Asymptotic formulas for q-regularly varying solutions of half-linear q-difference equations
Date
2021-06-08
Authors
Djordjevic, Katarina
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
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.
Description
Keywords
q-difference equation, non-oscillatory solution, asymptotic behavior, regular variation, q-regular variation, half-linear equation
Citation
Djordjević, K. S. (2021). Asymptotic formulas for q-regularly varying solutions of half-linear q-difference equations. <i>Electronic Journal of Differential Equations, 2021</i>(50), pp. 1-23.
Rights
Attribution 4.0 International