Electron. J. Differential Equations, Vol. 2021 (2021), No. 50, pp. 1-23.

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

Katarina S. Djordjevic

This article studies the asymptotic behavior of positive solutions of the q-difference half-linear equation

where q>1, Φ(x)=|x|αsgn 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.

Submitted February 28, 2021. Published June 8, 2021.
Math Subject Classifications: 26A12, 39A13, 39A22.
Key Words: q-difference equation; non-oscillatory solution; asymptotic behavior; regular variation; q-regular variation; half-linear equation.

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Katarina S. Djordjevic
Department of Mathematics
University of Nis, Faculty of Science and Mathematics
Visegradska 33, 18000 Nis, Serbia
email: katarina.kostadinov@pmf.edu.rs

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