×
Stević, Stevo

On a third-order system of difference equations. (English) Zbl 1243.39011

Appl. Math. Comput. 218, No. 14, 7649-7654 (2012).
Summary: We show that the system of difference equations \[ x_{n+1}=\frac{a_1x_{n-2}}{b_1y_nz_{n-1}x_{n-2}+c_1},\;y_{n+1}=\frac{a_2y_{n-2}}{b_2z_nx_{n-1}y_{n-2}+c_2},\;z_{n+1}=\frac{a_3z_{n-2}}{b_3x_ny_{n-1}z_{n-2}+c_3},\quad n\neq\mathbb{N}_0, \] where the parameters \(a_i,b_i,c_i\), \(i\in\{1,2,3\}\), and initial values \(x_{-j},y_{-j},z_{-j}\), \(j\in\{0,1,2\}\), are real numbers, can be solved, developing further the results in the literature.

Cited in 1 Review
Cited in 55 Documents

MSC:

39A20 Multiplicative and other generalized difference equations

Keywords:

third-order system of difference equations; explicit solutions; Riccati difference equation
PDF BibTeX XML Cite
\textit{S. Stević}, Appl. Math. Comput. 218, No. 14, 7649--7654 (2012; Zbl 1243.39011)
Full Text: DOI

References:

[1] Aloqeili, M., Dynamics of a \(k\) th order rational difference equation, Appl. Math. Comput., 181, 1328-1335 (2006) · Zbl 1108.39004
[2] Aloqeili, M., Dynamics of a rational difference equation, Appl. Math. Comput., 176, 768-774 (2006) · Zbl 1100.39002
[3] Andruch-Sobilo, A.; Migda, M., Further properties of the rational recursive sequence \(x_{n + 1} = ax_{n - 1} /(b + cx_n x_{n - 1})\), Opuscula Math., 26, 3, 387-394 (2006) · Zbl 1131.39003
[4] Andruch-Sobilo, A.; Migda, M., On the rational recursive sequence \(x_{n + 1} = ax_{n - 1} /(b + cx_n x_{n - 1})\), Tatra Mt. Math. Publ., 43, 1-9 (2009) · Zbl 1212.39008
[5] Bajo, I.; Liz, E., Global behaviour of a second-order nonlinear difference equation, J. Differ. Eq. Appl., 17, 10, 1471-1486 (2011) · Zbl 1232.39014
[7] Berg, L., On the asymptotics of nonlinear difference equations, Z. Anal. Anwendungen, 21, 4, 1061-1074 (2002) · Zbl 1030.39006
[8] Berg, L.; Stević, S., On difference equations with powers as solutions and their connection with invariant curves, Appl. Math. Comput., 217, 7191-7196 (2011) · Zbl 1260.39002
[9] Berg, L.; Stević, S., On some systems of difference equations, Appl. Math. Comput., 218, 1713-1718 (2011) · Zbl 1243.39009
[10] Berg, L.; Stević, S., On the asymptotics of the difference equation \(y_n(1 + y_{n - 1} \cdots y_{n - k + 1}) = y_{n - k}\), J. Differ. Eq. Appl., 17, 4, 577-586 (2011) · Zbl 1220.39011
[11] Cinar, C., On the positive solutions of difference equation, Appl. Math. Comput., 150, 1, 21-24 (2004) · Zbl 1050.39005
[12] Grove, E. A.; Ladas, G., Periodicities in Nonlinear Difference Equations (2005), Chapman & Hall: Chapman & Hall CRC Press, Boca Raton · Zbl 1078.39009
[13] Iričanin, B.; Stević, S., Some systems of nonlinear difference equations of higher order with periodic solutions, Dynam. Contin. Discrete Impuls. Syst., 13 a, 3-4, 499-508 (2006) · Zbl 1098.39003
[14] Iričanin, B.; Stević, S., On some rational difference equations, Ars Combin., 92, 67-72 (2009) · Zbl 1224.39014
[15] Kent, C. M., Convergence of solutions in a nonhyperbolic case, Nonlinear Anal., 47, 4651-4665 (2001) · Zbl 1042.39507
[16] Levy, H.; Lessman, F., Finite Difference Equations (1992), Dover Publications, Inc.: Dover Publications, Inc. New York · Zbl 0092.07702
[17] Papaschinopoulos, G.; Radin, M.; Schinas, C. J., Study of the asymptotic behavior of the solutions of three systems of difference equations of exponential form, Appl. Math. Comput., 218, 5310-5318 (2012) · Zbl 1245.39011
[18] Papaschinopoulos, G.; Schinas, C. J., On a system of two nonlinear difference equations, J. Math. Anal. Appl., 219, 2, 415-426 (1998) · Zbl 0908.39003
[19] Papaschinopoulos, G.; Schinas, C. J., On the behavior of the solutions of a system of two nonlinear difference equations, Commun. Appl. Nonlinear Anal., 5, 2, 47-59 (1998) · Zbl 1110.39301
[20] Papaschinopoulos, G.; Schinas, C. J., Invariants for systems of two nonlinear difference equations, Differ. Eq. Dynam. Syst., 7, 2, 181-196 (1999) · Zbl 0978.39014
[21] Papaschinopoulos, G.; Schinas, C. J., Invariants and oscillation for systems of two nonlinear difference equations, Nonlinear Anal. TMA, 46, 7, 967-978 (2001) · Zbl 1003.39007
[22] Papaschinopoulos, G.; Schinas, C. J., On the system of two difference equations \(x_{n + 1} = \sum_{i = 0}^k A_i / y_{n - i}^{p_i} y_{n + 1} = \sum_{i = 0}^k B_i / x_{n - i}^{q_i}\), J. Math. Anal. Appl., 273, 2, 294-309 (2002) · Zbl 1014.39016
[23] Papaschinopoulos, G.; Schinas, C. J.; Stefanidou, G., On the nonautonomous difference equation \(x_{n + 1} = A_n +(x_{n - 1}^p / x_n^q)\), Appl. Math. Comput., 217, 5573-5580 (2011) · Zbl 1221.39013
[24] Papaschinopoulos, G.; Stefanidou, G., Trichotomy of a system of two difference equations, J. Math. Anal. Appl., 289, 216-230 (2004) · Zbl 1045.39008
[25] Stević, S., On the recursive sequence \(x_{n + 1} = x_{n - 1} / g(x_n)\), Taiwanese J. Math., 6, 3, 405-414 (2002) · Zbl 1019.39010
[26] Stević, S., More on a rational recurrence relation, Appl. Math. E-Notes, 4, 80-85 (2004) · Zbl 1069.39024
[27] Stević, S., Global stability and asymptotics of some classes of rational difference equations, J. Math. Anal. Appl., 316, 60-68 (2006) · Zbl 1090.39009
[28] Stević, S., On positive solutions of a \((k + 1)\)-th order difference equation, Appl. Math. Lett., 19, 5, 427-431 (2006) · Zbl 1095.39010
[30] Stević, S., Existence of nontrivial solutions of a rational difference equation, Appl. Math. Lett., 20, 28-31 (2007) · Zbl 1131.39009
[31] Stević, S., On the recursive sequencex \(x_{n + 1} = \max \{c, x_n^p / x_{n - 1}^p \}\), Appl. Math. Lett., 21, 8, 791-796 (2008) · Zbl 1152.39012
[32] Stević, S., Global stability of a max-type equation, Appl. Math. Comput., 216, 354-356 (2010) · Zbl 1193.39009
[33] Stević, S., Global stability of some symmetric difference equations, Appl. Math. Comput., 216, 179-186 (2010) · Zbl 1193.39008
[34] Stević, S., On a nonlinear generalized max-type difference equation, J. Math. Anal. Appl., 376, 317-328 (2011) · Zbl 1208.39014
[35] Stević, S., On a system of difference equations, Appl. Math. Comput., 218, 3372-3378. (2011) · Zbl 1242.39017
[36] Stević, S., On the difference equation \(x_n = x_{n - 2} /(b_n + c_n x_{n - 1} x_{n - 2})\), Appl. Math. Comput., 218, 4507-4513 (2011) · Zbl 1256.39009
[37] Stević, S., Periodicity of a class of nonautonomous max-type difference equations, Appl. Math. Comput., 217, 9562-9566 (2011) · Zbl 1225.39018
[38] Stević, S., On some solvable systems of difference equations, Appl. Math. Comput., 218, 5010-5018 (2012) · Zbl 1253.39011
[39] Voulov, H. D., Periodic solutions to a difference equation with maximum, Proc. Amer. Math. Soc., 131, 7, 2155-2160 (2003) · Zbl 1019.39005
[40] Yang, X.; Liao, X., On a difference equation with maximum, Appl. Math. Comput., 181, 1-5 (2006) · Zbl 1148.39303
[41] Yang, X.; Liu, X.; Wang, L., Stability of a generalized Putnam equation, Appl. Math. Lett., 22, 4, 565-568 (2009) · Zbl 1171.39003
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.