scholarly journals On a class of solvable difference equations generalizing an iteration process for calculating reciprocals

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Stevo Stević
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Iteration Process ◽  
Nonlinear Difference Equation ◽  
Nonlinear Difference Equations ◽  
First Order ◽  
Nonzero Real Number ◽  
Solvable In Closed Form ◽  
The Difference ◽  
Reciprocal Value

AbstractThe well-known first-order nonlinear difference equation $$ y_{n+1}=2y_{n}-xy_{n}^{2}, \quad n\in {\mathbb {N}}_{0}, $$ y n + 1 = 2 y n − x y n 2 , n ∈ N 0 , naturally appeared in the problem of computing the reciprocal value of a given nonzero real number x. One of the interesting features of the difference equation is that it is solvable in closed form. We show that there is a class of theoretically solvable higher-order nonlinear difference equations that include the equation. We also show that some of these equations are also practically solvable.

Oscillation Theorems of Nonlinear Difference Equations of Second Order

2003 ◽  
Vol 10 (2) ◽  
pp. 343-352
Author(s):  
S. H. Saker
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Second Order ◽  
Nonlinear Difference Equation ◽  
Riccati Transformation ◽  
Nonlinear Difference Equations ◽  
Transformation Techniques ◽  
Oscillation Criteria ◽  
Special Cases ◽  
Oscillation Theorems

Abstract Using the Riccati transformation techniques, we establish some new oscillation criteria for the second-order nonlinear difference equation Some comparison between our theorems and the previously known results in special cases are indicated. Some examples are given to illustrate the relevance of our results.

scholarly journals Behaviour of the First-order q-Dierence Equations

2021 ◽  
Vol 11 (1) ◽  
pp. 68-74
Author(s):  
Mahmoud Belaghi ◽  
Murat Sari
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Significant Contribution ◽  
First Order ◽  
Linear And Nonlinear ◽  
The Difference

Since the need to investigate many aspects of q-dierence equations cannot be ruled out, this article aims to explore response of the mechanism modelled by linear and nonlinear q-difference equations. Therefore, analysis of an important bundle of nonlinear q-difference equations, in particular the q-Bernoulli difference equation, has been developed. In this context, capturing the behaviour of the q-Bernoulli difference equation as well as linear q-difference equations are considered to be a significant contribution here. Illustrative examples related to the difference equations are also presented.

scholarly journals A nonlinear difference equation with two parameters

1985 ◽  
Vol 26 (4) ◽  
pp. 430-451 ◽  
Author(s):  
A. Brown
Keyword(s):  
Difference Equation ◽  
General Problem ◽  
Iteration Process ◽  
Nonlinear Difference Equation ◽  
Cubic Equation ◽  
The Difference ◽  
The Stability ◽  
Image Position ◽  
Two Parameters ◽  
Parameter Problem

AbstractThe paper is mainly concerned with the difference equationwhere k and m are parameters, with k > 0. This equation arises from a method proposed for solving a cubic equation by iteration and represents a standardised form of the general problem. In using the above equation it is essential to know when the iteration process converges and this is discussed by means of the usual stability criterion. Critical values are obtained for the occurrence of solutions with period two and period three and the stability of these solutions is also examined. This was done by considering the changes as k increases, for a give value of m, which makes it effectively a one-parameter problem, and it turns out that the change with k can differ strongly from the usual behaviour for a one-parameter difference equation. For m = 2, for example it appears that the usual picture of stable 2-cycle solutions giving way to stable 4-cycle solutions is valid for smaller values of k but the situation is recersed for larger values of k where stable 4-cycle solutions precede stable 2-cycle solutions. Similar anomalies arise for the 3-cycle solutions.

scholarly journals Meromorphic solutions of certain nonlinear difference equations

2020 ◽  
Vol 18 (1) ◽  
pp. 1292-1301
Author(s):  
Huifang Liu ◽  
Zhiqiang Mao ◽  
Dan Zheng
Keyword(s):  
Difference Equation ◽  
Finite Order ◽  
Difference Equations ◽  
Nonlinear Difference Equation ◽  
Meromorphic Solutions ◽  
Nonlinear Difference Equations ◽  
Zero Distribution ◽  
Forward Difference

Abstract This paper focuses on finite-order meromorphic solutions of nonlinear difference equation {f}^{n}(z)+q(z){e}^{Q(z)}{\text{Δ}}_{c}f(z)=p(z) , where p,q,Q are polynomials, n\ge 2 is an integer, and {\text{Δ}}_{c}f is the forward difference of f. A relationship between the growth and zero distribution of these solutions is obtained. Using this relationship, we obtain the form of these solutions of the aforementioned equation. Some examples are given to illustrate our results.

scholarly journals Distribution of iterates of first order difference equations

1980 ◽  
Vol 22 (1) ◽  
pp. 133-143 ◽  
Author(s):  
James B. McGuire ◽  
Colin J. Thompson
Keyword(s):  
Invariant Measure ◽  
Difference Equation ◽  
Lebesgue Measure ◽  
Difference Equations ◽  
Absolutely Continuous ◽  
Order Difference ◽  
First Order ◽  
Order Difference Equation ◽  
The Difference ◽  
Biological Context

An invariant measure which is absolutely continuous with respect to Lebesgue measure is constructed for a particular first order difference equation that has an extensive biological pedigree. In a biological context this invariant measure gives the density of the population whose growth is governed by the difference equation. Further asymptotically universal results are obtained for a class of difference equations.

scholarly journals Global Asymptotic Stability of a Family of Nonlinear Difference Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Maoxin Liao
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Difference Equations ◽  
Global Asymptotic Stability ◽  
Recent Literature ◽  
Nonlinear Difference Equation ◽  
Nonlinear Difference Equations

In this note, we consider global asymptotic stability of the following nonlinear difference equationxn=(∏i=1v(xn-kiβi+1)+∏i=1v(xn-kiβi-1))/(∏i=1v(xn-kiβi+1)-∏i=1v(xn-kiβi-1)),  n=0,1,…, whereki∈ℕ  (i=1,2,…,v),  v≥2,β1∈[-1,1],β2,β3,…,βv∈(-∞,+∞),x-m,x-m+1,…,x-1∈(0,∞), andm=max1≤i≤v{ki}. Our result generalizes the corresponding results in the recent literature and simultaneously conforms to a conjecture in the work by Berenhaut et al. (2007).

scholarly journals Note on constructing a family of solvable sine-type difference equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ahmed El-Sayed Ahmed ◽  
Bratislav Iričanin ◽  
Witold Kosmala ◽  
Stevo Stević ◽  
Zdeněk Šmarda
Keyword(s):  
General Solution ◽  
Closed Form ◽  
Difference Equations ◽  
First Order ◽  
Solvable In Closed Form

AbstractWe obtain a family of first order sine-type difference equations solvable in closed form in a constructive way, and we present a general solution to each of the equations.

scholarly journals Oscillation Conditions for Difference Equations with a Monotone or Nonmonotone Argument

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
G. M. Moremedi ◽  
I. P. Stavroulakis
Keyword(s):  
Positive Integer ◽  
Difference Equation ◽  
Difference Equations ◽  
Delay Difference Equation ◽  
Real Numbers ◽  
First Order ◽  
All Solutions ◽  
Delay Difference ◽  
Constant Argument

Consider the first-order delay difference equation with a constant argument Δxn+pnxn-k=0,  n=0,1,2,…, and the delay difference equation with a variable argument Δxn+pnxτn=0,  n=0,1,2,…, where p(n) is a sequence of nonnegative real numbers, k is a positive integer, Δx(n)=x(n+1)-x(n), and τ(n) is a sequence of integers such that τ(n)≤n-1 for all n≥0 and limn→∞τ(n)=∞. A survey on the oscillation of all solutions to these equations is presented. Examples illustrating the results are given.

scholarly journals Analytical Study of the Difference Equation xn+1 = xnxn-6 / xn-5 (±1 – xnxn-6)

2019 ◽  
pp. 1-12
Author(s):  
Abdualrazaq Sanbo ◽  
Elsayed M. Elsayed ◽  
Faris Alzahrani
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Analytical Study ◽  
Initial Conditions ◽  
Specific Form ◽  
Positive Real ◽  
Rational Difference Equations ◽  
Special Cases ◽  
The Difference

This paper is devoted to find the form of the solutions of a rational difference equations with arbitrary positive real initial conditions. Specific form of the solutions of two special cases of this equation are given.

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