scholarly journals Hyers-Ulam stability of the linear recurrence with constant coefficients

2005 ◽  
Vol 2005 (2) ◽  
pp. 407076 ◽  
Author(s):  
Dorian Popa
Keyword(s):  
Linear Recurrence ◽  
Ulam Stability ◽  
Constant Coefficients

scholarly journals Hyers-Ulam Stability of a System of First Order Linear Recurrences with Constant Coefficients

2015 ◽  
Vol 2015 ◽  
pp. 1-5 ◽  
Author(s):  
Bing Xu ◽  
Janusz Brzdęk
Keyword(s):  
Ulam Stability ◽  
Linear Difference Equations ◽  
Positive Real ◽  
Fixed Sequence ◽  
Linear Recurrences ◽  
Order Linear ◽  
First Order ◽  
Constant Coefficients ◽  
Complex Coefficients ◽  
Positive Real Constant

We study the Hyers-Ulam stability in a Banach spaceXof the system of first order linear difference equations of the formxn+1=Axn+dnforn∈N0(nonnegative integers), whereAis a givenr×rmatrix with real or complex coefficients, respectively, and(dn)n∈N0is a fixed sequence inXr. That is, we investigate the sequences(yn)n∈N0inXrsuch thatδ∶=supn∈N0yn+1-Ayn-dn<∞(with the maximum norm inXr) and show that, in the case where all the eigenvalues ofAare not of modulus 1, there is a positive real constantc(dependent only onA) such that, for each such a sequence(yn)n∈N0, there is a solution(xn)n∈N0of the system withsupn∈N0yn-xn≤cδ.

scholarly journals Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients: The Case When the Monodromy Matrix Has Simple Eigenvalues

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 339 ◽  
Author(s):  
Constantin Buşe ◽  
Donal O’Regan ◽  
Olivia Saierli
Keyword(s):  
Positive Integer ◽  
Unit Circle ◽  
Nonnegative Integer ◽  
Monodromy Matrix ◽  
Time Dependent ◽  
Linear Recurrence ◽  
Ulam Stability ◽  
Periodic Coefficients ◽  
Periodic Sequences

Let q ≥ 2 be a positive integer and let ( a j ) , ( b j ) and ( c j ) (with j nonnegative integer) be three given C -valued and q-periodic sequences. Let A ( q ) : = A q − 1 ⋯ A 0 , where A j is defined below. Assume that the eigenvalues x , y , z of the “monodromy matrix” A ( q ) verify the condition ( x − y ) ( y − z ) ( z − x ) ≠ 0 . We prove that the linear recurrence in C x n + 3 = a n x n + 2 + b n x n + 1 + c n x n , n ∈ Z + is Hyers–Ulam stable if and only if ( | x | − 1 ) ( | y | − 1 ) ( | z | − 1 ) ≠ 0 , i.e., the spectrum of A ( q ) does not intersect the unit circle Γ : = { w ∈ C : | w | = 1 } .

scholarly journals Hyers–Ulam Stability for Quantum Equations of Euler Type

2020 ◽  
Vol 2020 ◽  
pp. 1-10 ◽  
Author(s):  
Douglas R. Anderson ◽  
Masakazu Onitsuka
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Discrete Dynamics ◽  
Direct Connection ◽  
Ulam Stability ◽  
Arbitrary Integer ◽  
Step Size ◽  
First Order ◽  
Constant Coefficients ◽  
Quantum Equations

Many applications using discrete dynamics employ either q-difference equations or h-difference equations. In this work, we introduce and study the Hyers–Ulam stability (HUS) of a quantum (q-difference) equation of Euler type. In particular, we show a direct connection between quantum equations of Euler type and h-difference equations of constant step size h with constant coefficients and an arbitrary integer order. For equation orders greater than two, the h-difference results extend first-order and second-order results found in the literature, and the Euler-type q-difference results are completely novel for any order. In many cases, the best HUS constant is found.

From linear recurrence relations to linear ODEs with constant coefficients

2016 ◽  
Vol 15 (06) ◽  
pp. 1650109 ◽  
Author(s):  
L. Gatto ◽  
D. Laksov
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Ordinary Differential Equations ◽  
Recurrence Relations ◽  
Schubert Calculus ◽  
Linear Recurrence ◽  
Linear Ordinary Differential Equations ◽  
Constant Coefficients ◽  
Linear Odes

Linear Ordinary Differential Equations (ODEs) with constant coefficients are studied by looking in general at linear recurrence relations in a module with coefficients in an arbitrary [Formula: see text]-algebra. The bridge relating the two theories is the notion of formal Laplace transform associated to a sequence of invertibles. From this more economical perspective, generalized Wronskians associated to solutions of linear ODEs will be revisited, mentioning their relationships with Schubert Calculus for Grassmannians.

scholarly journals Fourier Transformation and Stability of a Differential Equation on L 1 ℝ

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Hamid Rezaei ◽  
Zahra Zafarasa ◽  
Lotfollah Karimi
Keyword(s):  
Differential Equation ◽  
Fourier Transform ◽  
Linear Differential Equation ◽  
Fourier Transformation ◽  
Ulam Stability ◽  
Constant Coefficients ◽  
Linear Differential ◽  
The Fourier Transform

In the present paper, by the Fourier transform, we show that every linear differential equation with constant coefficients of n -th order has a solution in L 1 ℝ which is infinitely differentiable in ℝ ∖ 0 . Moreover the Hyers–Ulam stability of such equations on L 1 ℝ is investigated.

scholarly journals A new method for solving the linear recurrence relation with nonhomogeneous constant coefficients in some cases

2021 ◽  
Vol 2113 (1) ◽  
pp. 012070
Author(s):  
Ben-Chao Yang ◽  
Xue-Feng Han
Keyword(s):  
Recurrence Relation ◽  
Computer Programming ◽  
Recursive Relation ◽  
New Method ◽  
Linear Recurrence ◽  
General Proof ◽  
Constant Coefficients ◽  
Linear Recurrence Relation ◽  
Almost All ◽  
General Method

Abstract Recursive relation mainly describes the unique law satisfied by a sequence, so it plays an important role in almost all branches of mathematics. It is also one of the main algorithms commonly used in computer programming. This paper first introduces the concept of recursive relation and two common basic forms, then starts with the solution of linear recursive relation with non-homogeneous constant coefficients, gives a new solution idea, and gives a general proof. Finally, through an example, the general method and the new method given in this paper are compared and verified.

scholarly journals Mahgoub transform and Hyers-Ulam stability of $ n^{th} $ order linear differential equations

2021 ◽  
Vol 7 (4) ◽  
pp. 4992-5014
Author(s):  
S. Deepa ◽  
◽  
S. Bowmiya ◽  
A. Ganesh ◽  
Vediyappan Govindan ◽  
...  
Keyword(s):  
Differential Equations ◽  
Linear Differential Equations ◽  
Ulam Stability ◽  
Transform Method ◽  
Order Linear ◽  
Constant Coefficients ◽  
Linear Differential

<abstract><p>The main aim of this paper is to investigate various types of Hyers-Ulam stability of linear differential equations of $ n^{th} $ order with constant coefficients using the Mahgoub transform method. We also show the Hyers-Ulam constants of these differential equations and give some main results.</p></abstract>

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