ABSTRACT
In this paper, we represent that the following three-dimensional system of difference equations
where the parameters a, b, c, α, β, γ and the initial values x−i, y−i, z−i, i ∈ {0, 1}, are real numbers, can be solved in closed form by using transformation. We analyzed the solutions in 10 different cases depending on whether the parameters are zero or nonzero. It is noteworthy to depict that the solutions of some particular cases of this system are presented in terms of generalized Fibonacci numbers. Note that our results considerably extend and improve some recent results in the literature.
REFERENCES
[1] Abo-Zeid, R.—Kamal, H.: Global behavior of two rational third order difference equations, Univers. J. Math. Appl. 2(4) (2019), 212-217.10.32323/ujma.626465Suche in Google Scholar
[2] Abo-Zeid, R.: Behavior of solutions of a second order rational difference equation, Math. Morav. 23(1) (2019), 11-25.10.5937/MatMor1901011ASuche in Google Scholar
[3] De Moivre, A.: The Doctrine of Chances. In: Landmark Writings in Western Mathematics, London, 1756, pp. 1640-1940.Suche in Google Scholar
[4] Dekkar, I.—Touafek, N.—Yazlik, Y.: 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. RACSAM 111(2) (2017), 325-347.10.1007/s13398-016-0297-zSuche in Google Scholar
[5] Elabbasy, E. M.—El-Metwally, H. A.—Elsayed, E. M.: Global behavior of the solutions of some difference equations, Adv. Differ. Equ. 2011(1) (2011), 1-16.10.1186/1687-1847-2011-28Suche in Google Scholar
[6] Elabbasy, E. M.—Elsayed, E. M.: Dynamics of a rational difference equation, Chin. Ann. Math. Ser. B 30(2) (2009), 187-198.10.1007/s11401-007-0456-9Suche in Google Scholar
[7] Elmetwally, H.: Solutions form for some rational systems of difference equations, Discrete Dyn. Nature Soc. (2013), Art. ID 903593, 10 pp.10.1155/2013/903593Suche in Google Scholar
[8] Elsayed, E. M.: Qualitative behavior of a rational recursive sequence, Indag. Math. 19(2) (2008), 189-201.10.1016/S0019-3577(09)00004-4Suche in Google Scholar
[9] Elsayed, E. M.: Qualitative properties for a fourth order rational difference equation, Acta Appl. Math. 110(2) (2010), 589-604.10.1007/s10440-009-9463-zSuche in Google Scholar
[10] Elsayed, E. M.: Solution for systems of difference equations of rational form of order two, Comput. Appl. Math. 33(3) (2014), 751-765.10.1007/s40314-013-0092-9Suche in Google Scholar
[11] Elsayed, E. M.: Expression and behavior of the solutions of some rational recursive sequences, Math. Methods Appl. Sci. 18(39) (2016), 5682-5694.10.1002/mma.3953Suche in Google Scholar
[12] Elsayed, E. M.: Dynamics of recursive sequence of order two, Kyungpook Math J. 50(4) (2010), 483-497.10.5666/KMJ.2010.50.4.483Suche in Google Scholar
[13] Falcón, S.—Plaza, A.: The k-Fibonacci sequence and the Pascal 2-triangle, Chaos Solitons Fractals 33(1) (2007), 38-49.10.1016/j.chaos.2006.10.022Suche in Google Scholar
[14] Folly-Gbetoula, M.—Manda, K.—Gadjagboui, B. B. I.: The invariance, formulas for solutions and periodicity of some recurrence equations, Int. J. Contemp. Math. Sci. 14(4) (2019), 201-210.10.12988/ijcms.2019.9820Suche in Google Scholar
[15] Halim, Y.—Touafek, N.—Yazlik, Y.: Dynamic behavior of a second-order nonlinear rational difference equation, Turkish J. Math. 39(6) (2015), 1004-1018.10.3906/mat-1503-80Suche in Google Scholar
[16] Halim, Y.—Bayram, M.: On the solutions of a higher-order difference equation in terms of generalized Fibonacci sequence, Math. Methods Appl. Sci. 39 (2016), 2974-2982.10.1002/mma.3745Suche in Google Scholar
[17] Halim, Y.—Rabago, J. F. T.: On the solutions of a second-order difference equation in terms of generalized Padovan sequences, Math. Slovaca 68(3) (2018), 625-638.10.1515/ms-2017-0130Suche in Google Scholar
[18] Kara, M.—Yazlik, Y.: Solvability of a system of nonlinear difference equations of higher order, Turkish J. Math. 43(3) (2019), 1533-1565.10.3906/mat-1902-24Suche in Google Scholar
[19] Kara, M.—Yazlik, Y.: On the system of difference equations
[20] Kara, M.—Touafek, N.—Yazlik, Y.: Well-defined solutions of a three-dimensional system of difference equations, Gazi Univ. J. Sci. 33(3) (2020), 676-778.10.35378/gujs.641441Suche in Google Scholar
[21] Kara, M.—Yazlik, Y.—Tollu, D. T.: Solvability of a system of higher order nonlinear difference equations, Hacet. J. Math. Stat. 49(5) (2020), 1566-1593.10.15672/hujms.474649Suche in Google Scholar
[22] Kara, M.—Yazlik, Y.: On a solvable three-dimensional system of difference equations, Filomat 34(4) (2020), 1167-1186.10.2298/FIL2004167KSuche in Google Scholar
[23] Koshy, T.: Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, 2019.10.1002/9781118742297Suche in Google Scholar
[24] Öcalan, Ö.: Oscillation of nonlinear difference equations with several coefficients, Commun. Math. Anal. 4(1) (2008), 35-44.Suche in Google Scholar
[25] Papaschinopoulos, G.—Stefanidou, G.: Asymptotic behavior of the solutions of a class of rational difference equations, Inter. J. Difference Equ. 5(2) (2010), 233-249.10.1186/1687-1847-2010-196920Suche in Google Scholar
[26] Rabago, J. F. T.—Bacani, J. B.: On a nonlinear difference equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 24 (2017), 375-394.Suche in Google Scholar
[27] Stević, S.: Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences, Electron. J. Qual. Theory Differ. Equ. 67 (2014), 1-15.10.14232/ejqtde.2014.1.67Suche in Google Scholar
[28] Stević, S.: Two ways for solving a class of rational second-order difference equations, Adv. Differ. Equ. (2019), Art. No. 230, 15 pp.10.1186/s13662-019-2173-6Suche in Google Scholar
[29] Stević, S.: On a class of systems of rational second-order difference equations solvable in closed form, Math. Methods Appl. Sci. 43 (2020), 1001-1016.10.1002/mma.5809Suche in Google Scholar
[30] Tasdemir, E.—Soykan, Y.: Stability of negative equilibrium of a non-linear difference equation, J. Math. Sci. Adv. Appl. 49(1) (2018), 51-57.10.18642/jmsaa_7100121927Suche in Google Scholar
[31] Tollu, D. T.—Yazlik, Y.—Taskara N.: On a solvable nonlinear difference equation of higher order, Turkish J. Math. 42(4) (2018), 1765-1778.10.3906/mat-1705-33Suche in Google Scholar
[32] Tollu, D. T.—Yazlik, Y.—Taskara, N.: Behavior of positive solutions of a difference equation, J. Appl. Math. Inform. 35 (2017), 217-230.10.14317/jami.2017.217Suche in Google Scholar
[33] Tollu, D. T.—Yazlik, Y.—Taskara, N.: On fourteen solvable systems of difference equations, Appl. Math. Comput. 233 (2014), 310-319.10.1016/j.amc.2014.02.001Suche in Google Scholar
[34] Tollu, D. T.—Yazlik, Y.—Taskara, N.: The solutions of four Riccati difference equations associated with Fibonacci numbers, Balkan J. Math. 2(1) (2014), 163-172.Suche in Google Scholar
[35] Touafek, N.: On a second order rational difference equation, Hacet. J. Math. Stat. 41(6) (2012), 867-874.Suche in Google Scholar
[36] Touafek, N.— Elsayed, E. M.: On a second order rational systems of difference equations, Hokkaido Math. J. 44(1) (2015), 29-45.10.14492/hokmj/1470052352Suche in Google Scholar
[37] Yalcinkaya, I.—Cinar, C.: Global asymptotic stability of a system of two nonlinear difference equations, Fasc. Math. 43 (2010), 171-180.Suche in Google Scholar
[38] Yalcinkaya, I.—Tollu, D. T.: Global behavior of a second order system of difference equations, Adv. Stud. Contemp. Math. 26(4) (2016), 653-667.Suche in Google Scholar
[39] Yazlik, Y.—Kara, M.: On a solvable system of difference equations of higher-order with period two coefficients, Commun. Fac. Sci. Univ. Ank. Ser. A Math. Stat. 68(2) (2019), 1675-1693.10.31801/cfsuasmas.548262Suche in Google Scholar
[40] Yazlik, Y.—Taskara, N.—Uslu, K.—Yilmaz, N.: The generalized (s,t)-sequence and its matrix sequence, AIP Conf. Proc. 1389(1) (2011), 381-384.10.1063/1.3636742Suche in Google Scholar
[41] Yazlik, Y.—Köme, C.—Madhusudanan, V.: A new generalization of Fibonacci and Lucas p-numbers, J. Comput. Anal. Appl. 25(4) (2018), 657-669.Suche in Google Scholar
[42] Yazlik, Y.—Tollu, D. T.—Taskara, N.: On the solutions of difference equation systems with Padovan numbers, Appl. Math. 4(12A) (2013), 1-15.10.1186/1687-1847-2013-174Suche in Google Scholar
© 2023 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- Parameters in Inversion Sequences
- On the Pecking Order Between Those of Mitsch and Clifford
- A Note on Cuts of Lattice-Valued Functions and Concept Lattices
- Trigonometric Sums and Riemann Zeta Function
- Notes on the Equation d(n) = d(φ(n)) and Related Inequalities
- Harmony of Asymmetric Variants of the Filbert and Lilbert Matrices in q-form
- Maximal Density and the Kappa Values for the Families {a, a + 1, 2a + 1, n} and {a, a + 1, 2a + 1, 3a + 1, n}
- Extensions in Time Scales Integral Inequalities of Jensen’s Type via Fink’s Identity
- A Note on Fractional Simpson Type Inequalities for Twice Differentiable Functions
- Extended Bromwich-Hansen Series
- Iterative Criteria for Oscillation of Third-Order Delay Differential Equations with p-Laplacian Operator
- Vallée-Poussin Theorem for Equations with Caputo Fractional Derivative
- On Oscillatory Behavior of Third Order Half-Linear Delay Differential Equations
- On Existence and Uniqueness of Solutions for Ordinary Differential Equations in Locally Convex Topological Linear Spaces
- Bounds on Blow-Up Time for a Higher-Order Non-Newtonian Filtration Equation
- On a Solvable System of Difference Equations in Terms of Generalized Fibonacci Numbers
- The Smallest and the Largest Families of Some Classes of 𝒜-Continuous Functions
- Type II Exponentiated Half-Logistic Gompertz-G Family of Distributions: Properties and Applications
- Strong Consistency of Least-Squares Estimators in the Simple Linear Errors-in-Variables Regression Model with Widely Orthant Dependent Random Variables
Artikel in diesem Heft
- Parameters in Inversion Sequences
- On the Pecking Order Between Those of Mitsch and Clifford
- A Note on Cuts of Lattice-Valued Functions and Concept Lattices
- Trigonometric Sums and Riemann Zeta Function
- Notes on the Equation d(n) = d(φ(n)) and Related Inequalities
- Harmony of Asymmetric Variants of the Filbert and Lilbert Matrices in q-form
- Maximal Density and the Kappa Values for the Families {a, a + 1, 2a + 1, n} and {a, a + 1, 2a + 1, 3a + 1, n}
- Extensions in Time Scales Integral Inequalities of Jensen’s Type via Fink’s Identity
- A Note on Fractional Simpson Type Inequalities for Twice Differentiable Functions
- Extended Bromwich-Hansen Series
- Iterative Criteria for Oscillation of Third-Order Delay Differential Equations with p-Laplacian Operator
- Vallée-Poussin Theorem for Equations with Caputo Fractional Derivative
- On Oscillatory Behavior of Third Order Half-Linear Delay Differential Equations
- On Existence and Uniqueness of Solutions for Ordinary Differential Equations in Locally Convex Topological Linear Spaces
- Bounds on Blow-Up Time for a Higher-Order Non-Newtonian Filtration Equation
- On a Solvable System of Difference Equations in Terms of Generalized Fibonacci Numbers
- The Smallest and the Largest Families of Some Classes of 𝒜-Continuous Functions
- Type II Exponentiated Half-Logistic Gompertz-G Family of Distributions: Properties and Applications
- Strong Consistency of Least-Squares Estimators in the Simple Linear Errors-in-Variables Regression Model with Widely Orthant Dependent Random Variables