Oscillation Properties for First-Order Non-Linear Advanced Difference Equations
D. Palanisamy
Department of Mathematics, Government Arts and Science College, Idappadi-637 102, Salem (Dt.), (Affiliated to Periyar University, Salem - 636 011), Tamil Nadu, India.
A. Murugesan *
Department of Mathematics, Government Arts College (Autonomous), Salem-636 007, (Affiliated to Periyar University, Salem-636 011), Tamil Nadu, India.
*Author to whom correspondence should be addressed.
Abstract
This paper investigates the qualitative behaviour of the following class of first-order non-linear advanced difference equations featuring multiple non-monotone advanced arguments
\[\begin{equation*}
\Delta\omega(\vartheta)-\sum_{i=1}^{k}\rho_i(\vartheta)\omega^{\alpha}(\phi_i(\vartheta))=0;\qquad \vartheta\geq\vartheta_0,
\end{equation*}\]
where \(\alpha\) is a ratio of odd positive integers such that \(\alpha\ge1\), and k is a positive integer, {pi (\(\vartheta\))} are sequences of nonnegative real numbers for 1 ≤ i ≤ k and {\(\phi_i(\vartheta)\)} are sequences of positive integers such that \(\phi_i(\vartheta)\) ≥ \(\vartheta\) + 2. Specifically, we establish new sufficient conditions for the oscillation of all solutions to the difference equation in which the advanced arguments are not assumed to be monotonic. By constructing appropriate iterative sequences and analysing their convergence properties, distinct oscillation criteria are derived for both the linear case (\(\alpha\) = 1 ). The established criteria provide easily verifiable conditions that enhance the existing qualitative theory of non-linear difference equations. Finally, the theoretical findings are illustrated and validated using concrete examples.
Keywords: Oscillation, non-linear, advanced, first order, difference equation