Lyapunov theorems for measure functional differential equations via Kurzweil-equations

Abstract

We consider measure functional differential equations (we write measure FDEs) of the form Dx = f ('X IND. T', t)Dg, where f is Perron–Stieltjes integrable, 'X IND. T' is given by 'X IND. T'(θ) = x(t + θ), θ ∈ [−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 IND. 0',∞) → 'R POT. N' and g : ['T IND. 0',∞) → R, 'T IND. 0' ∈ 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 ˇS. Schwabik andM. 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).