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Nonlinear recurrence approaches a-1

Source: Miklós Schweitzer 2019, Problem 6

December 27, 2019
recurrence relation

Problem Statement

Let dd be a positive integer and 1<a≤(d+2)/(d+1)1 < a \le (d+2)/(d+1). For given x0,x1,…,xd∈(0,a−1)x_0, x_1,\dots, x_d \in (0, a-1), let xk+1=xk(a−xk−d)x_{k+1} = x_k (a - x_{k-d}), k≥dk \ge d. Prove that lim⁡k→∞xk=a−1\lim_{k \to \infty} x_k = a-1.
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