scholarly journals A generalizedσ-porous set with a small complement

2005 ◽  
Vol 2005 (5) ◽  
pp. 535-541
Author(s):  
Jaroslav Tišer
Keyword(s):  
Banach Space ◽  
Measure Zero ◽  
Porous Set

We show that in every Banach space, there is ag-porous set, the complement of which is ofℋ1-measure zero on everyC1curve.

Cubic Functional Equations on Restricted Domains of Lebesgue Measure Zero

2017 ◽  
Vol 60 (1) ◽  
pp. 95-103 ◽  
Author(s):  
Chang-Kwon Choi ◽  
Jaeyoung Chung ◽  
Yumin Ju ◽  
John Rassias
Keyword(s):  
Banach Space ◽  
Functional Equation ◽  
Lebesgue Measure ◽  
Stability Problem ◽  
Functional Equations ◽  
Measure Zero ◽  
Stability Theorem ◽  
Cubic Functional Equation ◽  
Real Normed Space ◽  
Restricted Domains

AbstractLet X be a real normed space, Y a Banach space, and f : X → Y. We prove theUlam–Hyers stability theorem for the cubic functional equationin restricted domains. As an application we consider a measure zero stability problem of the inequalityfor all (x, y) in Γ ⸦ ℝ2 of Lebesgue measure 0.

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.

ON A MEASURE ZERO STABILITY PROBLEM OF A CYCLIC EQUATION

2015 ◽  
Vol 93 (2) ◽  
pp. 272-282 ◽  
Author(s):  
JAEYOUNG CHUNG ◽  
JOHN MICHAEL RASSIAS
Keyword(s):  
Banach Space ◽  
Functional Equation ◽  
Lebesgue Measure ◽  
Stability Problem ◽  
Cyclic Subgroup ◽  
Real Banach Space ◽  
Measure Zero ◽  
Stability Theorem ◽  
Image Position ◽  
Dimensional Lebesgue Measure

Let $G$ be a commutative group, $Y$ a real Banach space and $f:G\rightarrow Y$. We prove the Ulam–Hyers stability theorem for the cyclic functional equation $$\begin{eqnarray}\displaystyle \frac{1}{|H|}\mathop{\sum }_{h\in H}f(x+h\cdot y)=f(x)+f(y) & & \displaystyle \nonumber\end{eqnarray}$$ for all $x,y\in {\rm\Omega}$, where $H$ is a finite cyclic subgroup of $\text{Aut}(G)$ and ${\rm\Omega}\subset G\times G$ satisfies a certain condition. As a consequence, we consider a measure zero stability problem of the functional equation $$\begin{eqnarray}\displaystyle \frac{1}{N}\mathop{\sum }_{k=1}^{N}f(z+{\it\omega}^{k}{\it\zeta})=f(z)+f({\it\zeta}) & & \displaystyle \nonumber\end{eqnarray}$$ for all $(z,{\it\zeta})\in {\rm\Omega}$, where $f:\mathbb{C}\rightarrow Y,\,{\it\omega}=e^{2{\it\pi}i/N}$ and ${\rm\Omega}\subset \mathbb{C}^{2}$ has four-dimensional Lebesgue measure $0$.

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 Stability of a Quartic Functional Equation in Restricted Domains

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Jaeyoung Chung ◽  
Yu-Min Ju
Keyword(s):  
Banach Space ◽  
Functional Equation ◽  
Normed Space ◽  
Stability Problem ◽  
Measure Zero ◽  
Stability Theorem ◽  
Real Normed Space ◽  
Quartic Functional Equation ◽  
Restricted Domains

LetXbe a real normed space andYa Banach space andf:X→Y. We prove the Ulam-Hyers stability theorem for the quartic functional equationf(2x+y)+f(2x-y)-4f(x+y)-4f(x-y)-24f(x)+6f(y)=0in restricted domains. As a consequence we consider a measure zero stability problem of the above inequality whenf:R→Y.

scholarly journals Two large subsets of a functional space

1985 ◽  
Vol 8 (1) ◽  
pp. 189-191
Author(s):  
F. S. Cater
Keyword(s):  
Banach Space ◽  
Measure Zero ◽  
Functional Space ◽  
The Other ◽  
Differentiable Functions ◽  
Set Of Points

LetP1denote the Banach space composed of all bounded derivativesfof everywhere differentiable functions on[0,1]such that the set of points wherefvanishes is dense in[0,1]. LetD0consist of those functions inPthat are unsigned on every interval, and letD1consist of those functions inP1that vanish on dense subsets of measure zero. ThenD0andD1are denseGδ-subsets ofP1with void interior. NeitherD0norD1is a subset of the other.

Unavoidable Porous Sets and Nondifferentiable Maps

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Banach Space ◽  
Mean Value ◽  
Porous Set ◽  
Lipschitz Maps ◽  
Fréchet Differentiability ◽  
Frechet Derivatives ◽  
Fréchet Derivatives ◽  
Frechet Differentiability ◽  
Porous Sets ◽  
Derivatives Of

This chapter discusses Γ‎ₙ-nullness of sets porous “¹at infinity” and/or existence of many points of Fréchet differentiability of Lipschitz maps into n-dimensional spaces. The results reveal a σ‎-porous set whose complement is null on all n-dimensional surfaces and the multidimensional mean value estimates fail even for ε‎-Fréchet derivatives. Previous chapters have established conditions on a Banach space X under which porous sets in X are Γ‎ₙ-null and/or the the multidimensional mean value estimates for Fréchet derivatives of Lipschitz maps into n-dimensional spaces hold. This chapter investigates in what sense the assumptions of these main results are close to being optimal.

ON FULLY CONVEX AND LOCALLY FULLY CONVEX BANACH SPACE

1990 ◽  
Vol 10 (3) ◽  
pp. 327-343 ◽  
Author(s):  
Qiyuan Na
Keyword(s):  
Banach Space ◽  
Convex Banach Space

Three Kinds of Numerical Indices of a Banach Space

2012 ◽  
Vol 112 (1) ◽  
pp. 21-35 ◽  
Author(s):  
Sung Guen Kim
Keyword(s):  
Banach Space
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