Asymptotic formulas for q-regularly varying solutions of half-linear q-difference equations

Abstract

This article studies the asymptotic behavior of positive solutions of the q-difference half-linear equation Dq(p(t)Φ(Dq(x(t)))) + r(t)Φ(x(qt)) = 0, t ∈ qℕ0 := {qn : n ∈ ℕ0}, where q > 1, Φ(x) = |x|α sgn x, α > 0, p : qℕ0 → (0, ∞), r : qℕ0 → ℝ, in the framework of q-regular variation. In particular, if r is eventually of one sign, p and |r| are q-regularly varying functions such that tα+1 r(t)/p(t) → 0, as t → ∞, we obtain asymptotic formulas for the q-regularly varying solutions. Moreover, when p(t) ≡ 1 and r is an eventually positive or eventually negative function, we obtain an asymptotic formula of a q-slowly varying solution. Using generalized regularly varying sequences, we apply these results to the half-linear difference equation case. At the end, we illustrate the obtained results with examples.