scholarly journals On a solvable three-dimensional system of difference equations

Filomat ◽  
2020 ◽  
Vol 34 (4) ◽  
pp. 1167-1186
Author(s):  
Merve Kara ◽  
Yasin Yazlik
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equations ◽  
Three Dimensional ◽  
Asymptotic Behavior Of Solutions ◽  
Dimensional System ◽  
Real Numbers ◽  
System Of Difference Equations ◽  
Initial Values ◽  
Forbidden Set ◽  
Three Dimensional System

In this paper, we show that the following three-dimensional system of difference equations xn = zn-2xn-3/axn-3 + byn-1, yn = xn-2yn-3/cyn-3 + dzn-1, zn = yn-2zn-3/ezn-3+ fxn-1, n ? N0, where the parameters a, b, c, d, e, f and the initial values x-i, y-i, z-i, i ? {1, 2, 3}, are real numbers, can be solved, extending further some results in literature. Also, we determine the asymptotic behavior of solutions and the forbidden set of the initial values by using the obtained formulas.

Solvability of a nonlinear three-dimensional system of difference equations with constant coefficients

2021 ◽  
Vol 71 (5) ◽  
pp. 1133-1148
Author(s):  
Merve Kara ◽  
Yasin Yazlik
Keyword(s):  
Difference Equations ◽  
Three Dimensional ◽  
Complex Numbers ◽  
Dimensional System ◽  
System Of Difference Equations ◽  
Initial Values ◽  
Constant Coefficients ◽  
Forbidden Set ◽  
Three Dimensional System

Abstract In this paper, we show that the following three-dimensional system of difference equations x n + 1 = y n x n − 2 a x n − 2 + b z n − 1 , y n + 1 = z n y n − 2 c y n − 2 + d x n − 1 , z n + 1 = x n z n − 2 e z n − 2 + f y n − 1 , n ∈ N 0 , $$\begin{equation*} x_{n+1}=\frac{y_{n}x_{n-2}}{ax_{n-2}+bz_{n-1}}, \quad y_{n+1}=\frac{z_{n}y_{n-2}}{cy_{n-2}+dx_{n-1}}, \quad z_{n+1}=\frac{x_{n}z_{n-2}}{ez_{n-2}+fy_{n-1}}, \quad n\in \mathbb{N}_{0}, \end{equation*}$$ where the parameters a, b, c, d, e, f and the initial values x −i , y −i , z −i , i ∈ {0, 1, 2}, are complex numbers, can be solved, extending further some results in the literature. Also, we determine the forbidden set of the initial values by using the obtained formulas. Finally, an application concerning a three-dimensional system of difference equations are given.

scholarly journals On a Third-Order System of Difference Equations with Variable Coefficients

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Stevo Stević ◽  
Josef Diblík ◽  
Bratislav Iricanin ◽  
Zdenek Šmarda
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equations ◽  
Variable Coefficients ◽  
Order System ◽  
Asymptotic Behavior Of Solutions ◽  
Real Numbers ◽  
System Of Difference Equations ◽  
Third Order ◽  
Initial Values ◽  
Explicit Formulae

We show that the system of three difference equationsxn+1=an(1)xn-2/(bn(1)ynzn-1xn-2+cn(1)),yn+1=an(2)yn-2/(bn(2)znxn-1yn-2+cn(2)), andzn+1=an(3)zn-2/(bn(3)xnyn-1zn-2+cn(3)),n∈N0, where all elements of the sequencesan(i),bn(i),cn(i),n∈N0,i∈{1,2,3}, and initial valuesx-j,y-j,z-j,j∈{0,1,2}, are real numbers, can be solved. Explicit formulae for solutions of the system are derived, and some consequences on asymptotic behavior of solutions for the case when coefficients are periodic with period three are deduced.

scholarly journals On the Behavior of a System of Rational Difference Equationsxn+1=xn-1/(ynxn-1-1),yn+1=yn-1/(xnyn-1-1),zn+1=1/xnzn-1

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Liu Keying ◽  
Wei Zhiqiang ◽  
Li Peng ◽  
Zhong Weizhou
Keyword(s):  
Difference Equations ◽  
Three Dimensional ◽  
Dimensional System ◽  
Rational Difference Equations ◽  
Initial Values ◽  
Three Dimensional System

We are concerned with a three-dimensional system of rational difference equations with nonzero initial values. We present solutions of the system in an explicit way and obtain the asymptotical behavior of solutions.

scholarly journals ON A SOLVABLE SYSTEM OF NON-LINEAR DIFFERENCE EQUATIONS WITH VARIABLE COEFFICIENTS

2021 ◽  
Vol 21 (1) ◽  
pp. 145-162
Author(s):  
MERVE KARA ◽  
YASIN YAZLIK
Keyword(s):  
Closed Form ◽  
Periodic Solutions ◽  
Difference Equations ◽  
Variable Coefficients ◽  
Linear Difference Equations ◽  
System Of Difference Equations ◽  
Solvable System ◽  
Initial Values ◽  
Non Linear ◽  
Forbidden Set

In this paper, we show that the system of difference equations can be solved in the closed form. Also, we determine the forbidden set of the initial values by using the obtained formulas. Finally, we obtain periodic solutions of aforementioned system.

scholarly journals THREE-DIMENSIONAL SYSTEM OF GLOBALLY MODIFIED NAVIER–STOKES EQUATIONS WITH DELAY

2010 ◽  
Vol 20 (09) ◽  
pp. 2869-2883 ◽  
Author(s):  
TOMÁS CARABALLO ◽  
JOSÉ REAL ◽  
ANTONIO M. MÁRQUEZ
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Navier Stokes ◽  
Asymptotic Behavior Of Solutions ◽  
Pullback Attractor ◽  
Dimensional System ◽  
Navier Stokes Equations ◽  
Three Dimensional System ◽  
Equations With Delay

We prove the existence and uniqueness of strong solutions of a three-dimensional system of globally modified Navier–Stokes equations with delay in the locally Lipschitz case. The asymptotic behavior of solutions, and the existence of pullback attractor are also analyzed.

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