scholarly journals Oscillations of Difference Equations with Several Oscillating Coefficients

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
L. Berezansky ◽  
G. E. Chatzarakis ◽  
A. Domoshnitsky ◽  
I. P. Stavroulakis
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Difference Operator ◽  
Oscillatory Behavior ◽  
Oscillating Coefficients ◽  
The Difference

We study the oscillatory behavior of the solutions of the difference equationΔx(n)+∑i=1mpi(n)x(τi(n))=0,n∈N0[∇xn-∑i=1mpinxσin=0, n∈N]where(pi(n)),1≤i≤mare real sequences with oscillating terms,τi(n)[σi(n)],1≤i≤mare general retarded (advanced) arguments, andΔ[∇]denotes the forward (backward) difference operatorΔx(n)=x(n+1)-x(n)[∇x(n)=x(n)-x(n-1)]. Examples illustrating the results are also given.

Impact of different types of non linearity on the oscillatory behavior of higher order neutral difference equations

2021 ◽  
Vol 71 (4) ◽  
pp. 941-960
Author(s):  
Ajit Kumar Bhuyan ◽  
Laxmi Narayan Padhy ◽  
Radhanath Rath
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Difference Operator ◽  
Sufficient Conditions ◽  
Oscillatory Behavior ◽  
Unbounded Solution ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Forward Difference ◽  
Different Types

Abstract In this article, sufficient conditions are obtained so that every solution of the neutral difference equation Δ m ( y n − p n L ( y n − s ) ) + q n G ( y n − k ) = 0 , $$\begin{equation*}\Delta^{m}\big(y_n-p_n L(y_{n-s})\big) + q_nG(y_{n-k})=0, \end{equation*}$$ or every unbounded solution of Δ m ( y n − p n L ( y n − s ) ) + q n G ( y n − k ) − u n H ( y α ( n ) ) = 0 , n ≥ n 0 , $$\begin{equation*}\Delta^{m}\big(y_n-p_n L(y_{n-s})\big) + q_nG(y_{n-k})-u_nH(y_{\alpha(n)})=0,\quad n\geq n_0, \end{equation*}$$ oscillates, where m=2 is any integer, Δ is the forward difference operator given by Δy n = y n+1 − y n ; Δ m y n = Δ(Δ m−1 y n ) and other parameters have their usual meaning. The non linear function L ∈ C (ℝ, ℝ) inside the operator Δ m includes the case L(x) = x. Different types of super linear and sub linear conditions are imposed on G to prevent the solution approaching zero or ±∞. Further, all the three possible cases, p n ≥ 0, p n ≤ 0 and p n changing sign, are considered. The results of this paper generalize and extend some known results.

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.

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.

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.

scholarly journals Oscillation Tests for Fractional Difference Equations

2018 ◽  
Vol 71 (1) ◽  
pp. 53-64 ◽  
Author(s):  
George E. Chatzarakis ◽  
Palaniyappan Gokulraj ◽  
Thirunavukarasu Kalaimani
Keyword(s):  
Difference Equation ◽  
Difference Operator ◽  
Oscillatory Behavior ◽  
Riccati Transformation ◽  
Fractional Difference ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
Fractional Difference Equation ◽  
Positive Integers ◽  
Theoretical Results

Abstract In this paper, we study the oscillatory behavior of solutions of the fractional difference equation of the form $$\Delta \left( {r\left( t \right)g\left( {{\Delta ^\alpha }x(t)} \right)} \right) + p(t)f\left( {\sum\limits_{s = {t_0}}^{t - 1 + \alpha } {{{(t - s - 1)}^{( - \alpha )}}x(s)} } \right) = 0, & t \in {_{{t_0} + 1 - \alpha }},$$ where Δα denotes the Riemann-Liouville fractional difference operator of order α, 0 < α ≤ 1, ℕt0+1−α={t0+1−αt0+2−α…}, t0 > 0 and γ > 0 is a quotient of odd positive integers. We establish some oscillatory criteria for the above equation, using the Riccati transformation and Hardy type inequalities. Examples are provided to illustrate the theoretical results.

scholarly journals Boundary-Value Problems for Differential-Difference Equations with Incommeasurable Shifts of Arguments Reducible to Nonlocal Problems

2019 ◽  
Vol 65 (4) ◽  
pp. 613-622
Author(s):  
E. P. Ivanova
Keyword(s):  
Boundary Value Problems ◽  
Difference Equations ◽  
Original Problem ◽  
Difference Operator ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Nonlocal Problems ◽  
Higher Order Terms ◽  
The Difference

We consider boundary-value problems for differential-difference equations containing incommeasurable shifts of arguments in higher-order terms. We prove that in the case of finite orbits of boundary points generated by the set of shifts of the difference operator, the original problem is reduced to the boundary-value problem for differential equation with nonlocal boundary conditions.

FROM BOUNDARY VALUE PROBLEMS TO DIFFERENCE EQUATIONS: A METHOD OF INVESTIGATION OF CHAOTIC VIBRATIONS

1999 ◽  
Vol 09 (07) ◽  
pp. 1285-1306 ◽  
Author(s):  
E. YU. ROMANENKO ◽  
A. N. SHARKOVSKY
Keyword(s):  
Difference Equation ◽  
Boundary Value Problems ◽  
Difference Equations ◽  
Boundary Value ◽  
Dynamical Behavior ◽  
Time Behavior ◽  
One Dimensional ◽  
Long Time ◽  
The Difference ◽  
The One

Among evolutionary boundary value problems for partial differential equations, there is a wide class of problems reducible to difference, differential-difference and other relevant equations. Of especial promise for investigation are problems that reduce to difference equations with continuous argument. Such problems, even in their simplest form, may exhibit solutions with extremely complicated long-time behavior to the extent of possessing evolutions that are indistinguishable from random ones when time is large. It is owing to the reduction to a difference equation followed by the employment of the properties of the one-dimensional map associated with the difference equation, that, it is in many cases possible to establish mathematical mechanisms for one or other type of dynamical behavior of solutions. The paper presents the overall picture in the study of boundary value problems reducible to difference equations (on which the authors have a direct bearing over the last ten years) and demonstrates with several simplest examples the potentialities that such a reduction opens up.

scholarly journals Oscillation of third-order delay difference equations with negative damping term

2018 ◽  
Vol 72 (1) ◽  
pp. 19
Author(s):  
Martin Bohner ◽  
Srinivasan Geetha ◽  
Srinivasan Selvarangam ◽  
Ethiraju Thandapani
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Comparison Theorems ◽  
Asymptotic Behavior Of Solutions ◽  
Delay Difference Equation ◽  
Third Order ◽  
Negative Damping ◽  
The Difference ◽  
Delay Difference Equations ◽  
Delay Difference

The aim of this paper is to investigate the oscillatory and asymptotic behavior of solutions of a third-order delay difference equation. By using comparison theorems, we deduce oscillation of the difference equation from its relation to certain associated first-order delay difference equations or inequalities. Examples are given to illustrate the main results.<br /><br />

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