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 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.

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 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.

scholarly journals Existence of a Period-Two Solution in Linearizable Difference Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
E. J. Janowski ◽  
M. R. S. Kulenović
Keyword(s):  
Difference Equation ◽  
Linear Equation ◽  
Difference Equations ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Minimal Period ◽  
Real Numbers ◽  
Existence And Nonexistence ◽  
The Difference ◽  
Necessary And Sufficient

Consider the difference equationxn+1=f(xn,…,xn−k),n=0,1,…,wherek∈{1,2,…}and the initial conditions are real numbers. We investigate the existence and nonexistence of the minimal period-two solution of this equation when it can be rewritten as the nonautonomous linear equationxn+l=∑i=1−lkgixn−i,n=0,1,…,wherel,k∈{1,2,…}and the functionsgi:ℝk+l→ℝ. We give some necessary and sufficient conditions for the equation to have a minimal period-two solution whenl=1.

scholarly journals DIFFERENCE EQUATION OF THE COLORED JONES POLYNOMIAL FOR TORUS KNOT

2004 ◽  
Vol 15 (09) ◽  
pp. 959-965 ◽  
Author(s):  
KAZUHIRO HIKAMI
Keyword(s):  
Difference Equation ◽  
Jones Polynomial ◽  
Order Difference ◽  
Colored Jones Polynomial ◽  
First Order ◽  
Torus Knot ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Hypergeometric Type ◽  
The Difference

We prove that the N-colored Jones polynomial for the torus knot [Formula: see text] satisfies the second order difference equation, which reduces to the first order difference equation for a case of [Formula: see text]. We show that the A-polynomial of the torus knot can be derived from the difference equation. Also constructed is a q-hypergeometric type expression of the colored Jones polynomial for [Formula: see text].

Dynamics of a Higher Order Rational Difference Equation

2011 ◽  
Vol 216 ◽  
pp. 50-55 ◽  
Author(s):  
Yi Yang ◽  
Fei Bao Lv
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Initial Conditions ◽  
Higher Order ◽  
Rational Difference Equation ◽  
The Difference

In this paper, we address the difference equation xn=pxn-s+xn-t/q+xn-t n=0,1,... with positive initial conditions where s, t are distinct nonnegative integers, p, q > 0. Our results not only include some previously known results, but apply to some difference equations that have not been investigated so far.

Sharp explicit oscillation conditions for difference equations with several delays

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Kirill M. Chudinov
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Linear Difference Equation ◽  
First Order ◽  
Variable Delays ◽  
All Solutions ◽  
Oscillation Test ◽  
Several Delays ◽  
Lower Limits

Abstract We consider explicit sufficient conditions for all solutions of a first-order linear difference equation with several variable delays and non-negative coefficients to be oscillatory. The conditions have the form of inequalities bounding below the upper and lower limits of the sums of coefficients over a subset of the discrete semiaxis. Our main results are oscillation tests based on a new principle for composing the estimated sums of coefficients. We also give some results in the form of examples, including a counterexample to a wrong oscillation test cited in several recent papers.

scholarly journals The evaluation of certain continued fractions

1937 ◽  
Vol 30 ◽  
pp. vi-x
Author(s):  
C. G. Darwin
Keyword(s):  
Difference Equation ◽  
Hypergeometric Function ◽  
Difference Equations ◽  
Continued Fractions ◽  
Continued Fraction ◽  
The Difference ◽  
Image Position

1. If the approximate numerical value of e is expressed as a continued fraction the result isand it was in finding the proof that the sequence extends correctly to infinity that the following work was done. First the continued fraction may be simplified by setting down the difference equations for numerator and denominator as usual, and eliminating two out of every successive three equations. A difference equation is thus formed between the first, fourth, seventh, tenth … convergents , and this equation will generate another continued fraction. After a little rearrangement of the first two members it appears that (1) implies2. We therefore consider the continued fractionwhich includes (2), and also certain continued fractions which were discussed by Prof. Turnbull. He evaluated them without solving the difference equations, and it is the purpose here to show how the difference equations may be solved completely both in his cases and in the different problem of (2). It will appear that the work is connected with certain types of hypergeometric function, but I shall not go into this deeply.

scholarly journals Interval Model of the Efficiency of the Functioning of Information Web Resources for Services on Ecological Expertise

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2116
Author(s):  
Mykola Dyvak ◽  
Oleksandr Papa ◽  
Andrii Melnyk ◽  
Andriy Pukas ◽  
Nataliya Porplytsya ◽  
...  
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Model Building ◽  
Experimental Studies ◽  
Discrete Models ◽  
Model Parameters ◽  
Incorrect Choice ◽  
Web Resources ◽  
Web Resource ◽  
The Difference

Mathematical models of the efficiency dynamics of information web resources are considered in this paper. The application of interval discrete models in the form of difference equations is substantiated and the approach to estimation of the model parameters is proposed. The proposed approach is based on the artificial bee colony algorithm (ABCA). A number of experimental studies have been carried out based on data on the functioning of web resources related to environmental monitoring services. The indicator of an information web resource user’s activity has been investigated. Three cases of model building in the form of difference equations as interval discrete models (IDM) have been considered. They vary in the general kind of expression. As a result of the computational experiments, it is shown that the adequacy of a model depends on the expression of the difference equation. In the case of its incorrect choice, the proposed method of parameters’ identification may be ineffective. The obtained interval discrete model in the difference equation form, which describes the efficiency of a web resource, makes it possible to optimize business processes in an organization that uses this web resource, as well as optimally allocate organizational resources and the workload of employees of the administrative service center. Based on the conducted experiments, the efficiency of the proposed model’s application is confirmed.

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