A proof of Conjecture 1 in Kulenović, Ladas and Overdeep (2003)

In Kulenović et al. [Open problems and conjectures: Edited by Gerry Ladas, J. Differ. Equ. Appl. 9(11) (2003), pp. 1053–1056] posed a conjecture asserting that every positive solution of the rational second-order difference equation yn+1=yn(1+yn)2yn(1+yn)+(1+yn−1),n=0,1,…,converges to a finite limit. We confirm this conjecture by deriving a short identity showing that the sign of yn+1−yn 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 tn:=yn(1+yn)1+yn−1,which is monotone in the direction opposite to that of yn. It yields nested two-sided enclosures of the limit and an exact invariant-series formula. Writing gn=tn−yn and ρn=tn/(1+tn)2, we prove that In=yn+gn∑j=0∞11+tn+j∏m=0j−1ρn+mis independent of n and satisfies In=L=limk→∞yk. Hence the limiting equilibrium selected by the initial data is determined by the invariant value I0.