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

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.

A multiplicative eigenvalues perturbation and its application to natural frequency assignment in undamped second-order systems

In this article, a novel theorem for eigenvalue perturbation inspired by the results of Brauer is introduced. The new result is valid for single and multiple eigenvalue perturbations, with preservation—no spillover of the remaining spectrum. An application of the proposed results is given since the model updating and the partial eigenvalue assignment problems for second-order linear dynamics have gained crescent attention in the last decades. Furthermore, its effectiveness is illustrated with numerical examples.

ON REGULIRATION MULTIPLE EIGENVALUE BY REDUCTION PSEUDOPERTURBATION METHODS

On the base of the bifurcation theory methods it is considered the problem of the retaining multiple eigenvalues and relevant eigenvectors and roots elements. An approch is suggested which allows reduce algebraic multiple to unit, that is reduce the problem of the retaining multiple eigenvalues to simple. For construction iterated process applied pseudoperturbation methods.