:cons:linear_1}
 \begin{gathered}
 u_{t}(x,t) + \nabla_{x} \left[ k(t) u(x,t) \right]  = 0, \quad
 x \in \mathbb{Z}, \quad x > a \in \mathbb{Z},   \; t \in \mathbb{R}^{+},\\
 u(x,0) = \phi(x), \quad \phi: \mathbb{Z} \to \mathbb{R},\\
 u(a,t) = \xi(t), \quad \xi \in C(\mathbb{R}_{0}^{+}) \cap C^{1}(\mathbb{R}^{+}),
 \end{gathered}
\end{equation}
 where $ k(t) > 0 $.

\begin{corollary}[Uniform stability] \label{cor:cons:stability}
 Let $ u_{1}(x,t) $ be a solution of \eqref{eq:cons:linear_1} with 
initial-boundary conditions $ \phi_{1}(x) $ and $  \xi_{1}(x) $. 
Let $ u_{2}(x,t) $ be a solution of \eqref{eq:cons:linear_1} 
with initial-boundary conditions $ \phi_{2}(x) $  and $ \xi_{2}(x) $. 
Then
 \begin{equation} \label{eq:NTE_DS_uniform_stab_3}
 \sup_{\begin{array}{c}
 x \in \mathbb{Z},x \geq a\\
 t \in \mathbb{R}_{0}^{+}  \\
 \end{array}
 }  | u_{1}(x,t) - u_{2}(x,t) |  \leq 
  \sup_{
  \begin{array}{c}
 x \in \mathbb{Z},x \geq a\\
 t \in \mathbb{R}_{0}^{+} \\
 \end{array}
 } \lbrace | \phi_{1}(x) - \phi_{2}(x) |, | \xi_{1}(t) -  \xi_{2}(t) | \rbrace
 \end{equation}
holds.
\end{corollary}

\begin{proof}
 Define function $ v(x,t) = u_{1}(x,t) - u_{2}(x,t) $. 
Then $ v(x,t) $ solves I-BVP \eqref{eq:cons:linear_1} with the initial-boundary  
conditions $ \phi_{1}(x) - \phi_{2}(x) $ and $ \xi_{1}(t) - \xi_{2}(t) $.
 Assumptions of the maximum principle in Theorem  \ref{thrm:max:max_DF(u)} 
are satisfied and hence, we obtain
 \begin{equation} \label{eq:cons:stab_1}
 \begin{aligned}
 u_{1}(x,t) - u_{2}(x,t) = v(x,t) 
& \leq  \sup_{ \begin{array}{c}
  x \in \mathbb{Z},x \geq a\\
  t \in \mathbb{R}_{0}^{+} \\
 \end{array}
 }  \lbrace \phi_{1}(x) - \phi_{2}(x) , \xi_{1}(t) - \xi_{2}(t) \rbrace \\
  & \leq  \sup_{ \begin{array}{c}
  x \in \mathbb{Z},x \geq a\\
  t \in \mathbb{R}_{0}^{+} \\
 \end{array}
 }  \lbrace \vert \phi_{1}(x) - \phi_{2}(x) \vert , 
\vert \xi_{1}(t) - \xi_{2}(t)  \vert \rbrace. 
 \end{aligned}
 \end{equation}

Similarly, assumptions of the minimum principle in Theorem 
\ref{thrm:max:min_DF(u)} are satisfied and therefore,  there is
\begin{equation} \label{eq:cons:stab_2}
 \begin{aligned}
 u_{1}(x,t) - u_{2}(x,t) = v(x,t) 
& \geq  \inf_{ \begin{array}{c}
  x \in \mathbb{Z},x \geq a\\
  t \in \mathbb{R}_{0}^{+} \\
 \end{array}
 } \lbrace \phi_{1}(x) - \phi_{2}(x) , \xi_{1}(t) - \xi_{2}(t) \rbrace \\
 & \geq  -  \sup_{ \begin{array}{c}
  x \in \mathbb{Z},x \geq a\\
  t \in \mathbb{R}_{0}^{+} \\
 \end{array}
 } \lbrace \vert \phi_{1}(x) - \phi_{2}(x) \vert , \vert \xi_{1}(t) 
- \xi_{2}(t)  \vert\rbrace.
 \end{aligned}
 \end{equation}
Finally, inequalities in \eqref{eq:cons:stab_1} and \eqref{eq:cons:stab_2} 
yield    \eqref{eq:NTE_DS_uniform_stab_3}. 
\end{proof}

Corollary \ref{cor:cons:stability} directly implies the following claim.

\begin{corollary} \label{cor:cons:unif_converg}
 Let $\lbrace u_{n}\rbrace_{n=1}^{+\infty}  $ be a sequence of solutions 
$ u_{n}(x,t) $ of   \eqref{eq:cons:linear_1} with the initial-boundary
 conditions $ \phi_{n}(x) $ and $ \xi_{n}(t) $ such that
\[
 \phi_{n}(x) \rightrightarrows \phi(x) \text{ for } x \in \mathbb{Z}, x \geq a,
\quad \text{and} \quad   \xi_{n}(t) \rightrightarrows \xi(t) \text{ for }
t \in \mathbb{R}_{0}^{+}.
 \]
Assume that $ u(x,t) $ is a solution of \eqref{eq:cons:linear_1} with the 
initial-boundary conditions $  \phi(x) $ and $ \xi(t) $. Then
\[
 u_{n}(x,t) \rightrightarrows u(x,t) \quad \text{for}
\quad x \in \mathbb{Z}, x \geq a, \quad \text{and}  \quad
t \in \mathbb{R}_{0}^{+}.
 \]
\end{corollary}

\section{Similar problem with space difference inside nonlinearity} 
\label{sec:Similar_problem}

In this section we analyze a similar problem as \eqref{eq:pr_f:problem_DF(u)}. 
We consider the following I-BVP where the nonlinear function $ F $ depends on 
difference of $ u(x,t) $:
\begin{equation} \label{eq:pr_f:problem_F(Du)}
 \begin{gathered}
 u_{t}(x,t) + F(x,t, \nabla_{x} u(x,t)) = f(x,t), \quad
 x \in \mathbb{Z}, \; x > a \in \mathbb{Z},     \; t \in \mathbb{R}^{+},\\
 u(x,0) = \phi(x), \quad \phi: \mathbb{Z} \to \mathbb{R},\\
 u(a,t) = \xi(t), \quad \xi \in C(\mathbb{R}_{0}^{+}) \cap C^{1}(\mathbb{R}^{+}).
 \end{gathered}
\end{equation}

\begin{remark} \label{rem:pr_f:lower_upper_F(Du)} \rm
 We define lower and upper solutions of \eqref{eq:pr_f:problem_F(Du)} 
similarly as in Definition   \ref{df:pr_f:lower_upper_DF(u)}.
\end{remark}

The following two theorems are the maximum and minimum principles 
for \eqref{eq:pr_f:problem_F(Du)}. We let proofs to the reader because 
we can prove them by stairs method again.

\begin{theorem}[Maximum principle] \label{thrm:max:max_F(Du)}
 Assume that $ F(\chi,\tau,\omega) $ satisfies
 \begin{itemize}
 \item[(A9)] for all $ \chi \in \mathbb{Z}, \chi > a $, and for all 
$ \tau \in \mathbb{R}^{+} $, there is
\[
  F(\chi,\tau,\omega) 
  \begin{cases}
  >0, & \text{for } \omega > 0,\\
  <0, & \text{for } \omega < 0,\\
  =0, & \text{for } \omega = 0,
  \end{cases}
 \]
 \end{itemize}
and $ f(\chi,\tau) \leq 0 $ for all $ \chi \in \mathbb{Z}, \chi > a $,
 $ \tau \in \mathbb{R}^{+} $. Let $ u(x,t) $ be a lower solution of 
\eqref{eq:pr_f:problem_F(Du)}. Then
 \[
 u(x,t) \quad \leq   \sup_{  \begin{array}{c}
 x \in \mathbb{Z},x \geq a\\
 t \in \mathbb{R}_{0}^{+} \\
 \end{array}
 }  \lbrace \phi(x), \xi(t) \rbrace
 \]
 holds for all $ \chi \in \mathbb{Z} $, $ \chi \geq a $, and for all 
$ \tau \in \mathbb{R}^{+} $.
\end{theorem}

\begin{theorem}[Minimum principle] \label{thrm:max:min_F(Du)}
 Assume that $ F(\chi,\tau,\omega) $ satisfies {\rm (A9)} and\\
 $ f(\chi,\tau) \geq 0 $ for all \mbox{$ x \in \mathbb{Z}$}, $ x > a $, 
and for all $ t \in \mathbb{R}_{0}^{+} $. Let $ u(x,t) $ be an upper 
solution of \eqref{eq:pr_f:problem_F(Du)}. Then
\[
 \inf_{
  \begin{array}{c}
 x \in \mathbb{Z},x \geq a\\
 t \in \mathbb{R}_{0}^{+} \\
 \end{array}
 }\quad  \lbrace \phi(x), \xi(t) \rbrace \quad \leq \quad u(x,t)
 \]
holds for all $ x \in \mathbb{Z} $, $ x \geq a $, and for all 
$ t \in \mathbb{R}_{0}^{+} $.
\end{theorem}

Now, we introduce analogue results for \eqref{eq:pr_f:problem_F(Du)} 
as in Sections \ref{sec:Existence_and_uniqueness} and \ref{sec:Consequences}. 
We omit proofs again because they are also similar as for 
\eqref{eq:pr_f:problem_DF(u)}.

\begin{theorem}[Existence and uniqueness] \label{thrm:exist:exist_F(Du)}
 Suppose that {\rm (A4)}, {\rm (A5)} hold, function $ F(\chi,\tau,\omega) $ 
satisfies {\rm (A6)}--{\rm (A9)}. Then \eqref{eq:pr_f:problem_F(Du)} possesses 
a unique solution $ u(x,t) $ which is defined for all $ x \in \mathbb{Z} $, 
$ x \geq a $, and $ t \in \mathbb{R}_{0}^{+} $.
\end{theorem}

\begin{corollary}[Boundedness of solutions] \label{cor:cons:boundedness_F(Du)}
 Let $ F(\chi, \tau, \omega) $ satisfy Assumption {\rm (A9)}, 
$ f(\chi, \tau) = 0 $ 
identically, $ \phi(x) $ and $ \xi(t) $  be bounded and $ u(x,t) $ be a 
solution of \eqref{eq:pr_f:problem_F(Du)}. Then $ u(x,t) $ is bounded.
\end{corollary}

\begin{corollary}[Sign preservation] \label{cor:cons:sign_F(Du)}
 Let $ F(\chi, \tau, \omega) $ satisfy {\rm (A9)}, $ f(\chi,\tau) $ be nonnegative, 
$ \phi(x) $ and $ \xi(t) $ be nonnegative and $ u(x,t) $ be a solution of   
\eqref{eq:pr_f:problem_F(Du)}.  Then $ u(x,t) $ is nonnegative.
\end{corollary}

Finally, in contrast to previous sections about the problem 
\eqref{eq:pr_f:problem_DF(u)}, we are able to prove following assertions 
about nonlinear problem \eqref{eq:pr_f:problem_F(Du)}.

\begin{corollary}[Uniform stability] \label{cor:cons:stability_F(Du)}
 Consider a function $ F(\chi, \tau, \omega) $ for which the partial 
derivative $ F_{\omega}(\chi,\tau,\omega) $ is a continuous and positive function. 
Let $ u_{1}(x,t) $ be a solution of \eqref{eq:pr_f:problem_F(Du)} with 
initial-boundary conditions $ \phi_{1}(x) $ and $  \xi_{1}(t) $. 
Let $ u_{2}(x,t) $ be a solution of \eqref{eq:pr_f:problem_F(Du)} with 
initial-boundary conditions $ \phi_{2}(x) $  and $ \xi_{2}(t) $. Then
\[
 \sup_{   \begin{array}{c}
 x \in \mathbb{Z},x \geq a\\
 t \in \mathbb{R}_{0}^{+}  \\
 \end{array}
 } | u_{1}(x,t) - u_{2}(x,t) |  \leq 
  \sup_{
  \begin{array}{c}
 x \in \mathbb{Z},x \geq a\\
 t \in \mathbb{R}_{0}^{+} \\
 \end{array}
 } \lbrace | \phi_{1}(x) - \phi_{2}(x) |, | \xi_{1}(t) -  \xi_{2}(t) | \rbrace
 \]
 holds.
\end{corollary}

\begin{proof}
 We prove the statement with the help of maximum and minimum principles from 
Theorems \ref{thrm:max:max_F(Du)} and \ref{thrm:max:min_F(Du)}. 
Thanks to the assumption that $ u_{1}(x,t) $ and $ u_{2}(x,t) $ are solutions 
we get the equality
 \[
 (u_{1})_{t}(x,t) + F(x,t,\nabla_{x} u_{1}(x,t)) 
- (u_{2})_{t}(x,t) - F(x,t,\nabla_{x} u_{2}(x,t)) = 0.
 \]
Applying the mean value theorem we can rewrite it to the form
\[
 (u_{1})_{t}(x,t) - (u_{2})_{t}(x,t) + F_{\omega}(x,t,\theta(x,t)) 
\nabla_{x}(u_{1}(x,t) - u_{2}(x,t)) = 0,
 \]
where $ \theta(x,t) = \alpha \nabla_{x} u_{1}(x,t) + (1-\alpha)\nabla_{x} u_{2}(x,t) 
$, $ \alpha \in [0,1] $.
Let us define an auxiliary function $ v(x,t) = u_{1}(x,t) - u_{2}(x,t) $. 
Consequently, $ v(x,t) $ solves
 \begin{gather*}
 v_{t}(x,t) + F_{\omega}(x,t,\theta(x,t)) \nabla_{x} v(x,t) = 0,\\
 v(x,0) = \phi_{1}(x) - \phi_{2}(x),\\
 v(a,t) = \xi_{1}(t) - \xi_{2}(t),\\
 \end{gather*}
when the assumptions of Theorems \ref{thrm:max:max_F(Du)} and 
\ref{thrm:max:min_F(Du)} are satisfied. Thus,  from Theorem 
\ref{thrm:max:max_F(Du)} we obtain
 \begin{align*}
 u_{1}(x,t) - u_{2}(x,t) = v(x,t) 
& \leq  \sup_{
   \begin{array}{c}
  x \in \mathbb{Z},x \geq a\\
  t \in \mathbb{R}_{0}^{+} \\
 \end{array}
 } \lbrace \phi_{1}(x) - \phi_{2}(x) , \xi_{1}(t) - \xi_{2}(t) \rbrace \\
 & \leq  \sup_{ 
   \begin{array}{c}
  x \in \mathbb{Z},x \geq a\\
  t \in \mathbb{R}_{0}^{+} \\
 \end{array}
 } \lbrace \vert \phi_{1}(x) - \phi_{2}(x) \vert , \vert \xi_{1}(t) 
- \xi_{2}(t)  \vert \rbrace.
 \end{align*}
 Similarly, from Theorem \ref{thrm:max:min_F(Du)}, there is
\[
\begin{aligned}
 u_{1}(x,t) - u_{2}(x,t) = v(x,t) 
& \geq  \inf_{ 
   \begin{array}{c}
  x \in \mathbb{Z},x \geq a\\
  t \in \mathbb{R}_{0}^{+} \\
 \end{array}
 }  \lbrace \phi_{1}(x) - \phi_{2}(x) , \xi_{1}(t) - \xi_{2}(t)  \rbrace \\
  & \geq  - \sup_{ 
   \begin{array}{c}
  x \in \mathbb{Z},x \geq a\\
  t \in \mathbb{R}_{0}^{+} \\
 \end{array}
 } \lbrace \vert \phi_{1}(x) - \phi_{2}(x) \vert , \vert \xi_{1}(t)
 - \xi_{2}(t)  \vert \rbrace 
 \end{aligned}
\]
 which completes the proof. 
\end{proof}

\begin{corollary} \label{cor:cons:unif_converg_F(Du)}
 Consider a function $ F(\chi, \tau, \omega) $ for which the partial derivative
 $ F_{\omega}(\chi,\tau,\omega) $ is a continuous and positive function. 
Let $\lbrace u_{n} \rbrace_{n=1}^{+\infty}  $ be a sequence of solutions 
$ u_{n}(x,t) $ of   \eqref{eq:pr_f:problem_F(Du)} with the initial-boundary 
conditions $ \phi_{n}(x) $ and $ \xi_{n}(t) $ for that
 \[
 \phi_{n}(x) \rightrightarrows \phi(x) \text{ for } x \in \mathbb{Z}, 
x \geq a, \quad \text{and} \quad   \xi_{n}(t) \rightrightarrows \xi(t) 
\text{ for } t \in \mathbb{R}_{0}^{+}.
 \]
Assume that $ u(x,t) $ is a solution of \eqref{eq:pr_f:problem_F(Du)} 
with the initial-boundary conditions $\phi(x) $ and $ \xi(t) $. Then
 \[
 u_{n}(x,t) \rightrightarrows u(x,t) \quad \text{for } 
x \in \mathbb{Z},\;  x \geq a, \; t \in \mathbb{R}_{0}^{+}.
 \]
\end{corollary}

\begin{corollary}[Comparison theorem] \label{cor:cons:comparison_F(Du)}
 Consider a function $ F(\chi, \tau, \omega) $ for which the partial 
derivative $ F_{\omega}(\chi,\tau,\omega) $ is  continuous and positive 
function. Suppose, there exists a solution $ u(x,t) $ of 
\eqref{eq:pr_f:problem_F(Du)}. Moreover,  let $ v(x,t) $ be a lower solution 
and $ w(x,t) $ be an upper solution of \eqref{eq:pr_f:problem_F(Du)}. Then
\[
 v(x,t) \leq u(x,t) \leq w(x,t)
\]
is necessarily satisfied for all $ x \in \mathbb{Z} $, $ x \geq a $, 
and for all $ t \in \mathbb{R}_{0}  ^{+} $.
\end{corollary}

\begin{proof}
 We define two auxiliary functions $ \overline{v}(x,t) = u(x,t) - v(x,t) $ and 
$ \overline{w}(x,t) = w(x,t) - u(x,t) $ and investigate their sign.

(1) First, we study the function $ \overline{v}(x,t) $. Because $ v(x,t) $ 
is a lower solution we get
\[
 0 \leq u_{t}(x,t) + F(x,t, \nabla_{x} u(x,t)) - v_{t}(x,t) 
- F(x,t, \nabla_{x} v(x,t)).
 \]
Thanks to assumptions on $ F $ we can use the mean value theorem and
 we can continue with our   estimate,
 \begin{align*}
  0 & \leq  u_{t}(x,t) + F(x,t, \nabla_{x} u(x,t)) - v_{t}(x,t) 
 - F(x,t, \nabla_{x} v(x,t)) \\
  & =  \left[u(x,t) - v(x,t)\right]_{t} + F_{\omega}(x,t,\theta(x,t)) 
\left[ \nabla_{x} u(x,t) -    \nabla_{x} v(x,t) \right]\\
& =  \overline{v}_{t}(x,t) + F_{\omega}(x,t,\theta(x,t)) 
 \nabla_{x} \overline{v}(x,t).
 \end{align*}
for some $ \theta(x,t) = \alpha \nabla_{x} u(x,t) + (1-\alpha)\nabla_{x} v(x,t) $, 
$ \alpha \in [0,1] $.  For initial and boundary conditions we have
 \begin{gather*}
  \overline{v}(x,0) = u(x,0) - v(x,0) \geq 0,\\
  \overline{v}(a,t) = u(a,t) - v(a,t) \geq 0.\\
 \end{gather*}
Thus, assumptions of Theorem \ref{thrm:max:min_F(Du)} are satisfied 
for $ \overline{v}(x,t) $ which implies
\[
 \overline{v}(x,t) \geq 0, \quad \text{i.e.,} \quad v(x,t) \leq u(x,t).
\]

(2) For the function $ \overline{w}(x,t) $ it is similar.
 By the same procedure we get
 \[
 \overline{w}(x,t) \geq 0, \quad \text{i.e.,} \quad u(x,t) \leq w(x,t).
 \]
\end{proof}

\begin{remark} \label{rem:cons:NO_comparison_DF(u)}
 If we would like to prove the similar assertions for 
\eqref{eq:pr_f:problem_DF(u)} by the same procedure then proofs  would 
fail after using the mean value theorem. In that case, the backward difference 
operator $ \nabla_{x} $ would be applied on the partial derivative 
$ F_{\omega}(x,t,\theta(x,t)) $. Hence, we would not be able to satisfy 
assumptions of Theorems \ref{thrm:max:max_DF(u)} and \ref{thrm:max:min_DF(u)} 
because we would not know the behavior of the function $ \theta(x,t) $.
\end{remark}

\section{Concluding remarks} \label{sec:Conclusion}

In this paper we present some maximum and minimum principles for transport 
equations with discrete space and continuous time and derive several applications.
But there are still many open questions left.

First, we can try to find another maximum principles with distinct or weaker 
assumptions or we can try to derive another properties of solutions 
of \eqref{eq:pr_f:problem_DF(u)} and \eqref{eq:pr_f:problem_F(Du)}.
 Next, we should say that, although, we consider nonlinear function $ F $ 
as a function $ F(\chi,\tau,\omega) $ in our problems, in many cases we have 
to assume that $ F $ is not a function of $ \chi $. Therefore, we can try to 
improve it and find better conditions.

We study only initial--boundary value problems as well. We can ask what 
will change if we consider an initial value problem on the whole $ \mathbb{Z} $. 
One can show that in that case we cannot prove maximum or minimum principles in 
the same way by \textit{stairs method} as Theorem \ref{thrm:max:max_DF(u)}. 
Moreover, we cannot use mathematical induction to prove the existence of solution 
of IVP because we have not where to start.

Further, we could try to generalize our results for more general time and space 
structures as in \cite{St,StTh_1,StTh_2} (in these papers dynamic equations on 
time-scales are studied, for more information about time-scale calculus see 
\cite{BoPe,Hi}).

In this paper we analyze equations with one space variable and hence, we can 
state the question what happens if we consider more space variables as on 
finite-dimensional lattice dynamical systems in \cite{ChPa,Chow,MP}.

Another natural generalization is to study evolutionary equations of higher 
order, e.g., diffusion or wave-type equations on discrete-space domains as 
in \cite{Che,CheGu_1,CheGu_2,Zi}.


\subsection*{Acknowledgements} %\label{sec:Acknowledgements}
This work was supported by the Czech Science Foundation, Grant No. 201121757 
and by the European Regional Development Fund (ERDF), project "NTIS – 
New Technologies for the Information Society", European Centre of Excellence,
CZ.1.05/1.1.00/02.0090. The author thank Petr Stehl\'ik, Anton\'in Slav\'ik 
and Pavel \v{R}eh\'ak for their helpful advises.


\begin{thebibliography}{00}

 \bibitem{BoPe} M. Bohner, A. Peterson; 
\emph{Dynamic Equations on Time Scales: An  
Introduction with Applications}, Boston: Birkhäuser, 2001.

 \bibitem{Che} X. Chen et. al.;
 \emph{Traveling Waves in Discrete Periodic Media for Bistable   Dynamics}, 
Arch. Rational Mech. Anal. \textbf{189} (2008), pp. 189-236.

 \bibitem{CheGu_1} X. Chen, J.-S. Guo; 
\emph{Existence and Asymptotic Stability of Traveling  Waves of Discrete 
Quasilinear Monostable Equations}, J. Differential Eq. \textbf{184} (2002), 
pp. 549-569.

 \bibitem{CheGu_2} X. Chen, J.-S. Guo; 
\emph{Uniqueness and existence of traveling waves for  discrete quasilinear 
monostable dynamics}, Math. Ann. \textbf{326} (2003), pp. 123-146.

 \bibitem{Ch} S. S. Cheng; 
\emph{Partial Difference Equations}, London: Taylor and Francis, 2003.

 \bibitem{ChPa} S.-N. Chow, J. Mallet-Paret, W. Shen; 
\emph{Traveling Waves in   Lattice Dynamical Systems}, 
J. Differential Eq. \textbf{149} (1998), pp. 248-291.

 \bibitem{Chow} S.-N. Chow et al.; 
\emph{Dynamical Systems}, Berlin: Springer, 2003.

 \bibitem{Hi} S. Hilger; 
\emph{Analysis on measure chains -- A unified approach to continuous and  
discrete calculus}, Results Math. \textbf{18}(1-2) (1990), pp. 18-56.

 \bibitem{KePe} W. G. Kelley, A. C. Peterson; 
\emph{The Theory of Differential Equations:  Classical and Qualitative}, 
New York: Springer, 2010.

 \bibitem{LVe} R. J. Le Veque; 
\emph{Finite Volume Methods for Hyperbolic Problems} 
(Cambridge Texts in  Applied Mathematics), Cambridge: Cambridge University 
Press, 2002.

 \bibitem{Lo} J. D. Logan; 
\emph{An Introduction to Nonlinear Partial Differential Equations}, 
New  Yersey: John Wiley, 2008.

 \bibitem{MP} J. Mallet-Paret; 
\emph{The global structure of traveling waves in spatial discrete  dynamical systems},
 J. Dynam. Differential Eq. \textbf{11}(1) (1999), pp. 49-127.

 \bibitem{MaThTo} J. Mawhin, H. B. Thompson, E. Tonkes; 
\emph{Uniqueness for boundary  value problems for second order finite 
difference equations}, J. Difference Eq. Appl. \textbf{10}(8) (2004),  pp. 749-757.

 \bibitem{PrWe} M. H. Protter, H. F. Weinberger; 
\emph{Maximum Principles in Differential  Equations}, 
New York: Springer-Verlag, 1984.

 \bibitem{PuSe} P. Pucci, J. Serrin; 
\emph{The Maximum Principle}, Basel: Birkhäuser, 2007.

 \bibitem{Ro} E. Rothe; 
\emph{Zweidimensionale parabolische randwertaufgaben als grenzfall  
eindimensionaler randwertaufgaben}, Math. Ann. 102 (1930), pp. 650–670.

 \bibitem{St} P. Stehl\'ik; 
\emph{Maximum principles for elliptic dynamic equations}, 
Math. Comp. Mod.  \textbf{51}(9-10) (2010), pp. 1193-1201.

 \bibitem{StTh_1} P. Stehl\'ik, B. Thompson; 
\emph{Applications of maximum principles to dynamic  equations on time scales}, 
J. Difference Eq. Appl. \textbf{16}(4) (2010), pp. 373-388.

 \bibitem{StTh_2} P. Stehl\'ik, B. Thompson; 
\emph{Maximum principles for second order dynamic  equations on time scales}, 
J. Math. Anal. Appl. \textbf{331} (2007), pp. 913-926.

 \bibitem{StVo} P. Stehl\'ik, J. Volek; 
\emph{Transport equation on semidiscrete domains and  Poisson-Bernoulli processes},
 J. Difference Eq. Appl. \textbf{19}(3) (2013), pp. 439-456.

 \bibitem{Zi} B. Zinner; 
\emph{Existence of Traveling Wavefront Solutions for the Discrete Nagumo  Equation}, 
J. Differential Eq. \textbf{96} (1992), pp. 1-27.

\end{thebibliography}

\end{document}