This paper describes a new method of monotone interpolation and smoothing of multivariate scattered data. It is based on the assumption that the function to be approximated is Lipschitz continuous. The method provides the optimal approximation in the worst case scenario and tight error bounds. Smoothing of noisy data subject to monotonicity constraints is converted into a quadratic programming ...

The main result is that a Banach space X is not super-reflexive if and only if the diamond graphs Dn Lipschitz embed into X with distortions independent of n. One of the consequences of that and previously known results is that dimension reduction a-la Johnson–Lindenstrauss fails in any non super reflexive space with non trivial type. We also introduce the concept of Lipschitz (p, r)-summing ma...

The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points F (S) of a nonexpansive mapping S and the set of solutions ΩA of the variational inequality for a monotone, Lipschitz continuous mapping A. We introduce a hybrid extragradient-like approximation method which is based on the well-known extragradient method and a hybrid (or outer approxi...

We consider a class of Lipschitz vector fields S :Ω →Rn whose values lie in a suitable cone and we show that the trajectories of the system x′ = S(x) admit a parametrization that is invertible and Lipschitz with its inverse. As a consequence, every v in W1,1(Ω) admits a representative that is absolutely continuous on almost every trajectory of x′ = S(x). If S is an arbritrary Lipschitz field th...

We give a new self-contained proof of Bronshtein’s theorem, that any continuous root of a Cn−1,1-family of monic hyperbolic polynomials of degree n is locally Lipschitz, and obtain explicit bounds for the Lipschitz constant of the root in terms of the coefficients. As a by-product we reprove the recent result of Colombini, Orrú, and Pernazza, that a C-curve of hyperbolic polynomials of degree n...

Let Ω be a bounded domain in R, with surface measure dσ and outward unit normal n. Throughout the paper we shall assume that Ω is Lipschitz (i.e., ∂Ω can be locally described by means of graphs of Lipschitz continuous functions in appropriate systems of coordinates) and that ∂Ω is connected. Also, we set Ω+ := Ω and Ω− := R \ Ω̄. The classical Faraday-Amperé-Maxwell equations in Ω+ or Ω− read { ...

Steepest descent drives both theory and practice of nonsmooth optimization. We study slight relaxations of two influential notions of steepest descent curves — curves of maximal slope and solutions to evolution equations. In particular, we provide a simple proof showing that lower-semicontinuous functions that are locally Lipschitz continuous on their domains — functions playing a central role ...

In this paper, we consider an energy of the type Dirichlet energy plus potential term, for a map with values into a metric cone. We investigate the regularity of minimizers of such functionals, and prove that they are always locally Hölder continuous. We establish that Lipschitz continuity is achieved in some cases where the target space has non-positive curvature, and show examples for which t...

In this paper we present an observer for controlled nonlinear systems that are locally Lipschitz continuous in both the state and control variables. This observer is based on a recently introduced model of observer for autonomous Lypschitz continuous systems, and can be designed to realize an arbitrary, finite accuracy when both the state space and control variables evolve in bounded regions. K...

This paper deals with the regularity of solutions of the Hamilton-Jacobi Inequality which arises in H∞ control. It shows by explicit counterexamples that there are gaps between existence of continuous and locally Lipschitz (positive definite and proper) solutions, and between Lipschitz and continuously differentiable ones. On the other hand, it is shown that it is always possible to smooth-out ...