Opuscula Mathematica
Loading...
ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2025
Volume
Vol. 45
Number
No. 2
Description
Journal Volume
Opuscula Mathematica
Vol. 45 (2025)
Projects
Pages
Articles
Combined effects for a class of fractional variational inequalities
(Wydawnictwa AGH, 2025) Deng, Shengbing; Luo, Wenshan; Torres Ledesma, César E.
In this paper, we study the existence of a nonnegative weak solution to the following nonlocal variational inequality: $\textcolor{white}\$\int_{\mathbb{R}^N}(-\Delta)^{\frac{s}{2}} u (-\Delta)^{{\frac{s}{2}}}(v-u)dx+\int_{\mathbb{R}^N}(1+\lambda M(x))u(v-u)dx \geq \int_{\mathbb{R}^N}f(u)(v-u)dx, \textcolor{white}\$$ for all $v \in\mathbb{K}$, where $s\in (0,1)$ and $M$ is a continuous steep potential well on $\mathbb{R}^N$. Using penalization techniques from del Pino and Felmer, as well as from Bensoussan and Lions, we establish the existence of nonnegative weak solutions. These solutions localize near the potential well $\operatorname{Int}(M^{-1}(0))$.
Representation of solutions of linear discrete systems with constant coefficients and with delays
(Wydawnictwa AGH, 2025) Diblík, Josef
The paper surveys the results achieved in representing solutions of linear non-homogeneous discrete systems with constant coefficients and with delays and their fractional variants by using special matrices called discrete delayed-type matrices. These are used to express solutions of initial problems in a closed and often simple form. Then, results are briefly discussed achieved by such representations of solutions in stability, controllability and other fields. In addition, a similar topic is dealt with concerning linear non-homogeneous differential equations with delays and their variants. Moreover, some comments are given to this parallel direction pointing some important moments in the developing this theory. An outline of future perspectives in this direction is discussed as well.
Augmenting graphs to partition their vertices into a total dominating set and an independent dominating set
(Wydawnictwa AGH, 2025) Haynes, Teresa W.| Henning, Michael A.
A graph $G$ whose vertex set can be partitioned into a total dominating set and an independent dominating set is called a TI-graph. There exist infinite families of graphs that are not TI-graphs. We define the TI-augmentation number $\operatorname{ti}(G)$ of a graph $G$ to be the minimum number of edges that must be added to $G$ to ensure that the resulting graph is a TI-graph. We show that every tree $T$ of order $n \geq 5$ satisfies $\operatorname{ti}(T) \leq \frac{1}{5}n$. We prove that if $G$ is a bipartite graph of order $n$ with minimum degree $\delta(G) \geq 3$, then $\operatorname{ti}(G) \leq \frac{1}{4}n$, and if $G$ is a cubic graph of order $n$, then $\operatorname{ti}(G) \leq \frac{1}{3}n$. We conjecture that $\operatorname{ti}(G) \leq \frac{1}{6}n$ for all graphs $G$ of order $n$ with $\delta(G) \geq 3$, and show that there exist connected graphs $G$ of sufficiently large order $n$ with $\delta(G) \geq 3$ such that $\operatorname{ti}(T) \geq (\frac{1}{6} - \varepsilon) n$ for any given $\varepsilon \gt 0$.
Complete characterization of graphs with local total antimagic chromatic number 3
(Wydawnictwa AGH, 2025) Lau, Gee-Choon
A total labeling of a graph $G = (V, E)$ is said to be local total antimagic if it is a bijection $f: V\cup E \to\{1,\ldots,|V|+|E|\}$ such that adjacent vertices, adjacent edges, and pairs of an incident vertex and edge have distinct induced weights where the induced weight of a vertex $v$ is $w_f(v) = \sum f(e)$ with $e$ ranging over all the edges incident to $v$, and the induced weight of an edge $uv$ is $w_f(uv) = f(u) + f(v)$. The local total antimagic chromatic number of $G$, denoted by $\chi_{lt}(G)$, is the minimum number of distinct induced vertex and edge weights over all local total antimagic labelings of $G$. In this paper, we first obtain general lower and upper bounds for $\chi_{lt}(G)$ and sufficient conditions to construct a graph $H$ with $k$ pendant edges and $\chi_{lt}(H) \in\{\Delta(H)+1, k+1\}$. We then completely characterize graphs $G$ with $\chi_{lt}(G)=3$. Many families of (disconnected) graphs $H$ with $k$ pendant edges and $\chi_{lt}(H) \in\{\Delta(H)+1, k+1\}$ are also obtained.
Spectral analysis of infinite Marchenko-Slavin matrices
(Wydawnictwa AGH, 2025) Palafox, Sergio; Silva, Luis O.
This work tackles the problem of spectral characterization of a class of infinite matrices arising from the modeling of small oscillations in a system of interacting particles. The class of matrices under discussion corresponds to the infinite Marchenko-Slavin class. The spectral functions of these matrices are completely characterized, and an algorithm is provided for the reconstruction of the matrix from its spectral function. The techniques used in this work are based on recent results for the spectral characterization of infinite band symmetric matrices with so-called degenerations.