On the Fréchet differentiability of distance functions
On the geometric characterization of differentiability. I.
On the geometric characterization of differentiability. II.
On the multiplicity points of monotone operators on separable Banach spaces
On the open mapping principle and convex multivalued mappings
On the range of the derivative of a real-valued function with bounded support
We study the set f’(X) = f’(x): x ∈ X when f:X → ℝ is a differentiable bump. We first prove that for any C²-smooth bump f: ℝ² → ℝ the range of the derivative of f must be the closure of its interior. Next we show that if X is an infinite-dimensional separable Banach space with a -smooth bump b:X → ℝ such that is finite, then any connected open subset of X* containing 0 is the range of the derivative of a -smooth bump. We also study the finite-dimensional case which is quite different. Finally,...
On the range of the derivative of a smooth function and applications.
We survey recent results on the structure of the range of the derivative of a smooth real valued function f defined on a real Banach space X and of a smooth mapping F between two real Banach spaces X and Y. We recall some necessary conditions and some sufficient conditions on a subset A of L(X,Y) for the existence of a Fréchet-differentiable mapping F from X into Y so that F'(X) = A. Whenever F is only assumed Gâteaux-differentiable, new phenomena appear: we discuss the existence of a mapping F...
On the range of the derivative of a smooth mapping between Banach spaces.
On the Size of Balls Covered by Analytic Transformations.
On the size of the sets of gradients of bump functions and starlike bodies on the Hilbert space
We study the size of the sets of gradients of bump functions on the Hilbert space , and the related question as to how small the set of tangent hyperplanes to a smooth bounded starlike body in can be. We find that those sets can be quite small. On the one hand, the usual norm of the Hilbert space can be uniformly approximated by smooth Lipschitz functions so that the cones generated by the ranges of its derivatives have empty interior. This implies that there are smooth Lipschitz bumps...
On the stability of mappings and an answer to a problem of TH.M. Rassias
On the Structure of the Spaces ... .
On the structure of universal differentiability sets
A subset of is called a universal differentiability set if it contains a point of differentiability of every Lipschitz function . We show that any universal differentiability set contains a ‘kernel’ in which the points of differentiability of each Lipschitz function are dense. We further prove that no universal differentiability set may be decomposed as a countable union of relatively closed, non-universal differentiability sets.
On the Subdifferential of the Supremum of Two Convex Functions.
On the tangency of multifunctions
On Uniform Differentiability
We introduce the notion of uniform Fréchet differentiability of mappings between Banach spaces, and we give some sufficient conditions for this property to hold.
On α(·)-monotone multifunctions and differentiability of γ-paraconvex functions
Let (X,d) be a metric space. Let Φ be a family of real-valued functions defined on X. Sufficient conditions are given for an α(·)-monotone multifunction to be single-valued and continuous on a weakly angle-small set. As an application it is shown that a γ-paraconvex function defined on an open convex subset of a Banach space having separable dual is Fréchet differentiable on a residual set.
One counterexample concerning the Fréchet differentiability of convex functions on closed sets
One-parameter submonoids in locally complete differentiable monoids
Optimal integrability of the Jacobian of orientation preserving maps
Dato un qualsiasi spazio invariante per riordinamenti su un insieme aperto , si determina il più piccolo spazio invariante per riordinamenti con la proprietà che se è una applicazione che mantiene l'orientamento e , allora appartiene localmente a .