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Generalized ordinary differential equations (we write generalized ODEs), introduced by J. Kurzweil in 1957, are known to encompass several other types of equations as measure functional differential equations, for instance. In this paper, we obtain converse Lyapunov theorems for generalized ODEs and, in particular, for measure functional differential equations which, in turn, encompass impulsive functionals differential equations as well as functional dynamic equations on time scales. We also relate uniform stability to boundedness of solutions. As an application, we establish necessary and sufficient conditions for a system of non–homogeneous nonlinear generalized ODEs defined in a Banach space and for a system of non–homogeneous measure functional differential equations to be asymptotically controllable. We include an example which illustrates the main results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain converse Lyapunov theorems for measure functional differential equations which are known to encompass dynamic equations on time scales, impulsive differential equations and functional differential equations (see [7,8]). Our main tool is to consider the framework of generalized ordinary differential equations, introduced by Jaroslav Kurzweil, and to manage all other equations in this setting which has the advantage to handle well both highly oscillating functions and functions with many discontinuities. [ABSTRACT FROM AUTHOR]