The study
makes a significant contribution to the theoretical understanding of nonlinear
difference equations, a class of recursive equations with wide applications in
fields like population dynamics and economics. In [1] E. Tasdemir, et al.
proved that the positive equilibrium of the nonlinear discrete equation
\(x_n+1=1+p\left(\frac{x_{n-m}}{x_n^2} \right)\) is globally asymptotically
stable for \(p \in \left(0, \frac{1}{2} \right)\), 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. Finally, a conjecture is posed about the behaviour of the solutions
for m > 2 and \(p \in (\frac{3}{4}, 1\)). The advanced analytical methods
employed showcase techniques that can guide future research and developments in
the field.
Author(s) Details
George L. Karakostas
Department
of Mathematics, University of Ioannina, Ioannina 45110, Greece.
Please see
the book here:- https://doi.org/10.9734/bpi/rumcs/v8/3698G
