We survey recent results on the structure of the range of the derivative of a smooth mapping f between two Banach spaces X and Y . We recall some necessary conditions and some sufficient conditions on a subsetA of (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: for instance, ...

Proximal mappings, which generalize projection mappings, were introduced by Moreau and shown to be valuable in understanding the subgradient properties of convex functions. Proximal mappings subsequently turned out to be important also in numerical methods of optimization and the solution of nonlinear partial differential equations and variational inequalities. Here it is shown that, when a con...

‎In this paper, we provide a complete description of weighted composition operators between extended Lipschitz algebras on compact metric spaces. We give necessary and sufficient conditions for the injectivity and the sujectivity of these operators. We also obtain some sufficient conditions and some necessary conditions for a weighted composition operator between these spaces to be compact.

In the Heisenberg group, we prove that the boundary of sets with finite H-perimeter and having a bound on the measure theoretic normal is an H-Lipschitz graph. Then we show that if the normal is, on the boundary, the restriction of a continuous mapping, then the boundary is an H-regular surface.

Let [Formula: see text], [Formula: see text], and [Formula: see text] be a strongly monotone and Lipschitz mapping. A Krasnoselskii-type sequence is constructed and proved to converge strongly to the unique solution of [Formula: see text]. Furthermore, our technique of proo f is of independent interest.

There is a constant c > 0 such that for every ε ∈ (0, 1) and n > 1/ε, the following holds. Any mapping from the n-point star metric into l1 with bi-Lipschitz distortion 1 + ε requires dimension d > c log n ε log(1/ε) .

A projected subgradient method for solving a class of set-valued mixed variational inequalities (SMVIs) is proposed when the mapping is not necessarily Lipschitz. Under some suitable conditions, it can be proven that the sequence generated by the method can strongly converge to the unique solution to the problem in the Hilbert spaces.

We study a notion of an equivariant, Lipschitz, permutationinvariant centroid for triples of points in mapping class groups MCG(S), which satisfies a certain polynomial growth bound. A consequence (via work of Druţu-Sapir or Chatterji-Ruane) is the Rapid Decay Property for MCG(S).