Abstract
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.
Highlights
An open problem, arising naturally in [1], is a problem referring to the relationship between the Hyers-Ulam stability of a certain linear recurrence of order n with periodic coefficients and the exponential dichotomy of the monodromy matrix associated to the recurrence
This article can be seen as a new link in the chain of articles [1,2,3,4,5] which address the Hyers-Ulam stability of linear scalar recurrences
We denote the solution of the nonhomogeneous linear recurrence (3) initiated from Y0 by (φ(n, Y0, ( f k ))
Summary
An open problem, arising naturally in [1], is a problem referring to the relationship between the Hyers-Ulam stability of a certain linear recurrence of order n with periodic coefficients and the exponential dichotomy of the monodromy matrix associated to the recurrence. The corresponding problem for second-order recurrences was completed in [2], where second-order linear differential equations were analyzed. This article can be seen as a new link in the chain of articles [1,2,3,4,5] which address the Hyers-Ulam stability of linear scalar recurrences. The connections of this topic to those existing in the literature was already presented in [3], so we do not present them again here. For developments concerning differential equations with impulses see, [10,11,12,13], and the references therein
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