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

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Abstract Let q be a positive integer and let ( a n ) and ( b n ) 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 T q : = A q −1 · · · A 0 (i.e. the set of all its eigenvalues) does not intersect the unit circle Γ = { z ∈ ℂ: | z | = 1}, i.e. T q 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 I 2 is the identity matrix of order 2 and 0.5 $$A(t) = \left( {\matrix{ 0 & 1 \cr {b(t)} & {a(t)} \cr } } \right),\quad t\in {\open R}.$$