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Global attractivity in a higher order nonlinear difference equation. (English) Zbl 1004.39010

The nonlinear rational difference equations are of paramount importance in their own right, and furthermore for the development of the basic theory of the global behavior of nonlinear difference equations. The authors study the global attractivity of the rational recursive sequence

${X}_{n+1}=\left(a-b{X}_{n}\right)/\left(c-{X}_{n-k}\right),\phantom{\rule{1.em}{0ex}}n=0,1,\cdots ,\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $a>$ or equal $0,c>b>0$ are real numbers and $k>$ or equal 1 is an integer, and the initial conditions ${X}_{-k},{X}_{-k+1},\cdots ,{X}_{-1}$ and ${X}_{0}$ are arbitrary. They prove that the positive equilibrium of equation (1) is a global attractor with a basin that depends on certain conditions of the coefficients.

I think that the authors must be aware of relevant published papers on the same topic, for example W. A. Kosmala, M. R. S. Kulenovic, G. Ladas and C. T. Teixeira [J. Math. Anal. Appl. 251, No. 2, 571-586 (2000; Zbl 0967.39004)].

MSC:
 39A11 Stability of difference equations (MSC2000) 39B05 General theory of functional equations