On the Rational Recursive Sequence 𠑥 𠑛 + 1 = ( 𠛼 − 𠛽 𠑥 𠑛 ) / ( 𠛾 − 𠛿 𠑥 𠑛 − 𠑥 𠑛 − 𠑘 )

E. M. E. Zayed, A. B. Shamardan and T. A. Nofal

International Journal of Mathematics and Mathematical Sciences, 2008, vol. 2008, 1-15

Abstract:

We study the global stability, the periodic character, and the boundedness character of the positive solutions of the difference equation 𠑥 𠑛 + 1 = ( 𠛼 − 𠛽 𠑥 𠑛 ) / ( 𠛾 − 𠛿 𠑥 𠑛 − 𠑥 𠑛 − 𠑘 ) , 𠑛 = 0 , 1 , 2 , … , 𠑘 ∈ { 1 , 2 , … } , in the two cases: (i) 𠛿 ≥ 0 , 𠛼 > 0 , 𠛾 > 𠛽 > 0 ; (ii) 𠛿 ≥ 0 , 𠛼 = 0 , 𠛾 , 𠛽 > 0 , where the coefficients 𠛼 , 𠛽 , 𠛾 , and 𠛿 , and the initial conditions 𠑥 − 𠑘 , 𠑥 − 𠑘 + 1 , … , 𠑥 − 1 , 𠑥 0 are real numbers. We show that the positive equilibrium of this equation is a global attractor with a basin that depends on certain conditions posed on the coefficients of this equation.

Date: 2008
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:391265

DOI: 10.1155/2008/391265

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