On a conjecture for the difference equation $ x_{n+1} = 1+p\frac{x_{n-m}}{x_n^2} $
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}) $, {l...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-07-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20231156?viewType=HTML |
| _version_ | 1797768323058892800 |
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| author | George L. Karakostas |
| author_facet | George L. Karakostas |
| author_sort | George L. Karakostas |
| collection | DOAJ |
| description | 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) $. |
| first_indexed | 2024-03-12T20:51:50Z |
| format | Article |
| id | doaj.art-1a171c990cf240e5903139b832c2ca6e |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-03-12T20:51:50Z |
| publishDate | 2023-07-01 |
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| series | AIMS Mathematics |
| spelling | doaj.art-1a171c990cf240e5903139b832c2ca6e2023-08-01T01:27:09ZengAIMS PressAIMS Mathematics2473-69882023-07-01810227142272910.3934/math.20231156On a conjecture for the difference equation $ x_{n+1} = 1+p\frac{x_{n-m}}{x_n^2} $George L. Karakostas0Department of Mathematics, University of Ioannina, Ioannina 45110, GreeceIn <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) $.https://www.aimspress.com/article/doi/10.3934/math.20231156?viewType=HTMLdifference equationsasymptotic stabilityequilibriumperiodic solutions |
| spellingShingle | George L. Karakostas On a conjecture for the difference equation $ x_{n+1} = 1+p\frac{x_{n-m}}{x_n^2} $ AIMS Mathematics difference equations asymptotic stability equilibrium periodic solutions |
| title | On a conjecture for the difference equation $ x_{n+1} = 1+p\frac{x_{n-m}}{x_n^2} $ |
| title_full | On a conjecture for the difference equation $ x_{n+1} = 1+p\frac{x_{n-m}}{x_n^2} $ |
| title_fullStr | On a conjecture for the difference equation $ x_{n+1} = 1+p\frac{x_{n-m}}{x_n^2} $ |
| title_full_unstemmed | On a conjecture for the difference equation $ x_{n+1} = 1+p\frac{x_{n-m}}{x_n^2} $ |
| title_short | On a conjecture for the difference equation $ x_{n+1} = 1+p\frac{x_{n-m}}{x_n^2} $ |
| title_sort | on a conjecture for the difference equation x n 1 1 p frac x n m x n 2 |
| topic | difference equations asymptotic stability equilibrium periodic solutions |
| url | https://www.aimspress.com/article/doi/10.3934/math.20231156?viewType=HTML |
| work_keys_str_mv | AT georgelkarakostas onaconjectureforthedifferenceequationxn11pfracxnmxn2 |