A qualitative investigation of a system of third-order difference equations with multiplicative reciprocal terms

Year 2024, Volume: 6 Issue: 2, 30 - 44, 18.12.2024

Durhasan Turgut Tollu , İbrahim Yalçınkaya

https://doi.org/10.54286/ikjm.1524180

Abstract

In this paper, we study the system of third-order difference equations
\begin{equation*}
x_{n+1}=a+\frac{a_{1}}{y_{n}}+\frac{a_{2}}{y_{n-1}}+\frac{a_{3}}{y_{n-2}}%
,\quad y_{n+1}=b+\frac{b_{1}}{x_{n}}+\frac{b_{2}}{x_{n-1}}+\frac{b_{3}}{%
x_{n-2}},\quad n\in \mathbb{N}_{0},
\end{equation*}%
where the parameters $a$, $a_{i}$, $b$, $b_{i}$, $i=1,2,3$, and the initial
values $x_{-j}$, $y_{-j}$, $j=0,1,2$, are positive real numbers. We first
prove a general convergence theorem. By applying this convergence theorem to
the system, we show that positive equilibrium is a global attractor. We also
study the local asymptotic stability of the equilibrium and show that it is
globally asymptotically stable. Finally, we study the invariant set of
solutions.

Keywords

Boundedness, equilibrium point, system of difference equations.

References

• R. P. Agarwal, Difference Equations and Inequalities,Marcel Dekker, New York, (1992).
• N. Akgunes and A. S. Kurbanli, On the system of rational difference equations xn = f ¡ xn−a1 , yn−b1 ¢ , yn = g ¡ yn−b2 , zn−c1 ¢ , zn = g ¡ zn−c2 ,xn−a2 ¢ , Selcuk Journal of Applied Mathematics, 15(1), (2014), 1-8.
• Y. Akrour, M. Kara, N. Touafek and Y. Yazlik, Solutions formulas for some general systems of nonlinear difference equations, Miskolc Mathematical Notes, 22(2) (2021), 529–555.
• E. Camouzis and G. Ladas, Dynamics of third-order rational difference equations with open problems and conjectures, Chapman and Hall/CRC, (2007).
• I. Dekkar, N. Touafek and Y. Yazlik, Global stability of a third-order nonlinear system of difference equations with period-two coefficients, Revista de la real academia de ciencias exactas, físicas y naturales. Serie A. Matemáticas, 111(2), (2017) 325-347. Doi:10.1007/s13398-016-0297-z
• R. DeVault, G. Ladas and S. W. Schultz, Necessary and sufficient conditions for the boundedness of xn+1 = A/x p n +B/x q n−1 , Journal of Difference Equations and Applications, 3(3-4)(1997), 259-266.
• R. DeVault, G. Ladas and S. W. Schultz, On the recursive sequence xn+1 = A/xn +1/xn−2, Proceedings of the American Mathematical Society, 126(11)(1998), 3257-3261.
• S. Elaydi, An Introduction to Difference Equations, third edition, Undergraduate Texts in Mathematics, Springer, New York, (1999).
• H. El-Metwally, E. A. Grove and G. Ladas, A global convergence result with applications to periodic solutions, Journal of Mathematical Analysis and Applications, 245(2000), 161-170.
• H. El-Metwally, E. A. Grove, G. Ladas and H. D. Voulov, On the global attractivity and the periodic character of some difference equations, Journal of Difference Equations and Applications, 7(6)(2001), 837- 850.
• 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, Mathematical Methods in the Applied Sciences, 40(10), (2017), 3599-3607.
• N. Haddad, N. Touafek and J. F. T. Rabago, Well-defined solutions of a system of difference equations, Journal of Applied Mathematics and Computing, 56, (2018), 439-458, https://doi.org/10.1007/s12190- 017-1081-8
• E. Hatir, T. Mansour and I. Yalcinkaya, On a fuzzy difference equation, Utilitas Mathematica, 93(2014), 135-151.
• M. Kara, D. T. Tollu and Y. Yazlik, Global behavior of two-dimensional difference equations system with two periodic coefficients, Tbilisi Mathematical Journal, 13(4), (2020), 49-64.
• M. Kara, Y. Yazlik and D. T. Tollu, Solvability of a system of higher order nonlinear difference equations, Hacettepe Journal of Mathematics and Statistics, 49(5), (2020), 1566-1593.
• M. Kara and Y. Yazlik, On the solutions of three-dimensional system of difference equations via recursive relations of order two and applications, Journal of Applied Analysis and Computation, 12(2), (2022) 736-753.
• M. Kara and Y. Yazlik, Solvable three-dimensional system of higher-order nonlinear difference equations, Filomat, 36(10), (2022), 3449-3469.
• M. Kara and Y. Yazlik, On a solvable system of rational difference equations of higher order, Turkish Journal of Mathematics, 46(2), (2022) 587-611.
• V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, (1993).
• M.R.S. Kulenovi´c and O. Merino, Discrete Dynamical Systems and Difference Equations with Mathematica, New York, NY, USA, CRC Press, 2002.
• A. S. Kurbanli, C. Çinar and D. ¸Sim¸sek, On the periodicity of solutions of the system of rational difference equations, Applied Mathematics, 2, (2011), 410-413.
• G. Papaschinopoulos and C. J. Schinas, On a system of two nonlinear difference equations, Journal of Mathematical Analysis and Applications, 219 (2) (1998), 415-426.
• G. Papaschinopoulos and C. J. Schinas, Stability of a class of nonlinear difference equations, Journal of Mathematical Analysis and Applications, 230 (1999), 211-222.
• G. Papaschinopoulos and C. J. Schinas, Invariants and oscillation for systems of two nonlinear difference equations, Nonlinear Analysis: Theory, Methods and Applications, 46 (2001), 967–978.
• G. Papaschinopoulos and C. J. Schinas, Oscillation and asymptotic stability of two systems of difference equations of rational form, Journal of Difference Equations and Applications, 7 (2001), 601-617.
• G. Papaschinopoulos and C. J. Schinas, On the system of two difference equations xn+1 = Pk i=0 Ai /y pi n−i , yn+1 = Pk i=0 Bi /x qi n−i , Journal of Mathematical Analysis and Applications, 273 (2) (2002), 294-309.
• G. Papaschinopoulos and B. K. Papadopoulos, On the fuzzy difference equation xn+1 = A + B/xn, Soft Computing, 6(2002), 456-461.
• C. G. Philos, I. K. Purnaras and Y. G. Sficas, Global attractivity in a nonlinear difference equation, Applied Mathematics and Computation, 62(2-3)(1994), 249-258.
• C. J. Schinas, Invariants for difference equations and systems of difference equations of rational form, Journal of Mathematical Analysis and Applications, 216(1)(1997), 164-179.
• S. Stevi´c and D. T. Tollu, Solvability and semi-cycle analysis of a class of nonlinear systems of difference equations, Mathematical Methods in the Applied Sciences, 42, (2019), 3579-3615. https://doi.org/10.1002/mma.5600
• S. Stevi´c and D. T. Tollu, Solvability of eight classes of nonlinear systems of difference equations, Mathematical Methods in the Applied Sciences, 42, (2019), 4065-4112. https://doi.org/10.1002/mma.5625
• N. Taskara, D. T. Tollu, N. Touafek and Y. Yazlik, A solvable system of difference equations, Communications of the Korean Mathematical Society, 35 (1) (2020), 301-319.
• D. T. Tollu, Y. Yazlik and N. Taskara, On fourteen solvable systems of difference equations, Applied Mathematics and Computation, 233, (2014), 310-319.
• I. Yalçınkaya, H. El-Metwally and D. T. Tollu, On the fuzzy difference equation zn+1 = A+B/zn−m, Mathematical Notes, 113(2023), 292–302.
• I. Yalçınkaya, H. El-Metwally, M. Bayram, et al., On the dynamics of a higher-order fuzzy difference equation with rational terms, Soft Computing, 27(2023), 10469–10479. https://doi.org/10.1007/s00500-023- 08586-y
• Y. Yazlik, E. M. Elsayed and N. Taskara, On the behaviour of the solutions of difference equation systems, Journal of Computational Analysis and Applications, 16(5), (2014), 932-941.
• Y. Yazlik, D. T. Tollu and N. Taskara, On the solutions of a three-dimensional system of difference equations, Kuwait Journal of Science, 43(1) (2016), 95-111.