A Proof of Conjecture 1 in Kulenovi´c, Ladas and Overdeep (2003)

Abstract

In [1], Kulenovi'c, Ladas and Overdeep posed a conjecture asserting that every positive solution of the rational second-order difference equation \[ y_{n+1}=\frac{y_n(1+y_n)^2}{y_n(1+y_n)+(1+y_{n-1})},\qquad n=0,1,\ldots, \] converges to a finite limit. We confirm this conjecture by deriving a short identity showing that the sign of $y_{n+1}-y_n$ is invariant with respect to $n$, so every positive solution is monotone. A simple estimate then gives an explicit initial-data-dependent upper bound in the increasing case, while the decreasing case is bounded below by positivity. Hence every positive solution converges. In addition, we introduce the auxiliary sequence \[ t_n:=\frac{y_n(1+y_n)}{1+y_{n-1}}, \] which is monotone in the direction opposite to that of $y_n$. It yields nested two-sided enclosures of the limit and an exact invariant-series formula. Writing $g_n=t_n-y_n$ and $\rho_n=t_n/(1+t_n)^2$, we prove that \[ I_n=y_n+g_n\sum_{j=0}^{\infty}\frac{1}{1+t_{n+j}} \prod_{m=0}^{j-1}\rho_{n+m} \] is independent of $n$ and satisfies $I_n=L=\lim_{k\to\infty}y_k$. Hence the limiting equilibrium selected by the initial data is determined by the invariant value $I_0$.