In Kulenovi & cacute; 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 y(n+1) = y(n)(1 + y(n))(2) / y(n)(1 + y(n))+ (1 + y(n-1)), n = 0, 1, . . . , 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) := y(n)(1 + y(n)) /1 + y(n-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 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;(infinity)(j=0) 1 /1 + t(n+j) & prod;(j-1)(m=0) rho(n+m) is independent ofn and satisfies I-n = L = lim(k ->infinity) y(k). Hence the limiting equilibrium selected by the initial data is determined by the invariant value I-0.
