Tamkang Journal of Mathematics

Weak solutions for the fractional Kirchhoff-type problem via Young measures

2024-10-25T05:06:52+00:00

The aim of this paper is to investigate the existence of weak solutions to the following Kirchhoff-type problem:
$$\begin{cases}
M\left([u]_{s p}^p\right) (-\Delta)_p^s (u)=f(x,u) \quad &\text { in } \Omega,\\ u=0 \quad &\text { in } \mathbb{R}^n \backslash \Omega,
\end{cases}$$
where $\Omega\subset\mathbb{R}^n$, $0<s<1<p<\infty$, $[u]_{s p}$ is Gagliardo semi-norm, $M$ is a continuous function with value in $\mathbb{R}^+$, $f$ a given function and $(-\Delta)_p^s$ is the fractional $p$-Laplacian operator.
Under appropriate assumptions on the main functions, we
obtain the existence results by applying the Galerkin method combined with the theory of Young measures.

A new hybrid generalization of orthogonal polynomials

2025-08-18T06:52:14+00:00

In this paper, we introduce and study hybrinomials defined by application of orthogonal polynomials. Using selected orthogonal polynomials and hybrid numbers operators, we define Hermite, Laguerre, Legendre and Chebyshev type hybrinomials and present some properties of them.

The Legendrian Self-Expander in the Standard Contact Euclidean Five-space

2025-08-13T05:13:12+00:00

Based on the geometric correspondence between Lagrangian and Legendrian submanifolds, we construct Legendrian 2-submanifolds in the standard contact Euclidean Five-space $\mathbb{R}^{5} $ satisfying the self-similarity equation $H+\theta\xi=\alpha{F}^{\perp}(\alpha>0) $, with particular focus on their self-expander solutions under Legendrian mean curvature flow. This paper mainly generalizes Theorem C of the work by Joyce-Lee-Tsui.

On the Uniform Boundedness of a Class of Hypersingular Integral Operators on the Hardy Space

2025-08-25T05:48:52+00:00

For a class of hypersingular integral operators, we establish optimal uniform bounds for their norms on the Hardy space $H^1(\R)$. Our results extend the classical result of Fefferman-Stein for the phase function $1/y$ to phase functions of the form $1/P(y)$ where $P$ is an arbitrary real polynomial. It is revealed that the presence and absence of a constant term in $P$ play a crucial role in the outcome.

Blow-up results of a time fractional heat equation with a nonlinear Neumann boundary condition

2025-11-05T06:56:53+00:00

The study of blow-up problems for time Caputo fractional heat equations are of great wide-ranging interest for its multitude of applications and the fact that these kinds of problems are found in several areas of science and engineering. This article is concerned with the blow-up solutions of a time fractional heat equation subject to a nonlinear Neumann boundary condition of power type. Firstly, under some restricted conditions, it is proved that every positive solution blows up in finite time. Secondly, it is proved that the blow up phenomenon can only occurs at the boundary.

$q$- Analogue of the Karry-Kalim-Adnan transform with applications to q-differential equations

2026-01-05T08:42:05+00:00

This work analyzes certain features of the Karry-Kalim-Adnan transform and discusses its $q$-analogues in a quantum calculus theory. It discusses a number of characteristics of the $q$-Karry-Kalim-Adnan transform and its application to a wide range of functions, including $q$- trigonometric,$q$- hyperbolic and $q$-exponential functions and some $q$-polynomials. Additionally, it utilizes First- and second-order $q$-initial value problems to illustrate effectiveness and performance of our proposed $q$-transform analogues. Over and above, the paper proves the $q$-convolution theorem and provides a number of tables to further ease the $q$- transform technique in solving various ostensibly $q$-initial value problems.