On asymptotic behavior of solutions of \(n\)th order Emden-Fowler type difference equations with advanced argument (Q2052727)

The authors consider Emden-Fowler type difference equations of the form \[\Delta^{(n)}u(k)+p(k)|u(\sigma(k))|^\lambda\operatorname{sgn}u(\sigma(k))=0,\] with \(n\ge 2\), \(0<\lambda<1\), \(p\) positive and \(\sigma(k)\ge k+1\). They establish conditions guaranteeing that the equation has the so-called property A. The property A (which was originally introduced for functional-differential equations, see, e.g., [\textit{J. R. Graef} et al., J. Math. Anal. Appl. 306, No. 1, 136--160 (2005; Zbl 1069.34088)]) for the equation under consideration means that each of its (proper) solution is oscillatory when \(n\) is even and either is oscillatory or satisfies \(\lim_{k\to\infty}|\Delta^{(i)}u(k)|=0\), \(i=0,1,\dots,n-1\), when \(n\) is odd.