Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2025
Volume
Vol. 45
Number
No. 4
Description
Journal Volume
Opuscula Mathematica
Vol. 45 (2025)
Projects
Pages
Articles
Convex geometries yielded by transit functions
(Wydawnictwa AGH, 2025) Changat, Manoj; Sheela, Lekshmi Kamal K.; Peterin, Iztok; Shanavas, Ameera Vaheeda
Let $V$ be a finite nonempty set. A transit function is a map $R:V\times V\rightarrow 2^V$ such that $R(u,u)=\{u\}$, $R(u,v)=R(v,u)$ and $u\in R(u,v)$ holds for every $u,v\in V$. A set $K\subseteq V$ is $R$-convex if $R(u,v)\subset K$ for every $u,v\in K$ and all $R$-convex subsets of $V$ form a convexity $\mathcal{C}_R$. We consider the Minkowski-Krein-Milman property that every $R$-convex set $K$ in a convexity $\mathcal{C}_R$ is the convex hull of the set of extreme points of $K$ from axiomatic point of view and present a characterization of it. Later we consider several well-known transit functions on graphs and present the use of the mentioned characterizations on them.
Datko-type theorems concerning asymptotic behaviour of exponential type in mean
(Wydawnictwa AGH, 2025) Hai, Pham Viet
In this paper, we study the concept of exponential (in)stability in mean for stochastic skew-evolution semiflows, in which the exponential (in)stability in the classical sense is replaced by an average with respect to a probability measure. Our paper consists of three major results. The first is to obtain Datko-type characterizations for the exponential stability in mean of stochastic skew-evolution semiflows. Next, we acquire Datko-type characterizations for the exponential instability in mean by extending the stability techniques. The last is to extend Lyapunov-type equations to the case of exponential (in)stability in mean.
Tail probability of the hitting time of Brownian motion to a sphere with fixed hitting sites
(Wydawnictwa AGH, 2025) Hamana, Yuji
We consider $d$-dimensional Brownian motion $\{B_\mu(t)\}_{t\geqq0}$ with a drift $\mu\in\mathbb{R}^d$ and the first hitting time $\sigma_{r,\mu}^{(d)}$ to the sphere with radius $r$ centered at the origin. This article deals with asymptotic behavior of the probability that both $t\lt\sigma_{r,\mu}^{(d)}\lt\infty$ and $B_\mu(\sigma_{r,\mu}^{(d)})\in A$ occur simultaneously, and we obtain that this probability admits an asymptotic expansion in powers of $1/t$ if $d\geqq3$ and in that of $1/\log t$ if $d=2$ for large $t$. Moreover, we investigate the case of Brownian motion with no drift.
Asymptotic behavior of the solutions of operators that are sum of pseudo p-Laplace type
(Wydawnictwa AGH, 2025) Jana, Purbita
The article investigates a Poisson-type problem for operators that are finite sum of pseudo $p$-Laplace-type operators within long cylindrical domains. It establishes that the rate of convergence is exponential, which is considered optimal. In addition, the study analyzes the asymptotic behavior of the related energy functional. This research contributes to a deeper understanding of the mathematical properties and asymptotic analysis of solutions in this context.
Multiplicity result for mixed local and nonlocal Kirchhoff problems involving critical growth
(Wydawnictwa AGH, 2025) Tripathi, Vinayak Mani
In this paper, we study the multiplicity of nonnegative solutions for the following nonlocal elliptic problem $\textcolor{white}\$ \begin{cases}M\Big(\ \int_{\mathbb{R}^N}|\nabla u|^2dx+\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\Big)\mathcal{L}(u) \\ = \lambda {f(x)}|u|^{p-2}u+|u|^{2^*-2}u &\text{ in }\Omega, \\ u=0 &\text{ on }\mathbb R^N\setminus \Omega, \qquad \qquad \end{cases} \textcolor{white}\$$ where $\Omega\subset\mathbb{R}^N$ is bounded domain with smooth boundary, $1\lt p\lt 2\lt 2^*=\frac{2N}{N-2}$, $N\geq 3$, $\lambda\gt 0$, $M$ is a Kirchhoff coefficient and $\mathcal{L}$ denotes the mixed local and nonlocal operator. The weight function $f\in L^{\frac{2^*}{2^*-p}}(\Omega)$ is allowed to change sign. By applying variational approach based on constrained minimization argument, we show the existence of at least two nonnegative solutions.