Optimality conditions in nondifferentiable programming and their applications to best approximations

This paper studies constrained optimization problems in Banach spaces without usual differentiability and convexity assumptions on the functionals involved in the data. The aim is to give optimality conditions for the problems from which one can derive the characterization of best approximations. The objective and the inequality constraint functionals are assumed to have one-sided directional derivatives. First-order necessary conditions are given in terms of subdifferentials of the directional derivatives. The notion of max-pseudoconvexity weaker than pseudoconvexity is introduced for sufficiency. The optimality conditions are applied to linear and nonlinear Tchebycheff approximation problems to derive the characterization of best approximations.     

Differentiability of the mappings of Carnot-Carathéodory spaces in the Sobolev and <Emphasis Type="Italic">BV</Emphasis>-topologies

We prove differentiability of the mappings of the Sobolev classes and BV-mappings of Carnot-Carathéodory spaces in the topology of these classes. We infer from these results a generalization of the Calderón-Zygmund theorems for mappings of the Carnot-Carathéodory spaces and other facts.     

Paracompact locally Lindelöf spaces and <Emphasis Type=”Italic”>L</Emphasis>-systems

Summary We establish relationships between paracompact locally Lindel&ouml;f spaces and all kinds of spaces with L-systems by means of closed mappings, weakly perfect mappings and quotient mappings. In particular, we obtain some characterizations of paracompact locally Lindel&ouml;f spaces. Interestingly, they are also characterized as weakly perfect inverse images of locally separable metric spaces.     

Geometrische Bedingungen für die Integrabilität von Vektorfeldern auf Teilmengen des ℝn

We give a geometric characterisation for those vectorfields on a subset X ⁿ, wich are locally integrable, that is, which locally have sufficiently many integral curves on X. From this we deduce, that integrable spaces X (where each field of a fixed class of differentiability is locally integrable) are rigid under differentiable deformations in the sense of Kodaira-Kuranishi. We give a general construction for integrable spaces and obtain, that analytic varieties induce integrable spaces for each class of differentiability. Compact analytic varieties are therefore C-rigid, which extends [4], 3,1.     

Flache Konvexität und Schwache Differenzierbarkeit

For a convex continuous function f:XR, the smoothness of the set {xX|f(x)r} is characterized by a differentiability property of f. Questions of this kind arise from optimization problems and gauge functionals, e.g. in the theory of Orlicz spaces.     

On the relationship between differentiability and absolute continuity of measures on ℝn

Summary We give an elementary proof of the fact that a finite Borel measure on ⁿ is absolutely continuous with a C ¹ density if and only if it has directional derivatives which are continuous almost everywhere. The Radon-Nikodym derivative of a differentiable measure is given in terms of the directional derivatives.     

Higher derivatives of mappings of locally convex spaces

We establish sufficient conditions for n-fold bounded differentiability (b-differentiability) of mappings of locally convex spaces and sufficient conditions for n-fold Hyers-Lang differentiability (HL-differentiability) of mappings of pseudotopological linear spaces. We describe a class of locally convex spaces on which there exist everywhere infinitely b-differentiable real functions which are not everywhere continuous (and so are not everywhere HL-differentiable). Our results show, in particular, that for a wide class of locally convex spaces a significant number of the known definitions of C-mappings fall into two classes of equivalent definitions.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 729–744 November, 1977.     

On the near differentiability property of Banach spaces

Let μ be a scalar measure of bounded variation on a compact metrizable abelian group G. Suppose that μ has the property that for any measure σ whose Fourier-Stieltjes transform vanishes at ∞, the measure μ*σ has Radon-Nikodým derivative with respect to λ, the Haar measure on G. Then L. Pigno and S. Saeki showed that μ itself has Radon-Nikodým derivative. Such property is not shared by vector measures in general. We say that a Banach space X has the near differentiability property if every X-valued measure of bounded variation shares the above property. We prove that Banach spaces with the Radon-Nikodým property have the near differentiability property, while Banach spaces with the near differentiability property enjoy the near Radon-Nikodým property. We also show that the Banach spaces L¹[0,1] and have the near differentiability property. Lastly, we show that Banach spaces with the near differentiability property have type II-Λ-Radon-Nikodým property, whenever Λ is a Riesz subset of type 0 of .     

Porosity,Differentiability and Pansu’s Theorem

We use porosity to study differentiability of Lipschitz maps on Carnot groups. Our first result states that directional derivatives of a Lipschitz function act linearly outside a \(\sigma \)-porous set. The second result states that irregular points of a Lipschitz function form a \(\sigma \)-porous set. We use these observations to give a new proof of Pansu’s theorem for Lipschitz maps from a general Carnot group to a Euclidean space.     

Implicit function theorems for generalized equations

We show that Lipschitz and differentiability properties of a solution to a parameterized generalized equation 0 f(x, y) + F(x), wheref is a function andF is a set-valued map acting in Banach spaces, are determined by the corresponding Lipschitz and differentiability properties of a solution toz g(x) + F(x), whereg strongly approximatesf in the sense of Robinson. In particular, the inverse map (f + F)^–1 has a local selection which is Lipschitz continuous nearx ₀ and Fréchet (Gateaux, Bouligand, directionally) differentiable atx ₀ if and only if the linearization inverse (f (x ₀) + f (x₀) (× – x₀) + F(×))^–1 has the same properties. As an application, we study directional differentiability of a solution to a variational inequality.This work was supported by National Science Foundation Grant Number DMS 9404431.     

Almost Frechet Differentiability of Lipschitz Mappings Between Infinite-Dimensional Banach Spaces

We give several sufficient conditions on a pair of Banach spacesX and Y under which each Lipschitz mapping from a domain inX to Y has, for every > 0, a point of -Fréchet differentiability.Most of these conditions are stated in terms of the moduli ofasymptotic smoothness and convexity, notions which have appearedin the literature under a variety of names. We prove, for example,that for > r > p 1, every Lipschitz mapping from a domainin an l_r-sum of finite-dimensional spaces into an l_p-sum offinite-dimensional spaces has, for every > 0, a point of-Fréchet differentiability, and that every Lipschitzmapping from an asymptotically uniformly smooth space to a finite-dimensionalspace has such points. The latter result improves, with a simplerproof, an earlier result of the second and third authors. Wealso survey some of the known results on the notions of asymptoticsmoothness and convexity, prove some new properties, and presentsome new proofs of existing results. 2000 Mathematical Subject Classification: 46G05, 46T20.     

On smoothness conditions and convergence rates in the CLT in Banach spaces

Summary In Banach spaces the rate of convergence in the Central Limit Theorem is of orderO(n^–1/2) for sets which have regular boundaries with respect to the given covariance structure and which are three times differentiable. We show that in infinite dimensional spaces it is impossible to weaken this differentiability condition in general, whereas in finite dimensional spaces the assumption of convexity suffices. Similar results hold for the expectation of smooth functionals.Research supported by SFB 343 at Bielefeld and by the Alexander von Humboldt Foundation and completed at the University of Bielefeld, FRGResearch supported by the SFB 343 at Bielefeld     

A class of 4-pseudomanifolds similar to the lens spaces

A family of 4-dimensional pseudomanifolds is introduced using a standard graph-theoretical representation of lens spaces Some homeomorphisms between these lens-like spaces are established, the computation of their fundamental groups and of bounds for their genera are carried out     

On Differentiability of Metric Projections onto Moving Convex Sets

We consider properties of the metric projections onto moving convex sets in normed linear spaces. Under certain conditions about the norm, directional differentiability of first and higher order of the metric projections at boundary points is characterized. The conditions are formulated in terms of differentiability of multifunctions and properties of the set-derivatives are shown.     

Metric Density and Quasimobius Mappings

We study the notion of -density of metric spaces which was introduced by V. Aseev and D. Trotsenko. Interrelation between -density and homogeneous density is established. We also characterize -dense spaces as arcwise connected metric spaces in which arcs are the quasimobius images of the middle-third Cantor set. Finally, we characterize quasiconformal self-mappings of ⁿ in terms of -density.     

On connections between approximate second-order directional derivative and second-order Dini derivative for convex functions

For a real-valued convex functionf, the existence of the second-order Dini derivative assures that of the limit of the approximate second-order directional derivativef (x ₀;d, d) when 0⁺ and both values are the same. The aim of the present work is to show the converse of this result. It will be shown that upper and lower limits of the approximate second-order directional derivative are equal to the second-order upper and lower Dini derivatives, respectively. Consequently the existence of the limit of the approximate second-order directional derivative and that of second-order Dini derivative are equivalent.Dedicated to Professor N. Furukawa of Kyushu University for his 60th birthday.     

The missing Wendland functions

The Wendland radial basis functions (Wendland, Adv Comput Math 4:389–396, 1995) are piecewise polynomial compactly supported reproducing kernels in Hilbert spaces which are norm–equivalent to Sobolev spaces. But they only cover the Sobolev spaces

$\label{eqstartrep} H^{d/2+k+1/2}({\mathbf{R}}^d),\;k\in {\mathbf{N}} $

(1)

and leave out the integer order spaces in even dimensions. We derive the missing Wendland functions working for half-integer k and even dimensions, reproducing integer-order Sobolev spaces in even dimensions, but they turn out to have two additional non-polynomial terms: a logarithm and a square root. To give these functions a solid mathematical foundation, a generalized version of the “dimension walk” is applied. While the classical dimension walk proceeds in steps of two space dimensions taking single derivatives, the new one proceeds in steps of single dimensions and uses “halved” derivatives of fractional calculus.

     

Stability problem of Hyers-Ulam-Rassias for generalized forms of cubic functional equation

Let n≥2 be an integer number. In this paper, we investigate the generalized Hyers Ulam- Rassias stability in Banach spaces and also Banach modules over a Banach algebra and a C＊-algebra and the stability using the alternative fixed point of an n-dimensional cubic functional equation in Banach spaces：f（2∑j=1^n-1 xj＋xn）＋f（2∑j=1^n-1 xj-xn）＋4∑j=1^n-1f（xj）=16f（∑j=1^n-1 xj）＋2∑j=1^n-1（f（xj＋xn）＋f（xj-xn）