Asymptotic and Oscillatory Behaviour of First-Order Neutral Difference Equations With Non-Monotone Advanced Arguments
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 studies the oscillatory and asymptotic behaviour of a class of first-order nonlinear neutral difference equations with non-monotone advanced arguments of the form
\(\begin{equation}
\Delta\left[\Theta(\vartheta)-\sum\limits_{i=1}^r \rho_i(\vartheta) \Theta\left(\tau_i(\vartheta)\right)\right]-\sum\limits_{i=1}^k \phi_i(\vartheta) g_i\left(\Theta\left(\varphi_i(\vartheta)\right)\right)=0 ; \quad \vartheta \geq \vartheta_0 .
\end{equation}\) (a)
Using appropriate comparison methods, auxiliary sequences, and exponential inequalities, new sufficient conditions are established for the oscillation and convergence to zero of all solutions of the equation under investigation. Auxiliary lemmas are proved to examine the asymptotic characteristics of eventually positive and eventually negative solutions, and improved oscillation criteria are established in terms of the limit inferior and limit superior associated with equation (a). The obtained results improve and extend several known oscillation theorems in the literature by allowing multiple non-monotone advanced arguments and nonlinear neutral terms under weaker hypotheses.
Keywords: Oscillation, nonoscillation, asymptotic behaviour, neutral, first order, advanced arguments, difference equation