Asplund Spaces Research Articles

Calmness for L-Subsmooth Multifunctions in Banach Spaces

Using variational analysis techniques, we study subsmooth multifunctions in Banach spaces. In terms of the normal cones and coderivatives, we provide some characterizations for such multifunctions to be calm. Sharper results are obtained for Asplund spaces. We also present some exact formulas of the modulus of the calmness. As applications, we provide some error bound results on nonconvex inequalities, which improve and generalize the existing error bound results.

Properties of lush spaces and applications to Banach spaces with numerical index 1

The concept of lushness was introduced recently as a Banach space property, which ensures that the space has numerical index 1. We prove that for Asplund spaces lushness is actually equivalent to numerical index 1. We prove that every separable Banach space containing an isomorphic copy of c0 can be renormed equivalently to be lush, and thus to have numerical index 1. The rest of the paper is devoted to the study of lushness just as a property of Banach spaces. We prove that lushness is separably determined, is stable under ultraproducts, and we characterize those spaces of the form X = (R, ‖ · ‖) with absolute norm such that X-sum preserves lushness of summands, showing that this property is equivalent to lushness of X.

Boundaries of compact convex sets and fragmentability

If X is a compact convex set in a real locally convex space, B ⊂ X is said to be its boundary if every affine continuous function on X attains its maximum at some point of B. We study relations between fragmentability of B and the whole set X. As a byproduct we obtain a characterization of separable Asplund spaces. We also study the possibility of finding the Haar system in a boundary of a metrizable compact convex set.

IN ASPLUND SPACES, APPROXIMATELY CONVEX FUNCTIONS AND REGULAR FUNCTIONS ARE GENERICALLY DIFFERENTIABLE

We prove that an approximately convex function on an open subset of an Asplund space is generically Fr´echet differentiable, as are genuine convex functions. Thus, we give a positive answer to a question raised by S. Rolewicz. We also prove a more general result of that type for regular functions on an open subset of an Asplund space.

Mazur intersection property for Asplund spaces

The main result of the present note states that it is consistent with the ZFC axioms of set theory (relying on Martin's Maximum MM axiom), that every Asplund space of density character ω 1 has a renorming with the Mazur intersection property. Combined with the previous result of Jiménez and Moreno (based upon the work of Kunen under the continuum hypothesis) we obtain that the MIP renormability of Asplund spaces of density ω 1 is undecidable in ZFC.

Variational analysis of extended generalized equations via coderivative calculus in Asplund spaces

This paper is devoted to the development of variational analysis and generalized differentiation in the framework of Asplund spaces. We mainly concern the study of a special class of set-valued mapping given in the form S ( x ) = { y ∈ Y | 0 ∈ F ( x , y ) + Q ( x , y ) } , x ∈ X , where both F and Q are set-valued mappings between Asplund spaces. Models of this type are associated with solutions maps to the so-called (extended) generalized equations and play a significant role in many aspects of variational analysis and its applications to optimization, stability, control theory, etc. In this paper we conduct a local variational analysis of such extended solution maps S and their remarkable specifications based on dual-space generalized differential constructions of the coderivative type. The major part of our analysis revolves around coderivative calculus largely developed and implemented in this paper and then applied to establishing verifiable conditions for robust Lipschitzian stability of extended generalized equations and related objects.

On generalized convexity in Asplund spaces

In this paper some properties of nonsmooth quasiconvex and pseudoconvex functions in weakly compactly generated Asplund spaces are proved.

Restricted Weak Upper Semi-continuity of Subdifferentials of Convex Functions on Banach Spaces

Let X be a Banach space and f a continuous convex function on X. Suppose that for each x ∈ X and each weak neighborhood V of zero in X* there exists δ > 0 such that $$\partial f(y)\subset\partial f(x)+V\;\;{\rm for\;all}\;y\in X\;{\rm with}\;\|y-x\|<\delta. $$ Then every continuous convex function g with \(g \leqslant f\) on X is generically Frechet differentiable. If, in addition, \(\lim\limits_{\|x\|\rightarrow\infty}f(x)=\infty\), then X is an Asplund space.

The Fermat rule for multifunctions for super efficiency

Using variational analysis, we study the vector optimization problems with objectives being closed multifunctions on Banach spaces or in Asplund spaces. In terms of the coderivatives and normal cones, we present Fermat’s rules as necessary or sufficient conditions for a super efficient solution of the above problems.

IN ASPLUND SPACES, APPROXIMATELY CONVEX FUNCTIONS AND REGULAR FUNCTIONS ARE GENERICALLY DIFFERENTIABLE

We prove that an approximately convex function on an open subset of an Asplund space is generically Fr´echet differentiable, as are genuine convex functions. Thus, we give a positive answer to a question raised by S. Rolewicz. We also prove a more general result of that type for regular functions on an open subset of an Asplund space.

Asplund space: Another criterion

The theorem proved in this paper establishes conditions under which a Banach space X is an Asplund space (i.e., its dual space is a space with the Radon-Nikodym (RN) property). The theorem is formulated in terms of the existence of a supersequentially compact set in (B(X**), ω*), where B(X**) stands for the unit ball of the second dual of X and ω* for the weak topology on the ball. The example presented in the paper shows that one cannot get rid of some restrictive conditions in the theorem in general.

Error bounds for systems of lower semicontinuous functions in Asplund spaces

In this paper, using the Frechet subdifferential, we derive several sufficient conditions ensuring an error bound for inequality systems in Asplund spaces. As an application we obtain in the context of Banach spaces a global error bound for quadratic nonconvex inequalities and we derive necessary optimality conditions for optimization problems.

Normal coderivative for multifunctions and implicit function theorems

In the framework of the theory of normal coderivative for multifunctions, new implicit function theorems are obtained. The main tools of the proofs are the Ekeland variational principle, a nonsmooth version of Fermat's rule, a sum rule, and the differential estimate for marginal functions established by B.S. Mordukhovich and Y. Shao [B.S. Mordukhovich, Y. Shao, Nonsmooth sequential analysis in Asplund spaces, Trans. Amer. Math. Soc. 348 (1996) 1235–1280].

Generalized Euler identity for subdifferentials of homogeneous functions and applications

In this paper, we mainly consider subdifferentials and basic subdifferentials of homogeneous functions defined on real Banach space and Asplund space respectively, and obtain the generalized Euler identity. As applications, we consider constrained optimization problems and several geometric properties of Banach space.

SMOOTH NORMS AND APPROXIMATION IN BANACH SPACES OF THE TYPE 𝒞(K)

We prove two results about smoothness in Banach spaces of the type C(K). Both build upon earlier papers of the first named author [3,4]. We first establish a special case of a conjecture that remains open for general Banach spaces and concerns smooth approximation. We recall that a bump function on a Banach space X is a function β : X → R which is not identically zero, but which vanishes outside some bounded set. The existence of bump function of class C implies that the Banach space X is an Asplund space, which, in the case where X = C(K), is the same as saying that K is scattered. It is a major unsolved problem to determine whether every Asplund space has a C bump function. Another open problem is whether the existence of just one bump function of some class C on a Banach space X implies that all continuous functions on X may be uniformly approximated by functions of class C. It is to this question that we give a positive answer (Theorem 2) in the special case of X = C(K). Our second result represents some mild progress with a conjecture made by the second author in [7]. The analysis in that paper of compact spaces constructed using trees suggested that for a compact space K the existence of an equivalent norm on C(K) which is of class C (except at 0 of course) might imply the existence of such a norm which is of class C∞. Certainly, this is what happens with norms constructed using linear Talagrand operators as in [5,6,7]. The other important (and older) method of obtaining C norms is to construct a norm with locally uniformly rotund dual norm. What we show in Theorem 2 is that, whenever C(K) admits an equivalent norm with LUR dual norm, then there is also an infinitely differentiable equivalent norm on C(K). For background on smoothness and renormings in Banach spaces, including an account of Asplund spaces, we refer the reader to [1]. In particular, an account is given there of the connection between smooth approximability of continuous functions and the existence of smooth partitions of unity. Following what seems to be standard practice in the literature, we have chosen to state the formal version of our first theorem (Theorem 1) in terms of partitions of unity, rather than approximation.

A mean value theorem in Asplund spaces

In this paper a nonsmooth mean value theorem in Asplund spaces, under invexity, is provided.

Paraconvex analysis on -manifolds†

Let be real Banach spaces. Let f(·) be a real-valued locally uniformly approximate convex function defined on an open subset . Let be an open subset. Let σ (·) be a differentiable mapping of ΩX into ΩY such that the differentials of are locally uniformly continuous function of x. Then f(σ (·)) is also a locally uniformly approximate convex function. Therefore the function f(σ (·)) is Fréchet differentiable on a dense G δ-set, provided X is an Asplund space, and Gateaux differentiable on a dense G δ-set, provided X is separable. As a consequence, we obtain that a locally uniformly approximate convex function defined on a -manifold is Fréchet differentiable on a dense G δ-set, provided is an Asplund space, and Gateaux differentiable on a dense G δ-set, provided E is separable. †Dedicated to Professor Diethard Pallaschke on the occasion of his 65th birthday.

Variational Analysis of Evolution Inclusions

The paper is devoted to optimization problems of the Bolza and Mayer types for evolution systems governed by nonconvex Lipschitzian differential inclusions in Banach spaces under endpoint constraints described by finitely many equalities and inequalities with generally nonsmooth functions. We develop a variational analysis of such problems mainly based on their discrete approximations and the usage of advanced tools of generalized differentiation satisfying comprehensive calculus rules in the framework of Asplund (and hence any reflexive Banach) spaces. In this way we establish extended results on stability of discrete approximations (with the strong $W^{1,2}$-convergence of optimal solutions under consistent perturbations of endpoint constraints) and derive necessary optimality conditions for nonconvex discrete-time and continuous-time systems in the refined Euler–Lagrange and Weierstrass–Pontryagin forms accompanied by the appropriate transversality inclusions. In contrast to the case of geometric endpoint constraints in infinite dimensions, the necessary optimality conditions obtained in this paper do not impose any nonempty interiority/finite-codimension/normal compactness assumptions. The approach and results developed in the paper make a bridge between optimal control/dynamic optimization and constrained mathematical programming problems in infinite-dimensional spaces.

The Lagrange Multiplier Rule for Multifunctions in Banach Spaces

We study general constrained multiobjective optimization problems with objectives being closed multifunctions in Banach spaces. In terms of the coderivatives and normal cones, we provide generalized Lagrange multiplier rules as necessary optimality conditions of the above problems. In an Asplund space setting, sharper results are presented.

Linear regularity and ϕ-regularity of nonconvex sets

In this paper, we discuss some sufficient conditions for the linear regularity and bounded linear regularity (and their variations) of finitely many closed (not necessarily convex) sets in a normed vector space. The accompanying necessary conditions are also given in the setting of Asplund spaces.