scholarly journals The implications for differentiability of a weak index of non-compactness

1993 ◽  
Vol 48 (1) ◽  
pp. 75-91 ◽  
Author(s):  
John R. Giles ◽  
Warren B. Moors
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Functions ◽  
Baire Space ◽  
Continuity Property ◽  
Set Valued Mappings

In a recent paper the authors showed that certain set-valued mappings from a Baire space into subsets of a Banach space which have a continuity property defined in terms of Kuratowski's index of non-compactness have inherent single-valued properties. Here we generalise the continuity property to one defined in terms of a weak index of non-compactness and we show that this wider class of set-valued mappings also has significant implications for the differentiability of convex functions on Banach spaces.

scholarly journals A selection theorem for weak upper semi-continuous set-valued mappings

1996 ◽  
Vol 53 (2) ◽  
pp. 213-227 ◽  
Author(s):  
Warren B. Moors
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Functions ◽  
Weak Topology ◽  
Weak Compactness ◽  
Baire Space ◽  
Selection Theorem ◽  
Set Valued Mapping ◽  
Set Valued Mappings

Let Φ be a set-valued mapping from a Baire space T into non-empty closed subsets of a Banach space X, which is upper semi-continuous with respect to the weak topology on X. In this paper, we give a condition on T which is sufficient to ensure that Φ admits a selection which is norm continuous at each point of a dense and Gδ subset of T. We also derive a variation of James' characterisation of weak compactness, which we use in conjunction with our selection theorem, to deduce some differentiability results for continuous convex functions defined on dual Banach spaces.

scholarly journals On weak Hadamard differentiability of convex functions on Banach spaces

1996 ◽  
Vol 54 (1) ◽  
pp. 155-166 ◽  
Author(s):  
J.R. Giles ◽  
Scott Sciffer
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Functions ◽  
Directional Differentiability ◽  
Hadamard Differentiability ◽  
Hadamard Directional Differentiability

We study two variants of weak Hadamard differentiability of continuous convex functions on a Banach space, uniform weak Hadamard differentiability and weak Hadamard directional differentiability, and determine their special properties on Banach spaces which do not contain a subspace topologically isomorphic to l1.

scholarly journals NEAREST POINTS AND DELTA CONVEX FUNCTIONS IN BANACH SPACES

2015 ◽  
Vol 93 (2) ◽  
pp. 283-294
Author(s):  
JONATHAN M. BORWEIN ◽  
OHAD GILADI
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Functions ◽  
Closed Set ◽  
Nearest Point ◽  
Set Of Points ◽  
Nearest Points

Given a closed set$C$in a Banach space$(X,\Vert \cdot \Vert )$, a point$x\in X$is said to have a nearest point in$C$if there exists$z\in C$such that$d_{C}(x)=\Vert x-z\Vert$, where$d_{C}$is the distance of$x$from$C$. We survey the problem of studying the size of the set of points in$X$which have nearest points in$C$. We then turn to the topic of delta convex functions and indicate how it is related to finding nearest points.

Smoothness, Convexity, Porosity, and Separable Determination

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Functions ◽  
Equivalent Norm ◽  
Fréchet Differentiability ◽  
Smooth Norm ◽  
Frechet Differentiability ◽  
Porous Sets ◽  
Set Of Points

This chapter shows how spaces with separable dual admit a Fréchet smooth norm. It first considers a criterion of the differentiability of continuous convex functions on Banach spaces before discussing Fréchet smooth and nonsmooth renormings and Fréchet differentiability of convex functions. It then describes the connection between porous sets and Fréchet differentiability, along with the set of points of Fréchet differentiability of maps between Banach spaces. It also examines the concept of separable determination, the relevance of the σ‎-porous sets for differentiability and proves the existence of a Fréchet smooth equivalent norm on a Banach space with separable dual. The chapter concludes by explaining how one can show that many differentiability type results hold in nonseparable spaces provided they hold in separable ones.

scholarly journals Variational-Like Inequalities for Weakly Relaxed η-α Pseudomonotone Set-Valued Mappings in Banach Space

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Syed Shakaib Irfan ◽  
Mohammad Firdosh Khan
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Existence Of Solutions ◽  
Reflexive Banach Spaces ◽  
Set Valued Mappings

We introduce and study variational-like inequalities for generalized pseudomonotone set-valued mappings in Banach spaces. By using KKM technique, we obtain the existence of solutions for variational-like inequalities for generalized pseudomonotone set-valued mappings in reflexive Banach spaces. The results presented in this paper are generalizations and improvements of the several well-known results in the literature.

scholarly journals Generic continuity of minimal set-valued mappings

1997 ◽  
Vol 63 (2) ◽  
pp. 238-262 ◽  
Author(s):  
W. B. Moors ◽  
J. R. Giles
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Metric Space ◽  
Complete Metric Space ◽  
Minimal Set ◽  
Set Valued Mapping ◽  
Set Valued Mappings ◽  
Permanence Properties ◽  
Generic Continuity ◽  
Sufficiency Conditions

AbstractWe study classes of Banach spaces where every set-valued mapping from a complete metric space into subsets of the Banach space which satisfies certain minimal properties, is single-valued and norm upper semi-continuous at the points of a dense Gδ subset of its domain. Characterisations of these classes are developed and permanence properties are established. Sufficiency conditions for membership of these classes are defined in terms of fragmentability and σ-fragmentability of the weak topology. A characterisation of non membership is used to show that l∞ (N) is not a member of our classe of generic continuity spaces.

scholarly journals D-gap Functions for a Class of Equilibrium Problems in Banach Spaces

2003 ◽  
Vol 3 (2) ◽  
pp. 274-286 ◽  
Author(s):  
I. V. Konnov ◽  
O. V. Pinyagina
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Stationary Point ◽  
Convex Functions ◽  
General Class ◽  
Real Banach Space ◽  
Equilibrium Problems ◽  
Gap Functions ◽  
Constrained Equilibrium ◽  
Type Algorithm

AbstractWe consider rather a general class of equilibrium problems in a real Banach space, which involve nonsmooth convex functions. We apply the D-gap function approach to these problems and show that, under certain additional assumptions, they can be converted into a problem of finding a stationary point of a differentiable function. Based on this property, we suggest a descent type algorithm to find a solution to the initial problem. An example of applications to nonlinearly constrained equilibrium problems is also given.

scholarly journals On smoothness of the Banach space embedding

1975 ◽  
Vol 13 (1) ◽  
pp. 69-74 ◽  
Author(s):  
J.R. Giles
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Continuity Property ◽  
Reflexive Banach Spaces ◽  
Natural Embedding ◽  
Smooth Embedding

For a Banach space X, smoothness at a point of the natural embedding ◯ in X**, is characterised by a continuity property of the support mapping from X into X*. It then becomes clear that there are many non-reflexive Banach spaces with smooth embedding, a matter of interest raised by Ivan Singer [Bull. Austral. Math. Soc. 12 (1975), 407–416].

scholarly journals Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 150
Author(s):  
Andriy Zagorodnyuk ◽  
Anna Hihliuk
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Entire Functions ◽  
Dimensional Algebra ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Linear Subspaces ◽  
Infinite Dimensional Banach Space

In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained.

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