Oscillation and Asymptotic Behavior of Second-Order Half-Linear Noncanonical Difference Equations of Advanced Type

We established new single-condition criteria for the oscillation of all solutions to a second-order half-linear advanced equation of the form ∆(φ(ζ)(∆x(ζ))^ν) + ρ(ζ)x^ν(ζ + η) = 0; ζ ≥ ζ₀ under the conditions that $\sum\nolimits_{s = \zeta }^\infty {{1 \over {\varphi {1 \over v}\left( s \right)}} < \infty }$. We derive new single-condition constraints for the oscillation of all unimprovable constant solutions to equation. Even in the linear case, the significant result is new and improves all of the prior results to the best of our knowledge. The advantage of our technique is the simple proof, solely relying on the monotonicities of a positive solution sequentially improved. Examples of our conclusions are presented.