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ON A THREE-DIMENSIONAL SYSTEM OF DIFFERENCE EQUATIONS WITH VARIABLE COEFFICIENTS
KARA, MERVE,YAZLIK, YASIN,TOUAFEK, NOURESSADAT,AKROUR, YOUSSOUF The Korean Society for Computational and Applied M 2021 Journal of applied mathematics & informatics Vol.39 No.3
Consider the three-dimensional system of difference equations $x_{n+1}=\frac{{\prod_{j=0}^{k}}z_n-3j}{{\prod_{j=1}^{k}}x_n-(3j-1)\;\(a_n+b_n{\prod_{j=0}^{k}}z_n-3j\)}$, $y_{n+1}=\frac{{\prod_{j=0}^{k}}x_n-3j}{{\prod_{j=1}^{k}}y_n-(3j-1)\;\(c_n+d_n{\prod_{j=0}^{k}}x_n-3j\)}$, $z_{n+1}=\frac{{\prod_{j=0}^{k}}y_n-3j}{{\prod_{j=1}^{k}}z_n-(3j-1)\;\(e_n+f_n{\prod_{j=0}^{k}}y_n-3j\)}$, n ∈ ℕ<sub>0</sub>, where k ∈ ℕ<sub>0</sub>, the sequences $(a_n)_{n{\in}{\mathbb{N}}_0$, $(b_n)_{n{\in}{\mathbb{N}}_0$, $(c_n)_{n{\in}{\mathbb{N}}_0$, $(d_n)_{n{\in}{\mathbb{N}}_0$, $(e_n)_{n{\in}{\mathbb{N}}_0$, $(f_n)_{n{\in}{\mathbb{N}}_0$ and the initial values x<sub>-3k</sub>, x<sub>-3k+1</sub>, …, x<sub>0</sub>, y<sub>-3k</sub>, y<sub>-3k+1</sub>, …, y<sub>0</sub>, z<sub>-3k</sub>, z<sub>-3k+1</sub>, …, z<sub>0</sub> are real numbers. In this work, we give explicit formulas for the well defined solutions of the above system. Also, the forbidden set of solution of the system is found. For the constant case, a result on the existence of periodic solutions is provided and the asymptotic behavior of the solutions is investigated in detail.
BEHAVIOR OF POSITIVE SOLUTIONS OF A DIFFERENCE EQUATION
TOLLU, D.T.,YAZLIK, Y.,TASKARA, N. The Korean Society for Computational and Applied M 2017 Journal of applied mathematics & informatics Vol.35 No.3
In this paper we deal with the difference equation $$y_{n+1}=\frac{ay_{n-1}}{by_ny_{n-1}+cy_{n-1}y_{n-2}+d}$$, $$n{\in}\mathbb{N}_0$$, where the coefficients a, b, c, d are positive real numbers and the initial conditions $y_{-2}$, $y_{-1}$, $y_0$ are nonnegative real numbers. Here, we investigate global asymptotic stability, periodicity, boundedness and oscillation of positive solutions of the above equation.
On global behavior of a system of nonlinear dierence equations of order two
D. T. Tollu,Y. Yazlik,N. Taskara 장전수학회 2017 Advanced Studies in Contemporary Mathematics Vol.27 No.3
In this paper we deal with the system of difference equations xn+1 = a/1 + xnyn-1, yn+1 = b/1 + ynxn-1, n ∈ N0, where the parameters a, b are positive real numbers and the initial conditions x-1, x0, y-1, y0 are nonnegative real numbers. We study global behavior of the above system. Also, we give rate of convergence of the solution which tends to unique positive equilibrium point of the system and illustrate our theoretical results by means of some numerical examples.
BEHAVIOR OF POSITIVE SOLUTIONS OF A DIFFERENCE EQUATION
D.T. Tollu,Y. Yazlik,N. Ta¸skara 한국전산응용수학회 2017 Journal of applied mathematics & informatics Vol.35 No.3
In this paper we deal with the difference equation yn+1 =ayn−1bynyn−1 + cyn−1yn−2 + d, n ∈ N0, where the coefficients a, b, c, d are positive real numbers and the initial conditions y−2, y−1, y0 are nonnegative real numbers. Here, we investigate global asymptotic stability, periodicity, boundedness and oscillation of positive solutions of the above equation.
A SOLVABLE SYSTEM OF DIFFERENCE EQUATIONS
Taskara, Necati,Tollu, Durhasan T.,Touafek, Nouressadat,Yazlik, Yasin Korean Mathematical Society 2020 대한수학회논문집 Vol.35 No.1
In this paper, we show that the system of difference equations $x_n={\frac{ay^p_{n-1}+b(x_{n-2}y_{n-1})^{p-1}}{cy_{n-1}+dx^{p-1}_{n-2}}}$, $y_n={\frac{{\alpha}x^p_{n-1}+{\beta}(y_{n-2}x_{n-1})^{p-1}}{{\gamma}x_{n-1}+{\delta}y^{p-1}_{n-2}}}$, n ∈ ℕ<sub>0</sub> where the parameters a, b, c, d, α, β, γ, δ, p and the initial values x-2, x-1, y-2, y-1 are real numbers, can be solved. Also, by using obtained formulas, we study the asymptotic behaviour of well-defined solutions of aforementioned system and describe the forbidden set of the initial values. Our obtained results significantly extend and develop some recent results in the literature.