Well-Defined Solutions of a Three-Dimensional System of Difference Equations

Kara, Merve; Touafek, N; Yazlık, Yasin

doi:10.35378/gujs.641441

Cited by 10 publications

(1 citation statement)

References 24 publications

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“…It is clear that if we want to understand our models, we need to know the behavior of the solutions of the equations of the models, and this fact will be possible if we can solve in closed form these equations. One can find in the literature a lot of works on difference equations where explicit formulas of the solutions are given, see for instance [1], [2], [5], [8], [9], [10], [7], [12], [11], [13], [16], [15], [14], [17], [18], [21], [20], [22]. Such type of difference equations and systems is called solvable difference equations.…”

Section: Introductionmentioning

confidence: 99%

Solutions formulas for some general systems of difference equations

Akrour¹,

Kara²,

Touafek³

et al. 2021

MMN

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In this paper, we give explicit formulas of the solutions of the two general systems of difference equationswhere n ∈ N 0 , f , g : D −→ R are "1 − 1" continuous functions on D ⊆ R, the initial values x −i , y −i , i = 0, 1, 2, 3 are arbitrary real numbers in D and the parameters a, b, c and d are arbitrary real numbers. Our results considerably extend some existing results in the literature.

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Section: Introductionmentioning

confidence: 99%

Solutions formulas for some general systems of difference equations

Akrour¹,

Kara²,

Touafek³

et al. 2021

MMN

Self Cite

View full text Add to dashboard Cite

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On a Solvable System of Difference Equations in Terms of Generalized Fibonacci Numbers

Yüksel,

Yazlik

2023

Mathematica Slovaca

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In this paper, we represent that the following three-dimensional system of difference equations x n + 1 = α y n + a y n y n − β z n − 1 , y n + 1 = β z n + b z n z n − γ x n − 1 , z n + 1 = γ x n + c x n x n − α y n − 1 , n ∈ ℕ 0 , $$\matrix{{{x_{n + 1}} = \alpha {y_n} + {{a{y_n}} \over {{y_n} - \beta {z_{n - 1}}}},\quad {y_{n + 1}} = \beta {z_n} + {{b{z_n}} \over {{z_n} - \gamma {x_{n - 1}}}},\quad {z_{n + 1}} = \gamma {x_n} + {{c{x_n}} \over {{x_n} - \alpha {y_{n - 1}}}},\qquad n \in {{\mathbb N}_0},} \cr} $$ 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.

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Solvability of a third-order system of nonlinear difference equations via a generalized Fibonacci sequence

Hamioud,

Dekkar,

Touafek

2024

MMN

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In this paper, we solve in closed-form the following third-order system of nonlinear difference equations xn+1 = ynyn−1xn−1p xn(anyn−2q + bnynyn−1),yn+1 = xnxn−1yn−1q yn(cnxn−2p + dnxnxn−1),p,q ∈ ℕ,n ∈ ℕ0 where the initial values x−i,y−i,i = 0,1,2 and the parameters (an)n∈ℕ0,(bn)n∈ℕ0,(cn)n∈ℕ0, (dn)n∈ℕ0 are non-zero real numbers. The form of the solutions of the one dimensional case of our system and a more general system defined by one to one functions are also presented.

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Well-Defined Solutions of a Three-Dimensional System of Difference Equations

Abstract: Highlights • This paper focuses on a three-dimensional system of difference equations. • The three-dimensional system of difference equations were solved. • Asymptotic behavior of solutions were determined and periodic solutions were demonstrated.

Cited by 10 publications

References 24 publications

Solutions formulas for some general systems of difference equations

Solutions formulas for some general systems of difference equations

On a Solvable System of Difference Equations in Terms of Generalized Fibonacci Numbers

Solvability of a third-order system of nonlinear difference equations via a generalized Fibonacci sequence

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