European Journal of Mathematical Analysis

Majorizing Sequences for Newton-Like Method and Their Limit Points

Ioannis K. Argyros — 2025-04-01

A plethora of problems from diverse disciplines of Mathematics, Mathematical Biology, Chemistry, Medicine, physics and Engineering to mention a few reduce to solving nonlinear equations or systems of equations usually in the finite dimensional Euclidean or more general spaces. The solutions of such equations are numbers or vectors of functions and can be found in closed form only in special cases. That is why researchers and practitioners develop mostly iterative methods which generate sequences approximating the solutions. The least number of iterations to be carried out in order to obtain a pre-decided error tolerance on the distances between consecutive iterates as well as the choice of initial points ensuring the convergence of the methods is very important. These two objectives can be achieved by introducing real majorizing sequences which control the behaviour of the iterates. Moreover, the closed form of the limits of the real sequences determine the radius of the ball that contains the initial points. In this paper we contribute by introducing more precise majorizing sequences and limit points.

Neumann and Dirichlet Problems for the Cauchy–Riemann and the Poisson Equations in the Partial Eclipse Domain

Ali Darya — 2025-04-01

In this paper, we consider the Neumann boundary value problem and the Dirichlet boundary value problem for complex partial differential equations in the partial eclipse domain. First, By the parqueting–reflection principle and the Cauchy–Pompeiu formula, a modified integral representation formula in the partial eclipse domain is constructed. Then, we explicitly solve the Neumann problem for the homogeneous equation and discuss the solvability conditions. Moreover, we investigate the Dirichlet problem for the Poisson equation in the partial eclipse domain. In other words, with the help of the Green’s function, we provide a unique solution for the Dirichlet boundary value problem for the Poisson equation and consider boundary behavior.

Numerical Results for Gauss-Seidel Iterative Algorithm Based on Newton Methods for Unconstrained Optimization Problems

Nguyen Dinh Dung — 2025-04-01

Optimization problems play a crucial role in various fields such as economics, engineering, and computer science. They involve finding the best value (maximum or minimum) of an objective function. In unconstrained optimization problems, the goal is to find a point where the function’s value reaches a maximum or minimum without being restricted by any conditions. Currently, there are many different methods to solve unconstrained optimization problems, one of which is the Newton method. This method is based on using a second-order Taylor series expansion to approximate the objective function. By calculating the first derivative (gradient) and second derivative (Hessian matrix) of the function, the Newton method determines the direction and step size to find the extrema. This method has a very fast convergence rate when near the solution and is particularly effective for problems with complex mathematical structures. In this paper, we introduce a Gauss-Seidel-type algorithm implemented for the Newton and Quasi-Newton methods, which is an efficient approach for finding solutions to optimization problems when the objective function is a convex functional. We also present some computational results for the algorithm to illustrate the convergence of the method.

Uncertainty Principles and Extremal Functions for Bessel Multiplier Operators in Quantum Calculus

Ahmed Chana — 2025-04-01

Using the q-Jackson integral and some elements of the q-harmonic analysis associated with the q-Bessel operator for fixed 0 < q < 1, we introduce the q-Bessel multiplier operators and we give some new results related to these operators as Plancherel’s, Calderón’s reproducing formulas and Heisenberg’s, Donoho-Stark’s uncertainty principles. Next, using the theory of reproducing kernels we give best estimates and an integral representation of the extremal functions related to these operators on weighted Sobolev spaces.

A Proposal of New Extended Symmetric Cosine Distribution

Christophe Chesneau — 2025-02-18

This article presents an extended symmetric version of the cosine distribution. The corresponding probability density function is constructed by a special linear combination of cosine and sine functions. These trigonometric functions are activated by two adjustable parameters with the aim of generating modulable oscillatory shapes. This gives the new distribution greater flexibility and applicability than the cosine distribution. Its main characteristics are then examined, focusing on its functional properties, the key moment measures and the generation of distributions with different support. A new skewed version of the standard normal distribution is also derived. Potential applications in various fields are discussed. Two simulated data examples are presented and analyzed, showing the superior performance of the new distribution compared to another two-parameter extended version of the cosine distribution.

Schwarz Algorithms for Stokes-Stokes Coupling

Alexandros Kyriakis — 2025-02-18

In this article, we exhibit the behavior of the Schwarz algorithms for the Steady Stokes equation in the case of two unbounded subdomains at the continuous level. The Schwarz methods have received a lot of attention during the last decades with the vast development of parallel computing devices. Hermann Amandus Schwarz, a German analyst, is considered to be the pioneer of the Domain Decomposition methods. We will closely observe how the overlapping and non overlapping Schwarz methods work for the steady Stokes problem. This problem has immediate practical application, modeling the flow of an incompressible fluid. For the analysis, we rely on Fourier analysis techniques and we provide comparison of the exhibited methods.

Hybrid Iterative Methods for Solving Nonlinear Equations in Banach Spaces

Ioannis K. Argyros — 2025-02-18

The present article contributes to the solution of equations which carry the symmetry property of the problem or not. Iterative methods with in- verses generate sequences converging faster to a solution of an equation than methods without inverses. However, the implementation of these methods has drawbacks, since the analytical form of these inverse may be unavailable or computationally very expensive. This problem is addressed in this paper by replacing the inverse with a finite sum of linear opera- tors. A convergence analysis is developed for the hybrid methods. The numerical examples demonstrate that the number of iterates is essentially the same between the hybrid and the original method. This technique is also extended to solve generalized equations.

Correspondences Among Inner Functions, Functions with Non-Negative Real Parts and Conformal Mappings

Ronen Peretz — 2025-01-13

We study an interesting family of dynamical systems on the set of the singular inner functions (defined on the unit disk). Starting with an inner function \( S_0(z) \), we obtain new singular inner functions \( S_1(z), S_2(z), \ldots \). This sequence converges to a holomorphic self-map of the unit disk which we call \( S \). The convergence is proved with the aid of a fixed-point theorem, a special case of the Earle-Hamilton Theorem. The function \( S \) itself is not a singular inner function as \( zS(z) \) is a conformal map. This conformal map has the surprising property that its inverse (which is a priori defined on a proper subset of the disk) extends to the entire disk. The motivating question for this research is whether \( z \) times a singular inner function can have an omitted value in the unit disk. This question appears within a book on the Krzyz problem written by the author. This question is still open.

The Rellich-Kondrachov Theorem for Gelfand Pairs Over Hypergroups

Ky T. Bataka — 2025-01-13

Embedding results play important rôles in mathematical analysis. This paper addresses some embedding theorems in the context of Sobolev spaces theory on Gelfand pairs over hypergroups. Mainly, the analogue of the Rellich-Kondrachov theorem is proved.

On η-Local Functions in Ideal Topological Spaces

Junvon A. Almocera — 2025-01-13

This study introduces and investigates a new local function called \( \eta \)-local function in ideal topological space \( (X, \tau, I) \) by using the notion of \( \eta \)-open sets in topological space \( (X, \tau) \). The operator \( (\cdot)_\eta^* : \mathcal{P}(X) \rightarrow \mathcal{P}(X) \) is defined as \((\cdot)_\eta^* (A) = A_\eta^* = \big\{ x \in X \mid A \cap U \notin I \ \text{for every} \ U \in \eta\text{-}O(x) \big\}\) for each \( A \subseteq X \), where \( \eta \)-\( O(x) \) is the set of all \( \eta \)-open subsets of \( X \) containing \( x \). This study establishes some properties of \( A_\eta^* \) including its relationships to the local function and local function \( \Gamma^* \) in ideal topological space \( (X, \tau, I) \). This study also introduces a new type of closure called the \( \eta \)-local closure in ideal topological space \( (X, \tau, I) \) which is denoted by \( Cl_\eta^*(A) \) for each \( A \subseteq X \). Furthermore, this study establishes some properties of the \( \eta \)-local closure.

Stability Results of Positive Weak Solution for a Class of Chemically Reacting Systems

Salah A. Khafagy — 2025-01-13

This paper aims to study the existence and non-existence results of positive weak solution to the quasilinear elliptic system:

\begin{cases}

-\Delta_p u = \lambda a(x) \left[ f(u,v) - \dfrac{1}{u^\alpha} \right], & x \in \Omega, \\

-\Delta_q v = \lambda b(x) \left[ g(u,v) - \dfrac{1}{v^\beta} \right], & x \in \Omega, \\

u = 0 = v, & x \in \partial\Omega,

\end{cases}

where \(\Delta_r w = \operatorname{div}(|\nabla w|^{r-2} \nabla w)\) is the \(r\)-Laplacian (\(r = p, q\)), \(r > 1\), \(\alpha, \beta \in (0,1)\), \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) (\(N > 1\)) with smooth boundary \(\partial\Omega\) and \(\lambda\) is a positive parameter. Here \(f, g\) are \(C^1\) increasing functions such that \(f, g : \mathbb{R}^+ \times \mathbb{R}^+ \rightarrow \mathbb{R}^+\); \(f(\upsilon_1, \upsilon_2) > 0\), \(g(\upsilon_1, \upsilon_2) > 0\) for \(\upsilon_1, \upsilon_2 > 0\). With \(C^1\) sign-changing functions \(a(x)\), \(b(x)\) that perhaps have negative values nearby the boundary. We establish our results via the sub-supersolution method. In addition, we study the stability and instability results of positive weak solution with different choices of \(f\) and \(g\).