Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0706.1383

Fixed point theorem for discontinuous mappings on PN spaces

Mohd Rafi
View original source ↗Read source document ↗Export metadata ↓

Abstract

We present a study on strong t-continuity and measure of discontinuity on PN spaces. As an application, we prove a fixed point theorem for a self mapping on PN spaces by means of measure of discontinuity.

Explore related subjects

math.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Mohd Rafi. 2007-06-11. Fixed point theorem for discontinuous mappings on PN spaces. https://arxiv.org/abs/0706.1383

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A unified approach to the Dirac fine structures on the $S$-spectrum and a connection with Gegenbauer polynomials

This paper contributes to the recently introduced theory of fine structures on the $S$-spectrum. We study, in a unified way, the functional calculi for axially Poly-Analytic-Harmonic functions on the $S$-spectrum. Axially Poly-Analytic-Harmonic functions of type $(β, m)$, for $β, m \in \mathbb{N}_0$ belong to the kernel of the Dirac-Laplace operators $D^βΔ^m$ of type $(β, m)$ and contain as particular cases axially Poly-Analytic and axially Poly-Harmonic functions. By applying these operators to the Cauchy kernels $S^{-1}_L(s,x)$ of (left) slice hyperholomorphic functions, we obtain an integral representation for axially Poly-Analytic-Harmonic functions. We point out that the kernels $D^βΔ^mS^{-1}_L(s,x)$ have a remarkable connection with Gegenbauer polynomials. By replacing the paravector operator $T$ with commuting components in the kernels $D^βΔ^m S^{-1}_L(s,x)$, we obtain the associated resolvent operators. With these resolvent operators, denoted by $S^{-1}_{L, D^βΔ^m}(s,T)$, we define the associated functional calculi based on the $S$-spectrum and study their properties.

math.FA↗
arXiv2026

Localized locally convex topologies

Motivated by ill-posed PDEs such as $\mathrm{div} (v) = F$ we study locally convex topologies $\mathcal{T}_{\mathcal{C}}$ on real vector spaces $X$ that are a ``localized'' version of a locally convex topology $\mathcal{T}$ to members of a family $\mathcal{C}$ of convex subsets of $X$. The distributions $F$ arising as $\mathrm{div} (v)$ are expected to be the members of the dual of well-chosen $X$ with respect to an appropriate localized topology $\mathcal{T}_{\mathcal{C}}$. In this work, the emphasis is on studying the functional analytic properties of $\mathcal{T}_{\mathcal{C}}$, according to those of $\mathcal{T}$ and $\mathcal{C}$. For instance, we show that in all foreseen applications, $\mathcal{T}_{\mathcal{C}}$ is sequential but none of Fréchet-Urysohn, barrelled, and bornological. These awkward phenomena are illustrated explicitly on a specific example corresponding to the distributional divergence of continuous vector fields in $\mathbb{R}^m$. We also show that, essentially, $\mathcal{T}_{\mathcal{C}}$ is semireflexive if and only if members of $\mathcal{C}$ are $\mathcal{T}$-compact. This leads to an abstract existence theorem, thereby establishing a general scheme for characterizing those $F$ such that $\mathrm{div} (v) = F$ for various classes of regularity of $v$, various classes of domains, and various boundary conditions.

math.FA↗
arXiv2026

Tilings and coverings by balls in $\ell_1$

A famous result of Klee from 1981 is that the Banach space $\ell_1(κ)$ admits a disjoint tiling by balls of radius $1$, for all cardinals $κ$ with $κ^ω=κ$. Klee also observed that the smallest cardinal in which such a tiling might exist is $κ= 2^{\aleph_0}$, leaving open the question whether, for $κ< 2^{\aleph_0}$, $\ell_1(κ)$ might admit a tiling by balls at all. Our main result answers this question in the negative, proving in particular that $\ell_1$ does not admit any tiling by balls. We also give a companion result about star-$n$-finite coverings by balls of $\ell_1(κ)$ and we give a construction of a star-finite tiling of $\mathcal{X} \oplus_\infty c_{00}$, for each space $\mathcal{X}$ whose dimension is at most countable.

math.FA↗