arXiv subjects

Ahmed Ghatasheh

Publications and source records attributed to Ahmed Ghatasheh.

5 recordsLinked to original sources

arXiv2025

Mean Value Theorems and L'Hospital-Type Rules for Regulated Functions

We introduce a generalization of Cauchy's mean value theorem for regulated functions. Building on this, we extend both L'Hospital's rule and L'Hospital's monotone rule to quotients of regulated functions. We demonstrate that our extended L'Hospital's rule encompasses both the discrete case, known as the Stolz-Cesaro theorem, and the classical continuous case. In addition, we show that these extensions handle some problems that classical rules cannot address. Finally, we provide Lebesgue-Stieltjes versions of L'Hospital's rule and L'Hospital's monotone rule and compare them with our extensions.

math.HO↗

arXiv2025

Refuting a Recent Proof of the Invariant Subspace Problem

This article demonstrates that the recent proof of the invariant subspace problem, as presented by Khalil et al., is incorrect.

math.FA↗

arXiv2019

Sign-Changing Points of Solutions of Homogeneous Sturm-Liouville Equations with Measure-Valued Coefficients

In this paper we investigate sign-changing points of nontrivial real-valued solutions of homogeneous Sturm-Liouville differential equations of the form $-d(du/dα)+udβ=0$, where $dα$ is a positive Borel measure supported everywhere on $(a,b)$ and $dβ$ is a locally finite real Borel measure on $(a,b)$. Since solutions for such equations are functions of locally bounded variation, sign-changing points are the natural generalization of zeros. We prove that sign-changing points for each nontrivial real-valued solution are isolated in $(a,b)$. We also prove a Sturm-type separation theorem for two nontrivial linearly independent solutions, and conclude the paper by proving a Sturm-type comparison theorem for two differential equations with distinct potentials.

math.CA↗

arXiv2019

Spectral Theory for Systems of Ordinary Differential Equations with Distributional Coefficients

We study the spectral theory for the first-order system $Ju'+qu=wf$ of differential equations on the real interval $(a,b)$ when $J$ is a constant, invertible skew-Hermitian matrix and $q$ and $w$ are matrices whose entries are distributions of order zero with $q$ Hermitian and $w$ non-negative. Also, we do not pose the definiteness condition customarily required for the coefficients of the equation. Specifically, we construct minimal and maximal relations, and study self-adjoint restrictions of the maximal relation. For these we determine Green's function and prove the existence of a spectral (or generalized Fourier) transformation. We have a closer look at the special cases when the endpoints of the interval $(a,b)$ are regular as well as the case of a $2\times2$ system. Two appendices provide necessary details on distributions of order zero and the abstract spectral theory for relations.

math.CA↗

arXiv2017

On Leighton's Comparison Theorem

We give a simple proof of a fairly flexible comparison theorem for equations of the type $-(p(u'+su))'+rp(u'+su)+qu=0$ on a finite interval where $1/p$, $r$, $s$, and $q$ are real and integrable. Flexibility is provided by two functions which may be chosen freely (within limits) according to the situation at hand. We illustrate this by presenting some examples and special cases which include Schrödinger equations with distributional potentials as well as Jacobi difference equations.

math.CA↗