Hyers-Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems

In this paper, the concepts of Hyers-Ulam stability are generalized for non-autonomous linear differential systems. We prove that the k-periodic linear differential matrix system (Z) over dot(t) - A(t)Z(t), t is an element of R is Hyers-Ulam stable if and only if the matrix family L - E(k, 0) has no eigenvalues on the unit circle, i.e. we study the Hyers-Ulam stability in terms of dichotomy of the differential matrix system (Z) over dot(t) = A(t)Z(t), t is an element of R. Furthermore, we relate Hyers-Ulam stability of the system (Z) over dot(t) = A(t)Z(t), t is an element of Rto the boundedness of solution of the following Cauchy problem: {(Y) over dot(t) = A(t)Y(t) + rho(t), t >= 0 Y(0) = x - x(0), where A(t) is a square matrix for any t is an element of R, rho(t) is a bounded function and x, x(0) is an element of C-m. (C) 2015 Elsevier Inc. All rights reserved.