@article{Karakostas2023, 
author = {George L. Karakostas},
title = {On a conjecture for the difference equation        x          n      +      1        =  1  +  p            x              n        −        m                    x      n      2},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {10},
pages = {22714-22729},
keywords = {difference equations, asymptotic stability, equilibrium, periodic solutions},
url = {https://www.sciopen.com/article/10.3934/math.20231156},
doi = {10.3934/math.20231156},
abstract = {In [24], E. Tasdemir, et al. proved that the positive equilibrium of the nonlinear discrete equation        x          n      +      1        =  1  +  p            x              n        −        m                    x      n      2       is globally asymptotically stable for    p  ∈  (  0  ,      1    2    ), {locally} asymptotically stable for    p  ∈  (      1    2    ,      3    4    ) and it was { conjectured} that for any    p in the open interval    (      1    2    ,      3    4    ) the equilibrium is { globally} asymptotically stable. In this paper, we prove that this conjecture is true for the closed interval    [      1    2    ,      3    4    ]  . In addition, it is shown that for    p  ∈  (      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  ∈  (      3    4    ,  1  ).}
}