Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 50, pp. 1-23.

Title: Asymptotic formulas for q-regularly varying solutions of half-linear
       q-difference equations
       
Author: Katarina S. Djordjevic (Univ. of Nis, Nis, Serbia)

Abstract: 
 This article studies the asymptotic behavior of positive solutions of the q-difference
 half-linear equation
 $$
 D_q(p(t)\Phi(D_q(x(t)))) + r(t)\Phi(x(qt))=0, \quad
 t \in q^{\mathbb{N}_0}:=\{q^n : n \in \mathbb{N}_0\},
 $$
 where q&gt;1, &Phi;(x)=|x|<sup>&alpha;</sup>sgn x,
 &alpha; &gt;0, p:q<sup>N<sub>0</sub></sup> &rarr;  (0,&infin;),
 r: q<sup>N<sub>0</sub></sup> &rarr; 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 t<sup>&alpha;+1</sup> r(t)/p(t) &rarr; 0,
 as t &rarr; &infin;, 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 08, 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.