The Whitney extension problem and Lipschitz selections of set-valued mappings in jet-spaces

We study a variant of the Whitney extension problem (1934) for the space . We identify with a space of Lipschitz mappings from into the space of polynomial fields on equipped with a certain metric. This identification allows us to reformulate the Whitney problem for as a Lipschitz selection problem for set-valued mappings into a certain family of subsets of . We prove a Helly-type criterion for the existence of Lipschitz selections for such set-valued mappings defined on finite sets. With the help of this criterion, we improve estimates for finiteness numbers in finiteness theorems for due to C. Fefferman.

     

On directional derivatives of selections of set-valued mappings

The paper is devoted to the question of existence of directional derivatives of selections of some set-valued mappings with starlike graphs.     

Generating continuous mappings with Lipschitz mappings

If is a metric space, then and denote the semigroups of continuous and Lipschitz mappings, respectively, from to itself. The relative rank of modulo is the least cardinality of any set where generates . For a large class of separable metric spaces we prove that the relative rank of modulo is uncountable. When is the Baire space , this rank is . A large part of the paper emerged from discussions about the necessity of the assumptions imposed on the class of spaces from the aforementioned results.

     

On lengths,areas and Lipschitz continuity of polyharmonic mappings

In this paper, we continue our investigation of polyharmonic mappings in the complex plane. First, we establish two Landau type theorems. We also show a three circles type theorem and an area version of the Schwarz lemma. Finally, we study Lipschitz continuity of polyharmonic mappings with respect to the distance ratio metric.     

Lipschitz selections and stability for quasiconvex programs

Stability of ε-optimal solutions for quasiconvex programs is studied.     

Multivalued almost-periodic mappings and selections of them

Izhevsk and Voronezh. Translated fromSibirskii Matematicheskii Zhurnal, Vol. 32, No. 2, pp. 172–175, March–April, 1991.     

Diagonals of separately pointwise Lipschitz mappings

It is proved that, for a metric space X and a normed space Z, the diagonals of pointwise Lipschitz mappings f : X ²? →?Z are exactly stable pointwise limits of pointwise Lipschitz mappings. The joint Lipschitz property of separately pointwise Lipschitz mappings f : X?×?Y?→?Z, where X, Y, and Z are metric spaces, is investigated.     

Vector‐valued optimal Lipschitz extensions

Consider a bounded open set and a Lipschitz function . Does this function always have a canonical optimal Lipschitz extension to all of U? We propose a notion of optimal Lipschitz extension and address existence and uniqueness in some special cases. In the case n = m = 2, we show that smooth solutions have two phases: in one they are conformal and in the other they are variants of infinity harmonic functions called infinity harmonic fans. We also prove existence and uniqueness for the extension problem on finite graphs. © 2011 Wiley Periodicals, Inc.     

Carathéodory selections for multivalued mappings

We prove the existence of a Carathéodory selection for a set-valued mapping satisfying standard conditions. Also, an application to random fixed point for random operators is established.     

Implicit-function subdifferentiation formulas for Lipschitz mappings

Translated from Matematicheskie Zametki, Vol. 50, No. 2, pp. 152–153, August, 1991.     

On Picard iterations for strongly accretive and strongly pseudo-contractive Lipschitz mappings

In this note, we speed up the convergence of the Picard sequence of iterations for strongly accretive and strongly pseudo-contractive mappings. Our results improve the results of Chidume [C.E. Chidume, Picard iteration for strongly accretive and strongly pseudo-contractive Lipschitz maps, ICTP Preprint no. IC2000098; C.E. Chidume, Iterative Algorithms for Non-expansive Mappings and Some of Their Generalizations, in: Nonlinear Analysis and Applications: To V. Lakshmikantham on his 80th Birthday, vol. 1, 2, Kluwer Acad. Publ, Dordrecht, 2003, pp. 383–429], and Liu [L. Liu, Approximation of fixed points of a strictly pseudo-contractive mapping, Proc. Amer. Math. Soc. 125 (2) (1997) 1363–1366], and some other known results. The technique of the proof, presented in this paper, is different from the technique used by Chidume.     

Isometries on spaces of vector valued Lipschitz functions

This paper gives a characterization of a class of surjective isometries on spaces of Lipschitz functions with values in a finite dimensional complex Hilbert space.     

Semigroups of mappings with rigid Lipschitz constant

It is shown that two well-known uniformly fixed point free lipschitzian semigroups of mappings have minimal Lipschitz constant on the positive part of the unit ball of . This implies that a question raised by T. Kuczumow has a negative answer.

     

Analysis and optimization of Lipschitz continuous mappings

In this paper, four fundamental theorems for continuously differentiable mappings (the multiplier rule for equality constraints of Carathéodory, the inverse mapping theorem, the implicit mapping theorem, and the general multiplier rule for inequality and equality constraints of Mangasarian and Fromovitz) are shown to have natural extensions valid when the mappings are only Lipschitz continuous. Involved in these extensions is a compact, convex set of linear mappings called the generalized derivative, which can be assigned to any Lipschitz continuous mapping and point of its (open) domain and which reduces to the usual derivative whenever the mapping is continuously differentiable. After a brief calculus for this generalized derivative is presented in Part I, the connection between the ranks of the linear mappings in the generalized derivative and theinteriority of the given mapping is explored in Parts II and IV; this relationship is used in Parts III and IV to prove the extensions of the theorems mentioned above.This paper is lovingly dedicated to the author's wife, Nancy Arneson Pourciau.Each section of this paper has benefited from the consistently accurate advice and unflagging enthusiasm of the author's teacher, Professor Hubert Halkin.     

On the Lipschitz continuity of certain quasiregular mappings between smooth Jordan domains

We first investigate the Lipschitz continuity of (K,K’)-quasiregular C ² mappings between two Jordan domains with smooth boundaries, satisfying certain partial differential inequalities concerning Laplacian. Then two applications of the obtained result are given: As a direct consequence, we get the Lipschitz continuity of ρ-harmonic (K,K’)-quasiregular mappings, and as the other application, we study the Lipschitz continuity of (K,K’)- quasiconformal self-mappings of the unit disk, which are the solutions of the Poisson equation Δw = g. These results generalize and extend several recently obtained results by Kalaj, Mateljevi? and Pavlovi?.     

Absolutely minimal Lipschitz extension of tree-valued mappings

We prove that every Lipschitz function from a subset of a locally compact length space to a metric tree has a unique absolutely minimal Lipschitz extension (AMLE). We relate these extensions to a stochastic game called Politics??a generalization of a game called Tug of War that has been used in Peres et?al. (J Am Math Soc 22(1):167?C210, 2009) to study real-valued AMLEs.     

Absolute continuity for Banach space valued mappings

We consider the notion ofp, λ, δ-absolute continuity for Banach space valued mappings introduced in [2] for real valued functions and for λ=1. We investigate the validity of some basic properties that are shared byn, λ-absolutely continuous functions in the sense of Maly and hencl. We introduce the class δ-BV _λ,loc^p and we give a characterization of the functions belonging to this class.     

Absolute continuity for Banach space valued mappings

We consider the notion ofp, λ, δ-absolute continuity for Banach space valued mappings introduced in [2] for real valued functions and for λ=1. We investigate the validity of some basic properties that are shared byn, λ-absolutely continuous functions in the sense of Maly and hencl. We introduce the class δ-BV _λ,loc ^p and we give a characterization of the functions belonging to this class.