Oscillation of second order nonlinear difference equation with continuous variable (Q5929384)

The authors prove different results in which sufficient conditions are obtained to assure that all the solutions to the second order nonlinear difference equation with continuous variable \(\Delta_{\tau}x(t) + f(t,x(t-\sigma))=0\) are oscillatory. Here \(\Delta_{\tau}x(t) = x(t+\tau) - x(t)\), \(\tau >0\), \(\sigma >0\), \(f \in C([t_0,\infty) \times \mathbb{R},\mathbb{R})\) and \(f(t,u)/u \geq p(t)>0\) for \(u \neq 0\) with \(p \in C(\mathbb{R},\mathbb{R}^+)\).NEWLINENEWLINENEWLINEThey use iterated integral transformations, generalized Riccati transformations together with integrating factors. NEWLINENEWLINENEWLINEThey present two examples in which the given results are applied.