\documentclass[twoside]{article}
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\pagestyle{myheadings}
\markboth{\hfil Kamanev-type oscillation criteria \hfil EJDE--2002/68}
{EJDE--2002/68\hfil Samir H. Saker \hfil}

\begin{document}

\title{\vspace{-1in}
\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 68, pp. 1--9. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)} \vspace{\bigskipamount} \\
%
 Kamenev-type oscillation criteria for forced Emden-Fowler superlinear
 difference equations 
%
\thanks{\emph{Mathematics Subject Classifications:} 39A10. \hfil\break\indent
{\em Key words:} Oscillation, Emden-Fowler difference equations. 
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted March 04, 2002. Published July 22, 2002.} }
\date{}

\author{Samir H. Saker}
\maketitle

\begin{abstract}
  Using Riccati transformation techniques, we establish 
  oscillation criteria for forced second-order Emden-Fowler 
  superlinear difference equations. Our criteria are discrete 
  analogues of the criteria used for differential equations by 
  Kamanev \cite{k1}.
\end{abstract}

\newtheorem{theorem}{Theorem}[section] 
\newtheorem{corollary}[theorem]{Corollary} 
\numberwithin{equation}{section}

\section{Introduction}

Consider the forced second-order nonlinear difference equation 
\begin{equation}
\Delta ^{2}x_{n-1}+q_{n}x_{n}^{\gamma }=g_{n},  \label{1}
\end{equation}
where $\gamma $ is quotient of positive odd integers, $n$ is an integer in
the set $\mathbb{N}=\{1,2\dots \}$, $\{q_{n}\}_{n=1}^{\infty }$ and 
$\{g_{n}\}_{n=1}^{\infty }$ are sequences of positive real numbers, $\Delta $
denotes the forward difference operator $\Delta x_{n}=x_{n+1}-x_{n}$ and 
$\Delta ^{2}x_{n}=\Delta (\Delta x_{n})$. In the case $\gamma >1$, Equation 
\eqref{1} is the prototype of a wide class of nonlinear difference equations
called Emden-Fowler superlinear difference equations.

In recent years there has been an increasing interest in the asymptotic
behavior of second-order difference equations, see, e.g., the monographs 
\cite{a1,a2}. Following this trend, we study the oscillations of \eqref{1}.
It is interesting to study \eqref{1} because, it is the discrete version of
the second order Emden-Fowler differential equation that has several
physical applications \cite{w1}.

We consider only nontrivial solutions of \eqref{1}; i.e., solutions such
that for every $i \in \mathbb{N}$, $\sup\{| x_{n}| : n\geq i\}>0$. A
solution $\{x_{n}\}$ of \eqref{1} is said to be oscillatory if for every 
$n_{1}\geq 1$ there exists an $n\geq n_{1}$ such that $x_{n}x_{n+1}\leq 0$,
otherwise it is non-oscillatory.

The oscillation of forced second order difference equations has been the
subject of many publications; see for example \cite{g1,h1,m1,p1,t1,w3,z1}
and references therein. In [3], the authors considered the linear forced
difference equation and given some sufficient conditions for oscillation. In 
\cite{p1,w3}, the authors considered the nonlinear forced difference
equations and established some conditions for oscillation. Unfortunately,
the oscillation criteria in \cite{g1,p1,w3} impose assumptions on the
unknown solutions, which diminishes the applicability of the criteria. In 
\cite{z1}, the authors considered the forced nonlinear delay difference
equation when $\{q_{n}\}_{n=0}^{\infty }$ is a nonnegative sequence with a
positive subsequence, and there exists a sequence $\{G_{n}\}_{n=0}^{\infty }$
such that $\Delta ^{2}G_{n}=g_{n}$ to obtain sufficient conditions for
oscillations.

In the continuous case, the differential equation 
\begin{equation}
x^{\prime\prime}(t)+q(t)f(x(t))=0,\quad t \geq t_{0}  \label{2}
\end{equation}
has been studied by many authors; see the survey papers \cite{k2,w2} which
give over 300 references. In Kamenev \cite{k1}, the average function 
\begin{equation}
A_{\lambda }(t)=\frac{1}{t^{\lambda }}\int_{t_{0}}^{t}(t-s)^{\lambda
}q(s)ds,\quad \lambda \geq 1  \label{3}
\end{equation}
plays a crucial role in the oscillation criteria for \eqref{2}. Philos \cite
{p2} improved Kamenev's result by proving the following result: Suppose
there exist continuous functions $H$ and $h$ defined from $D=\{(t,s):t\geq
s\geq t_{0}\}$ to $\mathbb{R}$ such that:\newline
(i) $H(t,t)=0,$ for $t\geq t_{0}$\newline
(ii) $H(t,s)>0$ for $t>s\geq t_{0}$, and $H$ has a continuous and
non-positive partial derivative on $D$ with respect to the second variable
and satisfies 
\begin{equation}
-\frac{\partial H(t,s)}{\partial s}=h(t,s)\sqrt{H(t,s)}\geq 0.  \label{4}
\end{equation}
Further, suppose that 
\begin{equation}
\lim_{t\to \infty }\frac{1}{H(t,t_{0})}\int_{t_{0}}^{t}[H(t,s)q(s)-\frac{1}{4%
}h^{2}(t,s)]\,ds=\infty .  \label{5}
\end{equation}
Then every solution of \eqref{2} oscillates. \smallskip

Using Riccati transformation techniques, we establish some new oscillation
criteria, for \eqref{1}, that are discrete analogues of \eqref{3} and %
\eqref{5}. Our results generalized and extended the conditions \eqref{3} and %
\eqref{5} to the discrete case and improve the results presented in \cite
{g1,p1,w3,z1}.

\section{Main Result}

\begin{theorem} \label{thm1}
Assume that there exists a positive sequence $\{\rho_{n}\}_{n=1}^{\infty }$
 such that for every positive number $\lambda \geq 1$,
\begin{equation}
\lim_{m\to \infty }\sup \frac{1}{m^{\lambda }}\sum
_{n=1}^{m-1}(m-n)^{\lambda }\Big[ \rho_{n}Q_{n}-\frac{\left( \rho
_{n+1}\right) ^{2}}{4\rho_{n}}\Big( \frac{\Delta \rho_{n}}{\rho_{n+1}}-
\frac{\lambda (m-n-1)^{\lambda -1}}{(m-n)^{\lambda }}\Big) ^{2}\Big]
=\infty  \label{6}
\end{equation}
where
\begin{equation*}
Q_{n}=\gamma \big( \frac{1}{\gamma -1}\big) ^{1-\frac{1}{\gamma }}\left(
q_{n}\right) ^{\frac{1}{\gamma }}(g_{n})^{1-\frac{1}{\gamma }}.
\end{equation*}
Then every unbounded solution of \eqref{1} oscillates.
\end{theorem}

\paragraph{Proof}

Suppose to the contrary that $\{x_{n}\}_{n=1}^{\infty }$ is an unbounded
non-oscillatory solution of \eqref{1}. First, we may assume that $\left\{
x_{n}\right\} $ is a positive solution of \eqref{1} for $n\geq n_{1}\geq 1$.
Define the sequence $\{w_{n}\}$ by 
\begin{equation}
w_{n}=\rho _{n}\frac{\Delta x_{n-1}}{x_{n}}.  \label{7}
\end{equation}
Then in view of \eqref{1}, we have 
\begin{equation*}
\Delta w_{n}=-[q_{n}x_{n}^{\gamma -1}-\frac{g_{n}}{x_{n}}]+\frac{\Delta \rho
_{n}}{\rho _{n+1}}w_{n+1}-\frac{\rho _{n}}{x_{n}x_{n+1}}.
\end{equation*}
Since $x_{n}$ is positive and unbounded, there exists $n_{2}\geq n_{1}$ such
that $\Delta x_{n}\geq 0,$ for $n\geq n_{2}$, and $x_{n+1}\geq x_{n}$, so
that 
\begin{equation}
\Delta w_{n}\leq -[q_{n}x_{n}^{\gamma -1}-\frac{g_{n}}{x_{n}}]+\frac{\Delta
\rho _{n}}{\rho _{n+1}}w_{n+1}-\frac{\rho _{n}}{(\rho _{n+1})^{2}}%
w_{n+1}^{2}.  \label{8}
\end{equation}
Set 
\begin{equation*}
f(x)=q_{n}x^{\gamma -1}-\frac{g_{n}}{x}\,.
\end{equation*}
Using differential calculus, we see that 
\begin{equation*}
f(x)\geq \gamma \big(\frac{1}{\gamma -1}\big)^{1-\frac{1}{\gamma }}(q_{n})^{%
\frac{1}{\gamma }}(g_{n})^{1-\frac{1}{\gamma }},
\end{equation*}
this and \eqref{8} imply 
\begin{equation}
\Delta w_{n}\leq -Q_{n}+\frac{\Delta \rho _{n}}{\rho _{n+1}}w_{n+1}-\frac{%
\rho _{n}}{\left( \rho _{n+1}\right) ^{2}}w_{n+1}^{2}.  \label{9}
\end{equation}
Therefore, 
\begin{multline}
\sum_{n=n_{2}}^{m-1}(m-n)^{\lambda }\rho _{n}Q_{n}  \label{10} \\
\leq -\sum_{n=n_{2}}^{m-1}(m-n)^{\lambda }\Delta
w_{n}+\sum_{n=n_{2}}^{m-1}(m-n)^{\lambda }\frac{\Delta \rho _{n}}{\rho _{n+1}%
}w_{n+1}-\sum_{n=n_{2}}^{m-1}(m-n)^{\lambda }\frac{\rho _{n}}{\rho _{n+1}^{2}%
}w_{n+1}^{2},
\end{multline}
Now, after summing by parts, we have 
\begin{equation*}
\sum_{n=n_{2}}^{m-1}(m-n)^{\lambda }\Delta w_{n}=-(m-n_{2})^{\lambda
}w_{n_{2}}-\sum_{n=n_{2}}^{m-1}w_{n+1}\Delta _{2}(m-n)^{\lambda },
\end{equation*}
where $\Delta _{2}(m-n)^{\lambda }=(m-n-1)^{\lambda }-(m-n)^{\lambda }$.
Then 
\begin{equation*}
\sum_{n=n_{2}}^{m-1}(m-n)^{\lambda }\Delta w_{n}=-(m-n_{2})^{\lambda
}w_{n_{2}}+\sum_{n=n_{2}}^{m-1}w_{n+1}((m-n)^{\lambda }-(m-n-1)^{\lambda }).
\end{equation*}
Using the inequality, $x^{\beta }-y^{\beta }\geq \beta y^{\beta -1}(x-y)$
for all x$\geq y>0$ and $\beta \geq 1,$ we obtain 
\begin{equation*}
\sum_{n=n_{2}}^{m-1}(m-n)^{\lambda }\Delta w_{n}\geq -(m-n_{2})^{\lambda
}w_{n_{2}}+\sum_{n=n_{2}}^{m-1}\lambda w_{n+1}(m-n-1)^{\lambda -1}.
\end{equation*}
Substitute this expression in \eqref{10} to obtain 
\begin{align*}
\sum_{n=n_{2}}^{m-1}(m-n)^{\lambda }\rho _{n}Q_{n} \leq &(m-n_{2})^{\lambda
}w_{n_{2}}-\sum_{n=n_{2}}^{m-1}\lambda w_{n+1}(m-n-1)^{\lambda -1} \\
&+\sum_{n=n_{2}}^{m-1}(m-n)^{\lambda }\frac{\Delta \rho _{n}}{\rho _{n+1}}%
w_{n+1}-\sum_{n=n_{2}}^{m-1}(m-n)^{\lambda }\frac{\rho _{n}}{\rho _{n+1}^{2}}%
w_{n+1}^{2}.
\end{align*}
Then 
\begin{eqnarray*}
\lefteqn{\frac{1}{m^{\lambda }}\sum_{n=n_{2}}^{m-1}(m-n)^{\lambda }\rho
_{n}Q_{n}} \\
&\leq &(\frac{m-n_{2}}{m})^{\lambda }w_{n_{2}} \\
&&-\frac{1}{m^{\lambda }}\sum_{n=n_{2}}^{m-1}(m-n)^{\lambda }\Big[\frac{\rho
_{n}}{\rho _{n+1}^{2}}w_{n+1}^{2}-\big(\frac{\Delta \rho _{n}}{\rho _{n+1}}-%
\frac{\lambda (m-n-1)^{\lambda -1}}{(m-n)^{\lambda }}\big)w_{n+1}\Big] \\
&=&(\frac{m-n_{2}}{m})^{\lambda }w_{n_{2}} \\
&&-\frac{1}{m^{\lambda }}\sum_{n=n_{2}}^{m-1}(m-n)^{\lambda }\Big[\frac{%
\sqrt{\rho _{n}}}{\rho _{n+1}}w_{n+1}-\frac{\rho _{n+1}}{2\sqrt{\rho _{n}}}%
\big(\frac{\Delta \rho _{n}}{\rho _{n+1}}-\frac{\lambda (m-n-1)^{\lambda -1}%
}{(m-n)^{\lambda }}\big)\Big]^{2} \\
&&+\frac{1}{m^{\lambda }}\sum_{n=n_{2}}^{m-1}(m-n)^{\lambda }\frac{\left(
\rho _{n+1}\right) ^{2}}{4\rho _{n}}\big(\frac{\Delta \rho _{n}}{\rho _{n+1}}%
-\frac{\lambda (m-n-1)^{\lambda -1}}{(m-n)^{\lambda }}\big)^{2},
\end{eqnarray*}
which implies 
\begin{multline*}
\frac{1}{m^{\lambda }}\sum_{n=n_{2}}^{m-1}\kappa (m-n)^{\lambda }\rho
_{n}Q_{n} \\
<(\frac{m-n_{2}}{m})^{\lambda }w_{n_{2}}+\frac{1}{m^{\lambda }}%
\sum_{n=n_{2}}^{m-1}(m-n)^{\lambda }\frac{(\rho _{n+1})^{2}}{4\rho _{n}}\big(%
\frac{\Delta \rho _{n}}{\rho _{n+1}}-\frac{\lambda (m-n-1)^{\lambda -1}}{%
(m-n)^{\lambda }}\big)^{2}\,.
\end{multline*}
Then 
\begin{multline*}
\frac{1}{m^{\lambda }}\sum_{n=n_{2}}^{m-1}(m-n)^{\lambda }\Big[\rho
_{n}Q_{n}-\frac{\left( \rho _{n+1}\right) ^{2}}{4\rho _{n}}\left( \frac{%
\Delta \rho _{n}}{\rho _{n+1}}-\frac{\lambda (m-n-1)^{\lambda -1}}{%
(m-n)^{\lambda }}\right) ^{2}\Big] \\
<(\frac{m-n_{2}}{m})^{\lambda }w_{n_{2}},
\end{multline*}
which yields 
\begin{equation*}
\lim_{m\rightarrow \infty }\frac{1}{m^{\lambda }}\sum_{n=n_{2}}^{m-1}(m-n)^{%
\lambda }\Big[\rho _{n}Q_{n}-\frac{\left( \rho _{n+1}\right) ^{2}}{4\rho _{n}%
}\left( \frac{\Delta \rho _{n}}{\rho _{n+1}}-\frac{\lambda (m-n-1)^{\lambda
-1}}{(m-n)^{\lambda }}\right) ^{2}\Big]<\infty ,
\end{equation*}
which contradicts \eqref{6}. Next, we consider the case when $x_{n}<0$ for 
$n\geq n_{1}$. We use the transformation $y_{n}=-x_{n}$ is a positive
solution of the equation $\Delta ^{2}y_{n-1}+q_{n}y_{n}^{\gamma }=-g_{n}$.
Define the sequence $\{w_{n}\}$ by 
\begin{equation}
w_{n}=\rho _{n}\frac{\Delta y_{n-1}}{x_{n}}.  \label{11}
\end{equation}
then, $w_{n}>0$ and satisfies 
\begin{equation}
\Delta w_{n}\leq -[q_{n}x_{n}^{\gamma -1}+\frac{g_{n}}{x_{n}}]+\frac{\Delta
\rho _{n}}{\rho _{n+1}}w_{n+1}-\frac{\rho _{n}}{(\rho _{n+1})^{2}}%
w_{n+1}^{2}.  \label{12}
\end{equation}
Set 
\begin{equation*}
F(x)=q_{n}x^{\gamma -1}+\frac{g_{n}}{x}.
\end{equation*}
Using differential calculus, we see that 
\begin{equation*}
F(x)\geq \gamma \big(\frac{1}{\gamma -1}\big)^{1-\frac{1}{\gamma }}(q_{n})^{%
\frac{1}{\gamma }}(g_{n})^{1-\frac{1}{\gamma }}.
\end{equation*}
and then \eqref{9} holds. The remainder of the proof is similar to that of
the proof of the first part and hence is omitted. The proof is complete
\hfill $\diamondsuit $

\begin{corollary} \label{coro1}
Assume that all assumptions in Theorem \ref{thm1} hold, except the
condition \eqref{6} which is replaced by
\begin{equation*}
\lim_{m\to \infty }\sup \frac{1}{m^{\lambda }}\sum
_{n=1}^{m-1}(m-n)^{\lambda }\rho_{n}Q_{n}=\infty ,
\end{equation*}
and
\begin{equation*}
\lim_{m\to \infty }\frac{1}{m^{\lambda }}\sum_{n=1}^{m-1}(m-n)%
\frac{\left( \rho_{n+1}\right) ^{2}}{\rho_{n}}\big( \frac{\Delta \rho_{n}%
}{\rho_{n+1}}-\frac{\lambda (m-n-1)^{\lambda -1}}{(m-n)^{\lambda }}\big)
^{2}<\infty\,.
\end{equation*}
Then every unbounded solution of \eqref{1} oscillates.
\end{corollary}

\begin{theorem} \label{thm2}
Assume that there exists a positive sequence $\{\rho_{n}\}_{n=1}^{\infty }$.
Furthermore, we assume that there exists a double sequence $\{H_{m,n}:m\geq
n\geq 0\}$\ such that (i) $H_{m,m}=0$ for $m\geq 0$
(ii) $H_{m,n}>0$ for $m>n\geq 0$,
(iii) $\Delta_{2}H_{m,n}=H_{m,n+1}-H_{m,n}\leq 0$ for $m\geq n\geq 0$. If
\begin{equation}
\limsup_{m\to \infty }\frac{1}{H_{m,0}}\sum_{n=1}^{m-1}\Big[
H_{m,n}\rho_{n}Q_{n}-\frac{\rho_{n+1}^{2}}{4\rho_{n}}\big( h_{m,n}
-\frac{\Delta \rho_{n}}{\rho_{n+1}}\sqrt{H_{m,n}}\big) ^{2}\Big]
=\infty , \label{13}
\end{equation}
where
\begin{equation*}
h_{m,n}=-\frac{\Delta_{2}H_{m,n}}{\sqrt{H_{m,n}}},\quad m>n\geq 0,
\end{equation*}
then every unbounded solution of \eqref{1} oscillates.
\end{theorem}

\paragraph{Proof}

We proceed as in the proof of Theorem \ref{thm1}, we may assume that %
\eqref{1} has an unbounded non-oscillatory solution $\{x_{n}\}_{n=1}^{\infty
}$ such that $x_{n}>0$ for $n\geq n_{1}$. Define $\{w_{n}\}$ by \eqref{7} as
before, then $w_{n}>0$ and satisfies \eqref{9} for all $n\geq n_{2}$.
Therefore, 
\begin{multline*}
\sum_{n=n_{2}}^{m-1}H_{m,n}\rho _{n}Q_{n} \\
\leq -\sum_{n=n_{2}}^{m-1}H_{m,n}\Delta w_{n}+\sum_{n=n_{2}}^{m-1}H_{m,n}%
\frac{\Delta \rho _{n}}{\rho _{n+1}}w_{n+1}-\sum_{n=n_{2}}^{m-1}H_{m,n}\frac{%
\rho _{n}}{\rho _{n+1}^{2}}w_{n+1}^{2},
\end{multline*}
which yields, after summing by parts, 
\begin{eqnarray*}
\lefteqn{\sum_{n=n_{2}}^{m-1}H_{m,n}\rho _{n}Q_{n}} \\
&\leq &H_{m,n_{2}}w_{n_{2}}+\sum_{n=n_{2}}^{m-1}w_{n+1}\Delta
_{2}H_{m,n}+\sum_{n=n_{2}}^{m-1}H_{m,n}\frac{\Delta \rho _{n}}{\rho _{n+1}}
w_{n+1}\\
&&-\sum_{n=n_{2}}^{m-1}H_{m,n}\frac{\rho _{n}}{\rho _{n+1}^{2}}
w_{n+1}^{2} \\
&=&H_{m,n_{2}}w_{n_{2}}-\sum_{n=n_{2}}^{m-1}h_{m,n}\sqrt{H_{m,n}}%
w_{n+1}+\sum_{n=n_{2}}^{m-1}H_{m,n}\frac{\Delta \rho _{n}}{\rho _{n+1}}
w_{n+1}\\
&&-\sum_{n=n_{2}}^{m-1}H_{m,n}\frac{\rho _{n}}{\rho _{n+1}^{2}}
w_{n+1}^{2} \\
&=&H_{m,n_{2}}w_{n_{2}} \\
&&-\sum_{n=n_{2}}^{m-1}\Big[\frac{\sqrt{H_{m,n}\rho _{n}}}{\rho _{n+1}}%
w_{n+1}+\frac{\rho _{n+1}}{2\sqrt{H_{m,n}\rho _{n}}}\Big(h_{m,n}\sqrt{H_{m,n}%
}-\frac{\Delta \rho _{n}}{\rho _{n+1}}H_{m,n}\Big)\Big]^{2} \\
&&+\frac{1}{4}\sum_{n=n_{2}}^{m-1}\frac{\left( \rho _{n+1}\right) ^{2}}{\bar{%
\rho}_{n}}\big(h_{m,n}-\frac{\Delta \rho _{n}}{\rho _{n+1}}\sqrt{H_{m,n}}%
\big)^{2}.
\end{eqnarray*}
Then 
\begin{equation*}
\sum_{n=n_{2}}^{m-1}\Big[H_{m,n}\rho _{n}Q_{n}-\frac{\rho _{n+1}^{2}}{4\rho
_{n}}\big(h_{m,n}-\frac{\Delta \rho _{n}}{\rho _{n+1}}\sqrt{H_{m,n}}\big)^{2}%
\Big]<H_{m,n_{2}}w_{n_{2}}\leq H_{m,0}w_{n_{2}},
\end{equation*}
which implies 
\begin{equation*}
\sum_{n=1}^{m-1}\Big[H_{m,n}\rho _{n}Q_{n}-\frac{\rho _{n+1}^{2}}{4\rho _{n}}%
\big(h_{m,n}-\frac{\Delta \rho _{n}}{\rho _{n+1}}\sqrt{H_{m,n}}\big)^{2}\Big]%
<H_{m,0}\sum_{n=1}^{n_{2}-1}\rho _{n}Q_{n}+H_{m,0}w_{n_{2}}.
\end{equation*}
Hence 
\begin{equation*}
\limsup_{m\rightarrow \infty }\frac{1}{H_{m,0}}\sum_{n=1}^{m-1}\Big[%
H_{m,n}\rho _{n}Q_{n}-\frac{\rho _{n+1}^{2}}{4\rho _{n}}\big(h_{m,n}-\frac{%
\Delta \rho _{n}}{\rho _{n+1}}\sqrt{H_{m,n}}\big)^{2}\Big]<\infty ,
\end{equation*}
which contradicts \eqref{13}. Next, we consider the case when $x_{n}<0$ for 
$n\geq n_{1}$. We use the transformation $y_{n}=-x_{n}$ is a positive
solution of the equation $\Delta ^{2}y_{n-1}+q_{n}y_{n}^{\gamma }=-g_{n}$.
Define the sequence $\{w_{n}\}$ by \eqref{11}, then \eqref{12} holds. The
remainder of the proof is similar to that of the proof of the first case and
hence is omitted. The proof is complete. \hfill $\diamondsuit $

\begin{corollary} \label{coro2}
Assume that all the assumptions of Theorem \ref{thm2} hold, except the
condition \eqref{12} which is replaced by
\begin{equation*}
\lim_{m\to \infty }\sup \frac{1}{H_{m,0}}\sum_{n=0}^{m-1}H_{m,n}\rho_{n}Q_{n}
=\infty ,
\end{equation*}
and
\begin{equation*}
\lim_{m\to \infty }\sup \frac{1}{H_{m,0}}
\sum_{n=1}^{m-1}\frac{\rho_{n+1}^{2}}{4\rho_{n}}\big( h_{m,n}-
\frac{\Delta \rho_{n}}{\rho_{n+1}}\sqrt{H_{m,n}}\big) ^{2}<\infty\,.
\end{equation*}
Then every unbounded solution of \eqref{1} oscillates.
\end{corollary}

By choosing the sequence $\{H_{m,n}\}$ appropriately, we can derive several
oscillation criteria for \eqref{1}. For instance, consider the double
sequence 
\begin{equation}
H_{m,n}=\Big( \ln \big(\frac{m+1}{n+1}\big)\Big) ^{\lambda },\quad \lambda
\geq 1,\;m\geq n\geq 0.  \label{13b}
\end{equation}
Then $H_{m,m}=0$ for $m\geq 0$ and $H_{m,n}>0$ and $\Delta_{2}H_{m,n}\leq 0$
for $m>n\geq 0$. Hence we have the following result.

\begin{corollary} \label{coro3}
Assume that the assumptions in Theorem \ref{thm2} hold, except the condition \eqref{12}
which is replaced by
\begin{equation}
\limsup_{m\to \infty }\frac{1}{\ln ^{\lambda }(m+1)}
\sum_{n=0}^{m-1}\Big[ \Big( \ln \frac{m+1}{n+1}\Big) ^{\lambda
}\rho_{n}Q_{n}-B_{m,n}\Big] =\infty  \label{14}
\end{equation}
where
\begin{equation*}
B_{m,n}=\frac{\rho_{n+1}^{2}}{4\rho_{n}}\Big( \frac{\lambda }{n+1}\big(
\ln \frac{m+1}{n+1}\big) ^{\frac{\lambda -2}{2}}-\frac{\Delta \rho_{n}}{
\rho_{n+1}}\sqrt{\big( \ln \frac{m+1}{n+1}\big) ^{\lambda }}\Big) ^{2}
\end{equation*}
for every positive number $\lambda \geq 1$. Then every unbounded solution of
\eqref{1} oscillates.
\end{corollary}

Another choice for a sequence is 
\begin{equation*}
H_{m,n}=\phi (m-n),\quad m\geq n\geq 0,
\end{equation*}
where $\phi :[0,\infty )\to \lbrack 0,\infty )$ is a continuously
differentiable function which satisfies $\phi (0)=0$, $\phi (u)>0$, and 
$\phi^{\prime }(u)\geq 0$ for $u>0$.

Yet another choice for a sequence is 
\begin{equation*}
H_{m,n}=(m-n)^{(\lambda )}\quad \lambda >2,\;m\geq n\geq 0,
\end{equation*}
where $(m-n)^{(\lambda)}=(m-n)(m-n+1)\dots (m-n+\lambda -1)$ and 
\begin{equation*}
\Delta_{2}(m-n)^{(\lambda )}=(m-n-1)^{(\lambda )}-(m-n)^{(\lambda
)}=-\lambda (m-n)^{(\lambda -1)}.
\end{equation*}

For these two sequences we can state corollaries similar to the one above.
Note that our results can be extended to the equation 
\begin{equation*}
\Delta (a_{n}\Delta x_{n})+q_{n}x_{n}^{\gamma }=g_{n}
\end{equation*}
where \{$a_{n}\}_{n=1}^{\infty }$ is a sequence of positive real numbers.
However, our results can not be applied in the case when $\gamma =1$ and
also it remains an open problem to give sufficient conditions for the
oscillation of all bounded solutions in this case.

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\noindent \textsc{Samir H. Saker}\newline
Department of Mathematics, Faculty of Science,\newline
Mansoura University, Mansoura, 35516, Egypt\newline
e-mail shsaker@mum.mans.edu.eg\newline
Faculty of Mathematics and Computer Science, \newline
Adam Mickiewicz University, \newline
Matejki 48/49, 60-769 Poznan, Poland \newline
e-mail shsaker@amu.edu.pl

\bigskip
{\bf Addendum posted by a managing editor on June 13, 2012.}
\medskip

A reader informed us of two inaccuracies in this article:
In the proof of Theorem 2.1, the statement
\begin{quote}
Since $x_n$ is positive and unbounded, there exists $n_2\geq n_1$ such that
$\Delta x_n\geq 0$ for $n\geq n_2$
\end{quote}
is incorrect. The sequence $x_n=n+(-1){n+1}$ provides a counterexample.
Also in the same proof, the statement
\begin{equation*}
f(x)\geq \gamma \big(\frac{1}{\gamma -1}\big)^{1-\frac{1}{\gamma }}
(q_{n})^{\frac{1}{\gamma }}(g_{n})^{1-\frac{1}{\gamma }}
\end{equation*}
is incorrect.

Regarding these inaccuracies, Prof. Saker informed us that
the results in this paper have been corrected 
and improved in later publications, by the same author.

\end{document}