Zaj́ıček has recently shown that for a lower semi-continuous real-valued function on an Asplund space, the set of points where the function is Fréchet subdifferentiable but not Fréchet differentiable is first category. We introduce another variant of Fréchet differentiability, called Fréchet directional differentiability, and show that for any realvalued function on a normed linear space, the se...

A theory of differential calculus for nonlinear maps between general locally convex spaces is developed. All convergence notions are topological, and only familiarity with basic results from point set topology, differential calculus in Banach spaces, and locally convex space theory is assumed. The chain rule for continuous kth order differentiability, smoothness of inverse functions, and functi...

We prove a $$C^m$$ Lusin approximation theorem for horizontal curves in the Heisenberg group. This states that every absolutely continuous curve whose velocity is $$m-1$$ times $$L^1$$ differentiable almost everywhere coincides with except on set of small measure. Conversely, we show result no longer holds if differentiability replaced by approximate differentiability. shows our optimal and hig...

Abstract In this paper, the dynamic behavior of a class switched systems with internally forced switching (IFS) is investigated. By introducing definitions continuous dependence and differentiability, differentiability solution relative to control function are obtained. past studies, optimal problem given by IFS mainly focused on special controlled (the piece affine system). Our results lay goo...

In this note we give the continuous solutions of some functional equations of associative type that are strictly monotonic in each variable. Neither solvability nor differentiability conditions will be assumed.

such as was treated in the previous papers [3] and [4]. However, we shall show that we can replace the strong continuous differentiability of A(t)A(s)~ by its Holder continuity by means of a slight change of the proof. It is quite clear that the differentiability of A(t)A(s)~ is not necessary for the construction of the formal fundamental solution U(t, s) of (0. 1'). In the previous papers, how...

The well-known finite-dimensional first-order open mapping theorem says that a continuous map with a finite-dimensional target is open at a point if its differential at that point exists and is surjective. An identical result, due to Graves, is true when the target is infinite-dimensional, if “differentiability” is replaced by “strict differentiability.” We prove general theorems in which the l...