Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2025
Volume
Vol. 45
Number
No. 5
Description
Journal Volume
Opuscula Mathematica
Vol. 45 (2025)
Projects
Pages
Articles
Spreading in claw-free cubic graphs
(Wydawnictwa AGH, 2025) Brešar, Boštjan; Hedžet, Jaka; Henning, Michael A.
Let $p \in \mathbb{N}$ and $q \in \mathbb{N} \cup \lbrace \infty \rbrace$. We study a dynamic coloring of the vertices of a graph $G$ that starts with an initial subset $S$ of blue vertices, with all remaining vertices colored white. If a white vertex $v$ has at least $p$ blue neighbors and at least one of these blue neighbors of $v$ has at most $q$ white neighbors, then by the spreading color change rule the vertex $v$ is recolored blue. The initial set $S$ of blue vertices is a $(p,q)$-spreading set for $G$ if by repeatedly applying the spreading color change rule all the vertices of $G$ are eventually colored blue. The $(p,q)$-spreading set is a generalization of the well-studied concepts of $k$-forcing and $r$-percolating sets in graphs. For $q \geq 2$, a $(1,q)$-spreading set is exactly a $q$-forcing set, and the $(1,1)$-spreading set is a $1$-forcing set (also called a zero forcing set), while for $q = \infty$, a $(p,\infty)$-spreading set is exactly a $p$-percolating set. The $(p,q)$-spreading number, $\sigma_{(p,q)}(G)$, of $G$ is the minimum cardinality of a $(p,q)$-spreading set. In this paper, we study $(p,q)$-spreading in claw-free cubic graphs. While the zero-forcing number of claw-free cubic graphs was studied earlier, for each pair of values $p$ and $q$ that are not both $1$ we either determine the $(p,q)$-spreading number of a claw-free cubic graph $G$ or show that $\sigma_{(p,q)}(G)$ attains one of two possible values.
From set-valued dynamical processes to fractals
(Wydawnictwa AGH, 2025) Guzik, Grzegorz; Kleszcz, Grzegorz
We present a general theory of topological semiattractors and attractors for set-valued semigroups. Our results extend and unify those previously obtained by Lasota and Myjak. In particular, we naturally generalize the concept of semifractals for the systems acting on Hausdorff topological spaces. The main tool in our analysis is the notion of topological (Kuratowski) limits. We especially focus on the forward asymptotic behavior of discrete set-valued processes generated by sequences of iterated function systems. In this context, we establish sufficient conditions for the existence of fractal-type limit sets.
Nontrivial solutions for Neumann fractional p-Laplacian problems
(Wydawnictwa AGH, 2025) Li, Chun; Mugnai, Dimitri; Zhao, Tai-Jin
In this paper, we investigate some classes of Neumann fractional p-Laplacian problems. We prove the existence and multiplicity of nontrivial solutions for several different nonlinearities, by using variational methods and critical point theory based on cohomological linking.
A general elliptic equation with intrinsic operator
(Wydawnictwa AGH, 2025) Motreanu, Dumitru
Existence and bound of a solution is established for a general elliptic equation with intrinsic operator subject to Dirichlet boundary condition. This provides a sufficient condition to the fundamental question if there is a solution belonging to a prescribed ball in the function space. An application deals with an equation involving a convolution product.
A note on nonlocal discrete problems involving sign-changing Kirchhoff functions
(Wydawnictwa AGH, 2025) Ricceri, Biagio
In this note, we establish a multiplicity theorem for a nonlocal discrete problem of the type \[\begin{cases} -\left(a\sum_{m=1}^{n+1}|x_m-x_{m-1}|^2+b\right)(x_{k+1}-2x_k+x_{k-1})=h_k(x_k), &k=1,\ldots ,n, \\ x_0=x_{n+1}=0 \end{cases}\] assuming \(a\gt 0\) and (for the first time) \(b\gt 0\).