European Journal of Mathematical Analysis

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

Ronen Peretz — Mon, 13 Jan 2025 00:00:00 +0800

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, Yaogan Mensah — Mon, 13 Jan 2025 00:00:00 +0800

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, Lezel M. Tutanes — Mon, 13 Jan 2025 00:00:00 +0800

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, A. Ezzat Mohamed — Mon, 13 Jan 2025 00:00:00 +0800

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 \(g\).