Second-order noncanonical mixed type difference equations of unstable type: new oscillation criteria
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Abstract
This paper is concerned with the oscillatory properties of the second order noncanonical difference equation with a deviating arguments of the form
\begin{equation*}
\Delta(a_n\Delta y_n) = q_ny_{\sigma(n)}.
\end{equation*}
The authors first transform the noncanonical equation into canonical form so that the discrete Kneser theorem can be applied to classify the nonoscillatory solutions into two types. Some new monotonic properties of the nonoscillatory solutions are then obtained, and they are used to eliminate certain type of nonoscillatory solutions. This leads to the development of new oscillation criteria for the equation. The results obtained are new and complement those currently existing in the literature. Examples to illustrate the importance of the main results are also presented.
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