# On the Nature of Solutions of a System of Second Order Nonlinear Difference Equations

### Abstract

In this paper, we investigate the dynamical behaviors of a system of second order nonlinear difference equations We study local stability of the equilibrium point of the system of the second order rational difference equations and oscillation behaviour of positive solutions of the aforementioned system. Moreover, we establish that the system has unbounded solutions.

### References

Camouzis E, and Papaschinopoulos G (2004) Global Asymptotic Behavior of Positive Solutions on the System of Rational Difference Equations x_(n+1)=1+x_n/y_(n-m) ,y_(n+1)=1+y_n/x_(n-m) . Applied Mathematics Letters, 17 (6): 733-737.

El-Owaidy HM, Ahmed AM, and Mousa MS (2003) On the Asymptotic Behaviour of the Difference Equation x_(n+1)=α+(x_(n-1)^p)/(x_n^p ). Journal of Applied Mathematics and Computing, 12 (1-2): 31-37.

Göcen M, and Cebeci A (2018) On the Periodic Solutions of Some Systems of Higher Order Difference Equations. Rocky Mountain J. Math., 48(3): 845-858.

Göcen M, and Güneysu M (2018) The Global Attractivity of Some Rational Difference Equations. J. Comput. Anal. Appl., 25(7): 1233-1243.

Gümüş M, and Soykan Y (2016) Global Character of a Six-Dimensional Nonlinear System of Difference Equations. Discrete. Dyn. Nature Soc. 2016, Article ID 6842521.

Okumuş İ, and Soykan Y (2018) Dynamical Behavior of a System of Three-Dimensional Nonlinear Difference Equations. Advance in Difference Equations, 2018:223.

Okumuş İ, and Soykan Y (2018) On the Dynamics of a Higher Order Nonlinear System of Difference Equations. arXiv: 1810.07986v1, 2018.

Okumuş İ, and Soykan Y (2018) Some Technique to Show the Boundedness of Rational Difference Equations. Journal of Progressive Research in Mathematics, 13(2): 2246-2258.

Okumuş İ, and Soykan Y (2017) On the Stability of a Nonlinear Difference Equations. Asian Journal of Mathematics and Computer Research, 17(2): 88-110.

Kocic VL, and Ladas G (1993) Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Chapman & Hall/CRC, London.

Kulenovic MRS, and Ladas G (2002) Dynamics of Second-Order Rational Difference Equations with Open Problems and Conjecture. Chapman & Hall/CRC, London.

Papaschinopoulos G, and Schinas CJ (1998) On a System of Two Difference Equations. J. Mathematical Analysis and Applications, 219 (2): 415-426.

Papaschinopoulos G, and Schinas CJ (2000) On the System of Two Nonlinear Difference Equations x_(n+1)=A+x_(n-1)/y_n ,y_(n+1)=A+y_(n-1)/x_n . Int. J. Math. Mathematical Sci., 23(12), 839-848.

Taşdemir E, and Soykan Y (2016) On the Periodicies of the Difference Equation x_(n+1)=x_n x_(n-1)+α. Karaelmas Sci. Eng. J., 6(2): 329-333.

Taşdemir E, and Soykan Y (2017) Long-Term Behavior of Solutions of the Non-Linear Difference Equation x_(n+1)=x_(n-1) x_(n-3)-1. Gen. Math. Notes, 38(1): 13-31.

Taşdemir E, and Soykan Y (2018) Stability of Negative Equilibrium of a Non-Linear Difference Equation. J. Math. Sci. Adv. Appl., 49(1): 51-57.

Taşdemir E, and Soykan Y (2019) Dynamical Analysis of a Non-Linear Difference Equation. J. Comput. Anal. Appl., 26(2): 288-301.

Yang, X (2005) On the System of Rational Difference Equations x_(n+1)=A+y_(n-1)/(x_(n-p) y_(n-q) ),y_(n+1)=A+x_(n-1)/(x_(n-r) y_(n-s) ). J. Math. Anal. Appl., 307 (1): 305-311.

Zhang D, Ji W, Wang L, and Li X (2013) On the Symmetrical System of Rational Difference Equation x_(n+1)=A+y_(n-k)/y_n ,y_(n+1)=A+x_(n-k)/x_n . Applied Mathematics, 4 (05): 834-837.

Zhang Q, Liu J, and Luo Z (2015) Dynamical Behavior of a System of Third-Order Rational Difference Equation. Discrete Dynamic in Nature and Society, Article ID 530453, 6 p.

Zhang Y, Yang X, Evans DJ, and Zhu C (2007) On the nonlinear difference equation system x_(n+1)=A+y_(n-m)/x_n ,y_(n+1)=A+x_(n-m)/y_n . Computers and Mathematics with Applications, 53 (10): 1561-1566.

*Journal of Progressive Research in Mathematics*,

*14*(2), 2399-2407. Retrieved from http://scitecresearch.com/journals/index.php/jprm/article/view/1664

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