Oscillation Criteria for First-Order Non-Linear Neutral Difference Equations with Several Delay 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 investigates the oscillatory behaviour of the following first-order non-linear neutral delay difference equation involving several delay arguments that are not necessarily monotone:
\[\Delta\left[\omega(v)-\sum_{i=1}^r \rho_i(v) \omega\left(\tau_i(v)\right)\right]+\sum_{i=1}^k q_i(v) g_i\left(\omega\left(\phi_i(v)\right)\right)=0 ; \quad v \geq v_0 .(*)\]
where {pi (v) } (1\(\le\) i \(\le\) r) and {qi (v) } (1 \(\le\) i \(\le\) k ) are sequences of real numbers, and {Ti (v) } (1 \(\le\) i \(\le\) r} and {\(\phi\)i (v) } (1 \(\le\) i \(\le\) k) are sequences of positive integers. Under appropriate assumptions on the neutral coefficients, delay sequences, and non-linear factors, new sufficient conditions are established to guarantee that every solution of the considered equation is oscillatory. The study uses the development of suitable auxiliary sequences, comparison methods, and exponential estimates for eventually positive solutions. Several auxiliary lemmas are established to obtain the main oscillation theorems formulated in terms of lim inf and lim sup conditions. The criteria obtained generalise and improve a number of existing oscillation results in the literature by allowing the simultaneous presence of several non-monotone delay arguments and non-linear neutral terms.
Keywords: Oscillation, non-linear, delay, neutral, first-order, difference equation