On a conjecture for the difference equation xn+1 = 1

In [24], E. Tasdemir, et al. proved that the positive equilibrium of the nonlinear discrete x2 nis globally asymptotically stable for p & ISIN; (0, 21), locally asymptotically stable for p & ISIN; (21, 34) and it was conjectured that for any p in the open interval (21, 34) the equilibrium is globally asymptotically stable. In this paper, we prove that this conjecture is true for the closed interval [21, 34]. In addition, it is shown that for p & ISIN; (34, 1) the behaviour of the solutions depend on the delay m. Indeed, here we show that in case m = 1, there is an unstable equilibrium and an asymptotically stable 2-periodic solution. But, in case m = 2, there is an asymptotically stable equilibrium. These results are obtained by using linearisation, a method lying on the well known Perron's stability theorem ( [17], p. 18). Finally, a conjecture is posed about the behaviour of the solutions form > 2 and p & ISIN; (34, 1).