LAUSR

The half‐linear q‐difference equation Dq(p(t)Φ(Dq(x(t))))+r(t)Φ(x(qt))=0,t∈qℕ0={qn:n∈ℕ0},q>1, where Φ(x)=|x|αsgnx , α>0,p:qℕ0→(0,∞),r:qℕ0→ℝ , is analyzed in the framework of q‐regular variation. Necessary and sufficient conditions for the existence of q‐regularly varying solutions are given, under the assumption that p is a q‐regularly varying function and with no sign restriction on r. It is examined in the case when r is eventually negative, whether all positive solutions are q‐regularly varying. Using generalized regularly varying sequences, these results are applied to the half‐linear difference equation case.