Dynamics of a Quadratic Rational Delay Difference Equation| Chapter 10 | Research Updates in Mathematics and Computer Science Vol. 8

 

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