Oscillation of second order nonlinear dynamic equations on time scales (Q1767339)

The authors consider the nonlinear second order dynamic equation \[ (p(t)x^\Delta)^\Delta+q(t)(f\circ x^\sigma)=0,\tag{1} \] where \(p\) and \(q\) are positive, real-valued continuous functions, and the nonlinearity \(f:\mathbb{R}\to\mathbb{R}\) satisfies the sign condition \(xf(x)>0\) and the superlinearity condition \(f(x)>K x\) for some \(K>0\) and every \(x\neq 0\). Two cases, depending on the convergence of the integral \[ \int _1^\infty\frac 1{p(t)}\Delta t\tag{2} \] are discussed separately. New sufficient conditions involving the integral over the coefficients of equation (1) which guarantee that all solutions are oscillatory (in the case when (2) is divergent) or either oscillatory or convergent to zero (in the case of convergence of the integral (2)) are derived. The sharpness of these criteria is shown on the example of the Euler dynamic equation. The authors' main tool is the Riccati transformation.