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

Abstract

<abstract><p>In <sup>[<xref ref-type="bibr" rid="b24">24</xref>]</sup>, 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 (<sup>[<xref ref-type="bibr" rid="b17">17</xref>]</sup>, p. 18). Finally, a conjecture is posed about the behaviour of the solutions for $ m > 2 $ and $ p\in(\frac{3}{4}, 1) $.</p></abstract>