Lyapunov theorems for measure functional differential equations via Kurzweil-equations

摘要

We consider measure functional differential equations (we write measure FDEs) of the form Dx = f (x(t), t) Dg, where f is Perron-Stieltjes integrable, xt is given by x(t) (theta) = x(t +theta),theta is an element of [-r, 0], with r > 0, and Dx and Dg are the distributional derivatives in the sense of the distribution of L. Schwartz, with respect to functions x :[t(0), infinity) -> R-n and g : [t(0), infinity) -> R, t(0) is an element of R, and we present new concepts of stability of the trivial solution, when it exists, of this equation. The new stability concepts generalize, for instance, the variational stability introduced by. S. Schwabik and M. Federson for FDEs and yet we are able to establish a Lyapunov-type theorem for measure FDEs via theory of generalized ordinary differential equations (also known as Kurzweil equations). (C) 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim