scholarly journals A porosity result in convex minimization

2005 ◽  
Vol 2005 (3) ◽  
pp. 319-326
Author(s):  
P. G. Howlett ◽  
A. J. Zaslavski
Keyword(s):  
Banach Space ◽  
Minimization Problem ◽  
Metric Space ◽  
Convex Functions ◽  
Convex Sets ◽  
Reflexive Banach Space ◽  
Complete Metric Space ◽  
Baire Category ◽  
Countable Intersection ◽  
Porous Set

We study the minimization problemf(x)→min,x∈C, wherefbelongs to a complete metric spaceℳof convex functions and the setCis a countable intersection of a decreasing sequence of closed convex setsCiin a reflexive Banach space. Letℱbe the set of allf∈ℳfor which the solutions of the minimization problem over the setCiconverge strongly asi→∞to the solution over the setC. In our recent work we show that the setℱcontains an everywhere denseGδsubset ofℳ. In this paper, we show that the complementℳ\ℱis not only of the first Baire category but also aσ-porous set.

scholarly journals Compact reduction in Lipschitz-free spaces

2021 ◽  
Vol 151 (6) ◽  
pp. 1683-1699
Author(s):  
Ramón J. Aliaga ◽  
Camille Noûs ◽  
Colin Petitjean ◽  
Antonín Procházka
Keyword(s):  
Banach Space ◽  
Metric Space ◽  
General Principle ◽  
Metric Spaces ◽  
Approximation Property ◽  
Complete Metric Space ◽  
Infinite Dimensional ◽  
Schur Property ◽  
Free Spaces ◽  
Compact Subsets

We prove a general principle satisfied by weakly precompact sets of Lipschitz-free spaces. By this principle, certain infinite dimensional phenomena in Lipschitz-free spaces over general metric spaces may be reduced to the same phenomena in free spaces over their compact subsets. As easy consequences we derive several new and some known results. The main new results are: $\mathcal {F}(X)$ is weakly sequentially complete for every superreflexive Banach space $X$, and $\mathcal {F}(M)$ has the Schur property and the approximation property for every scattered complete metric space $M$.

scholarly journals SOLVABILITY OF THE CAUCHY PROBLEM FOR EQUATIONS WITH RIEMANN–LIOUVILLE’S FRACTIONAL DERIVATIVES

2018 ◽  
Vol 62 (4) ◽  
pp. 391-397 ◽  
Author(s):  
Petr P. Zabreiko ◽  
Svetlana V. Ponomareva
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Fixed Point ◽  
Unique Solution ◽  
Metric Space ◽  
Fractional Derivatives ◽  
Volterra Equation ◽  
Complete Metric Space ◽  
The Cauchy Problem ◽  
The Right

In this article we study the solvability of the analogue of the Cauchy problem for ordinary differential equations with Riemann–Liouville’s fractional derivatives with a nonlinear restriction on the right-hand side of functions in certain spaces. The conditions for solvability of the problem under consideration in given function spaces, as well as the conditions for existence of a unique solution are given. The study uses the method of reducing the problem to the second-kind Volterra equation, the Schauder principle of a fixed point in a Banach space, and the Banach-Cachoppoli principle of a fixed point in a complete metric space.

scholarly journals Diametrically contractive mappings

2004 ◽  
Vol 70 (3) ◽  
pp. 463-468 ◽  
Author(s):  
Hong-Kun Xu
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Compact Subset ◽  
Metric Space ◽  
Complete Metric Space ◽  
Contractive Mapping ◽  
Weakly Compact ◽  
Contractive Mappings

A contractive mapping on a complete metric space may fail to have a fixed point. Diametrically contractive mappings are introduced and it is shown that a diametrically contractive self-mapping of a weakly compact subset of a Banach space always has a fixed point.

scholarly journals Existence of solutions of minimization problems with an increasing cost function and porosity

2003 ◽  
Vol 2003 (11) ◽  
pp. 651-670 ◽  
Author(s):  
Alexander J. Zaslavski
Keyword(s):  
Banach Space ◽  
Cost Function ◽  
Minimization Problem ◽  
Closed Subset ◽  
Existence Of Solutions ◽  
Lower Semicontinuous ◽  
Ordered Banach Space ◽  
Porous Set ◽  
Minimization Problems ◽  
Semicontinuous Functions

We consider the minimization problemf(x)→min,x∈K, whereKis a closed subset of an ordered Banach spaceXandfbelongs to a space of increasing lower semicontinuous functions onK. In our previous work, we showed that the complement of the set of all functionsf, for which the corresponding minimization problem has a solution, is of the first category. In the present paper we show that this complement is also aσ-porous set.

scholarly journals Tykhonov triples and convergence results for hemivariational inequalities

2021 ◽  
Vol 26 (2) ◽  
pp. 271-292
Author(s):  
Rong Hu ◽  
Mircea Sofonea ◽  
Yi-Bin Xiao
Keyword(s):  
Mathematical Model ◽  
Banach Space ◽  
Elastic Body ◽  
Metric Space ◽  
Reflexive Banach Space ◽  
Convergence Result ◽  
Hemivariational Inequalities ◽  
Mathematical Tool ◽  
Abstract Problem ◽  
Convergence Results

Consider an abstract Problem P in a metric space (X; d) assumed to have a unique solution u. The aim of this paper is to compare two convergence results u'n → u and u''n → u, both in X, and to construct a relevant example of convergence result un → u such that the two convergences above represent particular cases of this third convergence. To this end, we use the concept of Tykhonov triple. We illustrate the use of this new and nonstandard mathematical tool in the particular case of hemivariational inequalities in reflexive Banach space. This allows us to obtain and to compare various convergence results for such inequalities. We also specify these convergences in the study of a mathematical model, which describes the contact of an elastic body with a foundation and provide the corresponding mechanical interpretations.

A Note on Locally Expansive and Locally Accretive Operators

1983 ◽  
Vol 26 (2) ◽  
pp. 228-232
Author(s):  
Ricardo Torrejon
Keyword(s):  
Banach Space ◽  
Open Subset ◽  
Metric Space ◽  
Complete Metric Space ◽  
Accretive Operators

AbstractLet X be a Banach space, D an open subset of X and Y a complete metric space. Assume that Y is metrically convex. For closed, locally m-expansive and mapping open subsets of D onto open subsets of Y, is is shown that y ∊ T(D) if and only if there exists x0 ∊ D such that d(Tx0, y) ≤ d(Tx, y) for all x ≤ ∂D.

Baire's category in existence problems for non-convex lower semicontinuous evolution differential inclusions

2011 ◽  
Vol 54 (3) ◽  
pp. 645-667
Author(s):  
F.S. De Blasi ◽  
G. Pianigiani
Keyword(s):  
Cauchy Problem ◽  
Metric Space ◽  
Infinitesimal Generator ◽  
Complete Metric Space ◽  
Mild Solutions ◽  
Lower Semicontinuous ◽  
Baire Category ◽  
Empty Interior ◽  
Image Position ◽  
Bounded Sets

AbstractThe existence of mild solutions to the non-convex Cauchy problemis investigated. Here A is the infinitesimal generator of a C0-semigroup in a reflexive and separable Banach space $\mathbb{E}$, F is a Pompeiu–Hausdorff lower semicontinuous multifunction whose values are closed convex and bounded sets with non-empty interior contained in $\mathbb{E}$, and ∂F(t, x(t)) denotes the boundary of F(t, x(t)). Our approach is based on the Baire category method, with appropriate modifications which are actually necessary because, under our assumptions, the underlying metric space that naturally enters in the Baire method, i.e. the solution set of the convexified Cauchy problem (CF), can fail to be a complete metric space.

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 Generic uniqueness of minimal configurations with rational rotation numbers in Aubry-Mather theory

2004 ◽  
Vol 2004 (8) ◽  
pp. 691-721 ◽  
Author(s):  
Alexander J. Zaslavski
Keyword(s):  
Rotation Number ◽  
Metric Space ◽  
Rational Number ◽  
Complete Metric Space ◽  
Countable Intersection ◽  
Mather Theory ◽  
Rotation Numbers ◽  
Minimal Configuration ◽  
Generic Uniqueness

We study(h)-minimal configurations in Aubry-Mather theory, wherehbelongs to a complete metric space of functions. Such minimal configurations have definite rotation number. We establish the existence of a set of functions, which is a countable intersection of open everywhere dense subsets of the space and such that for each elementhof this set and each rational numberα, the following properties hold: (i) there exist three different(h)-minimal configurations with rotation numberα; (ii) any(h)-minimal configuration with rotation numberαis a translation of one of these configurations.

TRANSLATING FINITE SETS INTO CONVEX SETS

2001 ◽  
Vol 33 (6) ◽  
pp. 711-714 ◽  
Author(s):  
EVA MATOUšKOVA
Keyword(s):  
Banach Space ◽  
Convex Subset ◽  
Separable Banach Space ◽  
Convex Sets ◽  
Reflexive Banach Space ◽  
Small Diameter ◽  
Empty Interior ◽  
Finite Sets ◽  
Bounded Set ◽  
Finite Set

Let X be a reflexive Banach space, and let C ⊂ X be a closed, convex and bounded set with empty interior. Then, for every δ > 0, there is a nonempty finite set F ⊂ X with an arbitrarily small diameter, such that C contains at most δ · |F| points of any translation of F. As a corollary, a separable Banach space X is reflexive if and only if every closed convex subset of X with empty interior is Haar null.

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