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

George L. Karakostas; George L. Karakostas

doi:10.3934/math.20231156

AIMS Mathematics

2023, Volume 8, Issue 10: 22714-22729. doi: 10.3934/math.20231156

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

George L. Karakostas ^,

Department of Mathematics, University of Ioannina, Ioannina 45110, Greece

Received: 27 April 2023 Revised: 29 June 2023 Accepted: 07 July 2023 Published: 17 July 2023
MSC : 39A10, 39A23, 39A30

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) $.
- difference equations,
- asymptotic stability,
- equilibrium,
- periodic solutions
Citation: George L. Karakostas. On a conjecture for the difference equation $ x_{n+1} = 1+p\frac{x_{n-m}}{x_n^2} $[J]. AIMS Mathematics, 2023, 8(10): 22714-22729. doi: 10.3934/math.20231156

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Abstract

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) $.

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