Electron. J. Differential Equations, Vol. 2021 (2021), No. 50, pp. 123.
Asymptotic formulas for qregularly varying solutions of halflinear qdifference equations
Katarina S. Djordjevic
Abstract:
This article studies the asymptotic behavior of positive solutions of the qdifference
halflinear equation
where q>1, Φ(x)=x^{α}sgn x,
α >0, p:q^{N0} → (0,∞),
r: q^{N0} → R, in the framework of
qregular variation. In particular, if r is eventually of one sign, p and r
are qregularly varying functions such that t^{α+1} r(t)/p(t) → 0,
as t → ∞, we obtain asymptotic formulas for the qregularly varying solutions.
Moreover, when p(t)= 1 and r is an eventually positive or
eventually negative function, we obtain an asymptotic formula of a qslowly varying solution.
Using generalized regularly varying sequences, we apply these results to the
halflinear 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: qdifference equation; nonoscillatory solution; asymptotic behavior;
regular variation; qregular variation; halflinear 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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