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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 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 Multiplicative and other generalized difference equations

Citations:

Zbl 0925.39004
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Full Text: DOI

References:

[1] Amleh, A. M.; Kruse, N.; Ladas, G., On a class of difference equations with strong negative feedback, J. Difference Equ. Appl., 5, 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); 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), Dt. Verlag Wiss.: Dt. Verlag Wiss. Berlin · Zbl 0165.36901
[4] Berg, L., On the asymptotics of nonlinear difference equations, Z. Anal. Anwendungen, 21, 4, 1061-1074 (2002) · Zbl 1030.39006
[5] Berg, L., Inclusion theorems for non-linear difference equations with applications, J. Difference Equ. Appl., 10, 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, 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, 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) · Zbl 0925.39004
[10] Exam, Putnam, Amer. Math. Monthly, 734-736 (1965)
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