Asymptotic Convergence of the Solutions of a Dynamic Equation on Discrete Time Scales

J. Diblík, M. Růžičková, Z. Å Marda and Z. Å utá

Abstract and Applied Analysis, 2012, vol. 2012, 1-20

Abstract:

The paper investigates a dynamic equation Δ 𠑦 ( 𠑡 𠑛 ) = 𠛽 ( 𠑡 𠑛 ) [ 𠑦 ( 𠑡 𠑛 − 𠑗 ) − 𠑦 ( 𠑡 𠑛 − 𠑘 ) ] for 𠑛 → ∞ , where 𠑘 and 𠑗 are integers such that 𠑘 > 𠑗 ≥ 0 , on an arbitrary discrete time scale 𠕋 ∶ = { 𠑡 𠑛 } with 𠑡 𠑛 ∈ ℠, 𠑛 ∈ ℤ ∞ 𠑛 0 − 𠑘 = { 𠑛 0 − 𠑘 , 𠑛 0 − 𠑘 + 1 , … } , 𠑛 0 ∈ ℕ , 𠑡 𠑛 < 𠑡 𠑛 + 1 , Δ 𠑦 ( 𠑡 𠑛 ) = 𠑦 ( 𠑡 𠑛 + 1 ) − 𠑦 ( 𠑡 𠑛 ) , and l i m 𠑛 → ∞ 𠑡 𠑛 = ∞ . We assume 𠛽 ∶ 𠕋 → ( 0 , ∞ ) . It is proved that, for the asymptotic convergence of all solutions, the existence of an increasing and asymptotically convergent solution is sufficient. Therefore, the main attention is paid to the criteria for the existence of an increasing solution asymptotically convergent for 𠑛 → ∞ . The results are presented as inequalities for the function 𠛽 . Examples demonstrate that the criteria obtained are sharp in a sense.

Date: 2012
References: Add references at CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://downloads.hindawi.com/journals/AAA/2012/580750.pdf (application/pdf)
http://downloads.hindawi.com/journals/AAA/2012/580750.xml (text/xml)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:580750

DOI: 10.1155/2012/580750

Access Statistics for this article

More articles in Abstract and Applied Analysis from Hindawi
Bibliographic data for series maintained by Mohamed Abdelhakeem ().