Riccati transformation and sublinear oscillation for second order neutral delay dynamic equations (Q2919929)

The second-order sublinear neutral delay dynamic equation NEWLINE\[CARRIAGE_RETURNNEWLINE\left(r(t)\left((y(t) + p(t) y(\alpha(t)))^\Delta\right)^\gamma\right)^\Delta + q(t) y^\gamma(\beta(t)) = 0CARRIAGE_RETURNNEWLINE\]NEWLINE on a time scale \(\mathcal{T}\) is studied. Riccati transformation technique is used to investigate the oscillatory and asymptotic behavior of the solutions to this equation. The exponent \(\gamma\) is the quotient of odd positive integers and \(0 < \gamma \leq 1\) and the positive \(\Delta\)-differentiable function \(r \in C'_{rd}[a,\infty)\) for \(a \in \mathbb{R},\) satisfies one of the assumptions NEWLINE\[CARRIAGE_RETURNNEWLINE\int_{t_0}^\infty \left( \frac{1}{r(t)}\right) ^{\frac{1}{\gamma}} \;\Delta t = \infty CARRIAGE_RETURNNEWLINE\]NEWLINE or NEWLINE\[CARRIAGE_RETURNNEWLINE\int_{t_0}^\infty \left( \frac{1}{r(t)}\right) ^{\frac{1}{\gamma}} \;\Delta t < \infty . CARRIAGE_RETURNNEWLINE\]NEWLINE \(\alpha\) and \(\beta\) are rd-continuous functions on \(\mathcal{T}\) such that \(\alpha(t),\beta(t) \leq t\) and \(\lim_{t \rightarrow \infty} \alpha(t) = \infty = \lim_{t \rightarrow \infty} \beta(t).\) Several sufficient conditions involving some technical assumptions on the rd-continuous functions \(p, q, r\) and \(\alpha, \beta\) are used to establish that the above dynamic equation is oscillatory. It is claimed that some of these results are also new in the oscillation theory of second-order nonlinear neutral differential and delay equations. A brief discussion on the form of the dynamic equation to specific time scales and an example are also presented.