2.1 Retarded difference equations
We present new sufficient conditions for the oscillation of all solutions of (ER), under the assumption that the sequences \(\tau_{i}\) are increasing for all \(i\in \{ 1,\ldots,m \}\).
Theorem 2.1
Assume that (1.1) holds, the sequences
\(\{\tau_{i}(n) \}_{n\in\mathbb{N}_{0}}\)
are increasing for all
\(i\in \{1,\ldots ,m \}\)
and the sequence
τ
is defined by (1.4). Suppose also that for each
\(i\in \{1,\ldots,m \}\)
there exists a sequence
\(\{n_{i}(j) \}_{j\in\mathbb{N}}\)
such that
\(\lim_{j\rightarrow\infty}n_{i}(j)=\infty\),
$$ p_{k}(n)>0,\quad n\in A=\bigcap _{i=1}^{m} \biggl\{ \bigcup _{j\in\mathbb{N}} \bigl[ \tau\bigl(\tau \bigl(n_{i}(j) \bigr)\bigr),n_{i}(j) \bigr] \cap\mathbb{N} \biggr\} \neq\emptyset,\quad 1 \leq k\leq m , $$
with
$$ \liminf_{n\rightarrow\infty} \bigl\{ p_{k}(n):n\in A \bigr\} >0,\quad 1\leq k\leq m. $$
(2.1)
If, moreover,
$$ \Biggl[\prod _{i=1}^{m} \Biggl(\sum _{\ell=1}^{m}\liminf_{j\rightarrow \infty}\sum _{k=\tau_{\ell}(n(j))}^{n(j)-1}p_{i}(k) \Biggr) \Biggr] ^{1/m}> \frac{1}{e}, $$
(2.2)
where
\(n(j)=\min \{ n_{i}(j):1\leq i\leq m \} \), then all solutions of (ER) oscillate.
Proof
Assume, for the sake of contradiction, that \(\{ x(n) \} _{n\geq-w}\) is an eventually positive solution of (ER). Then there exists \(j_{0}\in\mathbb{N}\) such that
$$\begin{aligned}& p_{k}(n)>0\quad \mbox{for all } n\in\bigcap _{i=1}^{m} \bigl[\tau\bigl(\tau \bigl(n_{i}(j_{0})\bigr) \bigr),n_{i}(j_{0}) \bigr] \cap\mathbb{N},\quad 1\leq k\leq m, \\& x\bigl(\tau_{k}(n)\bigr)>0 \quad\mbox{for all } n\in\bigcap _{i=1}^{m} \bigl[ \tau \bigl(\tau\bigl(n_{i}(j_{0}) \bigr)\bigr),n_{i}(j_{0}) \bigr] \cap\mathbb{N},\quad 1\leq k\leq m. \end{aligned}$$
Therefore, by (ER) we have
$$ x(n+1)-x(n)=-\sum_{i=1}^{m}p_{i}(n) x\bigl(\tau_{i}(n)\bigr)< 0, $$
for every \(n\in\bigcap _{i=1}^{m} [\tau(\tau (n_{i}(j_{0}))),n_{i}(j_{0}) ] \cap\mathbb{N}\). This guarantees that the sequence x is strictly decreasing on \(\bigcap _{i=1}^{m} [\tau(\tau(n_{i}(j_{0}))),n_{i}(j_{0}) ]\cap\mathbb{N}\).
Set
$$ n(j_{0})=\min \bigl\{ n_{i}(j_{0}):1\leq i\leq m \bigr\} $$
and
$$ z_{i}\bigl(n(j_{0})\bigr)= \frac{x(\tau_{i}(n(j_{0})))}{x(n(j_{0}))},\quad 1\leq i\leq m , $$
(2.3)
with
$$ \varphi_{i}=\liminf_{j_{0}\rightarrow\infty}z_{i} \bigl(n(j_{0})\bigr),\quad 1\leq i\leq m. $$
(2.4)
It is obvious that
$$ z_{i}\bigl(n(j_{0})\bigr)>1 \quad\mbox{and}\quad \varphi_{i}\geq1 \quad\mbox{for } i=1,2,\ldots,m. $$
Dividing both sides of (ER) by \(x(n(j_{0}))\), we obtain
$$ \frac{\Delta x(n(j_{0}))}{x(n(j_{0}))}+\sum_{i=1}^{m}p_{i} \bigl(n(j_{0})\bigr) \frac{x(\tau_{i}(n(j_{0})))}{x(n(j_{0}))}=0, $$
or
$$ \frac{\Delta x(n(j_{0}))}{x(n(j_{0}))} +\sum_{i=1}^{m}p_{i} \bigl(n(j_{0})\bigr)z_{i}\bigl(n(j_{0})\bigr)=0. $$
(2.5)
Summing up (2.5) from \(\tau_{\rho}(n(j_{0}))\) to \(n(j_{0})-1\) for \(\rho =1,2,\ldots,m\), we find
$$ \sum_{j=\tau_{\rho}(n(j_{0}))}^{n(j_{0})-1} \frac{\Delta x(j)}{x(j)} +\sum_{i=1}^{m}\sum _{j=\tau_{\rho}(n(j_{0}))}^{n(j_{0})-1}p_{i}(j)z_{i}(j)=0. $$
(2.6)
But
$$\begin{aligned} \sum_{j=\tau_{\rho}(n(j_{0}))}^{n(j_{0})-1}\frac{\Delta x(j)}{x(j)} =&\sum _{j=\tau_{\rho}(n(j_{0}))}^{n(j_{0})-1} \biggl( \frac{x(j+1)}{x(j)} -1 \biggr) \\ \geq&\sum_{j=\tau_{\rho}(n(j_{0}))}^{n(j_{0})-1} \biggl( 1+\ln \frac{x(j+1) }{x(j)}-1 \biggr) =\ln\frac{x(n(j_{0}))}{x ( \tau_{\rho }(n(j_{0})) )}, \end{aligned}$$
or
$$ \sum_{j=\tau_{\rho}(n(j_{0}))}^{n(j_{0})-1} \frac{\Delta x(j)}{x(j)}\geq -\ln z_{\rho}\bigl(n(j_{0})\bigr). $$
(2.7)
Combining (2.6) and (2.7), we obtain
$$ -\ln z_{\rho}\bigl(n(j_{0})\bigr)+\sum _{i=1}^{m}\sum_{j=\tau_{\rho }(n(j_{0}))}^{n(j_{0})-1}p_{i}(j)z_{i}(j) \leq0, $$
or
$$ \ln z_{\rho}\bigl(n(j_{0})\bigr)\geq\sum _{i=1}^{m}\sum_{j=\tau_{\rho }(n(j_{0}))}^{n(j_{0})-1}p_{i}(j)z_{i}(j),\quad \rho=1,2,\ldots,m. $$
(2.8)
Now we will show that \(\varphi_{i}<\infty\) for \(i=1,2,\ldots,m\). Indeed, assume that \(\varphi_{i}=\infty\) for some i, \(i=1,2,\ldots,m\). By using (2.5) we have
$$ \frac{x(n(j_{0})+1)}{x(n(j_{0}))}-1+ \sum_{i=1}^{m}p_{i} \bigl(n(j_{0})\bigr)z_{i}\bigl(n(j_{0})\bigr)=0, $$
or
$$ \frac{x(n(j_{0})+1)}{x(n(j_{0}))} +\sum_{i=1}^{m}p_{i} \bigl(n(j_{0})\bigr)z_{i}\bigl(n(j_{0})\bigr)=1. $$
Consequently,
$$ \liminf_{j_{0}\rightarrow\infty} \Biggl[ \frac{x(n(j_{0})+1)}{x(n(j_{0}))} +\sum _{i=1}^{m}p_{i}\bigl(n(j_{0}) \bigr)z_{i}\bigl(n(j_{0})\bigr) \Biggr] =1 $$
and therefore
$$ \liminf_{j_{0}\rightarrow\infty}\frac{x(n(j_{0})+1)}{x(n(j_{0}))} +\sum _{i=1}^{m}\liminf_{j_{0}\rightarrow\infty} \bigl( p_{i}\bigl(n(j_{0})\bigr)z_{i} \bigl(n(j_{0})\bigr) \bigr) \leq1, $$
or
$$ \liminf_{j_{0}\rightarrow\infty}\frac{x(n(j_{0})+1)}{x(n(j_{0}))} +\sum _{i=1}^{m}\liminf_{j_{0}\rightarrow\infty }p_{i}(n(j_{0}) \liminf_{j_{0}\rightarrow\infty}z_{i}\bigl(n(j_{0})\bigr) \leq1. $$
In view of (2.1) and taking into account the fact that \(\varphi _{i}=\infty\) for some i, the above inequality is false. Hence \(\varphi_{i}<\infty\) for \(i=1,2,\ldots,m\).
Taking the limit inferiors on both sides of the above inequalities (2.8), we obtain
$$ \ln\varphi_{\rho}\geq\sum_{i=1}^{m} \varphi_{i} \Biggl( \liminf_{j_{0}\rightarrow\infty}\sum _{k=\tau_{\rho }(n(j_{0}))}^{n(j_{0})-1}p_{i}(k) \Biggr),\quad \rho=1,2, \ldots,m , $$
(2.9)
and by adding we find
$$ \sum_{i=1}^{m}\ln\varphi_{i}\geq \sum_{i=1}^{m}\varphi_{i} \Biggl( \sum_{j=1}^{m}\liminf_{j_{0}\rightarrow\infty} \sum_{k=\tau _{j}(n(j_{0}))}^{n(j_{0})-1}p_{i}(k) \Biggr). $$
Set
$$ f ( \varphi_{1},\varphi_{2},\ldots,\varphi_{m} ) \equiv \sum_{i=1}^{m}\ln\varphi_{i}- \sum_{i=1}^{m}\varphi_{i} \Biggl( \sum_{j=1}^{m}\liminf_{j_{0}\rightarrow\infty} \sum_{k=\tau _{j}(n(j_{0}))}^{n(j_{0})-1}p_{i}(k) \Biggr). $$
Clearly
$$ f ( \varphi_{1},\varphi_{2},\ldots,\varphi_{m} ) \geq0 \quad\mbox{for all } \varphi_{1},\varphi_{2},\ldots, \varphi_{m}\geq1. $$
Since
$$ \frac{\partial f}{\partial\varphi_{i}}=\frac{1}{\varphi_{i}} -\sum_{j=1}^{m} \liminf_{j_{0}\rightarrow\infty}\sum_{k=\tau _{j}(n(j_{0}))}^{n(j_{0})-1}p_{i}(k)=0, $$
for
$$ \varphi_{i}=\frac{1}{\sum_{j=1}^{m}\liminf_{j_{0}\rightarrow\infty }\sum_{k=\tau_{j}(n(j_{0}))}^{n(j_{0})-1}p_{i}(k)},\quad i=1,2,\ldots,m, $$
the function f has a maximum at the critical point
$$ \biggl( \frac{1}{\sum_{j=1}^{m}\liminf_{j_{0}\rightarrow\infty}\sum_{k=\tau _{j}(n(j_{0}))}^{n(j_{0})-1}p_{1}(k)},\ldots,\frac{1}{\sum_{j=1}^{m} \liminf_{j_{0}\rightarrow\infty}\sum_{k=\tau _{j}(n(j_{0}))}^{n(j_{0})-1}p_{m}(k)} \biggr) $$
because the quadratic form
$$ \sum_{i,j=1}^{m}\frac{\partial^{2}f}{\partial\varphi_{i}\,\partial \varphi _{j}} \alpha_{i}\alpha_{j}=-\sum_{i=1}^{m} \frac{\alpha _{i}^{2}}{\varphi _{i}^{2}}< 0. $$
Since \(f ( \varphi_{1},\varphi_{2},\ldots,\varphi_{m} ) \geq0\), the maximum of f at the critical point should be nonnegative. That is,
$$ \sum_{i=1}^{m} \Biggl[ -\ln \Biggl( \sum _{j=1}^{m}\liminf_{j_{0}\rightarrow \infty}\sum _{k=\tau_{j}(n(j_{0}))}^{n(j_{0})-1}p_{i}(k) \Biggr) \Biggr] -m\geq0, $$
i.e.,
$$ \max_{\varphi_{i}\geq1}f ( \varphi_{1},\varphi_{2}, \ldots ,\varphi _{m} ) =-\ln\prod _{i=1}^{m} \Biggl( \sum_{j=1}^{m}\liminf _{j_{0}\rightarrow\infty}\sum_{k=\tau _{j}(n(j_{0}))}^{n(j_{0})-1}p_{i}(k) \Biggr) -m. $$
Hence
$$ \prod _{i=1}^{m} \Biggl( \sum _{j=1}^{m}\liminf_{j_{0}\rightarrow \infty }\sum _{k=\tau_{j}(n(j_{0}))}^{n(j_{0})-1}p_{i}(k) \Biggr) \leq \frac{1}{e^{m}}, $$
or
$$ \Biggl[ \prod _{i=1}^{m} \Biggl( \sum _{j=1}^{m}\liminf_{j_{0}\rightarrow \infty}\sum _{k=\tau_{j}(n(j_{0}))}^{n(j_{0})-1}p_{i}(k) \Biggr) \Biggr] ^{1/m}\leq\frac{1}{e}, $$
which contradicts (2.2).
The proof of the theorem is complete. □
Theorem 2.2
Assume that (1.1) holds, the sequences
\(\{\tau_{i}(n) \}_{n\in\mathbb{N}_{0}}\)
are increasing for all
\(i\in \{1,\ldots ,m \}\)
and the sequence
τ
is defined by (1.4). Suppose also that for each
\(i\in \{ 1,\ldots,m \}\)
there exists a sequence
\(\{ n_{i}(j) \} _{j\in\mathbb{N}}\)
such that
\(\lim_{j\rightarrow\infty}n_{i}(j)=\infty\),
$$ p_{k}(n)>0,\quad n\in A=\bigcap _{i=1}^{m} \biggl\{ \bigcup _{j\in\mathbb{N}} \bigl[ \tau\bigl(\tau \bigl(n_{i}(j) \bigr)\bigr),n_{i}(j) \bigr] \cap\mathbb{N} \biggr\} \neq\emptyset,\quad 1 \leq k\leq m , $$
with
$$ \liminf_{n\rightarrow\infty} \bigl\{ p_{k}(n):n\in A \bigr\} >0,\quad 1 \leq k\leq m. $$
If, moreover,
$$\begin{aligned} &\frac{1}{m}\sum_{i=1}^{m} \liminf_{j\rightarrow\infty}\sum_{k=\tau _{i}(n(j))}^{n(j)-1}p_{i}(k) \\ &\quad{}+ \frac{2}{m}\mathop{\sum_{i< \ell}}_{i,\ell =1 }^{m} \Biggl( \liminf_{j\rightarrow\infty}\sum_{k=\tau_{\ell }(n(j))}^{n(j)-1}p_{i}(k) \times\liminf_{j\rightarrow\infty}\sum_{k=\tau _{i}(n(j))}^{n(j)-1}p_{\ell}(k) \Biggr) ^{1/2}>\frac{1}{e}, \end{aligned}$$
(2.10)
where
\(n(j)=\min \{ n_{i}(j):1\leq i\leq m \}\), then all solutions of (ER) oscillate.
Proof
Assume, for the sake of contradiction, that \(\{ x(n) \} _{n\geq-w}\) is an eventually positive solution of (ER). Then there exists \(j_{0}\in\mathbb{N}\) such that
$$\begin{aligned}& p_{k}(n)>0 \quad\mbox{for all } n\in\bigcap _{i=1}^{m} \bigl[\tau\bigl(\tau \bigl(n_{i}(j_{0})\bigr) \bigr),n_{i}(j_{0}) \bigr] \cap\mathbb{N}, \quad 1\leq k\leq m, \\& x\bigl(\tau_{k}(n)\bigr)>0\quad\mbox{for all }n\in\bigcap _{i=1}^{m} \bigl[ \tau \bigl(\tau\bigl(n_{i}(j_{0}) \bigr)\bigr),n_{i}(j_{0}) \bigr] \cap\mathbb{N},\quad 1\leq k\leq m. \end{aligned}$$
Therefore, by (ER) we have
$$ x(n+1)-x(n)=-\sum_{i=1}^{m}p_{i}(n) x\bigl(\tau_{i}(n)\bigr)< 0, $$
for every \(n\in\bigcap _{i=1}^{m} [ \tau(\tau (n_{i}(j_{0}))),n_{i}(j_{0}) ] \cap\mathbb{N}\). This guarantees that the sequence x is strictly decreasing on \(\bigcap _{i=1}^{m} [ \tau(\tau(n_{i}(j_{0}))),n_{i}(j_{0}) ] \cap\mathbb{N}\).
Taking into account the fact that \(\varphi_{i}<\infty\) for \(i=1,2,\ldots ,m \) (see Theorem 2.1), by using (2.9) and the fact that
$$ \frac{1}{e}>\frac{\ln\varphi_{\rho}}{\varphi_{\rho}},\quad \rho =1,2,\ldots,m, $$
we obtain
$$ \frac{1}{e}>\sum_{i=1}^{m} \Biggl( \liminf_{j_{0}\rightarrow\infty }\sum_{k=\tau_{\ell}(n(j_{0}))}^{n(j_{0})-1}p_{i}(k) \Biggr) \frac {\varphi _{i}}{\varphi_{\ell}},\quad \ell=1,2,\ldots,m. $$
Adding these inequalities we have
$$\begin{aligned} \frac{m}{e} \geq&\sum_{i=1}^{m} \Biggl( \liminf_{j_{0}\rightarrow\infty }\sum_{k=\tau_{i}(n(j_{0}))}^{n(j_{0})-1}p_{i}(k) \Biggr) \\ &{}+\mathop{\sum_{i< \ell}}_{i,\ell=1}^{m} \Biggl[ \Biggl( \liminf _{j_{0}\rightarrow\infty}\sum_{k=\tau_{\ell }(n(j_{0}))}^{n(j_{0})-1}p_{i}(k) \Biggr) \frac{\varphi_{i}}{\varphi _{\ell} }+ \Biggl( \liminf_{j_{0}\rightarrow\infty}\sum _{k=\tau _{i}(n(j_{0}))}^{n(j_{0})-1}p_{\ell}(k) \Biggr) \frac{\varphi_{\ell}}{ \varphi_{i}} \Biggr] \\ \geq&\sum_{i=1}^{m} \Biggl( \liminf _{j_{0}\rightarrow\infty}\sum_{k=\tau _{i}(n(j_{0}))}^{n(j_{0})-1}p_{i}(k) \Biggr) \\ &{}+2\mathop{\sum_{i<\ell}}_{i,\ell=1}^{m} \Biggl[ \Biggl( \liminf _{j_{0}\rightarrow\infty}\sum_{k=\tau_{\ell }(n(j_{0}))}^{n(j_{0})-1}p_{i}(k) \Biggr) \times \Biggl( \liminf_{j_{0}\rightarrow\infty}\sum _{k=\tau _{i}(n(j_{0}))}^{n(j_{0})-1}p_{\ell}(k) \Biggr) \Biggr] ^{1/2}, \end{aligned}$$
or
$$ \sum_{i=1}^{m}\liminf_{j_{0}\rightarrow\infty} \sum_{k=\tau _{i}(n(j_{0}))}^{n(j_{0})-1}p_{i}(k) +2\mathop{\sum_{i< l}}_{i,l=1}^{m} \sqrt{\liminf _{j_{0}\rightarrow\infty}\sum_{k=\tau_{\ell }(n(j_{0}))}^{n(j_{0})-1}p_{i}(k) \times\liminf_{j_{0}\rightarrow\infty }\sum_{k=\tau_{i}(n(j_{0}))}^{n(j_{0})-1}p_{\ell}(k)} \leq\frac{m}{e}, $$
which contradicts (2.10).
The proof of the theorem is complete. □
A slight modification in the proofs of Theorem 2.1 or Theorem 2.2 leads to the following result about retarded difference inequalities.
Theorem 2.3
Assume that all conditions of Theorem
2.1
or Theorem
2.2
hold. Then
-
(i)
the difference inequality
$$ \Delta x(n)+\sum_{i=1}^{m}p_{i}(n)x \bigl(\tau_{i}(n)\bigr)\leq0,\quad n\in\mathbb{N}_{0}, $$
has no eventually positive solutions;
-
(ii)
the difference inequality
$$ \Delta x(n)+\sum_{i=1}^{m}p_{i}(n)x \bigl(\tau_{i}(n)\bigr)\geq0, \quad n\in\mathbb{N}_{0}, $$
has no eventually negative solutions.
2.2 Advanced difference equations
Similar oscillation theorems for the (dual) advanced difference equation (EA) can be derived easily. The proofs of these theorems are omitted, since they follow a similar procedure as in Section 2.1.
Theorem 2.4
Assume that (1.2) holds, the sequences
\(\{\sigma _{i}(n) \} _{n\in\mathbb{N}}\)
are increasing for all
\(i\in \{ 1,\ldots,m \}\)
and the sequence
σ
is defined by (1.7). Suppose also that for each
\(i\in \{ 1,\ldots,m \}\)
there exists a sequence
\(\{ n_{i}(j) \} _{j\in\mathbb{N}}\)
such that
\(\lim_{j\rightarrow\infty}n_{i}(j)=\infty\),
$$ p_{k}(n)\geq0,\quad n\in B=\bigcap _{i=1}^{m} \biggl\{ \bigcup _{j\in\mathbb{N}} \bigl[ n_{i}(j),\sigma\bigl( \sigma \bigl(n_{i}(j)\bigr)\bigr) \bigr] \cap\mathbb{N} \biggr\} \neq \emptyset,\quad 1\leq k\leq m, $$
with
$$ \liminf_{n\rightarrow\infty} \bigl\{ p_{k}(n):n\in B \bigr\} >0,\quad 1\leq k\leq m. $$
(2.11)
If, moreover,
$$ \Biggl[ \prod _{i=1}^{m} \Biggl( \sum _{\ell=1}^{m}\liminf_{j\rightarrow \infty}\sum _{k=n(j)+1}^{\sigma_{\ell}(n(j))}p_{i}(k) \Biggr) \Biggr] ^{1/m}>\frac{1}{e}, $$
(2.12)
where
\(n(j)=\max \{ n_{i}(j):1\leq i\leq m \}\), then all solutions of (EA) oscillate.
Theorem 2.5
Assume that (1.2) holds, the sequences
\(\{\sigma _{i}(n) \}_{n\in\mathbb{N}}\)
are increasing for all
\(i\in \{ 1,\ldots,m \}\)
and the sequence
σ
is defined by (1.7). Suppose also that for each
\(i\in \{1,\ldots,m \}\)
there exists a sequence
\(\{ n_{i}(j) \} _{j\in\mathbb{N}}\)
such that
\(\lim_{j\rightarrow\infty}n_{i}(j)=\infty\),
$$ p_{k}(n)\geq0,\quad n\in B=\bigcap _{i=1}^{m} \biggl\{ \bigcup _{j\in\mathbb{N}} \bigl[ n_{i}(j),\sigma\bigl( \sigma \bigl(n_{i}(j)\bigr)\bigr) \bigr] \cap\mathbb{N} \biggr\} \neq \emptyset,\quad 1\leq k\leq m, $$
with
$$ \liminf_{n\rightarrow\infty} \bigl\{ p_{k}(n):n\in B \bigr\} >0,\quad 1 \leq k\leq m. $$
If, moreover,
$$\begin{aligned} &\frac{1}{m}\sum_{i=1}^{m} \liminf_{j\rightarrow\infty }\sum_{k=n(j)+1}^{\sigma_{i}(n(j))}p_{i}(k) \\ &\quad{}+ \frac{2}{m}\mathop{\sum_{i< \ell}}_{i,\ell=1}^{m} \Biggl( \liminf_{j\rightarrow\infty }\sum_{k=n(j)+1}^{\sigma_{\ell}(n(j))}p_{i}(k) \times\liminf_{j\rightarrow \infty}\sum_{k=n(j)+1}^{\sigma_{i}(n(j))}p_{\ell}(k) \Biggr) ^{1/2}>\frac{1}{e}, \end{aligned}$$
(2.13)
where
\(n(j)=\max \{ n_{i}(j):1\leq i\leq m \}\), then all solutions of (EA) oscillate.
A slight modification in the proofs of Theorems 2.4 or Theorem 2.5 leads to the following result about advanced difference inequalities.
Theorem 2.6
Assume that all conditions of Theorem
2.4
or Theorem
2.5
hold. Then
-
(i)
the difference inequality
$$ \nabla x(n)-\sum_{i=1}^{m}p_{i}(n)x \bigl(\sigma_{i}(n)\bigr)\geq0,\quad n\in\mathbb{N}, $$
has no eventually positive solutions;
-
(ii)
the difference inequality
$$ \nabla x(n)-\sum_{i=1}^{m}p_{i}(n)x \bigl(\sigma_{i}(n)\bigr)\leq0,\quad n\in\mathbb{N}, $$
has no eventually negative solutions.
2.3 Special cases
In the case where \(p_{i}\), \(i=1,2,\ldots,m\), are oscillating real constants and \(\tau_{i}\) are constant retarded arguments of the form \(\tau _{i}(n)=n-k_{i}\), (\(\sigma_{i}\) are constant advanced arguments of the form \(\sigma_{i}(n)=n+k_{i}\)), \(k_{i}\in\mathbb{N}\), \(i=1,2,\ldots,m\), (ER) ((EA)) takes the form
$$ \Delta x(n)+\sum_{i=1}^{m}p_{i}x(n-k_{i})=0,\quad n\in\mathbb{N}_{0} \qquad\Biggl( \nabla x(n)-\sum _{i=1}^{m}p_{i}x(n+k_{i})=0,\quad n \in\mathbb{N} \Biggr) . $$
(E)
For this equation, as a consequence of Theorem 2.1 (Theorem 2.4) and Theorem 2.2 (Theorem 2.5), we have the following corollary.
Corollary 2.1
Assume that
$$ \Biggl[ \prod _{i=1}^{m}p_{i} \Biggr] ^{1/m} \Biggl( \sum_{i=1}^{m}k_{i} \Biggr) >\frac{1}{e}, $$
(2.14)
or
$$ \frac{1}{m} \Biggl( \sum_{i=1}^{m} \sqrt{p_{i}k_{i}} \Biggr) ^{2}>\frac{1}{e}. $$
(2.15)
Then all solutions of (E) oscillate.