Global stability of some symmetric difference equations

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.