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Global behavior of a rational second order difference equation

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Abstract

In this paper, we solve the difference equation

$$\begin{aligned} x_{n+1}=\frac{\alpha }{1-x_nx_{n-1}}, \quad n=0,1,\ldots , \end{aligned}$$

where \(\alpha >0\) and the initial values \(x_{-1}\), \(x_{0}\) are real numbers. We find invariant sets and discuss the global behavior of the solutions of that equation. We show that when \(\alpha <\frac{2}{3\sqrt{3}}\), one of the positive equilibrium points attracts all orbits with initials outside a set of Lebesgue measure zero. Also, when \(\alpha =\frac{2}{3\sqrt{3}}\), the unique positive equilibrium points attracts all orbits with initials outside a set of Lebesgue measure zero. Finally, we show that when \(\alpha >\frac{2}{3\sqrt{3}}\), under certain conditions there exist solutions that are either periodic or converging to periodic solutions and give some examples. We show also the existence of dense solutions in the real line.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science and Arts, Bülent Ecevit University, 67100, Zonguldak, Turkey

    M. Gümüş

  2. Department of Basic Science, The Higher Institute for Engineering and Technology, Al-Obour, Cairo, Egypt

    R. Abo-Zeid

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  1. M. Gümüş
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  2. R. Abo-Zeid
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Correspondence to M. Gümüş.

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Gümüş, M., Abo-Zeid, R. Global behavior of a rational second order difference equation. J. Appl. Math. Comput. 62, 119–133 (2020). https://doi.org/10.1007/s12190-019-01276-9

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  • DOI: https://doi.org/10.1007/s12190-019-01276-9

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