scholarly journals On a class of solvable higher-order difference equations

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 461-477 ◽  
Author(s):  
Stevo Stevic ◽  
Mohammed Alghamdi ◽  
Abdullah Alotaibi ◽  
Elsayed Elsayed
Keyword(s):  
Difference Equation ◽  
Closed Form ◽  
Difference Equations ◽  
Higher Order ◽  
Real Numbers ◽  
Order Difference ◽  
Initial Values ◽  
Long Term Behavior

Closed form formulas for well-defined solutions of the next difference equation xn = xn-2xn-k-2/xn-k(an + bnxn-2xn-k-2), n ? N0, where k ? N, (an)n?N0, (bn)n?N0, and initial values x-i, i = 1,k+2 are real numbers, are given. Long-term behavior of well-defined solutions of the equation when (an)n?N0 and (bn)n?N0 are constant sequences is described in detail by using the formulas. We also describe the domain of undefinable solutions of the equation. Our results explain and considerably improve some recent results in the literature.

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 On the Difference Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Candace M. Kent ◽  
Witold Kosmala ◽  
Stevo Stević
Keyword(s):  
Difference Equation ◽  
Real Numbers ◽  
Initial Values ◽  
The Difference ◽  
Long Term Behavior

The long-term behavior of solutions of the following difference equation: , , where the initial values , , are real numbers, is investigated in the paper.

scholarly journals Long-Term Behavior of Solutions of the Difference Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Candace M. Kent ◽  
Witold Kosmala ◽  
Stevo Stević
Keyword(s):  
Difference Equation ◽  
Real Numbers ◽  
Initial Values ◽  
The Difference ◽  
Long Term Behavior

We investigate the long-term behavior of solutions of the following difference equation: , , where the initial values , , and are real numbers. Numerous fascinating properties of the solutions of the equation are presented.

scholarly journals Unbounded Solutions of the Difference Equationxn=xn−lxn−k−1

2011 ◽  
Vol 2011 ◽  
pp. 1-8
Author(s):  
Stevo Stević ◽  
Bratislav Iričanin
Keyword(s):  
Difference Equation ◽  
General Result ◽  
General Character ◽  
Real Numbers ◽  
Unbounded Solutions ◽  
Initial Values ◽  
Easy Problem ◽  
The Difference ◽  
Long Term Behavior

The following difference equationxn=xn−lxn−k−1,n∈ℕ0, wherek,l∈ℕ,k<l,gcd(k,l)=1, and the initial valuesx-l,…,x-2,x-1are real numbers, has been investigated so far only for some particular values ofkandl. To get any general result on the equation is turned out as a not so easy problem. In this paper, we give the first result on the behaviour of solutions of the difference equation of general character, by describing the long-term behavior of the solutions of the equation for all values of parameterskandl, where the initial values satisfy the following conditionmin{x-l,…,x-2,x-1}.

scholarly journals A Global Convergence Result for a Higher Order Difference Equation

2007 ◽  
Vol 2007 ◽  
pp. 1-7 ◽  
Author(s):  
Bratislav D. Iricanin
Keyword(s):  
Global Convergence ◽  
Difference Equation ◽  
Bounded Solution ◽  
Convergence Result ◽  
Higher Order ◽  
Unbounded Solution ◽  
Order Difference ◽  
Initial Values ◽  
Order Difference Equation

Letf(z1,…,zk)∈C(Ik,I)be a given function, whereIis (bounded or unbounded) subinterval ofℝ, andk∈ℕ. Assume thatf(y1,y2,…,yk)≥f(y2,…,yk,y1)ify1≥max{y2,…,yk},f(y1,y2,…,yk)≤f(y2,…,yk,y1)ify1≤min{y2,…,yk}, andfis non- decreasing in the last variablezk. We then prove that every bounded solution of an autonomous difference equation of orderk, namely,xn=f(xn−1,…,xn−k),n=0,1,2,…,with initial valuesx−k,…,x−1∈I, is convergent, and every unbounded solution tends either to+∞or to−∞.

scholarly journals Periodic Solutions of a System of Nonlinear Difference Equations with Periodic Coefficients

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Durhasan Turgut Tollu
Keyword(s):  
Periodic Solution ◽  
Positive Solution ◽  
Periodic Solutions ◽  
Difference Equations ◽  
Real Numbers ◽  
Nonlinear Difference Equations ◽  
System Of Difference Equations ◽  
Positive Real ◽  
Long Term Behavior

This paper is dealt with the following system of difference equations x n + 1 = a n / x n + b n / y n , y n + 1 = c n / x n + d n / y n , where n ∈ ℕ 0 = ℕ ∪ 0 , the initial values x 0   and   y 0 are the positive real numbers, and the sequences a n n ≥ 0 , b n n ≥ 0 , c n n ≥ 0 , and d n n ≥ 0 are two-periodic and positive. The system is an extension of a system where every positive solution is two-periodic or converges to a two-periodic solution. Here, the long-term behavior of positive solutions of the system is examined by using a new method to solve the system.

scholarly journals Solutions of the Difference Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
Candace M. Kent ◽  
Witold Kosmala ◽  
Michael A. Radin ◽  
Stevo Stević
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Boundedness Of Solutions ◽  
Real Numbers ◽  
Unbounded Solutions ◽  
The Difference ◽  
Long Term Behavior

Our goal in this paper is to investigate the long-term behavior of solutions of the following difference equation: , where the initial conditions and are real numbers. We examine the boundedness of solutions, periodicity of solutions, and existence of unbounded solutions and how these behaviors depend on initial conditions.

scholarly journals On the Difference Equation

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
İbrahim Yalçinkaya
Keyword(s):  
Difference Equation ◽  
Higher Order ◽  
Real Numbers ◽  
Positive Real ◽  
Global Behaviour ◽  
Initial Values ◽  
The Difference

We investigate the global behaviour of the difference equation of higher order , where the parameters and the initial values and are arbitrary positive real numbers.

scholarly journals On the solutions of a class of max-min-type difference equation system

2019 ◽  
Vol 13 (1) ◽  
pp. 165-177
Author(s):  
Huili Ma ◽  
Haixia Wang
Keyword(s):  
Difference Equation ◽  
Periodic Solutions ◽  
Difference Equations ◽  
Equation System ◽  
Real Numbers ◽  
Positive Real ◽  
General Solutions ◽  
Initial Values

We mainly investigate the general solutions and periodic solutions to the following system of max-type difference equations xn+1 = max{y2n-1, An/yn-1}, yn+1 = min{x2n-1,Bn/xn-1}, where n ? N, (An)n?N and (Bn)n?N are positive real sequences, and the initial values x-1 = ?, x0= ?; y-1= ?, y0 = ? are real numbers.

scholarly journals Higher order – l alpha difference equations

10.26524/cm86 ◽  
2020 ◽  
Vol 4 (2) ◽  
Author(s):  
Dominic Babu G ◽  
Rejin Rose D
Keyword(s):  
Difference Equation ◽  
Closed Form ◽  
Difference Equations ◽  
Difference Operator ◽  
Higher Order ◽  
Preliminary Results

In this paper, we present some basic definitions and preliminary results –l alpha difference operator and inverse. We derive the sum of infinite –l alpha series and infinite –l alpha multi-series formulae by equating summation and closed form of the generalized higher order –l alpha difference equation.

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