scholarly journals Generic differentiability of order-bounded convex oparators

1986 ◽  
Vol 28 (1) ◽  
pp. 22-29 ◽  
Author(s):  
Jonathan M. Borwein
Keyword(s):  
Sufficient Conditions ◽  
Optimization Theory ◽  
Asplund Space ◽  
Range Space ◽  
Weakly Compact ◽  
Generic Differentiability ◽  
Fréchet Differentiable ◽  
Order Intervals ◽  
Bounded Convex ◽  
Convex Operators

We give sufficient conditions for order-bounded convex operators to be generically differentiable (Gâteaux or Fréchet). When the range space is a countably order-complete Banach lattice, these conditions are also necessary. In particular, every order-bounded convex operator from an Asplund space into such a lattice is generically Fréchet differentiable, if and only if the lattice has weakly-compact order intervals, if and only if the lattice has strongly-exposed order intervals. Applications are given which indicate how such results relate to optimization theory.

On Super Weakly Compact Convex Sets and Representation of the Dual of the Normed Semigroup They Generate

2013 ◽  
Vol 56 (2) ◽  
pp. 272-282 ◽  
Author(s):  
Lixin Cheng ◽  
Zhenghua Luo ◽  
Yu Zhou
Keyword(s):  
Banach Space ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Lower Semicontinuous ◽  
Weakly Compact ◽  
Bounded Set ◽  
Fréchet Differentiable ◽  
Bounded Convex

AbstractIn this note, we first give a characterization of super weakly compact convex sets of a Banach space X: a closed bounded convex set K ⊂ X is super weakly compact if and only if there exists a w* lower semicontinuous seminorm p with p ≥ σK ≌ supxєK 〈.,x〉 such that p2 is uniformly Fréchet differentiable on each bounded set of X*. Then we present a representation theoremfor the dual of the semigroup swcc(X) consisting of all the nonempty super weakly compact convex sets of the space X.

Measure generation by Euler functionals

1995 ◽  
Vol 27 (03) ◽  
pp. 606-626
Author(s):  
R. V. Ambartzumian
Keyword(s):  
Sufficient Conditions ◽  
Additive Functional ◽  
Signed Measure ◽  
Convex Polygons ◽  
Bounded Convex

Guided by analogy with Euler's spherical excess formula, we define a finite-additive functional on bounded convex polygons in ℝ2(the Euler functional). Under certain smoothness assumptions, we find some sufficient conditions when this functional can be extended to a planar signed measure. A dual reformulation of these conditions leads to signed measures in the space of lines in ℝ2. In this way we obtain two sets of conditions which ensure that a segment function corresponds to a signed measure in the space of lines. The latter conditions are also necessary.

scholarly journals Seminormal Structure and Fixed Points of Cyclic Relatively Nonexpansive Mappings

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Moosa Gabeleh ◽  
Naseer Shahzad
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Nonexpansive Mappings ◽  
Sufficient Conditions ◽  
Weakly Compact ◽  
Relatively Nonexpansive Mapping ◽  
Relatively Nonexpansive Mappings ◽  
Relatively Nonexpansive ◽  
Contractive Type

LetAandBbe two nonempty subsets of a Banach spaceX. A mappingT:A∪B→A∪Bis said to be cyclic relatively nonexpansive ifT(A)⊆BandT(B)⊆AandTx-Ty≤x-yfor all (x,y)∈A×B. In this paper, we introduce a geometric notion of seminormal structure on a nonempty, bounded, closed, and convex pair of subsets of a Banach spaceX. It is shown that if (A,B) is a nonempty, weakly compact, and convex pair and (A,B) has seminormal structure, then a cyclic relatively nonexpansive mappingT:A∪B→A∪Bhas a fixed point. We also discuss stability of fixed points by using the geometric notion of seminormal structure. In the last section, we discuss sufficient conditions which ensure the existence of best proximity points for cyclic contractive type mappings.Erratum to “Seminormal Structure and Fixed Points of Cyclic Relatively Nonexpansive Mappings”

scholarly journals Balls intersection properties of Banach spaces

1992 ◽  
Vol 45 (2) ◽  
pp. 333-342 ◽  
Author(s):  
Dongjian Chen ◽  
Zhibao Hu ◽  
Bor-Luh Lin
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sufficient Conditions ◽  
Intersection Property ◽  
Necessary And Sufficient Conditions ◽  
Asplund Space ◽  
Necessary And Sufficient

Necessary and sufficient conditions for a Banach space with the Mazur intersection property to be an Asplund space are given. It is proved that Mazur intersection property is determined by the separable subspaces of the space. Corresponding problems for a space to have the ball-generated property are considered. Some comments on possible renorming so that a space having the Mazur intersection property are given.

scholarly journals A CONTINUITY CHARACTERIZATION OF ASPLUND SPACES

2011 ◽  
Vol 83 (3) ◽  
pp. 450-455
Author(s):  
J. R. GILES
Keyword(s):  
Banach Space ◽  
Asplund Space ◽  
Weakly Compact ◽  
Asplund Spaces ◽  
Subdifferential Mapping

AbstractA Banach space is an Asplund space if every continuous gauge has a point where the subdifferential mapping is Hausdorff weak upper semi-continuous with weakly compact image. This contributes towards the solution of a problem posed by Godefroy, Montesinos and Zizler.

Composition results for strongly summing and dominated multilinear operators

2014 ◽  
Vol 12 (10) ◽  
Author(s):  
Dumitru Popa
Keyword(s):  
Tensor Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Bilinear Operator ◽  
Multilinear Operators ◽  
Weakly Compact ◽  
Necessary And Sufficient ◽  
Composition Theorem

AbstractIn this paper we prove some composition results for strongly summing and dominated operators. As an application we give necessary and sufficient conditions for a multilinear tensor product of multilinear operators to be strongly summing or dominated. Moreover, we show the failure of some possible n-linear versions of Grothendieck’s composition theorem in the case n ≥ 2 and give a new example of a 1-dominated, hence strongly 1-summing bilinear operator which is not weakly compact.

scholarly journals О разложении в ряды экспонент функций, аналитических на выпуклых локально замкнутых множествах

2011 ◽  
Author(s):  
S.N. Melikhov ◽  
S. Momm
Keyword(s):  
Analytic Functions ◽  
Closed Subset ◽  
Sufficient Conditions ◽  
Nonempty Interior ◽  
Continuous Linear ◽  
Exponential Series ◽  
Representation Operator ◽  
Bounded Convex

Let Qbe a bounded, convex, locally closed subset of CN with nonempty interior. For N>1 sufficient conditions are obtained that an operator of the representation of analytic functions on Q by exponential series has a continuous linear right inverse. For N=1 the criterions for the existence of a continuous linear right inverse for the representation operator are proved.

Global bifurcation at isolated singular points of the Hadamard derivative

2021 ◽  
Vol 379 (2191) ◽  
pp. 20190379
Author(s):  
C. A. Stuart
Keyword(s):  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Global Bifurcation ◽  
Singular Boundary ◽  
Global Information ◽  
Singular Boundary Value Problems ◽  
Connected Component ◽  
Theme Issue ◽  
Fréchet Differentiable ◽  
Hadamard Derivative

Consider F ∈ C ( R × X , Y ) such that F ( λ , 0) = 0 for all λ ∈ R , where X and Y are Banach spaces. Bifurcation from the line R × { 0 } of trivial solutions is investigated in cases where F ( λ , · ) need not be Fréchet differentiable at 0. The main results provide sufficient conditions for μ to be a bifurcation point and yield global information about the connected component of { ( λ , u ) : F ( λ , u ) = 0  and  u ≠ 0 } ∪ { ( μ , 0 ) } containing ( μ , 0). Some necessary conditions for bifurcation are also formulated. The abstract results are used to treat several singular boundary value problems for which Fréchet differentiability is not available. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

Aubry set for asymptotically sub-additive potentials

2016 ◽  
Vol 16 (02) ◽  
pp. 1660009 ◽  
Author(s):  
Eduardo Garibaldi ◽  
João Tiago Assunção Gomes
Keyword(s):  
Dynamical Systems ◽  
Spectral Radius ◽  
Sufficient Conditions ◽  
Optimization Theory ◽  
Joint Spectral Radius ◽  
Maximum Value ◽  
Ergodic Optimization ◽  
Aubry Set ◽  
Central Result ◽  
Topological Dynamical Systems

Given a topological dynamical systems [Formula: see text], consider a sequence of continuous potentials [Formula: see text] that is asymptotically approached by sub-additive families. In a generalized version of ergodic optimization theory, one is interested in describing the set [Formula: see text] of [Formula: see text]-invariant probabilities that attain the following maximum value [Formula: see text] For this purpose, we extend the notion of Aubry set, denoted by [Formula: see text]. Our central result provides sufficient conditions for the Aubry set to be a maximizing set, i.e. [Formula: see text] belongs to [Formula: see text] if, and only if, its support lies on [Formula: see text]. Furthermore, we apply this result to the study of the joint spectral radius in order to show the existence of periodic matrix configurations approaching this value.

scholarly journals On the range of the derivative of a smooth mapping between Banach spaces

2005 ◽  
Vol 2005 (5) ◽  
pp. 499-507
Author(s):  
Robert Deville
Keyword(s):  
Banach Spaces ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Differentiable Mapping ◽  
Smooth Mapping ◽  
Lipschitz Continuous ◽  
Fréchet Differentiable ◽  
New Phenomena

We survey recent results on the structure of the range of the derivative of a smooth mappingfbetween two Banach spacesXandY. We recall some necessary conditions and some sufficient conditions on a subsetAofℒ(X,Y)for the existence of a Fréchet differentiable mappingffromXintoYso thatf′(X)=A. Wheneverfis only assumed Gâteaux differentiable, new phenomena appear: for instance, there exists a mappingffromℓ1(ℕ)intoℝ2, which is bounded, Lipschitz-continuous, and so that for allx,y∈ℓ1(ℕ), ifx≠y, then‖f′(x)−f′(y)‖>1.

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