Eventually constant solutions of a rational difference equation. (English) Zbl 1178.39012

Iričanin, Bratislav; Stević, Stevo

Appl. Math. Comput. 215, No. 2, 854-856 (2009).

The authors consider the rational difference equation \[ x_{n+1} = {{x_n+x_{n-1}+x_{n-2}x_{n-3}}\over{x_nx_{n-1}+x_{n-2}+x_{n-3}}}. \] The note displays the following results (with proofs):
1. Every positive solution that eventually equals 1 has one of the following forms 7mm

(a)

\(\{a,1,c;1,1,1,\dots\}\)

(b)

\(\{1,b,1,d;1,1,\dots\}\)

(c)

\(\{a,1,1,d;1,1,\dots\}\)

(d)

\(\{a,b,1,1,(2+ab)/(1+a+b);1,1,\dots\}\), \(a\neq 1\neq b\)

(e)

\(\{1,b,c,1,1,(2+bc)/(1+b+c);1,1,1,\dots\}\), \(b\neq 1\neq c\)

where \(x_{-3}=a\), \(x_{-2}=b\), \(x_{-1}=c\), \(x_0=d\) for convenience.
2. For almost all initial values, positive solutions are not eventually equal to 1.

Reviewer: Vladimir Răsvan (Craiova)

MSC:

39A20

Multiplicative and other generalized difference equations

Keywords:

rational difference equation; positive solution; eventual equality to 1

References:

[1]	Amleh, A. M.; Kruse, N.; Ladas, G., On a class of difference equations with strong negative feedback, J. Differ. Equations Appl., 5, 6, 497-515 (1999) · Zbl 0951.39002
[2]	Berenhaut, K.; Foley, J.; Stević, S., The global attractivity of the rational difference equation \(y_n = (y_{n - k} + y_{n - m}) /(1 + y_{n - k} y_{n - m})\), Appl. Math. Lett., 20, 54-58 (2007) · Zbl 1131.39006
[3]	Berenhaut, K.; Stević, S., The global attractivity of a higher order rational difference equation, J. Math. Anal. Appl., 326, 2, 940-944 (2007) · Zbl 1112.39002
[4]	Berg, L., Inclusion theorems for non-linear difference equations with applications, J. Differ. Equations Appl., 10, 4, 399-408 (2004) · Zbl 1056.39003
[5]	Berg, L., Corrections to Inclusion theorems for non-linear difference equations with applications, from [3], J. Differ. Equations Appl., 11, 2, 181-182 (2005) · Zbl 1080.39002
[6]	Berg, L.; Stević, S., Linear difference equations mod 2 with applications to nonlinear difference equations, J. Differ. Equations Appl., 14, 7, 693-704 (2008) · Zbl 1156.39003
[7]	Cinar, C.; Stević, S.; Yalçinkaya, I., A note on global asymptotic stability of a family of rational equations, Rostock. Math. Kolloq., 59, 41-49 (2004) · Zbl 1083.39003
[8]	De la Sen, M.; Alonso-Quesada, S., A control theory point of view on BevertonHolt equation in population dynamics and some of its generalizations, Appl. Math. Comput., 199, 2, 464-481 (2008) · Zbl 1137.92034
[9]	Kruse, N.; Nesemann, T., Global asymptotic stability in some discrete dynamical systems, J. Math. Anal. Appl., 235, 151-158 (1999) · Zbl 0933.37016
[10]	Ladas, G., Open problems and conjectures: a problem from the Putnam exam, J. Differ. Equations Appl., 4, 497-499 (1998) · Zbl 0925.39004
[11]	Stević, S., Global stability and asymptotics of some classes of rational difference equations, J. Math. Anal. Appl., 316, 60-68 (2006) · Zbl 1090.39009
[13]	Stević, S., Existence of nontrivial solutions of a rational difference equation, Appl. Math. Lett., 20, 28-31 (2007) · Zbl 1131.39009
[14]	Stević, S., Nontrivial solutions of a higher-order rational difference equation, Mat. Zametki, 84, 5, 772-780 (2008) · Zbl 1219.39007
[17]	Yang, Y.; Yang, X., On the difference equation \(x_{n + 1} = (px_{n - s} + x_{n - t}) /(qx_{n - s} + x_{n - t})\), Appl. Math. Comput., 203, 2, 903-907 (2008) · Zbl 1162.39015