scholarly journals Solution of general inhomogeneous linear difference equations

1983 ◽  
Vol 24 (4) ◽  
pp. 466-472
Author(s):  
H. Van Hoa
Keyword(s):  
General Solution ◽  
Difference Equation ◽  
Difference Equations ◽  
Continued Fraction ◽  
Linear Difference Equation ◽  
Second Order ◽  
Linear Difference Equations ◽  
Homogeneous Case ◽  
Simple Product ◽  
Image Position

AbstractThe general solution of the rth inhomogeneous linear difference equation is given in the formThe coefficients , i = 2, …, r, and b(n−r)(n) can be evaluated from n values , k = 0, …, n − 1, which santisfy an rth order homogenous linear difference equation. In the rth order homogeneous case and if n ≥ 2r, the method requires the evaluation of r determinants of successive orders n − 2r + 1, n − 2r + 2, …, n − r. If r ≤ n ≤ 2r − 1, only n − r determinants are required, with orders varying from 1 to n − r. In the second order ihnomogenous case, can be evaluated from a continued fraction amd a simple product.

scholarly journals Solution of homogeneous linear difference equations

1980 ◽  
Vol 21 (3) ◽  
pp. 293-296 ◽  
Author(s):  
J. D. Love
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Continued Fraction ◽  
Second Order ◽  
Linear Difference Equations ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Simple Product ◽  
Homogeneous Linear

AbstractWhen the first two elements of a sequence satisfying a second order difference equation are prescribed, the remaining elements are evaluated from a continued fraction and a simple product.

scholarly journals Note on some representations of general solutions to homogeneous linear difference equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Stevo Stević ◽  
Bratislav Iričanin ◽  
Witold Kosmala ◽  
Zdeněk Šmarda
Keyword(s):  
General Solution ◽  
Difference Equation ◽  
Difference Equations ◽  
Fibonacci Sequence ◽  
Linear Difference Equation ◽  
Linear Difference Equations ◽  
Second Order Difference Equation ◽  
Constant Coefficients ◽  
Order Difference Equation ◽  
Homogeneous Linear

Abstract It is known that every solution to the second-order difference equation $x_{n}=x_{n-1}+x_{n-2}=0$ x n = x n − 1 + x n − 2 = 0 , $n\ge 2$ n ≥ 2 , can be written in the following form $x_{n}=x_{0}f_{n-1}+x_{1}f_{n}$ x n = x 0 f n − 1 + x 1 f n , where $f_{n}$ f n is the Fibonacci sequence. Here we find all the homogeneous linear difference equations with constant coefficients of any order whose general solution have a representation of a related form. We also present an interesting elementary procedure for finding a representation of general solution to any homogeneous linear difference equation with constant coefficients in terms of the coefficients of the equation, initial values, and an extension of the Fibonacci sequence. This is done for the case when all the roots of the characteristic polynomial associated with the equation are mutually different, and then it is shown that such obtained representation also holds in other cases. It is also shown that during application of the procedure the extension of the Fibonacci sequence appears naturally.

scholarly journals Sharp Lyapunov-type inequalities for second-order half-linear difference equations with different kinds of boundary conditions

2021 ◽  
Vol 115 (3) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Difference Equation ◽  
Boundary Conditions ◽  
Difference Equations ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Second Order ◽  
Linear Difference Equations ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation

AbstractIn this work, we establish optimal Lyapunov-type inequalities for the second-order difference equation with p-Laplacian $$\begin{aligned} \Delta (\left| \Delta u(k-1)\right| ^{p-2}\Delta u(k-1))+a(k)\left| u(k)\right| ^{p-2}u(k)=0 \end{aligned}$$ Δ ( Δ u ( k - 1 ) p - 2 Δ u ( k - 1 ) ) + a ( k ) u ( k ) p - 2 u ( k ) = 0 with Dirichlet, Neumann, mixed, periodic and anti-periodic boundary conditions.

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 Perron-Type Criterion for Linear Difference Equations with Distributed Delay

2007 ◽  
Vol 2007 ◽  
pp. 1-12
Author(s):  
Jehad O. Alzabut ◽  
Thabet Abdeljawad
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Trivial Solution ◽  
Distributed Delay ◽  
Linear Difference Equation ◽  
Asymptotically Stable ◽  
Linear Difference Equations ◽  
Type Criterion

It is shown that if a linear difference equation with distributed delay of the formΔx(n)=∑k=−d0Δkζ(n+1,k−1)x(n+k−1),n≥1, satisfies a Perron condition then its trivial solution is uniformly asymptotically stable.

scholarly journals Meromorphic Solutions of Linear Difference Equations with Polynomial Coefficients

2017 ◽  
Vol 59 (1) ◽  
pp. 159-168
Author(s):  
Y. Zhang ◽  
Z. Gao ◽  
H. Zhang
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Necessary Conditions ◽  
Meromorphic Solution ◽  
Linear Difference Equation ◽  
Meromorphic Solutions ◽  
Linear Difference Equations ◽  
Polynomial Coefficients ◽  
Difference Analogue ◽  
Borel Exceptional Values

AbstractWe study the growth of the transcendental meromorphic solution f(z) of the linear difference equation:where q(z), p0(z), ..., pn-(z) (n ≥ 1) are polynomials such that p0(z)pn(z) ≢ 0, and obtain some necessary conditions guaranteeing that the order of f(z) satisfies σ(f) ≥ 1 using a difference analogue of the Wiman-Valiron theory. Moreover, we give the form of f(z) with two Borel exceptional values when two of p0(z), ..., pn(z) have the maximal degrees.

scholarly journals Some roughness results concerning reducibility for linear difference equations

1988 ◽  
Vol 11 (4) ◽  
pp. 793-804 ◽  
Author(s):  
Garyfalos Papaschinopoulos
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Exponential Dichotomy ◽  
Coefficient Matrix ◽  
Linear Difference Equation ◽  
Almost Periodic ◽  
Linear Difference Equations ◽  
Joint Property

In this paper we prove first that the exponential dichotomy of linear difference equations is “rough”. Moreover we prove that if the coefficient matrix of a linear difference equation is almost periodic, then the Joint property of having an exponential dichotomy with a projectionPand being reducible withPby an almost periodic kinematics similarity is “rough”.

scholarly journals A Hille-Wintner type comparison theorem for second order difference equations

1983 ◽  
Vol 6 (2) ◽  
pp. 387-393 ◽  
Author(s):  
John W. Hooker
Keyword(s):  
Difference Equation ◽  
Comparison Theorem ◽  
Difference Equations ◽  
Linear Difference Equation ◽  
Second Order ◽  
Order Difference

For the linear difference equationΔ(cn−1Δxn−1)+anxn=0   with   cn>0, a non-oscillation comparison theorem given in terms of the coefficientscnand the series∑n=k∞an, has been proved.

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].

scholarly journals Instability of second-order nonhomogeneous linear difference equations with real-valued coefficients

2021 ◽  
Vol 37 (3) ◽  
pp. 489-495
Author(s):  
MASAKAZU ONITSUKA ◽  
◽  
Keyword(s):  
Difference Equation ◽  
Linear Difference Equation ◽  
Second Order ◽  
Ulam Stability ◽  
Necessary And Sufficient Condition ◽  
Linear Difference Equations ◽  
Real Numbers ◽  
Step Size ◽  
Order Linear ◽  
Necessary And Sufficient

In J. Comput. Anal. Appl. (2020), pp. 152--165, the author dealt with Hyers--Ulam stability of the second-order linear difference equation $\Delta_h^2x(t)+\alpha \Delta_hx(t)+\beta x(t) = f(t)$ on $h\mathbb{Z}$, where $\Delta_hx(t) = (x(t+h)-x(t))/h$ and $h\mathbb{Z} = \{hk|\,k\in\mathbb{Z}\}$ for the step size $h>0$; $\alpha$ and $\beta$ are real numbers; $f(t)$ is a real-valued function on $h\mathbb{Z}$. The purpose of this paper is to clarify that the second-order linear difference equation has no Hyers--Ulam stability when the step size $h>0$ and the coefficients $\alpha$ and $\beta$ satisfy suitable conditions. Finally, a necessary and sufficient condition for Hyers--Ulam stability is obtained.

Sign in / Sign up

Export Citation Format

Share Document