On a conjecture for the difference equation $x_{n+1}=1+p\frac{x_{n-m}}{x_n^2}$

In ^[24], E. Tasdemir, et al. proved that the positive equilibrium of the nonlinear discrete equation $x_{n+1}=1+p\frac{x_{n-m}}{x_n^2}$ is globally asymptotically stable for $p\in (0,\frac{1}{2})$, {locally} asymptotically stable for $p\in (\frac{1}{2},\frac{3}{4})$ and it was { conjectured} that for any $p$ in the open interval $(\frac{1}{2},\frac{3}{4})$ the equilibrium is { globally} asymptotically stable. In this paper, we prove that this conjecture is true for the closed interval $[\frac{1}{2},\frac{3}{4}].$ In addition, it is shown that for $p\in (\frac{3}{4},1)$ 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 for $m>2$ and $p\in (\frac{3}{4},1)$.