# American Institute of Mathematical Sciences

doi: 10.3934/era.2021017

## On a general homogeneous three-dimensional system of difference equations

 1 LMAM Laboratory, Department of Mathematics, Mohamed Seddik Ben Yahia University, Jijel, Algeria 2 Department of Mathematics and Computer Sciences, Necmettin Erbakan University, Konya, Turkey 3 ENS Assia Djebar Constantine and LMAM Laboratory, Mohamed Seddik Ben Yahia University, Jijel, Algeria

* Corresponding author: ntouafek@gmail.com

Received  October 2020 Revised  January 2021 Published  March 2021

In this work, we study the behavior of the solutions of following three-dimensional system of difference equations
 $\begin{equation*} x_{n+1} = f(y_{n}, y_{n-1}), \, y_{n+1} = g(z_{n}, z_{n-1}), \, z_{n+1} = h(x_{n}, x_{n-1}) \end{equation*}$
where
 $n\in \mathbb{N}_{0}$
, the initial values
 $x_{-1}$
,
 $x_{0}$
,
 $y_{-1}$
,
 $y_{0}$
 $z_{-1}$
,
 $z_{0}$
are positive real numbers, the functions
 $f, \, g, \, h:\, \left(0, +\infty\right)^{2}\rightarrow\left(0, +\infty\right)$
are continuous and homogeneous of degree zero. By proving some general convergence theorems, we have established conditions for the global stability of the corresponding unique equilibrium point. We give necessary and sufficient conditions on existence of prime period two solutions of the above mentioned system. Also, we prove a result on oscillatory solutions. As applications of the obtained results, some particular systems of difference equations defined by homogeneous functions of degree zero are investigated. Our results generalize some existing ones in the literature.
Citation: Nouressadat Touafek, Durhasan Turgut Tollu, Youssouf Akrour. On a general homogeneous three-dimensional system of difference equations. Electronic Research Archive, doi: 10.3934/era.2021017
##### References:
 [1] M. A. E. Abdelrahman, On the difference equation $z_{m+1} = f(z_m, z_{m-1}, ..., z_{m-k})$, J. Taibah Univ. Sci., 13 (2019), 1014-1021.   Google Scholar [2] A. M. Amleh, E. A. Grove, G. Ladas and D. A. Georgiou, On the recursive seqience $x_{n+1} = \alpha+\frac{x_{n-1}}{ x_{n}}$, J. Math. Anal. Appl., 233 (1999), 790-798.  doi: 10.1006/jmaa.1999.6346.  Google Scholar [3] K. C. Border, Euler's Theorem for homogeneous functions, 2017. Available from: http://www.its.caltech.edu/ kcborder/Courses/Notes/EulerHomogeneity.pdf. Google Scholar [4] I. Dekkar, N. Touafek and Y. Yazlik, Global stability of a third-order nonlinear system of difference equations with period-two coefficients, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 111 (2017), 325-347.  doi: 10.1007/s13398-016-0297-z.  Google Scholar [5] R. DeVault and S. W. Scultz, On the dynamics of $x_{n+1} = {\beta x_n+\gamma x_{n-1}\over Bx_n+Dx_{n-2}}$, Commun. Appl. Nonlinear Anal., 12 (2005), 35-39.   Google Scholar [6] S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer, 2005.  Google Scholar [7] E. M. Elsayed, New method to obtain periodic solutions of period two and three of a rational difference equation, Nonlinear Dynam., 79 (2015), 241-250.  doi: 10.1007/s11071-014-1660-2.  Google Scholar [8] E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, Advances in Discrete Mathematics and Applications Volume 4, Chapman and hall/CRC, 2005.  Google Scholar [9] M. Gümüş, The global asymptotic stability of a system of difference equations, J. Difference Equ. Appl., 24 (2018), 976-991.  doi: 10.1080/10236198.2018.1443445.  Google Scholar [10] N. Haddad, N. Touafek and J. F. T. Rabago, Solution form of a higher-order system of difference equations and dynamical behavior of its special case, Math. Methods Appl. Sci., 40 (2017), 3599-3607.  doi: 10.1002/mma.4248.  Google Scholar [11] Y. Halilm, N. Touafek and Y. Yazlik, Dynamic behavior of a second-order nonlinear rational difference equation, Turkish J. Math., 39 (2015), 1004-1018.  doi: 10.3906/mat-1503-80.  Google Scholar [12] T. F. Ibrahim and N. Touafek, On a third order rational difference equation with variable coeffitients, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 20 (2013), 251-264.   Google Scholar [13] V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Mathematics and its Applications, Kluwer Academic Publisher, volume 256, 1993. doi: 10.1007/978-94-017-1703-8.  Google Scholar [14] A. S. Kurbanli, C. Çinar and I. Yalçinkaya, On the behavior of positive solutions of the system of rational difference equations $x_{n+1} = x_{n-1}/(y_{n}x_{n+1}+1)$, $y_{n+1} = y_{n-1}/(x_{n}y_{n+1}+1)$, Math. Comput. Modelling, 53 (2011), 1261–1267. doi: 10.1016/j.mcm.2010.12.009.  Google Scholar [15] O. Moaaz, Comment on New method to obtain periodic solutions of period two and three of a rational difference equation [Nonlinear Dyn 79:241-250], Nonlinear Dyn., 88 (2017), 1043-1049.   Google Scholar [16] O. Moaaz, Dynamics of difference equation $x_{n+1} = f(x_{n-l}, x_{n-k})$, Adv. Difference Equ., 2018 (2018), Paper No. 447, 14 pp. doi: 10.1186/s13662-018-1896-0.  Google Scholar [17] O. Moaaz, D. Chalishajar and O. Bazighifan, Some qualitative behavior of solutions of general class of difference equations, Mathematics, 7 (2019), Article 585, 12 pp. Google Scholar [18] O. Özkan, A. S. Kurbanli, On a system of difference equations, Discrete Dyn. Nat. Soc., 2013 (2013), Art. ID 970316, 7 pp. doi: 10.1155/2013/970316.  Google Scholar [19] S. Stević, Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences, Electron. J. Qual. Theory Differ. Equ., 2014 (2014), Article no. 67, 15 pp. doi: 10.14232/ejqtde.2014.1.67.  Google Scholar [20] N. Taskara, D. T. Tollu, N. Touafek and Y. Yazlik, A solvable system of difference equations, Commun. Korean Math. Soc., 35 (2020), 301-319.  doi: 10.4134/CKMS.c180472.  Google Scholar [21] D. T. Tollu and I. Yalçinkaya, Global behavior of a three-dimensional system of difference equations of order three, Commun. Fac. Sci. Univ. Ank., Ser. A1, Math. Stat., 68 (2019), no. 1, 1–16. doi: 10.31801/cfsuasmas.443530.  Google Scholar [22] N. Touafek, On a second order rational difference equation, Hacet. J. Math. Stat., 41 (2012), 867-874.   Google Scholar [23] N. Touafek, On a general system of difference equations defined by homogeneous functions, Math. Slovaca, to appear. Google Scholar [24] C. Wang, X. Jing, X. Hu and R. Li, On the periodicity of a max-type rational difference equation, J. Nonlinear Sci. Appl., 10 (2017), 4648-4661.  doi: 10.22436/jnsa.010.09.08.  Google Scholar [25] I. Yalcinkaya, C. Cinar and D. Simsek, Global asymptotic stability of a system of difference equations, Appl. Anal., 87 (2008), 677-687.  doi: 10.1080/00036810802140657.  Google Scholar [26] I. Yalcinkaya and D. T. Tollu, Global behavior of a second-order system of difference equations, Adv. Stud. Contemp. Math. Kyungshang, 26 (2016), 653-667.   Google Scholar [27] Y. Yazlik, E. M. Elsayed and N. Taskara, On the behaviour of the solutions of difference equation systems, J. Comput. Anal. Appl., 16 (2014), 932-941.   Google Scholar [28] Y. Yazlik and M. Kara, On a solvable system of difference equations of higher-order with period two coefficients, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 68 (2019), 1675-1693.  doi: 10.31801/cfsuasmas.548262.  Google Scholar [29] Y. Yazlik, D. T. Tollu and N. Taskara, On the solutions of a three-dimensional system of difference equations, Kuwait J. Sci., 43 (2016), 95-111.   Google Scholar

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##### References:
 [1] M. A. E. Abdelrahman, On the difference equation $z_{m+1} = f(z_m, z_{m-1}, ..., z_{m-k})$, J. Taibah Univ. Sci., 13 (2019), 1014-1021.   Google Scholar [2] A. M. Amleh, E. A. Grove, G. Ladas and D. A. Georgiou, On the recursive seqience $x_{n+1} = \alpha+\frac{x_{n-1}}{ x_{n}}$, J. Math. Anal. Appl., 233 (1999), 790-798.  doi: 10.1006/jmaa.1999.6346.  Google Scholar [3] K. C. Border, Euler's Theorem for homogeneous functions, 2017. Available from: http://www.its.caltech.edu/ kcborder/Courses/Notes/EulerHomogeneity.pdf. Google Scholar [4] I. Dekkar, N. Touafek and Y. Yazlik, Global stability of a third-order nonlinear system of difference equations with period-two coefficients, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 111 (2017), 325-347.  doi: 10.1007/s13398-016-0297-z.  Google Scholar [5] R. DeVault and S. W. Scultz, On the dynamics of $x_{n+1} = {\beta x_n+\gamma x_{n-1}\over Bx_n+Dx_{n-2}}$, Commun. Appl. Nonlinear Anal., 12 (2005), 35-39.   Google Scholar [6] S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer, 2005.  Google Scholar [7] E. M. Elsayed, New method to obtain periodic solutions of period two and three of a rational difference equation, Nonlinear Dynam., 79 (2015), 241-250.  doi: 10.1007/s11071-014-1660-2.  Google Scholar [8] E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, Advances in Discrete Mathematics and Applications Volume 4, Chapman and hall/CRC, 2005.  Google Scholar [9] M. Gümüş, The global asymptotic stability of a system of difference equations, J. Difference Equ. Appl., 24 (2018), 976-991.  doi: 10.1080/10236198.2018.1443445.  Google Scholar [10] N. Haddad, N. Touafek and J. F. T. Rabago, Solution form of a higher-order system of difference equations and dynamical behavior of its special case, Math. Methods Appl. Sci., 40 (2017), 3599-3607.  doi: 10.1002/mma.4248.  Google Scholar [11] Y. Halilm, N. Touafek and Y. Yazlik, Dynamic behavior of a second-order nonlinear rational difference equation, Turkish J. Math., 39 (2015), 1004-1018.  doi: 10.3906/mat-1503-80.  Google Scholar [12] T. F. Ibrahim and N. Touafek, On a third order rational difference equation with variable coeffitients, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 20 (2013), 251-264.   Google Scholar [13] V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Mathematics and its Applications, Kluwer Academic Publisher, volume 256, 1993. doi: 10.1007/978-94-017-1703-8.  Google Scholar [14] A. S. Kurbanli, C. Çinar and I. Yalçinkaya, On the behavior of positive solutions of the system of rational difference equations $x_{n+1} = x_{n-1}/(y_{n}x_{n+1}+1)$, $y_{n+1} = y_{n-1}/(x_{n}y_{n+1}+1)$, Math. Comput. Modelling, 53 (2011), 1261–1267. doi: 10.1016/j.mcm.2010.12.009.  Google Scholar [15] O. Moaaz, Comment on New method to obtain periodic solutions of period two and three of a rational difference equation [Nonlinear Dyn 79:241-250], Nonlinear Dyn., 88 (2017), 1043-1049.   Google Scholar [16] O. Moaaz, Dynamics of difference equation $x_{n+1} = f(x_{n-l}, x_{n-k})$, Adv. Difference Equ., 2018 (2018), Paper No. 447, 14 pp. doi: 10.1186/s13662-018-1896-0.  Google Scholar [17] O. Moaaz, D. Chalishajar and O. Bazighifan, Some qualitative behavior of solutions of general class of difference equations, Mathematics, 7 (2019), Article 585, 12 pp. Google Scholar [18] O. Özkan, A. S. Kurbanli, On a system of difference equations, Discrete Dyn. Nat. Soc., 2013 (2013), Art. ID 970316, 7 pp. doi: 10.1155/2013/970316.  Google Scholar [19] S. Stević, Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences, Electron. J. Qual. Theory Differ. Equ., 2014 (2014), Article no. 67, 15 pp. doi: 10.14232/ejqtde.2014.1.67.  Google Scholar [20] N. Taskara, D. T. Tollu, N. Touafek and Y. Yazlik, A solvable system of difference equations, Commun. Korean Math. Soc., 35 (2020), 301-319.  doi: 10.4134/CKMS.c180472.  Google Scholar [21] D. T. Tollu and I. Yalçinkaya, Global behavior of a three-dimensional system of difference equations of order three, Commun. Fac. Sci. Univ. Ank., Ser. A1, Math. Stat., 68 (2019), no. 1, 1–16. doi: 10.31801/cfsuasmas.443530.  Google Scholar [22] N. Touafek, On a second order rational difference equation, Hacet. J. Math. Stat., 41 (2012), 867-874.   Google Scholar [23] N. Touafek, On a general system of difference equations defined by homogeneous functions, Math. Slovaca, to appear. Google Scholar [24] C. Wang, X. Jing, X. Hu and R. Li, On the periodicity of a max-type rational difference equation, J. Nonlinear Sci. Appl., 10 (2017), 4648-4661.  doi: 10.22436/jnsa.010.09.08.  Google Scholar [25] I. Yalcinkaya, C. Cinar and D. Simsek, Global asymptotic stability of a system of difference equations, Appl. Anal., 87 (2008), 677-687.  doi: 10.1080/00036810802140657.  Google Scholar [26] I. Yalcinkaya and D. T. Tollu, Global behavior of a second-order system of difference equations, Adv. Stud. Contemp. Math. Kyungshang, 26 (2016), 653-667.   Google Scholar [27] Y. Yazlik, E. M. Elsayed and N. Taskara, On the behaviour of the solutions of difference equation systems, J. Comput. Anal. Appl., 16 (2014), 932-941.   Google Scholar [28] Y. Yazlik and M. Kara, On a solvable system of difference equations of higher-order with period two coefficients, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 68 (2019), 1675-1693.  doi: 10.31801/cfsuasmas.548262.  Google Scholar [29] Y. Yazlik, D. T. Tollu and N. Taskara, On the solutions of a three-dimensional system of difference equations, Kuwait J. Sci., 43 (2016), 95-111.   Google Scholar
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