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

Part of: Stability theory Differential and difference algebra

Published online by Cambridge University Press:  20 March 2019

Constantin Buşe ,
Vasile Lupulescu  and
Donal O'Regan
Constantin Buşe
Affiliation:
Politehnica University of Timişoara, Department of Mathematics, Timişoara, România ([email protected])
Vasile Lupulescu
Affiliation:
Constantin Brâncuşi University, Târgu Jiu, România ([email protected])
Donal O'Regan
Affiliation:
National University of Ireland, School of Mathematics, Statistics and Applied Mathematics, Galway, Ireland ([email protected])
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Abstract

Let q be a positive integer and let (an) and (bn) be two given ℂ-valued and q-periodic sequences. First we prove that the linear recurrence in ℂ 0.1

$$x_{n + 2} = a_nx_{n + 1} + b_nx_n,\quad n\in {\open Z}_+ $$
is Hyers–Ulam stable if and only if the spectrum of the monodromy matrix Tq: = Aq−1 · · · A0 (i.e. the set of all its eigenvalues) does not intersect the unit circle Γ = {z ∈ ℂ: |z| = 1}, i.e. Tq is hyperbolic. Here (and in as follows) we let 0.2
$$A_n = \left( {\matrix{ 0 & 1 \cr {b_n} & {a_n} \cr } } \right)\quad n\in {\open Z}_+ .$$
Secondly we prove that the linear differential equation 0.3
$${x}^{\prime \prime}(t) = a(t){x}^{\prime}(t) + b(t)x(t),\quad t\in {\open R},$$
 (where a(t) and b(t) are ℂ-valued continuous and 1-periodic functions defined on ℝ) is Hyers–Ulam stable if and only if P(1) is hyperbolic; here P(t) denotes the solution of the first-order matrix 2-dimensional differential system 0.4
$${X}^{\prime}(t) = A(t)X(t),\quad t\in {\open R},\quad X(0) = I_2,$$
where I2 is the identity matrix of order 2 and 0.5
$$A(t) = \left( {\matrix{ 0 & 1 \cr {b(t)} & {a(t)} \cr } } \right),\quad t\in {\open R}.$$

Keywords

Difference and differential equationsDiscrete dichotomyHyers–Ulam stability

MSC classification

Primary: 12H20: Abstract differential equations
Secondary: 34D09: Dichotomy, trichotomy 39B82: Stability, separation, extension, and related topics
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 5 , October 2020 , pp. 2175 - 2188
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Copyright
Copyright © Royal Society of Edinburgh 2019

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