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General cycles of potential form. (English) Zbl 1202.39002

Summary: We consider a general difference equation of order \(k\) of the form \[ x_{n+k} = F (x_{n+k-1}, \cdots, x_{n+2}, x_{n+1}) f_0 (x_n) \] with \(f_{0} : (0, \infty ) \rightarrow (0, \infty ), F : (0, \infty )^{k-1} \rightarrow (0, \infty )\) continuous maps and initial conditions \(x_{i} \in (0, \infty ), i = 0, 1, \cdots , k-1\). Under the hypothesis that \(f_{x}(y) := F(x, x, \cdots , x)f_{0}(y)\) is an involution, we provide all the \((k + 1)\)-cycles of the above-mentioned type, and we prove that if \(F\) separates variables, then all the \((k + 2)\)-cycles have a potential form \[ x_{n+k} = cx^{\alpha_{k-1}}_{n+k-1} \cdots x^{\alpha_1}_{n+1} x^{\alpha_0}_n, \quad n= 0,1, \dots, \] where \(c > 0, \alpha _{i} \in \mathbb R, x_{i} \in (0, \infty ), i = 0, 1, \cdots , k-1\). For this reason, we also describe all the \(p\)-cycles of order \(k \geq 2\) having the above potential form. Moreover, taking the particular case \(k = 3\) and \(p = 8\) disproves Conjecture 2.1 by E. A. Grove and G. Ladas [Advances in Discrete Mathematics and Applications 4. Boca Raton, FL: Chapman & Hall/CRC. xiii, 379 p. $ 99.00; £ 56.99 (2005; Zbl 1078.39009)].

MSC:

39A10 Additive difference equations
39A23 Periodic solutions of difference equations

Citations:

Zbl 1078.39009
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References:

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