In this paper we address some of the most fundamental questions regarding the differentiability structure of locally Lipschitz functions defined on separable Banach spaces. For example, we examine the relationship between integrability, D-representability, and strict differentiability. In addition to this, we show that on any separable Banach space there is a significant family of locally Lipsc...

We provide solution of first order linear fuzzy differential equations by variation of constant formula. The differentiability concept used in this paper is the strongly generalized differentiability since a fuzzy differential equation under this differentiability can have decreasing legth of support function. Under some conditions we provide solution with decreasing support and so the behaviou...

Peano differentiability generalizes ordinary differentiability to higher order. There are two ways to define Peano differentiability for functions defined on non-open sets. For both concepts, we investigate the question under which conditions a function defined on a closed set can be extended to a Peano differentiable function on the ambient space if the sets and functions are definable in an o...

Adomian decomposition method has been applied to solve many functional equations so far. In this article, we have used this method to solve the fuzzy differential equation under generalized differentiability. We interpret a fuzzy differential equation by using the strongly generalized differentiability. Also one concrete application for ordinary fuzzy differential equation with fuzzy input data...

We consider the typical behaviour of random dynamical systems order-preserving interval homeomorphisms with a positive Lyapunov exponent condition at endpoints. Our study removes any requirement for continuous differentiability save existence finite derivatives endpoints interval. construct suitable Baire space structure this class systems. Generically within our space, we show that stationary ...

We study distribution dependent stochastic differential equation driven by a continuous process, without any specification on its law, following the approach initiated in [17]. provide several criteria for existence and uniqueness of solutions which go beyond classical globally Lipschitz setting. In particular we show well-posedness equation, as well almost sure convergence associated particle ...

• Banach-Alaoglu: compactness of polars • Variant Banach-Steinhaus/uniform boundedness • Second polars • Weak boundedness implies boundedness • Weak-to-strong differentiability The comparison of weak and strong differentiability is due to Grothendieck, although the original sources are not widely available.

According to a new mean-value theorem, under the conditions of a function f ensuring the existence and uniqueness of Lagrange’s mean, there exists a unique mean M such that f(x)− f(y) x− y = M ( f ′(x), f ′(y) ) . The main result says that, in this equality, M is a power mean if, and only if, M is either geometric, arithmetic or harmonic. A Cauchy relevant type result is also presented. Introdu...

In this paper we investigate the approximation of continuous functions on Wasserstein space by smooth functions, with smoothness meant in sense Lions differentiability. particular, case a Lipschitz function are able to construct sequence infinitely differentiable having same constant as original function. This solves an open problem raised [12]. For (resp. twice) continuously function, show tha...

We present a systematic study of causality theory on Lorentzian manifolds with continuous metrics. Examples are given which show that some standard facts in smooth Lorentzian geometry, such as light-cones being hypersurfaces, are wrong when metrics which are merely continuous are considered. We show that existence of time functions remains true on domains of dependence with continuous metrics, ...