Eurasian Mathematical Journal

Similar transformation of one class of well-defined restrictions

2025-05-11T10:57:24+00:00

In this paper there is considered the description of all well-defined restrictions of a maximal operator in a Hilbert space. A class of well-defined restrictions is found for which a similar transformation has the domain of the fixed well-defined restriction. The resulting theorem is applied to the study of \( n \)-order differentiation operator and the Laplace operator.

Some new approaches in the theory of trigonometric series with monotone coefficients

2025-05-11T10:59:00+00:00

In this paper, we observe the latest results concerning the trigonometric series whose coefficients are monotone or fractional monotone. We study the asymptotic properties of the sums for both classes of series and also the problems of convergence and integrability for series with fractional monotone coefficients.

Constructive method for solving of one class of curvilinear integral equations of the first kind

2025-05-11T11:00:54+00:00

A new method for the construction of a quadrature formula for the normal derivative of the double-layer potential is developed and a method for calculating the approximate solution of the integral equation of the first kind for Dirichlet boundary value problems for the Helmholtz equation in the two-dimensional space is presented in this work.

Measure of noncompactness approach to nonlinear fractional pantograph differential equations

2025-05-11T11:03:34+00:00

The aim of this manuscript is to explore the existence and uniqueness of solutions for a class of nonlinear \( \Psi \)-Caputo fractional pantograph differential equations subject to nonlocal conditions. The proofs rely on key results in topological degree theory for condensing maps, coupled with the method of measures of noncompactness and essential tools in \( \Psi \)-fractional calculus. To support the theoretical findings, a nontrivial example is presented as an application.

Two-weight Hardy inequality on topological measure spaces

2025-05-11T11:04:47+00:00

We consider a Hardy type integral operator \( T \) associated with a family of open subsets \( \Omega(t) \) of an open set \( \Omega \) in a Hausdorff topological space \( X \). In the inequality

\[\left( \int_\Omega |Tf(x)|^q u(x)\,d\mu(x) \right)^{1/q} \leq C \left( \int_\Omega |f(x)|^p v(x)\,d\nu(x) \right)^{1/p},\]

the measures \( \mu, \nu \) are \( \sigma \)-additive Borel measures; the weights \( u, v \) are positive and finite almost everywhere, \( 1 < p < \infty \), \( 0 < q < \infty \), and \( C > 0 \) is independent of \( f, u, v, \mu, \nu \). We find necessary and sufficient conditions for the boundedness and compactness of the operator \( T \) and obtain two-sided estimates for its approximation numbers. All results are proved using domain partitions, thus providing a roadmap for generalizing many one-dimensional results to a Hausdorff topological space.

One-dimensional integral Rellich type inequalities

2025-05-11T11:07:08+00:00

The motive of this note is twofold. Inspired by the recent development of a new kind of Hardy inequality, here we discuss the corresponding Hardy–Rellich and Rellich inequality versions in the integral form. The obtained sharp Hardy–Rellich type inequality improves the previously known result. Meanwhile, the established sharp Rellich type integral inequality seems new.