Authors: Stević, Stevo 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Global stability of some symmetric difference equations
Journal: Applied Mathematics and Computation
Volume: 216
Issue: 1
First page: 179
Last page: 186
Issue Date: 1-Mar-2010
Rank: M21
ISSN: 0096-3003
DOI: 10.1016/j.amc.2010.01.029
Abstract: 
Suppose r ∈ (0, 1], m ∈ N and 1 ≤ k1 < k2 < ⋯ < k2 m + 1, and let S2 m + 1 = {1, 2, ..., 2 m + 1}. We show that every positive solution to the difference equationyn = frac(P2 m + 12 m + 1 (yn - k1r, yn - k2r, ..., yn - k2 m + 1r), P2 m2 m + 1 (yn - k1r, yn - k2r, ..., yn - k2 m + 1r)), n ∈ N0,whereP2 m + 12 m + 1 (x1, x2, ..., x2 m + 1) = underover(∑, frac(r = 1, r odd), 2 m + 1) under(∑, frac({t1, t2, ..., tr} ⊆ S2 m + 1, t1 < t2 < ⋯ < tr)) xt1 xt2 ⋯ xtrandP2 m2 m + 1 (x1, x2, ..., x2 m + 1) = 1 + underover(∑, frac(r = 2, r even), 2 m) under(∑, frac({t1, t2, ..., tr} ⊂ S2 m + 1, t1 < t2 < ⋯ < tr)) xt1 xt2 ⋯ xtr,converges to one. This result confirms a quite recent conjecture posed by Liu and Yang (2010) in [10]. We also prove another result regarding a related equation.
Keywords: Positive solution | Rational difference equation | Stability | Symmetry
Publisher: Elsevier

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