Characterization of Ulam-Hyers stability of linear differential equations with periodic coefficients

Abstract

Let X be a Banach space over R or C. We study Ulam–Hyers stability on the time interval R of linear differential equations with continuous coefficients, considering solutions with values in X for higher order equations, and solutions with values in Xn for first-order systems. Roughly speaking, Ulam–Hyers stability of an equation is its property of having an exact solution close to an approximate solution. For a first-order system x′=Ax, we obtain that its Ulam–Hyers stability is equivalent to another important property, namely for each continuous and bounded function g:R→Xn there exists a bounded solution of x′=Ax+g. Using this result, theory of Jordan forms of matrices and Floquet theory, we characterize the Ulam–Hyers stability of a system x′=Ax with periodic coefficients in terms of its characteristic multipliers. Our main result states that the Ulam–Hyers stability of a linear equation with periodic coefficients of order n with unknown x is equivalent to the Ulam–Hyers stability of the corresponding system satisfied by (x,x′,…,x(n−1)).