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

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
$$A(t) = \left( {\matrix{ 0 & 1 \cr {b(t)} & {a(t)} \cr } } \right),\quad t\in {\open R}.$$

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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
• ISSN: 0308-2105
• EISSN: 1473-7124
• URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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