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 Exact solution of a difference approximation to Duffing's equation

1981 ◽  
Vol 23 (1) ◽  
pp. 64-77 ◽  
Author(s):  
Renfrey B. Potts
Keyword(s):  
Exact Solution ◽  
Closed Form ◽  
Periodic Solutions ◽  
Difference Equations ◽  
Difference Approximation ◽  
Linear Difference Equations ◽  
Duffing's Equation ◽  
Duffing’S Equation ◽  
Non Linear

AbstractDuffing's equation, in its simplest form, can be approximated by various non-linear difference equations. It is shown that a particular choice can be solved in closed form giving periodic solutions.

Solvability of a product-type system of difference equations with six parameters

2016 ◽  
Vol 8 (1) ◽  
pp. 29-51 ◽  
Author(s):  
Stevo Stević
Keyword(s):  
Closed Form ◽  
Difference Equations ◽  
Type System ◽  
Product Type ◽  
System Of Difference Equations ◽  
Initial Values ◽  
Complex Valued

AbstractClosed form formulas for well-defined complex-valued solutions to a product-type system of difference equations of interest with six parameters are presented. The form of the solutions is described in detail in terms of the parameters and initial values.

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.

On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers

2020 ◽  
Vol 70 (3) ◽  
pp. 641-656
Author(s):  
Amira Khelifa ◽  
Yacine Halim ◽  
Abderrahmane Bouchair ◽  
Massaoud Berkal
Keyword(s):  
General Solution ◽  
Closed Form ◽  
Difference Equations ◽  
Fibonacci Numbers ◽  
Higher Order ◽  
Real Numbers ◽  
Lucas Numbers ◽  
Rational Difference Equations ◽  
Initial Values ◽  
Fibonacci And Lucas Numbers

AbstractIn this paper we give some theoretical explanations related to the representation for the general solution of the system of the higher-order rational difference equations$$\begin{array}{} \displaystyle x_{n+1} = \dfrac{1+2y_{n-k}}{3+y_{n-k}},\qquad y_{n+1} = \dfrac{1+2z_{n-k}}{3+z_{n-k}},\qquad z_{n+1} = \dfrac{1+2x_{n-k}}{3+x_{n-k}}, \end{array}$$where n, k∈ ℕ0, the initial values x−k, x−k+1, …, x0, y−k, y−k+1, …, y0, z−k, z−k+1, …, z1 and z0 are arbitrary real numbers do not equal −3. This system can be solved in a closed-form and we will see that the solutions are expressed using the famous Fibonacci and Lucas numbers.

scholarly journals OSCILLATIONS IN CERTAIN DIFFERENCE EQUATIONS

1997 ◽  
Vol 27 (3) ◽  
pp. 257-265
Author(s):  
ZDZISLAW SZAFRANSKI ◽  
BLAZEJ SZMANDA
Keyword(s):  
Difference Equations ◽  
Sufficient Conditions ◽  
Variable Coefficients ◽  
Linear Difference Equations ◽  
All Solutions

We obtain sufficient conditions for the oscillation of all solutions of some linear difference equations with variable coefficients.

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