Qualitative Properties for a Fourth Order Rational Difference Equation

In this paper we show that the system of difference equations x n = ay n−k + dy n−k x n−(k+l) bx n−(k+l) + cy n−l , y n = αx n−k + δx n−k y n−(k+l) βy n−(k+l) + γx n−l , where n ∈ N 0 , k and l are positive integers, the parameters a, b, c, d, α, β, γ, δ are real numbers and the initial values x −j , y −j , j = 1, k + l, are real numbers, can be solved in the closed form. We also determine the asymptotic behavior of solutions for the case l = 1 and describe the forbidden set of the initial values using the obtained formulas. Our obtained results significantly extend and develop some recent results in the literature.

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“…Some particular cases of the extension of Eq. (1.2) have been studied recently in papers [11][12][13][14]33]. More precisely, Eq.…”

Section: Introductionmentioning

confidence: 99%

“…More precisely, Eq. (1.2) in the case α = 0, δ = 1, β = 1, γ = ±1 was studied in [11,12] and in [33] the case α = 0, δ = 1 was studied, while in [13,14] the case k = 1, l = 1 and k = 2, l = 2 were studied. Eq.…”

Section: Introductionmentioning

confidence: 99%

Solvability of a system of higher order nonlinear difference equations

Kara

Yazlık

Tollu

2020

Hacettepe Journal of Mathematics and Statistics

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“…Motivated by the studies of [7,8,14,25], in this paper we deal with the following higher-order nonlinear difference equation…”

Section: Introductionmentioning

confidence: 99%

On a solvable nonlinear difference equation of higher order

Tollu¹,

Yazlık²,

Taşkara³

2018

Turk J Math

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In this paper we consider the following higher-order nonlinear difference equation xn = αx n−k + δx n−k x n−(k+l) βx n−(k+l) + γx n−l , n ∈ N0, where k and l are fixed natural numbers, and the parameters α , β , γ , δ and the initial values x−i , i = 1, k + l , are real numbers such that β 2 + γ 2 ̸ = 0. We solve the above-mentioned equation in closed form and considerably extend some results in the literature. We also determine the asymptotic behavior of solutions and the forbidden set of the initial values using the obtained formulae for the case l = 1 .

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“…Karamanoglu Mehmetbey University, Kamil Ozdag Science Faculty, Department of Mathematics, 70100 Karaman, Turkey. E mail: mervekara@kmu.edu.tr 2. Nevsehir Haci Bektas Veli University, Faculty of Science and Arts, Department of Mathematics, 50300 Nevsehir, Turkey.…”

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confidence: 99%