Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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A GLOBAL BEHAVIOR OF THE POSITIVE SOLUTIONS OF xn+1=βxn+ xn-2 ⁄ A+Bxn + xn-2

  • Park, Jong-An (DEPARTMENT OF MATHEMATICS KANGWON NATIONAL UNIVERSITY)
  • Published : 2008.01.31
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Abstract

In this paper we prove that every positive solution of the third order rational difference equation $$x_{n+1}\;=\;\frac{{\beta}x_n\;+\;x_{n-2}}{A\;+\;Bx_n\;+\;x_{n-2}}$ converges to the positive equilibrium point $$\bar{x}\;=\;\frac{{\beta}\;+\;1\;-\;A}{B\;+\;1}$, where $0\;<\;{\beta}\;{\leq}\;B$, $1\;<\;A\;<\;{\beta}\;+\;1$

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References

  1. E. Camouzis, Global analysis of solutions of $x_{n+1} = \frac{\beta{x_{n}}+\delta{x_{n-2}}}{A+B{x_{n}+C{x_{n-1}}$, J. Math. Anal. Appl. 316 (2006), 616-627 https://doi.org/10.1016/j.jmaa.2005.05.008
  2. M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equation, with Open Problems and Conjectures, Chapman and Hall/CRC, 2002
  3. R. D. Nussbaum, Global stability, two conjectures and maple, Nonlinear Anal. 66 (2007), 1064-1090 https://doi.org/10.1016/j.na.2006.01.005

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