\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 190, pp. 1--14.\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-2014/190\hfil Difference equations with continuous arguments]
{Oscillation of solutions of linear impulsive partial
difference equations with \\ continuous variables}

\author[B. Karpuz \hfil EJDE-2014/190\hfilneg]
{Ba\c{s}ak Karpuz}  % in alphabetical order

\address{Ba\c{s}ak Karpuz \newline
Dokuz Eyl\"{u}l University, T{\i}naztepe Campus,
Faculty of Science, Department of Mathematics,
Buca, 35160 \.{I}zmir, Turkey}
\email{bkarpuz@gmail.com}
%\urladdr{http://www2.aku.edu.tr/\string~bkarpuz}


\thanks{Submitted July 22, 2014. Published September 16, 2014.}
\subjclass[2000]{39B22, 39A14, 39A21}
\keywords{Oscillation; partial difference equations with continuous arguments;
\hfill\break\indent partial difference equations with discrete arguments; impulse effects}

\begin{abstract}
 This article studies oscillation of linear partial difference equations
 with continuous arguments under impulse perturbations through both variables.
 The results improve on previously established results; furthermore a new
 connection is established between impulsive partial difference equations
 with continuous arguments and the more developed area of partial difference
 equations with discrete variables.
\end{abstract}

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

\section{Introduction}\label{intro}

The theory of impulsive equations is an important area of scientific activity.
Since every nonimpulsive equation can be regarded as an impulsive equation with 
no impulse effect, this fact makes the theory of impulsive equations more 
interesting than the corresponding theory of nonimpulsive equations.
Moreover, such equations appear in the modeling of several real-world phenomena 
in many areas such as physics, biology and engineering.

To the best of our knowledge, first paper on impulsive equations was published 
on the oscillation differential equations \cite{MR0991929}.
From the publication of this paper up to the present time, impulsive delay 
differential equations started receiving attention of many mathematicians and 
numerous papers have been published on various types of equations.
Most of the publications are devoted to first-order impulsive delay differential 
equations and there is just a few works in the direction of impulsive partial 
difference equations with continuous arguments (IPDEWCA).
In \cite{MR2583416}, the authors studied the oscillation of solutions to IPDEWCA
\begin{equation}
\begin{gathered}
\begin{aligned}
&p_1z(t+a,s+b)+p_2z(t+a,s)+p_3z(t,s+b)\\
&-p_4z(t,s)+p(t,s)z(t-\tau,s-\sigma)=0 \quad
\text{for } (t,s)\in\big(\mathbb{R}_0^{+}\backslash\{t_k\}_{k\in\mathbb{N}_0})\big)
\times\mathbb{R}_0^{+}
\end{aligned}\\
z(t_k^{+},s)=\alpha_kz(t_k^{-},s)\quad\text{for }
 k\in\mathbb{N}_0 \text{ and } s\in\mathbb{R}_0^{+},
\end{gathered}\label{introeq1}
\end{equation}
where the impulse points $\{t_k\}_{k\in\mathbb{N}_0}$ are assumed to be
placed equidistancely through the first axes.
Note that only the first variable of the unknown function is exposed to 
impulse effects.

In this article, we shall draw our attention to the qualitative behavior 
of solutions to IPDEWCA by introducing impulse effects to both of the variables 
of the unknown function, which is more compatible with the nature of partial 
difference equations with continuous arguments.
Here, we shall adopt a method, which is similar to that of \cite{MR1652915}, 
for building a bridge between the solutions of IPDEWCA and the solutions 
of partial difference equations with continuous arguments (PDEWCA),
and combining this technique with the one in \cite{MR2548024},
we will be able to establish a new connection between oscillation of IPDEWCA 
and difference equations with discrete arguments (PDEWDA).
Therefore, the detailed process taking place in the proofs for IPDEWCA will 
be cleared away.
Finally, we would like to mention that our main result is also new even for 
the nonimpulsive case, improves the ones in \cite{MR2583416} for the autonomous 
case, and includes the results of \cite{MR1742457}.
For fundamental results in the theory of PDEWDA and PDEWCA, the readers are 
referred to the books \cite{MR2193620,MR2388616} and the survey \cite{MR2000596} 
devoted to the study of various (including qualitative) properties of the solutions.

This article is organized as follows:
In Section \ref{oda}, we study the oscillation of PDEWDA by removing a 
condition in the well-known oscillation criteria introduced in \cite{MR2388616};
in Section \ref{ocavda}, we construct a connection between the oscillation 
of PDEWCA and the oscillation of PDEWDA, which extends almost all of the
 oscillation results given for PDEWDA to PDEWCA.
In Section \ref{oicavda}, we relate the oscillation of IPDEWCA with the 
oscillation of PDEWCA, so that the results for the oscillation of PDEWDA 
can be also applied to reveal the oscillation of IPDEWCA.
Finally in Section~\ref{dfc}, we make our final comments and compare our 
results with the ones introduced in \cite{MR2583416}, and an illustrative 
example concerning the autonomous case is given to mention the importance 
of our results.

\section{Oscillation of PDEWDA}\label{oda}

In this section, we confine our attention to the difference inequality
 with discrete arguments
\begin{equation}
\begin{split}
&p_1A(m+1,n+1)+p_2A(m+1,n)+p_3A(m,n+1)\\
&-p_4A(m,n)+p(m,n)A(m-\kappa,n-\ell)\leq0
\quad\text{for } (m,n)\in\mathbb{Z}_0\times\mathbb{Z}_0,
\end{split} \label{odaeq1}
\end{equation}
where $\mathbb{Z}_k:=\{n\in\mathbb{Z}:\ n\geq k\}$ for $k\in\mathbb{Z}$, under the following conditions:
\begin{itemize}
\item[(A1)] $p_1\geq0$ and $p_2,p_3,p_4>0$;
\item[(A2)] $\kappa,\ell\in\mathbb{Z}_0$;
\item[(A3)] $p:\mathbb{N}_0\times\mathbb{N}_0\to\mathbb{R}_0^{+}$.
\end{itemize}

\begin{definition}\label{odadf1} \rm
A double sequence $A:\mathbb{Z}_{-\kappa}\times\mathbb{Z}_{-\ell}\to\mathbb{R}$ satisfying
\eqref{odaeq1} identically on $\mathbb{Z}_0\times\mathbb{Z}_0$ is called a solution
 of \eqref{odaeq1}.
\end{definition}

\begin{definition}\label{odadf2} \rm
A solution $A$ of \eqref{odaeq1} is called eventually positive if 
there exists $(m_0,n_0)\in\mathbb{Z}_0\times\mathbb{Z}_0$ such that $A(m,n)>0$
for all $(m,n)\in\mathbb{Z}_{m_0}\times\mathbb{Z}_{n_0}$.
and is called eventually negative if negative of $A$ is eventually positive.
A solution neither eventually positive nor negative is called oscillatory.
If every solution of \eqref{odaeq1} oscillates, then it is called oscillatory.
\end{definition}

\begin{theorem}[{\cite[Theorem~2.15]{MR2388616}}] \label{thma}
Assume that {\rm (A1)--(A3)} hold, and that
\begin{itemize}
\item[(i)] $\limsup_{\substack{m\to\infty\\ n\to\infty}}p(m,n)>0$;
\item[(ii)] 
    \begin{itemize}
    \item[(a)]  If $\kappa\geq\ell\geq1$, then
        \begin{align*}
 &\limsup_{\substack{m\to\infty\\ n\to\infty}}
   \sup_{\lambda\in E}\Big\{\lambda \prod_{i=1}^{\ell}
 \big[p_4-\lambda p(m-i,n-i)\big]\\
       &\times\prod_{j=1}^{\kappa-\ell}\big[p_4-\lambda p(m-\ell-j,n-\ell)\big]\Big\}
<\Big(p_1+2\frac{p_2p_3}{p_4}\Big)^{\ell}p_2^{\kappa-\ell},
        \end{align*}
where
\[
        E:=\big\{\lambda>0:\ p_4-\lambda p(m,n)>0 \text{ for all large }
 (m,n)\in\mathbb{Z}_0\times\mathbb{Z}_0\big\}.
\]

    \item[(b)] If $\ell\geq\kappa\geq1$, then
\begin{align*}
     &\limsup_{\substack{m\to\infty\\ n\to\infty}}
\sup_{\lambda\in E}\Big\{\lambda\prod_{i=1}^{\kappa}\big[p_4-\lambda p(m-i,n-i)\big]\\
    &\times\prod_{j=1}^{\ell-\kappa}\big[p_4-\lambda p(m-\kappa,n-\kappa-j)\big]\Big\}
<\Big(p_1+2\frac{p_2p_3}{p_4}\Big)^{k}p_3^{\ell-\kappa}.
\end{align*}
    \end{itemize}
\end{itemize}
Then \eqref{odaeq1} has no eventually positive solutions.
\end{theorem}

The proof of Theorem~\ref{thma} uses the property that the set $E$ is bounded,
 which is ensured by (i) in Theorem~\ref{thma}.
In the following result, we shall remove the requirement for (i)
in Theorem~\ref{thma} by introducing a new proof.
To this end, we need to introduce
\begin{equation}
\Lambda(m,n):=\big\{\lambda>0:\ p_4-\lambda p(i,j)>0\ \text{for all}\ (i,j)\in[m-\kappa,m)\times[n-\ell,n)\cap\mathbb{Z}_0\times\mathbb{Z}_0\big\}.
\end{equation}

\begin{theorem}\label{odathm1}
Assume that {\rm (A1)--(A3)} hold.
Assume also that
\begin{equation}
\begin{split}
&\limsup_{\substack{m\to\infty\\ n\to\infty}}
\sup_{\lambda\in\Lambda(m,n)}\Big\{\lambda\prod_{i=1}^{\ell}\big[p_4-\lambda p(m-i,n-i)\big]\\
&\times\prod_{j=1}^{\kappa-\ell}\big[p_4-\lambda p(m-\ell-j,n-\ell)\big]
\Big\}<p_2^{\kappa-\ell}\Big(p_1+2\frac{p_2p_3}{p_4}\Big)^{\ell}\quad
\text{if } \kappa\geq\ell\geq1,
\end{split}\label{odathm1eq1}
\end{equation}
or
\begin{align*}
&\limsup_{\substack{m\to\infty\\ n\to\infty}}\sup_{\lambda\in\Lambda(m,n)}
\Big\{\lambda\prod_{i=1}^{\kappa}\big[p_4-\lambda p(m-i,n-i)\big]\\
&\times\prod_{j=1}^{\ell-\kappa}\big[p_4-\lambda p(m-\kappa,n-\kappa-j)\big]\Big\}
<p_3^{\ell-\kappa}\Big(p_1+2\frac{p_2p_3}{p_4}\Big)^{k}\quad\text{if }
 \ell\geq\kappa\geq1.
\end{align*}
Then \eqref{odaeq1} has no eventually positive solutions.
\end{theorem}

\begin{proof}
We shall only give a proof for the case $\kappa\geq\ell$ since the remaining 
case follows by using similar arguments.
Let $A$ be an eventually positive solution of \eqref{odaeq1}.
We may find $(m_0,n_0)\in\mathbb{Z}_0\times\mathbb{Z}_0$ such that $A(m,n)>0$
and $A(m-\kappa,n-\ell)>0$ for all $(m,n)\in\mathbb{Z}_{m_0}\times\mathbb{Z}_{n_0}$.
Now, define
\begin{equation}
B(m,n):=\frac{A(m-\kappa,n-\ell)}{A(m,n)}>0\quad\text{for }
 (m,n)\in\mathbb{Z}_{m_0}\times\mathbb{Z}_{n_0}.\label{odathm1prfeq1}
\end{equation}
We shall first show that
\begin{equation}
\lim_{\substack{m\to\infty\\ n\to\infty}}B(m,n)=\infty.\label{odathm1prfeq2}
\end{equation}
Using \eqref{odathm1prfeq1}, we may rewrite \eqref{odaeq1} as
\begin{equation}
\begin{aligned}
&p_1A(m+1,n+1)+p_2A(m+1,n)+p_3A(m,n+1)\\
&\leq\big[p_4-B(m,n)p(m,n)\big]A(m,n)
\end{aligned} \label{odathm1prfeq3}
\end{equation}
for all $(m,n)\in\mathbb{Z}_{m_0}\times\mathbb{Z}_{n_0}$.
It follows from \eqref{odaeq1} that
\begin{equation}
A(m+1,n)\leq\frac{p_4}{p_2}A(m,n)\quad\text{and}\quad 
A(m,n+1)\leq\frac{p_4}{p_3}A(m,n)
\label{odathm1prfeq4}
\end{equation}
for all $(m,n)\in\mathbb{Z}_{m_0}\times\mathbb{Z}_{n_0}$.
Using \eqref{odathm1prfeq4} in \eqref{odathm1prfeq3}, we obtain
\[
\Big(p_1+2\frac{p_2p_3}{p_4}\Big)A(m+1,n+1)\leq
\big[p_4-B(m,n)p(m,n)\big]A(m,n)
\]
for all $(m,n)\in\mathbb{Z}_{m_0}\times\mathbb{Z}_{n_0}$,
or
\[
A(m+1,n+1)\leq\Big(p_1+2\frac{p_2p_3}{p_4}\Big)^{-1}
\big[p_4-B(m,n)p(m,n)\big]A(m,n)
\]
for all $(m,n)\in\mathbb{Z}_{m_0}\times\mathbb{Z}_{n_0}$.
This yields
\begin{equation}
\begin{aligned}
&A(m,n)\\
&\leq\Big(p_1+2\frac{p_2p_3}{p_4}\Big)^{-1}\big[p_4-B(m-1,n-1)p(m-1,n-1)\big]
A(m-1,n-1) \\
&\leq \Big(p_1+2\frac{p_2p_3}{p_4}\Big)^{-2}\big[p_4-B(m-1,n-1)p(m-1,n-1)\big] \\
&\quad \times\big[p_4-B(m-2,n-2)p(m-2,n-2)\big]A(m-2,n-2) \\
&\quad \dots \\
&\leq \Big(p_1+2\frac{p_2p_3}{p_4}\Big)^{-\ell}
\Big(\prod_{i=1}^{\ell}\big[p_4-B(m-i,n-i)p(m-i,n-i)\big]\Big)
A(m-\ell,n-\ell)
\end{aligned} \label{odathm1prfeq7}
\end{equation}
for all $(m,n)\in\mathbb{Z}_{m_0+\ell}\times\mathbb{Z}_{n_0+\ell}$.
Using \eqref{odathm1prfeq3}, we obtain
\[
A(m+1,n)<\frac{1}{p_2}\big[p_4-B(m,n)p(m,n)\big]A(m,n)\quad\text{for all }
 (m,n)\in\mathbb{Z}_{m_0}\times\mathbb{Z}_{n_0},
\]
which yields
\begin{equation}
\begin{aligned}
&A(m-\ell,n-\ell)\\
&<\frac{1}{p_2}\big[p_4-B(m-\ell-1,n-\ell)p(m-\ell-1,n-\ell)\big]
A(m-\ell-1,n-\ell) \\
&\quad \dots \\
&\leq \frac{1}{p_2^{\kappa-\ell}}
\Big(\prod_{j=1}^{\kappa-\ell}\big[p_4-B(m-\ell-j,n-\ell)p(m-\ell-j,n-\ell)\big]
\Big)A(m-\kappa,n-\ell)
\end{aligned} \label{odathm1prfeq9}
\end{equation}
for all $(m,n)\in\mathbb{Z}_{m_0+\kappa}\times\mathbb{Z}_{n_0+\kappa}$.
Define
\begin{equation}
C(m,n):=\min\big\{B(i,j):\ (i,j)\in[m-\kappa,m]
\times[n-\ell,n]\cap\mathbb{Z}^{2}\big\}
\label{odathm1prfeq10}
\end{equation}
for $(m,n)\in\mathbb{Z}_{m_0}\times\mathbb{Z}_{n_0}$.
Using \eqref{odathm1prfeq7}, \eqref{odathm1prfeq9} and \eqref{odathm1prfeq10},
we obtain
\begin{align*}
A(m,n)&\leq\frac{1}{p_2^{\kappa-\ell}}
\Big(p_1+2\frac{p_2p_3}{p_4}\Big)^{-\ell}
\Big(\prod_{i=1}^{\ell}\big[p_4-B(m-i,n-i)p(m-i,n-i)\big]\Big)\\
&\quad \times\Big(\prod_{j=1}^{\kappa-\ell}
\big[p_4-B(m-\ell-j,n-\ell)p(m-\ell-j,n-\ell)\big]\Big)A(m-\kappa,n-\ell)
 \\
&\leq \frac{1}{p_2^{\kappa-\ell}}\Big(p_1+2\frac{p_2p_3}{p_4}\Big)^{-\ell}
\Big(\prod_{i=1}^{\ell}\big[p_4-C(m,n)p(m-i,n-i)\big]\Big)\\
&\quad \times\Big(\prod_{j=1}^{\kappa-\ell}
\big[p_4-C(m,n)p(m-\ell-j,n-\ell)\big]\Big)A(m-\kappa,n-\ell)
\end{align*}
or equivalently
\begin{equation}
\begin{split}
1&\leq \frac{1}{p_2^{\kappa-\ell}}\Big(p_1+2\frac{p_2p_3}{p_4}\Big)^{-\ell}
 \Big(\prod_{i=1}^{\ell}\big[p_4-C(m,n)p(m-i,n-i)\big]\Big)\\
&\quad \times\Big(\prod_{j=1}^{\kappa-\ell}\big[p_4-C(m,n)p(m-\ell-j,n-\ell)\big]
\Big)B(m,n)
\end{split}\label{odathm1prfeq11}
\end{equation}
for all $(m,n)\in\mathbb{Z}_{m_0+2\kappa}\times\mathbb{Z}_{n_0+2\kappa}$.
It follows from \eqref{odathm1prfeq3} and \eqref{odathm1prfeq10} that
\begin{equation}
C(m,n)\in\Gamma(m,n)\quad\text{for all }
 (m,n)\in\mathbb{Z}_{m_0+2\kappa}\times\mathbb{Z}_{n_0+2\kappa}.
\label{odathm1prfeq12}
\end{equation}
Then from \eqref{odathm1prfeq11} and \eqref{odathm1prfeq12}, we have
\begin{equation}
\begin{split}
C(m,n)&\leq \frac{1}{p_2^{\kappa-\ell}}\Big(p_1+2\frac{p_2p_3}{p_4}\Big)^{-\ell}
C(m,n)\Big(\prod_{i=1}^{\ell}\big[p_4-C(m,n)p(m-i,n-i)\big]\Big)\\
&\quad \times\Big(\prod_{j=1}^{\kappa-\ell}\big[p_4-C(m,n)p(m-\ell-j,n-\ell)\big]
\Big)B(m,n)
\end{split}\label{odathm1prfeq13}
\end{equation}
for all $(m,n)\in\mathbb{Z}_{m_0+2\kappa}\times\mathbb{Z}_{n_0+2\kappa}$.
The condition \eqref{odathm1eq1} implies existence of a constant $\mu<1$
and $(m_1,n_1)\in\mathbb{Z}_{m_0+2\kappa}\times\mathbb{Z}_{n_0+2\kappa}$ such that
\begin{equation}
\begin{split}
&\frac{1}{p_2^{\kappa-\ell}}\Big(p_1+2\frac{p_2p_3}{p_4}\Big)^{-\ell}
C(m,n)\prod_{i=1}^{\ell}\big[p_4-C(m,n)p(m-i,n-i)\big]\\
&\times\prod_{j=1}^{\kappa-\ell}\big[p_4-C(m,n)p(m-\ell-j,n-\ell)\big]<\mu
\end{split}\label{odathm1prfeq14}
\end{equation}
for all $(m,n)\in\mathbb{Z}_{m_1}\times\mathbb{Z}_{n_1}$.
Using \eqref{odathm1prfeq14} in \eqref{odathm1prfeq13}, we obtain
\begin{equation}
C(m,n)\leq\mu B(m,n)\quad\text{for all } (m,n)\in\mathbb{Z}_{m_1}\times\mathbb{Z}_{n_1}.\label{odathm1prfeq15}
\end{equation}
To prove that \eqref{odathm1prfeq2} is true assume the contrary; i.e.,
\begin{equation}
\rho:=\liminf_{\substack{m\to\infty\\ n\to\infty}}B(m,n)<\infty.
\end{equation}
Note that from \eqref{odathm1prfeq10}, we have
\begin{equation}
\liminf_{\substack{m\to\infty\\ n\to\infty}}C(m,n)=\rho.
\end{equation}
First assume that $\rho>0$.
Then taking $\liminf$ on both sides of \eqref{odathm1prfeq15},
we obtain $\rho\leq\mu\rho$, which is a contradiction since $\mu<1$.
Next assume that $\rho=0$, then there exists $(m_2,n_2)\in\mathbb{Z}_{m_1}\times\mathbb{Z}_{n_1}$
 such that
\begin{equation}
\min\{B(i,j): (i,j)\in[m_1,m_2]\times[n_1,n_2]\cap\mathbb{Z}_0\times\mathbb{Z}_0\}=B(m_2,n_2).\label{odathm1prfeq18}
\end{equation}
Then \eqref{odathm1prfeq10}, \eqref{odathm1prfeq15} and \eqref{odathm1prfeq18}
 yield that
\begin{align*}
\mu B(m_2,n_2)&\geq C(m_2,n_2)\\
&\geq\min\{B(i,j): (i,j)\in[m_1,m_2]\times[n_1,n_2]\cap\mathbb{Z}_0\times\mathbb{Z}_0\}
=B(m_2,n_2),
\end{align*}
which is a contradiction since $\mu<1$.
Therefore, we have just proved that \eqref{odathm1prfeq2} is true.
Now, we show that \eqref{odathm1eq1} implies
\begin{equation}
\limsup_{\substack{m\to\infty\\ n\to\infty}}\sum_{i=m-\kappa}^{m-1}\sum_{j=n-\ell}^{n-1}p(i,j)>0.\label{odathm1prfeq20}
\end{equation}
If \eqref{odathm1prfeq20} is not true, then we have
\begin{equation}
\lim_{\substack{m\to\infty\\ n\to\infty}}p(m,n)=0.
\end{equation}
We may find $(m_2,n_2)\in\mathbb{Z}_{m_1}\times\mathbb{Z}_{n_1}$ such that
\begin{equation}
p(m,n)<\varepsilon\quad\text{for all } (m,n)\in\mathbb{Z}_{m_2}\times\mathbb{Z}_{n_2},\label{odathm1prfeq22}
\end{equation}
where
\begin{equation}
\varepsilon:=\frac{1}{2\kappa}\Big(\frac{p_4\kappa}{\kappa+1}\Big)^{\kappa+1}
\frac{1}{p_2^{\kappa-\ell}}\Big(p_1+2\frac{p_2p_3}{p_4}\Big)^{-\ell}>0.\label{odathm1prfeq23}
\end{equation}
From \eqref{odathm1prfeq22}, we have
\begin{equation}
p_4-\frac{p_4}{\varepsilon}p(m,n)>0\quad\text{for all }
 (m,n)\in\mathbb{Z}_{m_2}\times\mathbb{Z}_{n_2},\label{odathm1prfeq24}
\end{equation}
which implies $(0,p_4/\varepsilon)\subset\Lambda(m,n)$ for all
$(m,n)\in\mathbb{Z}_{m_2}\times\mathbb{Z}_{n_2}$.
Then, using \eqref{odathm1prfeq23} and \eqref{odathm1prfeq24}, we have
\begin{align*}
&\sup_{\lambda\in\Lambda(m,n)}\Big\{\lambda\prod_{i=1}^{\ell}\big[p_4-\lambda p(m-i,n-i)\big]\prod_{j=1}^{\kappa-\ell}\big[p_4-\lambda p(m-\ell-j,n-\ell)\big]\Big\} \\
&\geq \sup_{\lambda\in(0,p_4/\varepsilon)}\Big\{\lambda\big(p_4-\lambda\varepsilon\big)^{\kappa}\Big\}=\frac{1}{\varepsilon\kappa}\Big(\frac{p_4\kappa}{\kappa+1}\Big)^{\kappa+1} \\
&\geq 2p_2^{\kappa-\ell}\Big(p_1+2\frac{p_2p_3}{p_4}\Big)^{\ell}
\end{align*}
for all $(m,n)\in\mathbb{Z}_{m_2}\times\mathbb{Z}_{n_2}$.
This contradicts \eqref{odathm1eq1}, and proves that \eqref{odathm1prfeq20} is true.
Now, we will use \eqref{odathm1prfeq20} to show
\begin{equation}
\liminf_{\substack{m\to\infty\\ n\to\infty}}B(m,n)<\infty.\label{odathm1prfeq25}
\end{equation}
The proof will be completed if we can show that \eqref{odathm1prfeq25} holds,
which contradicts \eqref{odathm1prfeq2}.
From \eqref{odathm1prfeq20}, there exist a constant $\delta>0$ and an
increasing divergent double sequence
$\{(\xi_{r},\zeta_{r})\}_{r\in\mathbb{N}_0}\subset\mathbb{Z}_{m_2}\times\mathbb{Z}_{n_2}$ such that
\begin{equation}
\sum_{i=\xi_{r}-\kappa}^{\xi_{r}-1}\sum_{j=\zeta_{r}-\ell}^{\zeta_{r}-1}p(i,j)
\geq\delta\quad\text{for all }r\in\mathbb{N}_0.
\end{equation}
Dirichlet's Pigeonhole principle implies existence of a  sequence
$\{(\alpha_{r},\beta_{r})\}_{r\in\mathbb{N}_0}\subset\mathbb{Z}_0\times\mathbb{Z}_0$ such that
\begin{equation}
\xi_{r}-\kappa\leq\alpha_{r}<\xi_{r},\quad
\zeta_{r}-\ell\leq\beta_{r}<\zeta_{r},\quad
p(\alpha_{r},\beta_{r})>\frac{\delta}{\kappa\ell}\quad\text{for all } r\in\mathbb{N}_0.
\end{equation}
From \eqref{odaeq1}, for all $r\in\mathbb{N}_0$, we obtain
\begin{align*}
0&\geq p_1A(\alpha_{r}+1,\beta_{r}+1)+p_2A(\alpha_{r}+1,\beta_{r})
 +p_3A(\alpha_{r},\beta_{r}+1)\\
&\quad -p_4A(\alpha_{r},\beta_{r})+p(\alpha_{r},\beta_{r})
 A(\alpha_{r}-\kappa,\beta_{r}-\ell) \\
&>-p_4A(\alpha_{r},\beta_{r})+\frac{\delta}{\kappa\ell}
A(\alpha_{r}-\kappa,\alpha_{r}-\ell),
\end{align*}
which yields
\begin{equation}
B(\alpha_{r},\beta_{r})
=\frac{A(\alpha_{r}-\kappa,\beta_{r}-\ell)}{A(\alpha_{r},\beta_{r})}
\leq\frac{1}{\delta}p_4\kappa\ell\quad\text{for all}\ r\in\mathbb{N}_0.\label{odathm1prfeq27}
\end{equation}
Letting $r\to\infty$ in \eqref{odathm1prfeq27}, we reach at \eqref{odathm1prfeq25},
and thus the proof is complete.
\end{proof}

\begin{remark}\label{odarmk1} \rm
Under the assumptions of Theorem~\ref{odathm1}, every solution of the following PDEWDA
\begin{align*}
&p_1A(m+1,n+1)+p_2A(m+1,n)+p_3A(m,n+1)\\
&-p_4A(m,n)+p(m,n)A(m-\kappa,n-\ell)=0
\quad\text{for } (m,n)\in\mathbb{Z}_0\times\mathbb{Z}_0
\end{align*}
is oscillatory.
This result therefore improves Theorem \ref{thma}.
\end{remark}

\section{Oscillation of PDEWCA via PDEWDA}\label{ocavda}

In this section, we reduce the oscillation of PDEWCA to the oscillation of PDEWDA,
 which is a relatively more developed area.
Let us consider the PDEWCA
\begin{equation}
\begin{split}
&p_1z(t+a,s+b)+p_2z(t+a,s)+p_3z(t,s+b)\\
&-p_4z(t,s)+p(t,s)z(t-\tau,s-\sigma)\leq0
\quad\text{for } (t,s)\in\mathbb{R}_0^{+}\times\mathbb{R}_0^{+}
\end{split}\label{ocavdaeq1}
\end{equation}
under the following assumptions:
\begin{itemize}
\item[(A4)] $a,b>0$ and $\tau,\sigma>0$;
\item[(A5)] $p_2,p_3\geq p_4$;
\item[(A6)] $p:\mathbb{R}_0^{+}\times\mathbb{R}_0^{+}\to\mathbb{R}_0^{+}$ is a continuous function.
\end{itemize}

\begin{definition}\label{ocavdadf1} \rm
A continuous function $z:[-\tau,\infty)\times[-\sigma,\infty)\to\mathbb{R}$
satisfying \eqref{ocavdaeq1} identically on $\mathbb{R}_0^{+}\times\mathbb{R}_0^{+}$
is called a solution of \eqref{ocavdaeq1}.
\end{definition}

\begin{definition}\label{ocavdadf2} \rm
A solution $z$ of \eqref{ocavdaeq1} is called eventually positive if there 
exists $(t_0,s_0)\in\mathbb{R}_0^{+}\times\mathbb{R}_0^{+}$ such that $z(t,s)>0$ for all
$(t,s)\in[t_0,\infty)\times[s_0,\infty)$.
and is called eventually negative if negative of $z$ is eventually positive.
A solution neither eventually positive nor negative is called oscillatory.
If every solution of \eqref{ocavdaeq1} oscillates, then it is called oscillatory.
\end{definition}

We define the minimized function $q$ of $p$ by
\begin{equation}
q(t,s):=\min\big\{p(\eta,\zeta): (\eta,\zeta)\in[t,t+a]\times[s,s+b]\big\}
\quad\text{for } (t,s)\in\mathbb{R}_0^{+}\times\mathbb{R}_0^{+}.\label{ocavdaeq2}
\end{equation}
For simplicity of notation, we let
\begin{equation}
\upsilon:=\Big\lfloor\frac{\tau}{a}\Big\rfloor\quad\text{and}\quad
\nu:=\Big\lfloor\frac{\sigma}{b}\Big\rfloor,\label{ocavdaeq3}
\end{equation}
where $\lfloor\cdot\rfloor:\mathbb{R}\to\mathbb{Z}$ denotes the least integer function.

Below, we quote one of the most important results for oscillation of 
the PDEWCA
\begin{equation}
\begin{split}
&p_1z(t+a,s+b)+p_2z(t+a,s)+p_3z(t,s+b)\\
&-p_4z(t,s)+p(t,s)z(t-\tau,s-\sigma)=0
\quad\text{for } (t,s)\in\mathbb{R}_0^{+}\times\mathbb{R}_0^{+}.
\end{split}\label{ocavdaeq3a}
\end{equation}

\begin{theorem}[{\cite[Theorem~2.37]{MR2388616}}] \label{thmb}
Assume that {\rm (A4)--(A6)} hold, and that
\begin{itemize}
\item[(i)] 
$ \limsup_{\substack{t\to\infty\\ s\to\infty}}q(t,s)>0$;

\item[(ii)]
    \begin{itemize}
    \item[(a)] If $\upsilon\geq\nu\geq1$, then
        \begin{align*}
       &\limsup_{\substack{t\to\infty\\ s\to\infty}}\sup_{\lambda\in E}
 \Big\{\lambda \prod_{i=1}^{\ell}\big[p_4-\lambda q(t-ai,s-bi)\big]\\
       &\times\prod_{j=1}^{\kappa-\ell}\big[p_4-\lambda q(t-a\ell-aj,n-b\ell)\big]
\Big\}<\Big(p_1+2\frac{p_2p_3}{p_4}\Big)^{\ell}p_2^{\kappa-\ell},
        \end{align*}
   where
        \begin{equation}
        E:=\big\{\lambda>0:\ p_4-\lambda q(t,s)>0\ \text{for all large}\ (t,s)\in\mathbb{R}_0^{+}\times\mathbb{R}_0^{+}\big\}.
        \end{equation}
    \item[(b)] If $\nu\geq\upsilon\geq1$, then
        \begin{align*}
       &\limsup_{\substack{t\to\infty\\ s\to\infty}}\sup_{\lambda\in E}
\Big\{\lambda \prod_{i=1}^{\kappa}\big[p_4-\lambda p(t-ai,s-bi)\big]\\
        &\times\prod_{j=1}^{\ell-\kappa}\big[p_4-\lambda p(t-a\kappa,s-b\kappa-bj)\big]\Big\}<\Big(p_1+2\frac{p_2p_3}{p_4}\Big)^{k}p_3^{\ell-\kappa}.
        \end{align*}
     \end{itemize}
\end{itemize}
Then \eqref{ocavdaeq3a} is oscillatory.
\end{theorem}

The removal of the condition (i) of Theorem~\ref{thmb}
 is just a simple corollary of the main result of this section.

\begin{lemma}[\protect{\cite[Lemma~2.36]{MR2388616}}]\label{ocavdalm1}
Assume that {\rm (A1), (A4)--(A6)} hold.
If $z$ is an eventually positive solution of the partial difference inequality 
with continuous variables
\begin{equation}
\begin{split}
&p_1z(t+a,s+b)+p_2z(t+a,s)+p_3z(t,s+b)\\
&-p_4z(t,s)+p(t,s)z(t-\tau,s-\sigma)\leq0
\quad\text{for }(t,s)\in\mathbb{R}_0^{+}\times\mathbb{R}_0^{+},
\end{split}\label{ocavdalm1eq1}
\end{equation}
then $z$ eventually satisfies the partial difference inequality with 
continuous variables
\begin{equation}
\begin{split}
&p_1z(t+a,s+b)+p_2z(t+a,s)+p_3z(t,s+b)\\
&-p_4z(t,s)+q(t,s)z(t-a\upsilon,s-b\nu)\leq0
\quad\text{for } (t,s)\in\mathbb{R}_0^{+}\times\mathbb{R}_0^{+}.
\end{split}\label{ocavdalm1eq2}
\end{equation}
\end{lemma}

\begin{corollary}\label{ocavdacrl1}
Assume that {\rm (A1), (A4)--(A6)} hold.
If \eqref{ocavdalm1eq2} has no eventually positive solutions,
then \eqref{ocavdalm1eq1} also has no eventually positive solutions.
\end{corollary}

For the next result, which builds a bridge between the oscillation 
of PDEWCA and of PDEWDA, we introduce
\begin{equation}
r_{t,s}(m,n):=q(t+am,s+bn)\quad\text{for } (m,n)\in\mathbb{Z}_0\times\mathbb{Z}_0,\label{ocavdaeq4}
\end{equation}
where $(t,s)\in\mathbb{R}_0^{+}\times\mathbb{R}_0^{+}$.

\begin{theorem}\label{ocavdathm1}
Assume that {\rm (A1), (A4)--(A6)} hold.
Moreover, assume that for some fixed $(\eta,\zeta)\in[0,a)\times[0,b)$,
the partial difference inequality with discrete variables
\begin{equation}
\begin{split}
&p_1A(m+1,n+1)+p_2A(m+1,n)+p_3A(m,n+1)\\
&-p_4A(m,n)+r_{\eta,\zeta}(m,n)A(m-\upsilon,n-\nu) \leq0
\quad\text{for } (m,n)\in\mathbb{N}_0\times\mathbb{N}_0
\end{split}\label{ocavdathm1eq1}
\end{equation}
has no eventually positive solutions.
Then, \eqref{ocavdalm1eq2} has no eventually positive solutions.
\end{theorem}

\begin{proof}
Assume the contrary that $z$ is an eventually positive solution of 
\eqref{ocavdalm1eq2}.
Now we define the double sequence
\[
A(m,n):=z(\eta+am,\zeta+bn)\quad\text{for }(m,n)\in\mathbb{N}_0\times\mathbb{N}_0.
\]
Then $A$ is eventually positive.
Substituting $(t,s)=(\eta+am,\zeta+bn)$ for $(m,n)\in\mathbb{N}_0\times\mathbb{N}_0$
into \eqref{ocavdalm1eq2},
we see that the double sequence $A$ satisfies \eqref{ocavdathm1eq1}.
This is a contradiction.
\end{proof}

\begin{corollary}\label{ocavdacrl2}
Assume that {\rm (A1), (A4)--(A6)} hold.
Assume also that there exists $(\eta,\zeta)\in[0,a)\times[0,b)$ such that
\begin{equation}
\begin{split}
&\limsup_{\substack{m\to\infty\\ n\to\infty}}
\sup_{\lambda\in\Lambda_{\eta,\zeta}}\Big\{\lambda
\prod_{i=1}^{\nu}\big[p_4-\lambda r_{\eta,\zeta}(m-i,n-i)\big]\\
&\times\prod_{j=1}^{\upsilon-\nu}\big[p_4-\lambda r_{\eta,\zeta}(m-\nu-j,n-\nu)\big]
\Big\}<p_2^{\upsilon-\nu}\Big(p_1+2\frac{p_2p_3}{p_4}\Big)^{\nu}
\end{split}\label{ocavdacrl2eq1}
\end{equation}
if $\upsilon\geq\nu\geq1$,
or
\begin{equation}
\begin{split}
&\limsup_{\substack{m\to\infty\\ n\to\infty}}
\sup_{\lambda\in\Lambda_{\eta,\zeta}}\Big\{\lambda
\prod_{i=1}^{\upsilon}\big[p_4-\lambda r_{\eta,\zeta}(m-i,n-i)\big]\\
&\times\prod_{j=1}^{\nu-\upsilon}\big[p_4-\lambda r_{\eta,\zeta}
(m-\upsilon,n-\upsilon-j)\big]\Big\}<p_3^{\nu-\upsilon}
\Big(p_1+2\frac{p_2p_3}{p_4}\Big)^{k}
\end{split}\label{ocavdacrl2eq2}
\end{equation}
if $\nu\geq\upsilon\geq1$,
where
\[
\Lambda_{t,s}:=\{\lambda>0:\ p_4-\lambda r_{t,s}(m,n)>0\text{f or all large }
 (m,n)\in\mathbb{N}_0\times\mathbb{N}_0\}
\]
for $(t,s)\in\mathbb{R}_0^{+}\times\mathbb{R}_0^{+}$.
Then, \eqref{ocavdalm1eq2} has no eventually positive solutions.
\end{corollary}

\begin{proof}
Under the conditions of the corollary, \eqref{ocavdathm1eq1} 
has no eventually positive solutions by Theorem~\ref{odathm1}.
Therefore, it follows from Theorem~\ref{ocavdathm1} that \eqref{ocavdalm1eq2} 
has no eventually positive solutions.
Finally, an application of Corollary \ref{ocavdacrl1} completes the proof.
\end{proof}

\begin{remark}\label{ocavdarmk1} \rm
Note that for all $(\eta,\zeta)\in[0,a)\times[0,b)$, we have
\begin{align*}
&\limsup_{\substack{m\to\infty\\ n\to\infty}}\sup_{\lambda\in\Lambda_{\eta,\zeta}}
\Big\{\lambda\prod_{i=1}^{\nu}\big[p_4-\lambda r_{\eta,\zeta}(m-i,n-i)\big]
\prod_{j=1}^{\upsilon-\nu}\big[p_4-\lambda r_{\eta,\zeta}(m-\nu-j,n-\nu)\big]\Big\}\\
&\leq\limsup_{\substack{t\to\infty\\ s\to\infty}}\sup_{\lambda\in E}
\Big\{\lambda\prod_{i=1}^{\nu}\big[p_4-\lambda q(t-ai,s-bi)\big]
\prod_{j=1}^{\upsilon-\nu}\big[p_4-\lambda q(t-a\nu-aj,s-b\nu)\big]\Big\},
\end{align*}
where
\begin{equation}
E:=\big\{\lambda>0:\ p_4-\lambda q(t,s)>0 \text{ for all large }
 (t,s)\in\mathbb{R}_0^{+}\times\mathbb{R}_0^{+}\big\}.
\end{equation}
Hence, Corollary~\ref{ocavdacrl2} improves Theorem~\ref{thmb}.
\end{remark}

\section{Oscillation of IPDEWCA via PDEWDA}\label{oicavda}

The primary assumptions of this section are as follows:
\begin{itemize}
\item[(A7)]  $\{t_k\}_{k\in\mathbb{N}_0}$ and $\{s_{\ell}\}_{\ell\in\mathbb{N}_0}$
 are increasing divergent sequences of nonnegative reals;
\item[(A8)]  $\{\alpha_k\}_{k\in\mathbb{N}_0}$ and $\{\beta_{\ell}\}_{\ell\in\mathbb{N}_0}$
 are sequences of real numbers, which involve no zero terms;
\item[(A9)] $p:\mathbb{R}_0^{+}\times\mathbb{R}_0^{+}\to\mathbb{R}$ is a continuous function.
\end{itemize}
For simplicity of notation, we define 
\[
D:=\big((\mathbb{R}_0^{+}\backslash\{t_k\}_{k\in\mathbb{N}_0})\times\mathbb{R}_0^{+}\big)
\cup\big(\mathbb{R}_0^{+}\times(\mathbb{R}_0^{+}\backslash\{s_{\ell}\}_{\ell\in\mathbb{N}_0})\big).
\]
This section is concerned with the oscillation of solutions of the IPDEWCA
\begin{equation}
\begin{gathered}
\begin{aligned}
&p_1z(t+a,s+b)+p_2z(t+a,s)+p_3z(t,s+b)\\
&-p_4z(t,s)+p(t,s)z(t-\tau,s-\sigma)=0
\quad\text{for }(t,s)\in D \end{aligned}\\
z(t_k^{+},s)=\alpha_kz(t_k^{-},s)\quad\text{for }k\in\mathbb{N}_0 \text{ and }
 s\in\mathbb{R}_0^{+}\\
z(t,s_{\ell}^{+})=\beta_{\ell}z(t,s_{\ell}^{-})\quad\text{for }\ell\in\mathbb{N}_0
 \text{ and } t\in\mathbb{R}_0^{+}.
\end{gathered}\label{oicavdaeq1}
\end{equation}

\begin{definition}\label{oicavdadf1} \rm
A function $z:[-\tau,\infty)\times[-\sigma,\infty)\to\mathbb{R}$ is called a solution
of \eqref{oicavdaeq1} provided that each of the following conditions are satisfied:
\begin{itemize}
\item[(i)]  $z$ is continuous on each of the intervals of the form
 $(t_{k-1},t_k]\times(s_{\ell-1},s_{\ell}]$ for each $k,\ell\in\mathbb{N}$;
\item[(ii)] the limit value $z(t_k^{-},\cdot)$ for each $k\in\mathbb{N}_0$
and the limit value $z(\cdot,s_{\ell}^{-})$ for each $\ell\in\mathbb{N}_0$ exist
and are finite;
\item[(iii)] $z$ satisfies
    \begin{itemize}
    \item[(a)] the first equation in \eqref{oicavdaeq1} if 
$(t,s)\in(\mathbb{R}_0^{+}\times\mathbb{R}_0^{+})\backslash\{(t_k,s_{\ell})\}_{k,\ell\in\mathbb{N}_0}$;
    \item[(b)] the first equation in \eqref{oicavdaeq1} together with the second 
one if $(t,s)\in D$ with $t\in\{t_k\}_{k\in\mathbb{N}}$;
    \item[(c)] the first equation in \eqref{oicavdaeq1} together with the 
third one if $(t,s)\in D$ with $s\in\{s_{\ell}\}_{\ell\in\mathbb{N}}$;
    \item[(d)] the last two equations in \eqref{oicavdaeq1} 
if $(t,s)\in\{(t_k,s_{\ell})\}_{k,\ell\in\mathbb{N}_0}$.
    \end{itemize}
\end{itemize}
\end{definition}

Oscillation and nonoscillation of solutions to \eqref{oicavdaeq1} 
are defined similar to Definition~\ref{odadf2}.
Below, we quote the first result on the oscillation of IPDEWCA.

\begin{theorem}[\protect{\cite[Theorem~1]{MR2583416}}]\label{thmc}
Assume that {\rm (A1), (A4)--(A6)} and {\rm (i)} of Theorem~\ref{thmb} hold, 
and that
\begin{itemize}
\item[(i)]  $\{\alpha_k\}_{k\in\mathbb{N}_0}$ is a sequence of reals such that
$\alpha_k>1$ for all $k\in\mathbb{N}_0$;
\item[(ii)]  $\sum_{k=0}^{\infty}(\alpha_k-1)<\infty$;
\item[(iii)] 
    \begin{itemize}
    \item[(a)]  If $\upsilon\geq\nu\geq1$, then
        \begin{align*}
       &\limsup_{\substack{t\to\infty\\ s\to\infty}}\sup_{\lambda\in E}
\Big\{\lambda \prod_{i=1}^{\ell}\Big[p_1+2\frac{p_2p_3}{p_4}
 \prod_{0<t_k<t-a(i-2)}\frac{1}{\alpha_k}\Big]^{-1}\\
        &\times\prod_{j=1}^{\ell}\Big[p_4\prod_{0<t_k\leq t-a(i-1)}\alpha_k
-\lambda q(t-ai,s-bi)\Big]\\
        &\times\prod_{j=1}^{\ell-\kappa}\Big[p_4\prod_{0<t_k
\leq t-a(\ell+j-1)}\alpha_k-\lambda q(t-a\ell-aj,s-b\ell)\Big]\Big\}
<p_2^{\kappa-\ell},
        \end{align*}
  where $q$, $\upsilon$ and $\nu$ are defined as in \eqref{ocavdaeq2} 
and \eqref{ocavdaeq3}, and
\[
    E:=\big\{\lambda>0:\ p_4\prod_{0<t_k<t+a}\alpha_k-\lambda q(t,s)>0
 \text{ for all large } (t,s)\in\mathbb{R}_0^{+}\times\mathbb{R}_0^{+}\big\}.
\]

\item[(b)] If $\nu\geq\upsilon\geq1$, then
\begin{align*}
 &\limsup_{\substack{t\to\infty\\ s\to\infty}}\sup_{\lambda\in E}\Big\{\lambda
\prod_{i=1}^{\kappa}\Big[p_1+2\frac{p_2p_3}{p_4}\prod_{0<t_k<t-a(i-2)}
\frac{1}{\alpha_k}\Big]^{-1}\\
        &\times\prod_{j=1}^{\kappa}\Big[p_4\prod_{0<t_k\leq t-a(i-1)}\alpha_k
 -\lambda q(t-ai,s-bi)\Big]\\
        &\times\prod_{j=1}^{\ell-\kappa}\Big[p_4\prod_{0<t_k\leq t-a(\ell+j-1)}
\alpha_k-\lambda q(t-a\kappa,s-b\kappa-bj)\Big]\Big\}<p_3^{\kappa-\ell}.
        \end{align*}
    \end{itemize}
\end{itemize}
Then \eqref{introeq1} is oscillatory.
\end{theorem}

Now consider the following companion partial difference inequality with continuous
 arguments
\begin{equation}
\begin{split}
&p_1\bigg(\prod_{\substack{t\leq t_{i}<t+a\\ s\leq s_{j}<s+b}}\alpha_{i}\beta_{j}\bigg)
\omega(t+a,s+b)+p_2\bigg(\prod_{t\leq t_{i}<t+a}\alpha_{i}\bigg)\omega(t+a,s)\\
&+p_3\Big(\prod_{s\leq s_{j}<s+b}\beta_{j}\Big)\omega(t,s+b)-p_4\omega(t,s) \\
&+\bigg(\prod_{\substack{t-\tau\leq t_{i}<t\\ s-\sigma\leq s_{j}<s}}
\frac{1}{\alpha_{i}\beta_{j}}\bigg)p(t,s)\omega(t-\tau,s-\sigma)=0
\quad\text{for }(t,s)\in\mathbb{R}_0^{+}\times\mathbb{R}_0^{+}.
\end{split}\label{oicavdaeq2}
\end{equation}

\begin{lemma}\label{oicavdalm1}
Assume that {\rm (A1), (A4), (A7)--(A9)} hold.
\begin{itemize}
\item[(i)] If $z$ is a solution of \eqref{oicavdaeq1}, then the companion function 
$\omega$ defined by
    \begin{equation}
    \omega(t,s):=\bigg(\prod_{\substack{0\leq t_{i}<t\\ 0\leq s_{j}<s}}
\frac{1}{\alpha_{i}\beta_{j}}\bigg)z(t,s)
\quad\text{for }(t,s)\in\mathbb{R}_0^{+}\times\mathbb{R}_0^{+}\label{oicavdalm1eq1}
    \end{equation}
    is a solution of \eqref{oicavdaeq2};

\item[(ii)] If $\omega$ is a solution of \eqref{oicavdaeq2}, then the function 
$z$ defined by
    \begin{equation}
    z(t,s):=\bigg(\prod_{\substack{0\leq t_{i}<t\\ 0\leq s_{j}<s}}\alpha_{i}
\beta_{j}\bigg)\omega(t,s)\quad\text{for }(t,s)\in\mathbb{R}_0^{+}\times\mathbb{R}_0^{+}\label{oicavdalm1eq2}
    \end{equation}
    is a solution of \eqref{oicavdaeq1}.
\end{itemize}
\end{lemma}

\begin{proof}
We only give the proof of the part (i) since the part (ii)
can be proved similarly.
Assume that $z$ is a solution of \eqref{oicavdaeq1}.
First, we have to show that $\omega$ defined by \eqref{oicavdalm1eq1} is continuous.
Let $(t,s)\in(t_{k-1},t_k)\times(s_{\ell-1},s_{\ell})$ for some $k,\ell\in\mathbb{N}$.
Then using \eqref{oicavdaeq1}, we obtain
\begin{align*}
\omega(t_k^{+},s)&=\bigg(\prod_{\substack{0\leq t_{i}\leq t_k\\ 0\leq s_{j}<s}}
\frac{1}{\alpha_{i}\beta_{j}}\bigg)z(t_k^{+},s)
=\bigg(\prod_{\substack{0\leq t_{i}\leq t_k\\ 0\leq s_{j}<s}}
\frac{1}{\alpha_{i}\beta_{j}}\bigg)\alpha_kz(t_k,s)\\
&=\bigg(\prod_{\substack{0\leq t_{i}<t_k\\ 0\leq s_{j}<s}}
\frac{1}{\alpha_{i}\beta_{j}}\bigg)z(t_k,s)=\omega(t_k,s),
\end{align*}
for $k\in\mathbb{N}_0$ and
\begin{align*}
\omega(t,s_{\ell}^{+})
&=\bigg(\prod_{\substack{0\leq t_{i}<t\\ 0\leq s_{j}\leq s_{\ell}}}
 \frac{1}{\alpha_{i}\beta_{j}}\bigg)z(t,s_{\ell}^{+})
=\bigg(\prod_{\substack{0\leq t_{i}<t\\ 0\leq s_{j}\leq s_{\ell}}}
 \frac{1}{\alpha_{i}\beta_{j}}\bigg)\beta_{\ell}z(t,s_{\ell})\\
&=\bigg(\prod_{\substack{0\leq t_{i}<t_k\\ 0\leq s_{j}<s}}
 \frac{1}{\alpha_{i}\beta_{j}}\bigg)z(t,s_{\ell})=\omega(t,s_{\ell})
\end{align*}
for $\ell\in\mathbb{N}_0$.
Combining the conclusion above we have 
$\omega(t_k^{+},s_{\ell}^{+})=\omega(t_k,s_{\ell})$ for all $k,\ell\in\mathbb{N}$.
Therefore, we have just verified that $\omega$ is continuous.
Now, we show that $\omega$ solves \eqref{oicavdaeq1}.
For all $(t,s)\in\mathbb{R}_0^{+}\times\mathbb{R}_0^{+}$, we have
\begin{align*}
&p_1\bigg(\prod_{\substack{t\leq t_{i}<t+a\\ s\leq s_{j}<s+b}}
 \alpha_{i}\beta_{j}\bigg)\omega(t+a,s+b)
+p_2\bigg(\prod_{t\leq t_{i}<t+a}\alpha_{i}\bigg)\omega(t+a,s)\\
&+p_3\bigg(\prod_{s\leq s_{j}<s+b}\beta_{j}\bigg)\omega(t,s+b)-p_4\omega(t,s)
+\bigg(\prod_{\substack{t-\tau\leq t_{i}<t\\ s-\sigma\leq s_{j}<s}}
\frac{1}{\alpha_{i}\beta_{j}}\bigg)p(t,s)\omega(t-\tau,s-\sigma)
\\
&=p_1\bigg(\prod_{\substack{t\leq t_{i}<t+a\\ s\leq s_{j}<s+b}}
 \alpha_{i}\beta_{j}\bigg)\bigg(\prod_{\substack{0\leq t_{i}<t+a\\ 0\leq s_{j}<s+b}}
 \frac{1}{\alpha_{i}\beta_{j}}\bigg)z(t+a,s+b) \\
&\quad +p_2\bigg(\prod_{t\leq t_{i}<t+a}\alpha_{i}\bigg)
 \bigg(\prod_{\substack{0\leq t_{i}<t+a\\ 0\leq s_{j}<s}}
 \frac{1}{\alpha_{i}\beta_{j}}\bigg)z(t+a,s)\\
&\quad +p_3\Big(\prod_{s\leq s_{j}<s+b}\beta_{j}\Big)
 \bigg(\prod_{\substack{0\leq t_{i}<t\\ 0\leq s_{j}<s+b}}
 \frac{1}{\alpha_{i}\beta_{j}}\bigg)z(t,s+b)
 -p_4\bigg(\prod_{\substack{0\leq t_{i}<t\\ 0\leq s_{j}<s}}
 \frac{1}{\alpha_{i}\beta_{j}}\bigg)z(t,s)\\
&+\bigg(\prod_{\substack{t-\tau\leq t_{i}<t\\ s-\sigma\leq s_{j}<s}}
 \frac{1}{\alpha_{i}\beta_{j}}\bigg)
 \bigg(\prod_{\substack{0\leq t_{i}<t-\tau\\ 0\leq s_{j}<s-\sigma}}
 \frac{1}{\alpha_{i}\beta_{j}}\bigg)p(t,s)z(t-\tau,s-\sigma),
\end{align*}
which is equal to
\begin{align*}
&\bigg(\prod_{\substack{0\leq t_{i}<t\\ 0\leq s_{j}<s}}
 \frac{1}{\alpha_{i}\beta_{j}}\bigg)\Big[p_1z(t+a,s+b)+p_2z(t+a,s)+p_3z(t,s+b)\\
&-p_4z(t,s)+p(t,s)z(t-\tau,s-\sigma)\Big]=0.
\end{align*}
This completes the proof of the part (i).
\end{proof}

Next, to give a result on the oscillation we introduce the following assumption.
\begin{itemize}
\item[(A10)] $\{\alpha_k\}_{k\in\mathbb{N}_0}$ and $\{\beta_{\ell}\}_{\ell\in\mathbb{N}_0}$
 are sequences of positive reals.
\end{itemize}

\begin{theorem}\label{oicavdathm1}
Assume that {\rm (A1), (A4), (A7), (A9), (A10)} hold.
Then, \eqref{oicavdaeq1} is oscillatory if and only if so is \eqref{oicavdaeq2}.
\end{theorem}

\begin{proof}
Clearly, (A10) implies that the transforms in \eqref{oicavdalm1eq1}
and \eqref{oicavdalm1eq2} are oscillation invariant.
\end{proof}

\begin{itemize}
\item[(A11)] There exist positive constants $q_1,q_2,q_3,q_4$ such that
    \begin{gather*}
    \bigg(\prod_{\substack{t\leq t_{i}<t+a\\ s\leq s_{j}<s+b}}
\alpha_{i}\beta_{j}\bigg)\geq q_1,\quad
\Big(\prod_{t\leq t_{i}<t+a}\alpha_{i}\Big)\geq q_2,\quad 
\Big(\prod_{s\leq s_{j}<s+b}\beta_{j}\Big)\geq q_3,\\
\bigg(\prod_{\substack{t-\tau\leq t_{i}<t+a\\ s-\sigma\leq s_{j}<s+b}}
\frac{1}{\alpha_{i}\beta_{j}}\bigg)\geq q_4
\end{gather*}
    for all sufficiently large $(t,s)\in\mathbb{R}_0^{+}\times\mathbb{R}_0^{+}$;

\item[(A12)]  $p_2q_2,p_3q_3\geq p_4>0$.
\end{itemize}

\begin{lemma}\label{oicavdalm2}
Assume that {\rm (A1), (A4), (A6), (A7), (A10)--(A12)} hold.
If \eqref{oicavdaeq1} is nonoscillatory, then the following difference 
inequality with continuous arguments
\begin{equation}
\begin{split}
&p_1q_1\omega(t+a,s+b)+p_2q_2\omega(t+a,s)+p_3q_3\omega(t,s+b)\\
&-p_4\omega(t,s)+q_4q(t,s)\omega(t-\upsilon a,s-\nu b)\leq0
\quad\text{for }(t,s)\in\mathbb{R}_0^{+}\times\mathbb{R}_0^{+},
\end{split} \label{oicavdalm2eq1}
\end{equation}
where $q$, $\upsilon$ and $\nu$ are defined as in \eqref{ocavdaeq2} 
and \eqref{ocavdaeq3}, has an eventually positive solution.
\end{lemma}

\begin{proof}
The proof follows from Theorem~\ref{oicavdathm1} and (A11), we learn that
\begin{align*}
&p_1q_1\omega(t+a,s+b)+p_2q_2\omega(t+a,s)+p_3q_3\omega(t,s+b) \\
&-p_4\omega(t,s)+q_4q(t,s)\omega(t-\tau,s-\sigma) \leq0
\quad\text{for }(t,s)\in\mathbb{R}_0^{+}\times\mathbb{R}_0^{+}
\end{align*}
admits an eventually positive solution.
An application of Lemma~\ref{ocavdalm1} shows that \eqref{oicavdalm2eq1}
 has an eventually positive solution.
The proof is complete.
\end{proof}

\begin{lemma}\label{oicavdalm3}
Assume that {\rm (A1), (A4), (A6), (A7), (A10)--(A12)} hold.
Moreover assume that for some fixed $\eta,\zeta\in[0,a)\times[0,b)$, 
the partial difference inequality with discrete variables
\begin{align*}
&p_1q_1A(m+1,n+1)+p_2q_2A(m+1,n)+p_3q_3A(m,n+1) \\
&-p_4A(m,n)+q_4r_{\eta,\zeta}(m,n)A(m-\upsilon,n-\nu) \leq0
\quad\text{for }(m,n)\in\mathbb{Z}_0\times\mathbb{Z}_0,
\end{align*}
where $r$ is defined as in \eqref{ocavdaeq4}, has no eventually positive solutions.
Then, \eqref{oicavdaeq1} is oscillatory.
\end{lemma}

The proof of the above lemma follows from Theorem~\ref{ocavdathm1} and 
Lemma~\ref{oicavdalm2}.

\begin{corollary}\label{oicavdacrl1}
Assume that {\rm (A1), (A4), (A6), (A7), (A10)--(A12)} hold.
Moreover, assume that there exists $\eta,\zeta\in[0,a)\times[0,b)$ 
such that \eqref{ocavdacrl2eq1} holds if $\upsilon\geq\nu\geq1$ or 
\eqref{ocavdacrl2eq2} holds if $\nu\geq\upsilon\geq1$.
Then, \eqref{oicavdaeq1} is oscillatory.
\end{corollary}

The proof of the above corollary follows from Corollary~\ref{ocavdacrl2} 
and Lemma~\ref{oicavdalm3}.

\section{Discussion and final comments}\label{dfc}

In this section, we restrict our attention to the autonomous case for 
emphasizing the significance of results.
In \cite{MR2583416}, authors assumed that $\alpha_k>1$ for all
$k\in\mathbb{N}_0$ and $\sum_{k\in\mathbb{N}_0}(\alpha_k-1)<\infty$, which is equivalent
to the condition $\prod_{k\in\mathbb{N}_0}\alpha_k<\infty$
(see \cite[Theorem~7.4.6]{ha/ep96}).
A necessary condition for this condition is $\lim_{k\to\infty}\alpha_k=1$
(see \cite[Corollary~7.4.3]{ha/ep96}), which is strong and not required in 
our results.

\begin{example}\label{dfcex1} \rm
Consider the autonomous IDEWCA
\begin{equation}
\begin{gathered}
\begin{aligned}
&p_1z(t+a,s+b)+p_2z(t+a,s)+p_3z(t,s+b)-p_4z(t,s)\\
&+pz(t-\upsilon a,s-\nu b)=0
\quad\text{for }(t,s)\in\big(\mathbb{R}_0^{+}\backslash\mathbb{N}_0\big)
\times\big(\mathbb{R}_0^{+}\backslash\mathbb{N}_0\big)
\end{aligned}\\
z(k^{+},s)=\alpha z(k^{-},s)\quad\text{for }k\in\mathbb{N}_0 \text{ and } s\in\mathbb{R}_0^{+}\\
z(t,\ell^{+})=\beta z(t,\ell^{-})\quad\text{for }\ell\in\mathbb{N}_0 \text{ and }
 t\in\mathbb{R}_0^{+},
\end{gathered}\label{dfcex1eq1}
\end{equation}
where $p_1,p_2,p_3,p_4>0$, $p>0$, $a,b>0$, $\upsilon,\nu\in\mathbb{N}$ and 
$\alpha,\beta>0$ with $p_2\alpha,p_3\beta\geq p_4$.
Then Corollary~\ref{oicavdacrl1} implies that every solution of
 \eqref{dfcex1eq1} oscillates if
\[
p> \begin{cases}
\dfrac{(\alpha\beta)^{\nu-1}p_2^{\nu-\upsilon}}{\upsilon
 \big(p_1+2\alpha^{\upsilon}\beta^{\nu}p_2p_3\big)^{\nu}}
\Big(\dfrac{\upsilon p_4}{\upsilon+1}\Big)^{\upsilon+1}
&\text{if } \upsilon\geq\nu\\[10pt]
\dfrac{(\alpha\beta)^{\upsilon-1}p_3^{\upsilon-\nu}}
{\nu\big(p_1+2\alpha^{\upsilon}\beta^{\nu}p_2p_3\big)^{\upsilon}}
\Big(\dfrac{\nu p_4}{\nu+1}\Big)^{\nu+1}
&\text{if } \nu\geq\upsilon.
\end{cases}
\]
If we let $\beta=1$, then \eqref{dfcex1eq1} reduces to a particular case 
of the equation studied in \cite{MR2583416}.
But unfortunately all the results therein fail to apply to this equation 
because of the condition $\alpha>1$ (see (ii) of Theorem~\ref{thmc}).
\end{example}

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\end{document}