## On the solutions of systems of difference equations.(English)Zbl 1146.39023

Summary: We show that every solution of the following system of difference equations
$x_{n+1}^{(1)}=x_n^{(2)}/(x_n^{(2)}-1),\;x_{n+1}^{(2)}= x_n^{(3)}/(x_n^{(3)-1}),\dots,x_{n+1}^{(k)}=x_n^{(1)}/(x_n^{(1)}-1)$ as well as of the system
$x_{n+1}^{(1)}=x_n^{(k)}/(x_n^{(k)}-1), x_{n+1}^{(2)}= x_n^{(1)}/(x_n^{(1)}-1),\dots,x_{n+1}(k)=x_n(k-1)/(x_n^{(k-1)}-1)$
is periodic with period $$2_k$$ if $$k\neq 0 \pmod 2$$, and with period $$k$$ if $$k=0 \pmod2$$ where the initial values are nonzero real numbers for $$x_0^{(1)},x_0^{(2)},\dots,x_0^{(k)}\neq 1$$.

### MSC:

 39A11 Stability of difference equations (MSC2000) 39A20 Multiplicative and other generalized difference equations
Full Text:

### References:

  Papaschinopoulos, G; Schinas, CJ, On a system of two nonlinear difference equations, Journal of Mathematical Analysis and Applications, 219, 415-426, (1998) · Zbl 0908.39003  Camouzis, E; Papaschinopoulos, G, Global asymptotic behavior of positive solutions on the system of rational difference equations [inlineequation not available: see fulltext.], [inlineequation not available: see fulltext.], Applied Mathematics Letters, 17, 733-737, (2004) · Zbl 1064.39004  Çinar, C, On the positive solutions of the difference equation system [inlineequation not available: see fulltext.], [inlineequation not available: see fulltext.], Applied Mathematics and Computation, 158, 303-305, (2004) · Zbl 1066.39006  Çinar, C; Yalçinkaya, İ, On the positive solutions of difference equation system [inlineequation not available: see fulltext.], [inlineequation not available: see fulltext.], [inlineequation not available: see fulltext.], International Mathematical Journal, 5, 521-524, (2004) · Zbl 1208.39030  Clark, D; Kulenović, MRS, A coupled system of rational difference equations, Computers & Mathematics with Applications, 43, 849-867, (2002) · Zbl 1001.39017  Grove, EA; Ladas, G; McGrath, LC; Teixeira, CT, Existence and behavior of solutions of a rational system, Communications on Applied Nonlinear Analysis, 8, 1-25, (2001) · Zbl 1035.39013  Özban, AY, On the positive solutions of the system of rational difference equations [inlineequation not available: see fulltext.], [inlineequation not available: see fulltext.], Journal of Mathematical Analysis and Applications, 323, 26-32, (2006) · Zbl 1115.39012  Özban, AY, On the system of rational difference equations [inlineequation not available: see fulltext.], [inlineequation not available: see fulltext.], Applied Mathematics and Computation, 188, 833-837, (2007) · Zbl 1123.39006  Papaschinopoulos, G; Schinas, CJ, On the behavior of the solutions of a system of two nonlinear difference equations, Communications on Applied Nonlinear Analysis, 5, 47-59, (1998) · Zbl 1110.39301  Papaschinopoulos, G; Schinas, CJ, Invariants for systems of two nonlinear difference equations, Differential Equations and Dynamical Systems, 7, 181-196, (1999) · Zbl 0978.39014  Papaschinopoulos, G; Schinas, CJ; Stefanidou, G, On a [inlineequation not available: see fulltext.]-order system of lyness-type difference equations, No. 2007, 13, (2007) · Zbl 1154.39013  Irićanin, B; Stević, S, Some systems of nonlinear difference equations of higher order with periodic solutions, Dynamics of Continuous, Discrete and Impulsive Systems, Series A Mathematical Analysis, 13, 499-507, (2006) · Zbl 1098.39003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.