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%Title of the contribution

% \begin{frontmatter}
\title{Summation comparison
  theorems\\ for half-linear\\ second order
  difference equations\\ on finite
  interval}


\author{{\normalfont R. {Ma\v r\'\i k}}}

% \address{Mendel University\\ Dpt. of Mathematics\\ 
% Zem\v ed\v elsk\' a 1\\Brno 613 00\\Czech Republic}

%\ead{marik@mendelu.cz}

 % \begin{keyword} 

 % \end{keyword}

\thanks{Author is supported by the Grant  P201/10/1032 of the Czech Science Foundation.}

\maketitle


\begin{center}
  \textbf{Received: May 16, 2011}

\bigskip
  \textbf{Abstract}
\end{center}

\begin{quote}
  \parindent=0pt

  \uppercase {Ma\v r\'ik}, R.:\textit
  {Summation comparison theorems for
    half-linear second order difference
    equations on finite interval.}

\bigskip
  In the paper, new comparison theorems
  for the half-linear difference equation
  $$
  \Delta\Bigl(R_k\Phi(\Delta
  z_k)\Bigr)+C_k\Phi(z_{k+1})=0,
  \quad\Phi(u)=|u|^{p-2}u, \quad p>1,
  $$
  are derived. We show that if a solution
  of this equation has a generalized zero
  on the discrete interval $[a,b]$, then
  the same holds for a solution of its
  majorant. % In contrast to typical
  % results, % the majorant is understood in
  % % the integral sense and
  % the solutions
  % under consideration satisfy nonzero
  % initial condition.
  The main tool used in the paper is the
  variational technique which relates
  nonexistence of a solution with a
  generalized zero with nonegativity of
  the $p$-degree functional defined on
  the suitable class of admissible
  functions.

\bigskip
  difference equation, second order,
  focal point, half-linear equation,
  $p$-degree functional, free end point

\bigskip

\leavevmode\hbox {{MSC2000:}
  } 34A10

\end{quote}

%\pagestyle{plain}
\markboth{\textit{R. {Ma\v
      r\'ik}}}{{Summation type
    comparison theorems}}

   % \textbf{Keywords:} 


% \end{frontmatter}

\section{Introduction}
Consider the second order half-linear
difference equation
\begin{equation}
 % L[z_k]\equiv
  \Delta\Bigl(R_k\Phi(\Delta x_k)\Bigr)+C_k\Phi(x_{k+1})=0,
%  \tag{E}
\label{marik:E}
\end{equation}
where $\Delta$ is the forward difference
operator, $\seq C$, $\seq R$ are real
sequences, $R_k\neq 0$ for
$k=0,\ldots,n+1,$ and
$\Phi(u)=|u|^{p-2}u$, $p>1$, is a power
type nonlinearity. The study of equation
\eqref{marik:E} has been initiated in \v
Reh\'ak (2001) and the most important
results are summarized in Chapter 8 of
the monograph Do\v sl\'y, \v Reh\'ak
(2005).

Despite the lack of linearity, a constant
multiple of any solution of
\eqref{marik:E} is also a solution and
equation \eqref{marik:E} has one half of
linearity properties.  It is well known
that there is a close similarity between
equation \eqref{marik:E} and the linear
second order difference equation.  In
particular, many results from oscillation
theory of second order linear difference
equations can be extended to
\eqref{marik:E}. These oscillation and
nonoscillation results are frequently
based on a comparison of two equations on
the infinite interval.  The aim of this
paper is to derive comparison theorems
which compare \eqref{marik:E} with
another half-linear difference 
equation
\begin{equation}
 % l[y_k]\equiv
  \Delta\Bigl(r_k\Phi(\Delta y_k)\Bigr)+c_k\Phi(y_{k+1})=0
  %\tag{e}
\label{marik:e}
\end{equation}
on a finite interval. 

First, let us recall the definition of a
generalized zero, which is (from the
point of view of the Sturmian comparison
theory) a natural replacement for zeros
of solutions to differential
equations. Remark that, unless stated
explicitly otherwise, under the interval
$[m,n]$ we actually mean the discrete set
$\{m, m+1, \dots n\}$. In a similar way
we work also with other intervals. 

\begin{definition}
  The interval $(m,m+1]$ is said to contain a
  {\it generalized zero} of a solution
  $x=\seq x$ of Eq. \eqref{marik:E} if
  $x_{m}\neq 0$ and $R_m x_m x_{m+1}\leq 0$.
\end{definition}

It is well known, see for example \v
Reh\' ak (2001) or Do\v sl\'y, \v Reh\'
ak (2005), that equation \eqref{marik:E}
tends to have more generalized zeros than
\eqref{marik:e}, if the inequalities
$R_i\leq r_i $ and $C_i\geq c_i$ are
satisfied. In contrast to the pointwise
comparison we formulate our results more
generally in terms of sums of the
coefficients $C_i$ and $c_i$. Our aim is
to derive a discrete version of the
following theorem due to Leighton.

\begin{Theorem}[Leighton (1983), Theorem 1.1]  
  \label{thA}
  Let $p(t)$ and $q(t)$ be piecewise
  continuous on $[a,f]$ with $q(x)\geq 0$
  there, and suppose that
\begin{equation*}
  \int_a^xq(t)\dt\leq \int_a^xp(t)\dt,\quad a\leq x\leq f;\quad p(x)\not\equiv q(x) 
\end{equation*}
holds. If equation
\begin{equation*}
  u''+q(t)u=0
\end{equation*}
has a solution $u(t)$ with the property that
$u'(a)=u(f)=0$, $u(x)>0$ on the interval of real numbers $[a,f)$, a
solution $v(t)$ of
\begin{equation*}
  v''+p(t)v=0
\end{equation*}
with $v(a)>0$, $v'(a)\leq 0$ must have a zero
on the interval of real numbers $(a,f)$.
\end{Theorem}

Another aspect which makes our results
different from those published in the
literature is that similarly as in
Theorem \ref{thA} we compare two
solutions which do not vanish at the left
end point of the interval. As far as the
author knows, the results are new even
for the linear diference equation.

The main tool used in the paper is the
variational technique which relates
equation \eqref{marik:E} and the
corresponding discrete scalar $p$-degree
functional
\begin{equation}
  J(\eta)\equiv A|\eta_0|^{p}+\sum_{k=0}^n\Bigl( R_k|\Delta
  \eta_{k}|^p-C_k|\eta_{k+1}|^p \Bigr ), \qquad A\in\R %\tag{J}
  \label{marik:J}
\end{equation}
defined on the class of nontrivial
sequences $\seq \eta_{k=0}^{n+1}$ such
that $\eta_{n+1}=0$. Note that since we
aim to compare the solutions which do not
vanish at the left end point of the
interval, we drop the usual requirement
$\eta_0=0$ from the definition of the
admissible sequences for functional $J$
and also include the term $A|\eta_0|^p$.
The relationship between the half-linear
difference equation and the $p$-degree
functional is established in the
following theorem.

\begin{Theorem}[Ma\v r\'ik (2000),
  Theorem 1] \label{marik:RT} The
  following statements are equivalent:
  \def\theenumi{\roman{enumi}}
  \def\labelenumi{(\theenumi)}
 \begin{enumerate}
 \item The solution $x=\seq x$ of
   Eq. \eqref{marik:E} given by
   $R_0\Phi\Bigl(\frac{ \Delta x_0}{x_0
   }\Bigr)=A$ has no generalized zero on
   $(0,n+1]$.
 \item Functional \eqref{marik:J} is positive
   definite on the class of nontrivial
   sequences $\eta=\seq \eta_{k=0}^{n+1}$
   satisfying $\eta_{n+1}=0$
 \end{enumerate}
\end{Theorem}

The following result allows to compare
two solutions of two different equations
and it is an immediate consequence of
Theorem \ref{marik:RT}. The crucial
aspect of the proof of this theorem lies
in the fact that the functional $J$
vanishes for the sequence which solves
equation \eqref{marik:E} and satisfies
initial condition closely connected with
the value $a$.
\begin{Theorem}[Leighton type comparison
  theorem, {Ma\v r\'ik (2000), Corollary 1}]\label{marik:RTC}
  Let $y=\seq y$ be a solution of
  Eq. \eqref{marik:e}, such that
  $y_{n+1}=0\neq y_0$ and
  $a:=r_0\Phi\bigl({\Delta y_0\over
    y_0}\bigr)$.  Let A be such that
  $$
  V(y):=(A-a)|y_0|^p+\sum_{k=0}^n \Bigl[
  (R_k-r_k)|\Delta
  y_{k}|^p-(C_k-c_k)|y_{k+1}|^p \Bigr]\leq 0.
  $$
  Then the solution $x=\seq x$ of
  Eq. \eqref{marik:E} given by
  $R_0\Phi\bigl({\Delta x_0\over
    x_0}\bigr)=A$ has a generalized zero
  on $(0,n+1]$, i.e., there exists
  $i\in(0,n]$ such that $x_i\neq 0$ and
  $R_i x_i x_{i+1}\leq 0$ holds.
\end{Theorem}


\section{Main results}

This section contains the main results of
the paper. In the following theorem we
prove that if the solution of half-linear
differential equation \eqref{marik:e}
vanishes at the point $n+1$, then the
solution of the equation \eqref{marik:E}
with a sufficiently large coefficient
$C_i$ has a generalized zero on
$(0,n+1]$, if the initial difference is
negative and not too large. However, the
words ``sufficiently large'' are here in
the integral sense as \eqref{marik:ner}
shows. Hence the inequality $C_i\geq c_i$
is not necessary for all $i$.

Note the technical assumption on
nonegativity of $c_i$ which assures that
the solution which starts with positive
initial value and nonpositive initial
difference is nonincreasing.

\begin{theorem}\label{marik:th1}
  Let $r_k>0$ on $[0,n]$, $c_k\geq 0$ on
  $[0,n-1]$, $c_0>0$. Let $y$ be a solution of
  \eqref{marik:e} on $[0,n-1]$ such that
  $y_0\geq y_1>0$, $y_k>0$ on $[0,n]$ and
  $y_{n+1}=0$. Denote $a=r_0\Phi\left(\frac{\Delta y_0}{y_0}\right)$. Let $R_k\leq r_k$ on $[0,n]$,
  $C_0\geq c_0$, $A\leq a$ and
  \begin{equation}
    \left|\frac{y_0}{y_1}\right|^p\Bigl(a - A\Bigr)+
    \sum_{i=0}^{k}\Bigl(C_i-c_i\Bigr)\geq 0 \label{marik:ner}
  \end{equation}
  for $k\in [0,n-1]$. Then the solution $z=\seq z$
  of \eqref{marik:E} given by the conditions
  $z_0>0$, $A=R_0\Phi\left(\frac{\Delta
      z_0}{z_0}\right)$ has a generalized
  zero on $(0,n+1]$, i.e., there exists
  $i\in[0,n]$ such that $z_i\neq 0$ and $R_i
  z_i z_{i+1}\leq 0$.
\end{theorem}
\begin{proof}
  From $R_k\leq r_k$ we get
  \begin{equation*}
    V(y)\leq (A-a)|y_0|^p+\sum_{k=0}^n(c_k-C_k)|y_{k+1}|^p.
  \end{equation*}
  % We use Corollary \ref{marik:RTC} with $A=0=a$. It holds
  % $$
  % V(y_{k})\leq \sum_{k=0}^{n}\Bigl(c_k-C_k\Bigr)|y_{k+1}|^p=
  % \sum_{k=0}^{n-1}\Bigl(c_k-C_k\Bigr)|y_{k+1}|^p.
  % $$
  Further, from \eqref{marik:e} it follows
  $$
  \Phi(\Delta y_{k+1})={r_k \over
    r_{k+1}}\Phi(\Delta y_{k})- {c_k\over
    r_{k+1}}\Phi(y_{k+1})
  $$
  for $k\in[0,n-1]$ and
  $$\Phi(\Delta y_{k+1})<0$$ for $k\in[0,n-1]$.
  Hence $|y_{k+1}|^p$ is decreasing on
  $[0,n]$. Clearly there exists
  $\epsilon\in \R$, $\epsilon>0$, such
  that the intervals of real numbers
  $I_k:=(|y_{k}|^p-\epsilon,|y_{k}|^p+
  \epsilon )\subseteq \R^+$, $k\in[1,n]$,
  satisfy $I_j\cap I_k=\emptyset$ for
  $j\not= k$. In each $I_k$ let us choose
  $\alpha_k$, $\alpha_k\in I_k\cap \Q^+$,
  such that
  \begin{equation}\label{star}
  (c_k-C_k)|y_{k+1}|^p\leq (c_k-C_k)
  \alpha_{k+1}\qquad \text{for $k\in[0,n-1]$.}
  \end{equation}
  Denote by $\beta$
  the least common multiple of denominators
  of $\alpha_k$. Then the numbers $\beta_k$ defined by
  $\beta_k=\beta \alpha_k$ form a decreasing
  sequence for $k\in[1,n]$ and
  $\beta_k\in\N$.  Combining these
  computations with $y_{n+1}=0$ we obtain
  \begin{align*}
    V(y)&\leq(A-a)|y_0|^p+\sum_{k=0}^{n-1}(c_k-C_k)|y_{k+1}|^p\\
    &\leq(A-a)|y_0|^p+\sum_{k=0}^{n-1}(c_k-C_k)\alpha_{k+1}\\
    & \leq(A-a)|y_0|^p+{1\over
      \beta}\sum_{k=0}^{n-1}
    (c_k-C_k)\beta_{k+1} \\&
    =(A-a)|y_0|^p+{1\over
      \beta}\sum_{k=0}^{n-1}
    \sum_{i=1}^{\beta_{k+1}} (c_k-C_k) 
  \end{align*}
Changing the order of summation we get 
\begin{equation*}V(y)
    \leq(A-a)|y_0|^p+{1\over
      \beta}\sum_{i=1}^{\beta_{1}}\sum_{k=0}^{\gamma_i}
    (c_k-C_k),
\end{equation*}
  where $\gamma_i$ is a well defined number
  from the discrete interval $[0,n-1]$. More
  precisely, $\gamma_k$ denotes how many times the
  number $k$ appears in the double sum
  $\sum_{k=0}^{n-1}\sum_{i=1}^{\beta_{k+1}}k$.
  By \eqref{marik:ner}, we obtain
  \begin{align*}
    V(y)&\leq (A-a)|y_0|^p +{1\over
      \beta}\sum_{i=1}^{\beta_{1}}\left|\frac{y_0}{y_1}\right|^p(a-A)\\
    &=(A-a)|y_0|^p +\alpha_1\left|\frac{y_0}{y_1}\right|^p(a-A)
    \\&
    =
    |y_0|^p(a-A)\frac{\alpha_1-|y_1|^p}{|y_1|^p}.
  \end{align*}
  Since \eqref{star} and $C_0\geq c_0$ imply $\alpha_1\leq |y_1|^p$, we have $V(y)\leq 0$. 
  Now the statement follows from Theorem
  \ref{marik:RTC}.
\end{proof}


There is a variant of Theorem
\ref{marik:th1} which is based on the
nonegativity of slightly different sum
than \eqref{marik:ner}. Namely, the
coefficient $c_k$ has the weight
$\frac{R_{k+1}}{r_{k+1}}$ in this sum. To
derive this modification of Theorem
\ref{marik:th1} we need the following
Lemma \ref{lemma1}. This lemma is proved
in Ma\v r\'ik (2000), Corollary 3, as a
corollary of Theorem
\ref{marik:RT}. However, the original
version contains some misprints and for
this reason we restate this lemma with a
shorter proof than the proof presented in
Ma\v r\'ik (2000).
\begin{lemma} \label{lemma1}
  Let $y=\seq y$ be a solution of
  Eq. \eqref{marik:e} on $[0,n-1]$, such that
  $y_{n+1}=0\neq y_0$ and
  $a=r_0\Phi\bigl({\Delta y_0\over
    y_0}\bigr)$.  Let A be such that 
  \begin{equation*}
    \tilde V(y):=\Bigl(A-{R_0\over r_0}a\Bigr)|y_0|^p
    -\sum_{k=0}^{n-1}
    \Bigl\{\Delta\Bigl({R_k\over r_k}\Bigr)r_k \Phi(\Delta y_{k})
    y_{k+1} +\Bigl(C_k-{R_{k+1}\over r_{k+1}}c_{k}\Bigr)|y_{k+1}|^p
    \Bigr\}\leq 0.
  \end{equation*}
 Then the solution $z=\seq z$ of
 Eq. \eqref{marik:E} given by
 $R_0\Phi\bigl({\Delta z_0\over z_0}\bigr)=A$ has
 a generalized zero on $(0,n+1]$, i.e., there
 exists $i\in[0,n]$ such that $R_i z_i z_{i+1}\leq
 0$ holds.
\end{lemma}

\begin{proof}
  Let $y=\seq y$ be a solution of
  \eqref{marik:e} on $[0,n-1]$ which
  satisfies $y_{n+1}=0\neq y_0$ and
  $a=r_0\Phi\bigl(\frac{\Delta
    y_0}{y_0}\bigr)$. Then
 \begin{align}
  L[y_{k}]&\equiv\Delta\bigl(R_k \Phi(\Delta
  y_{k})\bigr)+C_k\Phi\bigl(y_{k+1}\bigr)=
  \Delta\Bigl({R_k \over r_k} r_k \Phi(\Delta
  y_{k})\Bigr)+C_k\Phi(y_{k+1})\notag\\
  &=
  \Delta\Bigl({R_k \over r_k}\Bigr)
  r_k \Phi(\Delta y_{k})+{R_{k+1} \over
  r_{k+1}}\Delta\Bigl(r_k\Phi(\Delta y_{k})\Bigr)+
  C_k\Phi(y_{k+1})\notag\\
 &=  \Delta\Bigl({R_k \over r_k}\Bigr)
  r_k \Phi(\Delta y_{k}) + \Phi(y_{k+1}) \Bigl[C_k-{R_{k+1}\over
  r_{k+1}}c_k\Bigr] \label{marik:lyk}
 \end{align}
 holds for $k\in[0,n-1]$.  Using summation by
 parts we get
\begin{align*}
  \sum_{k=0}^n y_{k+1}L[y_{k}]& = 
  \sum_{k=0}^n y_{k+1}\Bigl\{\Delta(R_k\Phi(\Delta y_k))+C_{k}|y_{k+1}|^p\Bigr\}
\\&=R_{n+1}\Phi(\Delta
  y_{n+1})y_{n+1} -R_0\Phi(\Delta y_0) y_0-\sum_{k=0}^n\Bigl[
  R_k|\Delta y_{k}|^p-C_k|y_{k+1}|^p\Bigr]
\end{align*}
Therefore in view of \eqref{marik:lyk} and
$y_{n+1}=0$, clearly
\begin{align*}
  J(y)&=|y_0|^p\Bigl[A-R_0\Phi\Bigl({\Delta
    y_0\over y_0}\Bigr)\Bigr]-
  \sum_{n=0}^{n} y_{k+1}L[y_{k}]\\
  &=\Bigl(A-{R_0\over r_0}a\Bigr)|y_0|^p-
  \sum_{n=0}^{n-1} y_{k+1}L[y_{k}]
  % \\
  % =|y_0|^p\Bigl[A-R_0\Phi\Bigl({\Delta
  % y_0\over y_0}\Bigr)\Bigr]- \sum
  \\&= \Bigl(A-{R_0\over
    r_0}a\Bigr)|y_0|^p-\sum_{k=0}^{n-1}
  \Bigl\{\Delta\Bigl({R_k\over r_k}\Bigr)r_k
  \Phi(\Delta y_{k}) y_{k+1}
  +\Bigl(C_k-{R_{k+1}\over
    r_{k+1}}c_{k}\Bigr)|y_{k+1}|^p
  \Bigr\}\\&\leq 0.
\end{align*}
and the statement follows from Theorem
\ref{marik:RT}.
\end{proof}


\begin{theorem}\label{marik:th2}
  Let $r_k>0$ on $[0,n]$, $c_k\geq 0$ on
  $[0,n-1]$, $C_0>\frac{R_1}{r_1}c_0$,
  $\Delta\frac{R_k}{r_k}\leq 0$. Let
  $y=\seq y$ be a solution of
  \eqref{marik:e} on $[0,n-1]$, such that
  $y_0\geq y_1>0$, $y_k>0$ on $[0,n]$,
  $y_{n+1}=0$. Suppose that
  $A>\frac{R_0}{r_0}a$,
  \begin{equation*}
    \left|\frac{y_0}{y_1}\right|^p\left(\frac{R_0}{r_0}a-A\right)+\sum_{i=0}^k\left(C_i-\frac{R_{i+1}}{r_{i+1}}c_i\right)\geq 0
  \end{equation*}
  for $k\in[0,n-1]$.  Then the solution
  $x=\seq x$ of \eqref{marik:E} given by
  the condition $x_0>0$,
  $A=R_0\Phi\left(\frac{\Delta
      x_0}{x_0}\right)$ has a generalized
  zero on $[0,n+1]$, i.e., there exists
  $i\in[0,n]$ such that $x_i\neq 0$ and
  $R_i x_i x_{i+1}\leq 0$.
\end{theorem}
\begin{proof}
  From the assumption $\Delta\frac{R_k}{r_k}\leq
  0$ we get
  \begin{equation*}
    \tilde V(y)\leq \Bigl(A-{R_0\over r_0}a\Bigr)|y_0|^p 
    - \sum_{k=0}^{n-1}
    \Bigl(C_k-{R_{k+1}\over r_{k+1}}c_{k}\Bigr)|y_{k+1}|^p 
  \end{equation*}
  The remaining part of the proof is
  essentially similar to the proof of
  Theorem~ \ref{marik:th1} where we
  replace $V(y)$, $a$ and $c_k$ by
  $\tilde V(y)$, $\frac{R_0}{r_0}a$ and
  $\frac{R_{k+1}}{r_{k+1}}c_k$,
  respectively.
\end{proof}

\section{Summary}

The classical results in the comparison
theory of half-linear differential and
difference equations deal with the
generalized zeros of solutions which
vanish at the left end point of the
interval. Focal points, i.e. generalized
zeros of solutions which start with zero
difference, can be considered as a
natural continuation of this
research. The results presented in this
paper include focal points if we choose
$A=a=0$ in Theorems \ref{marik:th1} and
\ref{marik:th2}.

Another companion of the conjugate point
and the focal point is also the so called
coupled point, the point associated with
functional defined on another class of
admissible functions, such as functional
with free end points. Theory of discrete
coupled points has been introduced in a
series of papers by Hilscher and Zeidan,
see Hilscher, Zeidan (2002, 2004, 2005)
and the references therein. The possible
extension of coupled point to half-linear
equation and possibility to formulate
comparison theorems in terms of coupled
points is still an open question and a
subject of the current research.

Further, there are results from the
theory of differential equations, which
allow to study nonoscillatory half-linear
differential equations as a perturbation
of another half-linear equation. This
technique has been started in the paper
Elbert, Schneider (2000) and extended in
a series of papers by Do\v sl\' y and
coauthors. Among others, it has been
shown that this method extends to
difference, see e.g. Do\v sl\'y
and Fi\v snarov\'a (2008, 2009), and can
be formulated in variational setting, see
Do\v sl\'y and Fi\v snarov\'a (2011).  We
hope that developing similar method for
functional \eqref{marik:J} instead of
functional with zero end points opens the
door to future extensions.

% \begin{thebibliography}{9}

\begin{center}
  \uppercase{References}
\end{center}

\def\bibitem#1#2\par{\par #2\medskip}

\parindent -1.5em
\leftskip 1.5em 

\bibitem{DF1} \uppercase{Do\v sl\'y}, O.,
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  Linearized Riccati technique and
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  438130: 1--18. ISSN 1687-1839.

\bibitem{DF2} \uppercase{Do\v sl\'y}, O.,
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  Perturbation principle in discrete
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\bibitem{DF3} \uppercase{Do\v sl\'y}, O.,
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  Variational technique and principal
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\begin{center}
  Address
\end{center}
doc. Mgr. Robert Ma\v r\'ik, Ph.D., \'Ustav matematiky, Mendelova univerzita v Brn\v e, Zem\v ed\v elsk\' a 1, 613 00 Brno, \v Cesk\' a republika, email: marik@mendelu.cz
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