Consider the first-order linear delay (advanced) differential equation$${x}^{\text{'}}\left(t\right)+p\left(t\right)x\left(\tau \left(t\right)\right)=0\phantom{\rule{1.0em}{0ex}}({x}^{\text{'}}\left(t\right)-q\left(t\right)x\left(\sigma \left(t\right)\right)=0),\phantom{\rule{1.0em}{0ex}}t\ge {t}_{0},$$
where $p$
$\left(q\right)$ is a continuous function of nonnegative real numbers and the argument $\tau \left(t\right)$
$\left(\sigma \right(t\left)\right)$ is not necessarily monotone. Based on an iterative technique, a new oscillation criterion is established when the well-known conditions$$\underset{t\to \infty}{lim\; sup}{\int}_{\tau \left(t\right)}^{t}p\left(s\right)\mathrm{d}s>1\phantom{\rule{1.0em}{0ex}}\left(\underset{t\to \infty}{lim\; sup}{\int}_{t}^{\sigma \left(t\right)}q\left(s\right)\mathrm{d}s>1\right)$$
and $$\underset{t\to \infty}{lim\; inf}{\int}_{\tau \left(t\right)}^{t}p\left(s\right)\mathrm{d}s>\frac{1}{\mathrm{e}}\phantom{\rule{1.0em}{0ex}}\left(\underset{t\to \infty}{lim\; inf}{\int}_{t}^{\sigma \left(t\right)}q\left(s\right)\mathrm{d}s>\frac{1}{\mathrm{e}}\right)$$
are not satisfied. An example, numerically solved in MATLAB, is also given to illustrate the applicability and strength of the obtained condition over known ones.

Sufficient oscillation conditions involving $lim\; sup$ and $lim\; inf$ for first-order differential equations with non-monotone deviating arguments and nonnegative coefficients are obtained. The results are based on the iterative application of the Grönwall inequality. Examples, numerically solved in MATLAB, are also given to illustrate the applicability and strength of the obtained conditions over known ones.

This paper is concerned with the oscillatory behavior of first-order nonlinear difference equations with variable deviating arguments. The corresponding difference equations of both retarded and advanced type are studied. Examples illustrating the results are also given.

Consider the difference equation $$\Delta x\left(n\right)+\sum _{i=1}^{m}{p}_{i}\left(n\right)x\left({\tau}_{i}\left(n\right)\right)=0,\phantom{\rule{1.0em}{0ex}}n\ge 0\phantom{\rule{1.0em}{0ex}}\left[\nabla x\left(n\right)-\sum _{i=1}^{m}{p}_{i}\left(n\right)x\left({\sigma}_{i}\left(n\right)\right)=0,\phantom{\rule{1.0em}{0ex}}n\ge 1\right],$$
where $\left({p}_{i}\left(n\right)\right)$, $1\le i\le m$ are sequences of nonnegative real numbers, ${\tau}_{i}\left(n\right)$ [${\sigma}_{i}\left(n\right)$], $1\le i\le m$ are general retarded (advanced) arguments and $\Delta $ [$\nabla $] denotes the forward (backward) difference operator $\Delta x\left(n\right)=x(n+1)-x\left(n\right)$ [$\nabla x\left(n\right)=x\left(n\right)-x(n-1)$]. New oscillation criteria are established when the well-known oscillation conditions $$\underset{n\to \infty}{lim\; sup}\sum _{i=1}^{m}\sum _{j=\tau \left(n\right)}^{n}{p}_{i}\left(j\right)>1\phantom{\rule{1.0em}{0ex}}\left[\underset{n\to \infty}{lim\; sup}\sum _{i=1}^{m}\sum _{j=n}^{\sigma \left(n\right)}{p}_{i}\left(j\right)>1\right]$$
and $$\underset{n\to \infty}{lim\; inf}\sum _{i=1}^{m}\sum _{j={\tau}_{i}\left(n\right)}^{n-1}{p}_{i}\left(j\right)>\frac{1}{\mathrm{e}}\phantom{\rule{1.0em}{0ex}}\left[\underset{n\to \infty}{lim\; inf}\sum _{i=1}^{m}\sum _{j=n+1}^{{\sigma}_{i}\left(n\right)}{p}_{i}\left(j\right)>\frac{1}{\mathrm{e}}\right]$$
are not satisfied. Here $\tau \left(n\right)={max}_{1\le i\le m}{\tau}_{i}\left(n\right)$
$[\sigma \left(n\right)={min}_{1\le i\le m}{\sigma}_{i}\left(n\right)]$. The results obtained essentially improve known results in the literature. Examples illustrating the results are also given.

We obtain some new sufficient conditions for the oscillation of the solutions of the second-order quasilinear difference equations with delay and advanced neutral terms. The results established in this paper are applicable to equations whose neutral coefficients are unbounded. Thus, the results obtained here are new and complement some known results reported in the literature. Examples are also given to illustrate the applicability and strength of the obtained conditions over the known ones.

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