we characterize compact composition operators on real banach spaces of complex-valued bounded lipschitz functions on metric spaces, not necessarily compact, with lipschitz involutions and determine their spectra.

This paper is devoted to the autonomous Lagrange problem of the calculus of variations with a discontinuous Lagrangian. We prove that every minimizer is Lipschitz continuous if the Lagrangian is coercive and locally bounded. The main difference with respect to the previous works in the literature is that we do not assume that the Lagrangian is convex in the velocity. We also show that, under so...

Given any positive integers m and d, we say the a sequence of points (xi)i∈I in Rm is Lipschitz-d-controlling if one can select suitable values yi (i ∈ I) such that for every Lipschitz function f : Rm → Rd there exists i with |f(xi)−yi| < 1. We conjecture that for every m ≤ d, a sequence (xi)i∈I ⊂ Rm is d-controlling if and only if sup n∈N |{i ∈ I : |xi| ≤ n}| nd =∞. We prove that this conditio...

where g satis…es a Lipschitz condition with the constant K: In (1.1) and (1.2) the constant 14 is best possible in the sense that it cannot be replaced by a smaller quantity. The same applies for the constant 1 12 in (1.3). The interested reader may …nd many other inequalities involving Lipschitzian functions in the papers [1]-[11], [13]-[20] and in the references therein. In this paper we esta...

The implicit function theorem asserts that there exists a ball of nonzero radius within which one can express a certain subset of variables, in a system of equations, as functions of the remaining variables. We derive a lower bound for the radius of this ball in the case of Lipschitz maps. Under a sign-preserving condition, we prove that an implicit function exists in the case of a set of inequ...

This work addresses the design of static output feedback control of discrete-time nonlinear systems satisfying a local Lipschitz continuity condition with time-varying uncertainties. The controller has also a guaranteed disturbance attenuation level (H∞ performance). Thanks to the linearity of the proposed LMIs in both the admissible Lipschitz constant of the system and the disturbance attenuat...

We consider the nonlinear delay differential evolution equation { u′(t) ∈ Au(t)+ f (t, ut ), t ∈ R+, u(t) = g(u)(t), t ∈ [−τ, 0], where τ ≥ 0, X is a real Banach space, A is the infinitesimal generator of a nonlinear semigroup of contractions whose Lipschitz seminorm decays exponentially as t → e−ωt when t → +∞ and f : R+×C([−τ, 0]; D(A)) → X is jointly continuous. We prove that if f Lipschitz ...

Let fX1; : : : ; Xng be a collection of binary valued random variables and let f : f0; 1g n ! R be a Lipschitz function. Under a negative dependence hypothesis known as the strong Rayleigh condition, we show that f Ef satis es a concentration inequality. The class of strong Rayleigh measures includes determinantal measures, weighted uniform matroids and exclusion measures; some familiar example...

In this paper, we consider a class of nonsmooth, nonconvex constrained optimization problems where the objective function may be not Lipschitz continuous and the feasible set is a general closed convex set. Using the theory of the generalized directional derivative and the Clarke tangent cone, we derive a first order necessary optimality condition for local minimizers of the problem, and define...

We consider a finite string where, at both end points, a homogeneous Dirichlet boundary condition holds. One boundary point is fixed, and the other is moving; hence the length of the string is changing in time. The string is controlled through the movement of this boundary point. We consider movements of the boundary that are Lipschitz continuous. Only movements for which at the given finite te...