In the first part of this paper, we prove that in a sense the class of bi-Lipschitz δ-convex mappings, whose inverses are locally δ-convex, is stable under finite-dimensional δ-convex perturbations. In the second part, we construct two δ-convex mappings from l1 onto l1, which are both bi-Lipschitz and their inverses are nowhere locally δ-convex. The second mapping, whose construction is more co...

It is shown here that if (Y, ‖ · ‖Y ) is a Banach space in which martingale differences are unconditional (a UMD Banach space) then there exists c = c(Y ) ∈ (0,∞) with the following property. For every n ∈ N and ε ∈ (0, 1/2], if (X, ‖·‖X) is an n-dimensional normed space with unit ball BX and f : BX → Y is a 1-Lipschitz function then there exists an affine mapping Λ : X → Y and a sub-ball B∗ = ...

This paper deals with precise mapping properties of the Bergman and Szegö projections of pseudo-convex domains of finite type in C whose Levi form are locally diagonalizable at every point of the boundary (see Section 2 for a precise definition). We obtain sharp estimates for these operators for usual Lpk Sobolev spaces, classical Lipschitz spaces Λα and nonisotropic Lipschitz spaces Γα related...

In this paper we introduce a hybrid relaxed-extragradient method for finding a common element of the set of common fixed points of N nonexpansive mappings and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. The hybrid relaxed-extragradient method is based on two well-known methods: hybrid and extragradient. We derive a strong convergence ...

The Skorokhod oblique reflection problem is studied in the case of n-dimensional convex polyhedral domains. The natural sufficient condition on the reflection directions is found, which together with the Lipschitz condition on the coefficients gives the existence and uniqueness of the solution. The continuity of the corresponding solution mapping is established. This property enables one to con...

In this note we prove that any W 1,2 mapping u in the plane that minimizes an appropriate quasiconvex energy functional subject to the Jacobian constraint det∇u = 1 a.e., are necessarily Lipschitz. Furthermore we show that the minimizers corresponding to uniformly convex energy are affine and give an example of non-affine minimizers subject to affine boundary data corresponding to a convex ener...

A.2. Contraction lemma The following is a version of Talagrand’s contraction lemma (Ledoux & Talagrand, 2011). Since our definition of Rademacher complexity does not use absolute values, we give an explicit proof below. Lemma 8. Let H be a hypothesis set of functions mapping X to R and Ψ1, . . . ,Ψm, μ-Lipschitz functions for some μ > 0. Then, for any sample S of m points x1, . . . , xm ∈ X , t...

We propose a strongly convergent algorithm for finding a common point in the solution set of a class of pseudomonotone equilibrium problems and the set of fixed points of nonexpansive mappings in a real Hilbert space. The proposed algorithm uses only one projection and does not require any Lipschitz condition for the bifunctions. AMS 2010 Mathematics subject classification: 65 K10, 65 K15, 90 C...

This contribution deals with numerical solution of shape optimization problems in frictional contact mechanics. The state problem in our case is given by 2D static Signorini problems with Tresca friction and a solution-dependent coefficient of friction. A suitable Lipschitz continuity assumption on the coefficient of friction is made, ensuring unique solvability of the discretized state problem...

We consider optimal estimation problems characterized by a state vector with i) dynamics described via a differential equation with Lipschitz nonlinearities, ii) partial information provided via a Lipschitz nonlinear mapping, and iii) an Lp norm measure of the estimation error to be minimized. An approximate solution of such optimal estimation problem is searched for by restricting the optimiza...