Unbounded Solutions of the Difference Equation 𠑥 𠑛 = 𠑥 𠑛 − 𠑙 𠑥 𠑛 − 𠑘 − 1

Stevo Stević and Bratislav Iričanin

Abstract and Applied Analysis, 2011, vol. 2011, 1-8

Abstract:

The following difference equation 𠑥 𠑛 = 𠑥 𠑛 − 𠑙 𠑥 𠑛 − 𠑘 − 1 , 𠑛 ∈ ℕ 0 , where 𠑘 , 𠑙 ∈ ℕ , 𠑘 < 𠑙 , g c d ( 𠑘 , 𠑙 ) = 1 , and the initial values 𠑥 − 𠑙 , … , 𠑥 − 2 , 𠑥 − 1 are real numbers, has been investigated so far only for some particular values of 𠑘 and 𠑙 . To get any general result on the equation is turned out as a not so easy problem. In this paper, we give the first result on the behaviour of solutions of the difference equation of general character, by describing the long-term behavior of the solutions of the equation for all values of parameters 𠑘 and 𠑙 , where the initial values satisfy the following condition  𠑥 m i n − 𠑙 , … , 𠑥 − 2 , 𠑥 − 1  .

Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:561682

DOI: 10.1155/2011/561682

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