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.

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.

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 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.

On the growth analysis of meromorphic solutions of finite ϕ-order of linear difference equations

Analysis ◽  
2020 ◽  
Vol 40 (4) ◽  
pp. 193-202
Author(s):  
Sanjib Kumar Datta ◽  
Nityagopal Biswas
Keyword(s):  
Difference Equation ◽  
Complex Plane ◽  
Difference Equations ◽  
Growth Analysis ◽  
Linear Difference Equation ◽  
Higher Order ◽  
Meromorphic Solutions ◽  
Linear Difference Equations ◽  
Unbounded Function ◽  
Growth Properties

AbstractIn this paper, we investigate some growth properties of meromorphic solutions of higher-order linear difference equationA_{n}(z)f(z+n)+\dots+A_{1}(z)f(z+1)+A_{0}(z)f(z)=0,where {A_{n}(z),\dots,A_{0}(z)} are meromorphic coefficients of finite φ-order in the complex plane where φ is a non-decreasing unbounded function. We extend some earlier results of Latreuch and Belaidi [Z. Latreuch and B. Belaïdi, Growth and oscillation of meromorphic solutions of linear difference equations, Mat. Vesnik 66 2014, 2, 213–222].

Asymptotic behavior in neutral difference equations with negative coefficients

2014 ◽  
Vol 64 (2) ◽  
Author(s):  
G. Chatzarakis ◽  
G. Miliaras
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Difference Equations ◽  
Difference Operator ◽  
Retarded Argument ◽  
Real Numbers ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Forward Difference ◽  
Deviated Argument

AbstractIn this paper, we study the asymptotic behavior of the solutions of a neutral difference equation of the form $\Delta [x(n) + cx(\tau (n))] - p(n)x)(\sigma (n)) = 0,$, where τ(n) is a general retarded argument, σ(n) is a general deviated argument, c ∈ ℝ, (−p(n))n≥0 is a sequence of negative real numbers such that p(n) ≥ p, p ∈ ℝ+, and Δ denotes the forward difference operator Δx(n) = x(n+1)−x(n).

scholarly journals Existence of almost periodic solutions to some nonautonomous higher-order stochastic difference equations

2020 ◽  
Vol 7 (1) ◽  
pp. 72-80
Author(s):  
Paul H. Bezandry
Keyword(s):  
Difference Equation ◽  
Periodic Solutions ◽  
Difference Equations ◽  
Higher Order ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Stochastic Difference Equation ◽  
Stochastic Difference Equations

AbstractThe paper studies the existence of almost periodic solutions to some nonautonomous higher-order stochastic difference equation of the form:X\left( {t + n} \right) + \sum\limits_{r = 1}^{n - 1} {{A_r}\left( t \right)X\left( {t + r} \right) + {A_0}\left( t \right)X\left( t \right) = f\left( {t,X\left( t \right)} \right),}n ∈ 𝕑, by means of discrete dichotomy techniques.

Oscillatory and asymptotic behavior of solutions of nonlinear neutral-type difference equations

1996 ◽  
Vol 38 (2) ◽  
pp. 163-171 ◽  
Author(s):  
John R. Graef ◽  
Agnes Miciano ◽  
Paul W. Spikes ◽  
P. Sundaram ◽  
E. Thandapani
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Difference Equations ◽  
Neutral Type ◽  
Higher Order ◽  
Asymptotic Behavior Of Solutions ◽  
Delay Difference Equation ◽  
Neutral Delay ◽  
Image Position ◽  
Delay Difference

AbstractThe authors consider the higher-order nonlinear neutral delay difference equationand obtain results on the asymptotic behavior of solutions when (pn) is allowed to oscillate about the bifurcation value –1. We also consider the case where the sequence {pn} has arbitrarily large zeros. Examples illustrating the results are included, and suggestions for further research are indicated.

scholarly journals Oscillation for Super Linear/Linear Second Order Neutral Difference Equations with Variable Several Delays

2020 ◽  
Vol 5 (4) ◽  
pp. 663-681
Author(s):  
Chittaranjan Behera ◽  
Radhanath Rath ◽  
Prayag Prasad Mishra
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Difference Operator ◽  
Sufficient Conditions ◽  
Second Order ◽  
Multiple Delays ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Forward Difference ◽  
Several Delays

This article, is concerned with finding sufficient conditions for the oscillation and non oscillation of the solutions of a second order neutral difference equation with multiple delays under the forward difference operator, which generalize and extend some existing results.This could be possible by extending an important lemma from the literature.

scholarly journals Growth of meromorphic solutions of some difference equations

2010 ◽  
Vol 4 (2) ◽  
pp. 309-321 ◽  
Author(s):  
Xiu-Min Zheng ◽  
Zong-Xuan Chen ◽  
Tu Jin
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Higher Order ◽  
Meromorphic Solutions ◽  
Entire Solutions ◽  
Order Difference ◽  
Difference Analogue ◽  
Algebraic Difference ◽  
The Difference

We investigate higher order difference equations and obtain some results on the growth of transcendental meromorphic solutions, which are complementary to the previous results. Examples are also given to show the sharpness of these results. We also investigate the growth of transcendental entire solutions of a homogeneous algebraic difference equation by using the difference analogue of Wiman-Valiron Theory.

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