Oscillation of Second-Order Half-linear Retarded Difference Equations via a Canonical Transform

Abstract

Abstract The aim of this paper is to investigate the second order half-linear retarded difference equation Δ ( μ ( n ) ( Δ η ( n ) ) α ) + δ ( n ) η α ( σ ( n ) ) = 0 \Delta \left( {\mu \left( n \right){{\left( {\Delta \eta \left( n \right)} \right)}^\alpha }} \right) + \delta \left( n \right){\eta ^\alpha }\left( {\sigma \left( n \right)} \right) = 0 under the condition ∑ n = n 0 ∞ μ − 1 α ( n ) < ∞ \sum\limits_{n = {n_0}}^\infty {{\mu ^{ - {1 \over \alpha }}}} \left( n \right) < \infty \, (i.e., nonconanical form). Unlike most existing results, the oscillatory behavior of solutions of this equation is attained by transforming it into an equation in canonical form. Particular examples are provided to show the significance of our main results.