Sufficient conditions are established for the asymptotic stability of the zero solution of the equation (1.1) with $p\equiv 0$ and the boundedness of all solutions of the equation (1.1) with $p\ne 0$. Our result includes and improves several results in the literature ([4], [5], [8]).

In the paper the concept of a controllable continuous flow in a metric space is introduced as a generalization of a controllable system of differential equations in a Banach space, and various kinds of stability and of boundedness of this flow are defined. Theorems stating necessary and sufficient conditions for particular kinds of stability and boundedness are formulated in terms of Ljapunov functions.

The generalized linear differential equation $dx=d\left[a\right(t\left)\right]x+df$ where $A,f\in B{V}_n^{loc}\left(J\right)$ and the matrices $I-{\Delta }^{-}A\left(t\right),I+{\Delta }^{+}A\left(t\right)$ are regular, can be transformed $\frac{dy}{ds}=B\left(s\right)y+g\left(s\right)$ using the notion of a logarithimc prolongation along an increasing function. This method enables to derive various results about generalized LDE from the well-known properties of ordinary LDE. As an example, the variational stability of the generalized LDE is investigated.