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

 dc.contributor.author Djordjevic, Katarina dc.date.accessioned 2021-08-27T14:02:03Z dc.date.available 2021-08-27T14:02:03Z dc.date.issued 2021-06-08 dc.description.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. dc.description.department Mathematics dc.format Text dc.format.extent 23 pages dc.format.medium 1 file (.pdf) dc.identifier.citation Djordjević, K. S. (2021). Asymptotic formulas for q-regularly varying solutions of half-linear q-difference equations. Electronic Journal of Differential Equations, 2021(50), pp. 1-23. dc.identifier.issn 1072-6691 dc.identifier.uri https://hdl.handle.net/10877/14460 dc.language.iso en dc.publisher Texas State University, Department of Mathematics dc.rights Attribution 4.0 International dc.rights.uri https://creativecommons.org/licenses/by/4.0/ dc.source Electronic Journal of Differential Equations, 2021, San Marcos, Texas: Texas State University and University of North Texas. dc.subject q-difference equation dc.subject non-oscillatory solution dc.subject asymptotic behavior dc.subject regular variation dc.subject q-regular variation dc.subject half-linear equation dc.title Asymptotic formulas for q-regularly varying solutions of half-linear q-difference equations dc.type Article

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