Global stability, two conjectures and Maple

Consider the second order difference equation u(-1) > 0, u(0) > 0 and u(n+1) = f (u(n-1), u(n)) for n >= 0, where either (a) f (u, v) = (u+pv)/(u+qv) or (b) f (u, v) = (p+qv)/(1+u). If 0 <= q < p in case (a) or p > 0 and q > 0 case (b), it has been conjectured (see [M.R.S. Kulenovi6, G. Ladas, Dynamics of Second Order Rational Difference Equations, Chapman and Hall/CRC Press, 2001]) that lim(n ->infinity) u(n) exists and equals L, where L > 0 and L = f (L, L). We prove this conjecture in case (a) and significantly extend the range of p and q for which it is known in case (b). In cases (a) and (b), these questions are equivalent to global stability of the fixed point (L, L) of the planar map 0(u, v) = (v, f (u, v)). For 0 as in case (a), we consider natural four dimensional extensions T of Phi(3) and S of Phi(2). For 0 <= q < p, we prove that (L, L, L, L) is a global stable fixed point of T, but we also describe precisely a range of parameters 0 <= q < p for which S has at least three distinct fixed points in the positive orthant. We describe (Section 3) some general principles underlying our arguments. Symbolic calculations using Maple play a crucial role in our arguments in Section 4. (c) 2006 Published by Elsevier Ltd.