scholarly journals Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 512 ◽  
Author(s):  
Constantin Buşe ◽  
Donal O’Regan ◽  
Olivia Saierli
Keyword(s):  
Positive Integer ◽  
Unit Circle ◽  
Monodromy Matrix ◽  
Time Dependent ◽  
Multiple Eigenvalue ◽  
Linear Recurrence ◽  
Ulam Stability ◽  
Periodic Coefficients ◽  
Periodic Sequences ◽  
Linear Scalar

Let q ≥ 2 be a positive integer and let ( a j ) , ( b j ) , and ( c j ) (with j a non-negative integer) be three given C -valued and q-periodic sequences. Let A ( q ) : = A q − 1 ⋯ A 0 , where A j is as is given below. Assuming that the “monodromy matrix” A ( q ) has at least one multiple eigenvalue, we prove that the linear scalar recurrence 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 the spectrum of A ( q ) does not intersect the unit circle Γ : = { w ∈ C : | w | = 1 } . Connecting this result with a recently obtained one it follows that the above linear recurrence is Hyers-Ulam stable if and only if the spectrum of A ( q ) does not intersect the unit circle.

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 } .

Hyers–Ulam stability for equations with differences and differential equations with time-dependent and periodic coefficients

2019 ◽  
Vol 150 (5) ◽  
pp. 2175-2188 ◽  
Author(s):  
Constantin Buşe ◽  
Vasile Lupulescu ◽  
Donal O'Regan
Keyword(s):  
Monodromy Matrix ◽  
Time Dependent ◽  
Linear Recurrence ◽  
Ulam Stability ◽  
Identity Matrix ◽  
Periodic Functions ◽  
Periodic Sequences ◽  
First Order ◽  
Linear Differential ◽  
Image Position

AbstractLetqbe a positive integer and let (an) and (bn) be two given ℂ-valued andq-periodic sequences. First we prove that the linear recurrence in ℂ0.1$$x_{n + 2} = a_nx_{n + 1} + b_nx_n,\quad n\in {\open Z}_+ $$is Hyers–Ulam stable if and only if the spectrum of the monodromy matrixTq: =Aq−1· · ·A0(i.e. the set of all its eigenvalues) does not intersect the unit circle Γ = {z∈ ℂ: |z| = 1}, i.e.Tqis hyperbolic. Here (and in as follows) we let0.2$$A_n = \left( {\matrix{ 0 & 1 \cr {b_n} & {a_n} \cr } } \right)\quad n\in {\open Z}_+ .$$Secondly we prove that the linear differential equation0.3$${x}^{\prime \prime}(t) = a(t){x}^{\prime}(t) + b(t)x(t),\quad t\in {\open R},$$(wherea(t) andb(t) are ℂ-valued continuous and 1-periodic functions defined on ℝ) is Hyers–Ulam stable if and only ifP(1) is hyperbolic; hereP(t) denotes the solution of the first-order matrix 2-dimensional differential system0.4$${X}^{\prime}(t) = A(t)X(t),\quad t\in {\open R},\quad X(0) = I_2,$$whereI2is the identity matrix of order 2 and0.5$$A(t) = \left( {\matrix{ 0 & 1 \cr {b(t)} & {a(t)} \cr } } \right),\quad t\in {\open R}.$$

scholarly journals Stability of a Functional Equation Deriving from Quadratic and Additive Functions in Non-Archimedean Normed Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Abasalt Bodaghi ◽  
Sang Og Kim
Keyword(s):  
Functional Equation ◽  
Positive Integer ◽  
General Solution ◽  
Functional Equations ◽  
Quadratic Functional Equation ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Additive Functions ◽  
Normed Spaces

We obtain the general solution of the generalized mixed additive and quadratic functional equationfx+my+fx−my=2fx−2m2fy+m2f2y,mis even;fx+y+fx−y−2m2−1fy+m2−1f2y,mis odd, for a positive integerm. We establish the Hyers-Ulam stability for these functional equations in non-Archimedean normed spaces whenmis an even positive integer orm=3.

On the Fourier series of a finitely described convex curve and a conjecture of H. S. Shapiro

1985 ◽  
Vol 98 (3) ◽  
pp. 513-527 ◽  
Author(s):  
T. Sheil-Small
Keyword(s):  
Fourier Series ◽  
Analytic Function ◽  
Positive Integer ◽  
Unit Circle ◽  
Fourier Coefficients ◽  
Convex Curve

AbstractLet F(eis) denote a homeomorphism of the positively oriented unit circle onto a convex curve Γ and let f (eit) = F(eiΦ(t)), where Φ(t) is a non-decreasing function such that Φ(2π) – Φ(0) ≤ 2πN (N a positive integer). If f (eit) has Fourier coefficients cn, we show that is either constant or an N -valent analytic function in {|z| < 1}. We prove that where d is the distance from 0 to Γ and δ(N) > 0 depends only on N. This settles affirmatively a conjecture of H. S. Shapiro.

scholarly journals Mathematical Modeling of a Class of Symmetrical Islamic Design

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 517
Author(s):  
Mostafa ZAHRI
Keyword(s):  
Mathematical Modeling ◽  
Unit Circle ◽  
Convergence Properties ◽  
Periodic Sequences ◽  
New Model ◽  
One Dimensional ◽  
Geometric Patterns ◽  
Interesting Class ◽  
The One

In this paper, we present a new model for simulating an interesting class of Islamic design. Based on periodic sequences on the one-dimensional manifolds, and from emerging numbers, we construct closed graphs with edges on the unit circle. These graphs build very nice shapes and lead to a symmetrical class of geometric patterns of so-called Islamic design. Moreover, we mathematically characterize and analyze some convergence properties of the used up-down sequences. Finally, four planar type of patterns are simulated.

scholarly journals On the fourth-order linear recurrence formula related to classical Gauss sums

2017 ◽  
Vol 15 (1) ◽  
pp. 1251-1255 ◽  
Author(s):  
Chen Zhuoyu ◽  
Zhang Wenpeng
Keyword(s):  
Positive Integer ◽  
Fourth Order ◽  
Recurrence Formula ◽  
Character Sums ◽  
Gauss Sums ◽  
Linear Recurrence ◽  
Analytic Method ◽  
Order Linear ◽  
Computational Problem

Abstract Let p be an odd prime with p ≡ 1 mod 4, k be any positive integer, ψ be any fourth-order character mod p. In this paper, we use the analytic method and the properties of character sums mod p to study the computational problem of G(k, p) = τk(ψ)+τk(ψ), and give an interesting fourth-order linear recurrence formula for it, where τ(ψ) denotes the classical Gauss sums.

Solution and stability of an n-dimensional functional equation

Analysis ◽  
2019 ◽  
Vol 39 (3) ◽  
pp. 107-115 ◽  
Author(s):  
Sandra Pinelas ◽  
V. Govindan ◽  
K. Tamilvanan
Keyword(s):  
Banach Space ◽  
Functional Equation ◽  
Fixed Point ◽  
Positive Integer ◽  
General Solution ◽  
Functional Equations ◽  
Ulam Stability ◽  
Fixed Point Methods ◽  
Fixed Positive Integer

AbstractIn this paper, we prove the general solution and generalized Hyers–Ulam stability of n-dimensional functional equations of the form\sum_{\begin{subarray}{c}i=1\\ i\neq j\neq k\end{subarray}}^{n}f\biggl{(}-x_{i}-x_{j}-x_{k}+\sum_{% \begin{subarray}{c}l=1\\ l\neq i\neq j\neq k\end{subarray}}^{n}x_{l}\biggr{)}=\biggl{(}\frac{n^{3}-9n^{% 2}+20n-12}{6}\biggr{)}\sum_{i=1}^{n}f(x_{i}),where n is a fixed positive integer with \mathbb{N}-\{0,1,2,3,4\}, in a Banach space via direct and fixed point methods.

Invariant curves and time-dependent potentials

1991 ◽  
Vol 11 (2) ◽  
pp. 365-378 ◽  
Author(s):  
Stephane Laederich ◽  
Mark Lev
Keyword(s):  
Hamiltonian Systems ◽  
Time Dependent ◽  
Invariant Curves ◽  
Periodic Coefficients ◽  
Polynomial Potential ◽  
The Stability ◽  
Image Position ◽  
Limiting Case

AbstractIn this paper we prove the existence of invariant curves and thus stability for all time for a class of Hamiltonian systems with time-dependent potentials, namely, for systems of the formwhereis a superquadratic polynomial potential with periodic coefficients. As a limiting case, a proof of the stability of Ulam's problem of a particle bouncing between two periodicially moving walls is given.

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