Abstract and Applied Analysis
Volume 2011 (2011), Article ID 986343, 12 pages
http://dx.doi.org/10.1155/2011/986343
Research Article

Asymptotic Behavior of Solutions to Half-Linear 𝑞 -Difference Equations

Pavel Řehák

Institute of Mathematics, Academy of Sciences of the Czech Republic, Žižkova 22, 61662 Brno, Czech Republic

Received 11 October 2010; Accepted 18 November 2010

Academic Editor: Elena Braverman

Copyright © 2011 Pavel Řehák. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We derive necessary and sufficient conditions for (some or all) positive solutions of the half-linear 𝑞 -difference equation 𝐷 𝑞 ( Φ ( 𝐷 𝑞 𝑦 ( 𝑡 ) ) ) + 𝑝 ( 𝑡 ) Φ ( 𝑦 ( q t ) ) = 0 , 𝑡 ∈ { 𝑞 𝑘 ∶ 𝑘 ∈ ℕ 0 } with 𝑞 > 1 , Φ ( 𝑢 ) = | 𝑢 | 𝛼 − 1 s g n 𝑢 with 𝛼 > 1 , to behave like 𝑞 -regularly varying or 𝑞 -rapidly varying or 𝑞 -regularly bounded functions (that is, the functions 𝑦 , for which a special limit behavior of 𝑦 ( q t ) / 𝑦 ( 𝑡 ) as 𝑡 → ∞ is prescribed). A thorough discussion on such an asymptotic behavior of solutions is provided. Related Kneser type criteria are presented.