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Article: On boundedness of the solutions of the difference equation [x.sub.n+1] = [x.sub.n-1]/(p + [x.sub.n]).
- Article from:
- Discrete Dynamics in Nature and Society
- Article date:
- January 1, 2006
- Author:
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We study the difference equation [x.sub.n+1] = [x.sub.n-1]/(p + [x.sub.n]), n = 0, 1, ..., where initial values [x.sub.-1],[x.sub.0] [member of] (0,+ [infinity]) and 0
Kulenovic and Ladas in [2] (also see [1]) studied the following difference equation:
[x.sub.n+1] = [x.sub.n-1]/ p + [x.sub.n] , n = 0, 1, ..., (1)
where initial values [x.sub.-1],[x.sub.0] [member of] (0, + [infinity]) and p [member of] (0, + [infinity]), and obtained the following theorem.
Theorem 1. (i) If p > 1, then the unique equilibrium 0 of (1) is globally asymptotically stable.
(ii) If p = 1, then every positive solution of (1) converges to a period-two solution.
(iii) If 0