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Existence of nontrivial solutions of a rational difference equation. (English) Zbl 1131.39009
The author determines the asymptotic behaviour of a special solution of $$x_{n+1}= (x_n+ x_{n-1}+ x_{n-2} x_{n-3})/(x_n x_{n-1}+ x_{n-2}+ x_{n-3})$$ which confirms a conjecture of {\it L. Ladas} [J. Difference Equ. Appl. 4, No. 5, 497--499 (1998; Zbl 0925.39004)] concerning the existence of a solution being not eventually constant.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 39A20 Generalized difference equations
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##### References:
 [1] Amleh, A. M.; Kruse, N.; Ladas, G.: On a class of difference equations with strong negative feedback. J. difference equ. Appl. 5, No. 6, 497-515 (1999) · Zbl 0951.39002 [2] K. Berenhaut, S. Stević, The global attractivity of a higher order rational difference equation, J. Math. Anal. Appl. (in press) · Zbl 1112.39002 [3] Berg, L.: Asymptotische darstellungen und entwicklungen. (1968) · Zbl 0165.36901 [4] Berg, L.: On the asymptotics of nonlinear difference equations. Z. anal. Anwendungen 21, No. 4, 1061-1074 (2002) · Zbl 1030.39006 [5] Berg, L.: Inclusion theorems for non-linear difference equations with applications. J. difference equ. Appl. 10, No. 4, 399-408 (2004) · Zbl 1056.39003 [6] Berg, L.: Corrections to ”inclusion theorems for non-linear difference equations with applications,” from [3]. J. difference equ. Appl. 11, No. 2, 181-182 (2005) · Zbl 1080.39002 [7] Berg, L.; Wolfersdorf, L. V.: On a class of generalized autoconvolution equations of the third kind. Z. anal. Anwendungen 24, No. 2, 217-250 (2005) · Zbl 1104.45001 [8] Kruse, N.; Nesemann, T.: Global asymptotic stability in some discrete dynamical systems. J. math. Anal. appl. 235, 151-158 (1999) · Zbl 0933.37016 [9] Ladas, G.: Open problems and conjectures. J. difference equ. Appl. 4, 497-499 (1998) [10] Exam, Putnam: Amer. math. Monthly. 734-736 (1965)