Global stability, two conjectures and Maple Original Research Article

Consider the second order difference equation u−1>0,u0>0u−1>0,u0>0 and un+1=f(un−1,un)un+1=f(un−1,un) for n≥0n≥0, where either (a) View the MathML sourcef(u,v)=u+pvu+qv or (b) View the MathML sourcef(u,v)=p+qv1+u. If 0≤q0p>0 and q>0q>0 in case (b), it has been conjectured (see [M.R.S. Kulenović, G. Ladas, Dynamics of Second Order Rational Difference Equations, Chapman and Hall/CRC Press, 2001]) that limn→∞unlimn→∞un exists and equals LL, where L>0L>0 and L=f(L,L)L=f(L,L). We prove this conjecture in case (a) and significantly extend the range of pp and qq 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)(L,L) of the planar map Φ(u,v)=(v,f(u,v))Φ(u,v)=(v,f(u,v)). For ΦΦ as in case (a), we consider natural four dimensional extensions TT of Φ3Φ3 and SS of Φ2Φ2. For 0≤q