Let $(X,d)$ be an infinite compact metric space, let $(B,parallel . parallel)$ be a unital Banach space, and take $alpha in (0,1).$ In this work, at first we define the big and little $alpha$-Lipschitz vector-valued (B-valued) operator algebras, and consider the little $alpha$-lipschitz $B$-valued operator algebra, $lip_{alpha}(X,B)$. Then we characterize its second dual space.

The aim of this paper is to obtain new coincidence and common fixed point theorems by using Lipschitz type conditions of hybrid maps (not necessarily continuous) on a metric space. As applications, we demonstrate the existence of common fixed points from the set of best approximations. Our work sets analogues, unifies and improves various known results existing in the literature. 2000 Mathemati...

We prove that a compact metric space (or more generally an analytic subset of a complete separable metric space) of Hausdorff dimension bigger than k can be always mapped onto a k-dimensional cube by a Lipschitz map. We also show that this does not hold for arbitrary separable metric spaces. As an application we essentially answer a question of Urbański by showing that the transfinite Hausdorff...

A well-known open question is whether every countable collection of Lipschitz functions on a Banach space X with separable dual has a common point of Fréchet differentiability. We show that the answer is positive for some infinite-dimensional X. Previously, even for collections consisting of two functions this has been known for finite-dimensional X only (although for one function the answer is...

We construct noninvertible maps on every compact surface and study their chaotic properties from both the measure theoretic and topological points of view. We use some topological techniques employed by others for diffeomorphisms and extend to the noninvertible case.

This note is devoted to the differentiability properties of H-Lipschitz maps defined on abstract Wiener spaces and with values in metric spaces, so we start by recalling some basic definitions related to the Wiener space structure. Let (E, ‖ · ‖) be a separable Banach space endowed with a Gaussian measure γ. Recall that a Gaussian measure γ on E equipped with its Borel σ−algebra B is a probabil...

A rigid map u : Ω ⊂ R → R is a Lipschitz-continuous map with the property that at every x ∈ Ω where u is differentiable then its gradient Du(x) is an orthogonal m × n matrix. If Ω is convex, then u is globally a short map, in the sense that |u(x) − u(y)| ≤ |x − y| for every x, y ∈ Ω; while locally, around any point of continuity of the gradient, u is an isometry. Our motivation to introduce Lip...

Let E be a q-uniformly smooth Banach space possessing a weakly sequentially continuous duality map (e.g., `p, 1 < p < ∞). Let T be a Lipschitzian pseudocontractive selfmapping of a nonempty closed convex and bounded subset K of E and let ω ∈ K be arbitrary. Then the iteration sequence {zn} defined by z0 ∈ K, zn+1 = (1 − μn+1)ω + μn+1yn; yn = (1 − αn)zn + αnTzn, converges strongly to a fixed poi...

It is well-known that on quasi-pseudometric space $(X,q)$, every $q^s$-Cauchy sequence left (or right) $K$-Cauchy but the converse does not hold in general. In this article, we study a class of maps preserve (right) sequences call sequentially-regular maps. Moreover, characterize totally bounded sets terms and right uniformly locally semi-Lipschitz

First we prove a Littlewood-Paley diagonalization result for bi-Lipschitz perturbations of the identity map on the real line. This result entails a number of corollaries for the Hilbert transform along lines and monomial curves in the plane. Second, we prove a square function bound for a single scale directional operator. As a corollary we give a new proof of part of a theorem of Katz on direct...