Oscillation of second order unstable neutral difference equations with continuous arguments
Various conditions for oscillatory behavior of the bounded solutions of \[ \Delta^2_\tau(x(t)-px(t-\sigma))=f(t,x(g(t))) \] are given. Here \(\tau>0\), \(\sigma\in \mathbb R\), \(f(t,u)/u\geq q(t)>0\), \(q\in C((t_0,\infty),\mathbb R_+)\), \(g\in C^1((t_0,\infty),\mathbb R_+)\),\(0<g(t)<t\), \(\liminf_{t\rightarrow\infty}(g(t+\tau)-g(t))>\tau\), \(\Delta_\tau(x(t))=x(t+\tau)-x(t)\). The oscillatory behavior is valid for all bounded solutions of the equation and for \(0\leq p<1\), \(p=1\), \(p<0\), \(p>0\) respectively. The criteria are expressed in terms of equations coefficients via various iterated integral transformations.
scientific article; zbMATH DE number 5007517