×

zbMATH — the first resource for mathematics

On a system of difference equations with period two coefficients. (English) Zbl 1256.39008
Considered is the following system of difference equations \[ x_{n+1}=\frac{a_nx_{n-1}}{b_ny_nx_{n-1}+c_n}, \qquad y_{n+1}=\frac{\alpha_n y_{n-1}}{\beta_n x_ny_{n-1}+\gamma_n}, \qquad n\in \mathbb{N}_0, \] where the sequences \(a_n\), \(b_n\), \(c_n\), \(\alpha_n\), \(\beta_n\), \(\gamma_n\) are two-periodic and the initial values \(x_{-1}\), \(x_0\), \(y_{-1}\), \(y_0\) are real numbers. For the all possible cases, the author derives explicit formulae for all well-defined solutions. These results extend those in his paper [Appl. Math. Comput. 218, No. 14, 7649–7654 (2012; Zbl 1243.39011)].

MSC:
39A20 Multiplicative and other generalized difference equations
39A23 Periodic solutions of difference equations
Citations:
Zbl 1243.39011
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aloqeili, M., Global stability of a rational symmetric difference equation, Appl. math. comput., 215, 3, 950-953, (2009) · Zbl 1177.39019
[2] Berenhaut, K.; Stević, S., The behaviour of the positive solutions of the difference equation xn=A+(xn−2/xn−1)p, J. differ. equat. appl., 12, 9, 909-918, (2006) · Zbl 1111.39003
[3] Berg, L.; Stević, S., Periodicity of some classes of holomorphic difference equations, J. differ. equat. appl., 12, 8, 827-835, (2006) · Zbl 1103.39004
[4] Berg, L.; Stević, S., Linear difference equations mod 2 with applications to nonlinear difference equations, J. differ. equat. appl., 14, 7, 693-704, (2008) · Zbl 1156.39003
[5] Berg, L.; Stević, S., On the asymptotics of the difference equation yn(1+yn−1⋯yn−k+1)=yn−k, J. differ. equat. appl., 17, 4, 577-586, (2011) · Zbl 1220.39011
[6] Iričanin, B., Global stability of some classes of higher-order nonlinear difference equations, Appl. math. comput., 216, 4, 1325-1328, (2010) · Zbl 1194.39012
[7] Iričanin, B., The boundedness character of two stevic-type fourth-order difference equations, Appl. math. comput., 217, 5, 1857-1862, (2010) · Zbl 1219.39006
[8] 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
[9] Iričanin, B.; Stević, S., Eventually constant solutions of a rational difference equation, Appl. math. comput., 215, 854-856, (2009) · Zbl 1178.39012
[10] Iričanin, B.D.; Stević, S., On a class of third-order nonlinear difference equations, Appl. math. comput., 213, 479-483, (2009) · Zbl 1178.39011
[11] Karakostas, G.L., Asymptotic 2-periodic difference equations with diagonally self-invertible responces, J. differ. equat. appl., 6, 329-335, (2000) · Zbl 0963.39020
[12] Kent, C.M., Convergence of solutions in a nonhyperbolic case, Nonlinear anal., 47, 4651-4665, (2001) · Zbl 1042.39507
[13] Kent, C.M.; Kosmala, W.; Radin, M.A.; Stević, S., Solutions of the difference equation xn+1=xnxn−1−1, Abstr. appl. anal., 2010, 13, (2010), (Article ID 469683) · Zbl 1198.39017
[14] Kent, C.M.; Kosmala, W.; Stević, S., On the difference equation xn+1=xnxn−2−1, Abstr. appl. anal. vol., 2011, 15, (2011), (Article ID 815285)
[15] Papaschinopoulos, G.; Schinas, C.J., On the behavior of the solutions of a system of two nonlinear difference equations, Comm. appl. nonlinear anal., 5, 2, 47-59, (1998) · Zbl 1110.39301
[16] Papaschinopoulos, G.; Schinas, C.J., Invariants for systems of two nonlinear difference equations, Diff. equat. dynam. syst., 7, 2, 181-196, (1999) · Zbl 0978.39014
[17] 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
[18] 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, 55735580, (2011) · Zbl 1221.39013
[19] Papaschinopoulos, G.; Stefanidou, G., Trichotomy of a system of two difference equations, J. math. anal. appl., 289, 216-230, (2004) · Zbl 1045.39008
[20] Stević, S., On the recursive sequence xn+1=xn−1/g(xn), Taiwanese J. math., 6, 3, 405-414, (2002)
[21] Stević, S., More on a rotional recurrence relation, Appl. math. E-notes, 4, 80-85, (2004) · Zbl 1069.39024
[22] Stević, S., Global stability and asymptotics of some classes of rational difference equations, J. math. anal. appl., 316, 60-68, (2006) · Zbl 1090.39009
[23] Stević, S., On positive solutions of a (k+1)th order difference equation, Appl. math. lett., 19, 5, 427-431, (2006) · Zbl 1095.39010
[24] Stević, S., A short proof of the cushing – henson conjecture, Discrete dyn. nat. soc., 2006, 5, (2006), (Article ID 37264) · Zbl 1149.39300
[25] Stević, S., Existence of nontrivial solutions of a rational difference equation, Appl. math. lett., 20, 28-31, (2007) · Zbl 1131.39009
[26] Stević, S., Nontrivial solutions of a higher-order rational difference equation, Math. notes, 84, 5-6, 718-724, (2008) · Zbl 1219.39007
[27] Stević, S., On the recursive sequence \(x_{n + 1} = \max \{c, x_n^p / x_{n - 1}^p \}\), Appl. math. lett., 21, 8, 791-796, (2008) · Zbl 1152.39012
[28] Stević, S., Boundedness character of a class of difference equations, Nonlinear anal. TMA, 70, 839-848, (2009) · Zbl 1162.39011
[29] Stević, S., Global stability of a difference equation with maximum, Appl. math. comput., 210, 525-529, (2009) · Zbl 1167.39007
[30] Stević, S., Global stability of a MAX-type equation, Appl. math. comput., 216, 354-356, (2010) · Zbl 1193.39009
[31] Stević, S., Global stability of some symmetric difference equations, Appl. math. comput., 216, 179-186, (2010) · Zbl 1193.39008
[32] Stević, S., On a generalized MAX-type difference equation from automatic control theory, Nonlinear anal. TMA, 72, 1841-1849, (2010) · Zbl 1194.39007
[33] Stević, S., Periodicity of MAX difference equations, Util. math., 83, 69-71, (2010) · Zbl 1236.39018
[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., Periodicity of a class of nonautonomous MAX-type difference equations, Appl. math. comput., 217, 9562-9566, (2011) · Zbl 1225.39018
[36] Stević, S., On the difference equation xn=xn−2/(bn+cnxn−1xn−2), Appl. math. comput., 218, 8, 4507-4513, (2011) · Zbl 1256.39009
[37] Stević, S., On a system of difference equations, Appl. math. comput., 218, 7, 3372-3378, (2011) · Zbl 1242.39017
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.