Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2021
Volume
Vol. 41
Number
No. 4
Description
Journal Volume
Opuscula Mathematica
Vol. 41 (2021)
Projects
Pages
Articles
Total connected domination game
(Wydawnictwa AGH, 2021) Bujtás, Csilla; Henning, Michael A.; Iršič, Vesna; Klavžar, Sandi
The (total) connected domination game on a graph $G$ is played by two players, Dominator and Staller, according to the standard (total) domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of $G$. If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the (total) connected game domination number ($\gamma_{\rm tcg}(G)$) $\gamma_{\rm cg}(G)$ of $G$. We show that $\gamma_{\rm tcg}(G) \in \{\gamma_{\rm cg}(G),\gamma_{\rm cg}(G) + 1,\gamma_{\rm cg}(G) + 2\}$, and consequently define $G$ as Class $i$ if $\gamma_{\rm tcg}(G) = \gamma_{\rm cg} + i$ for $i \in \{0,1,2\}$. A large family of Class $0$ graphs is constructed which contains all connected Cartesian product graphs and connected direct product graphs with minumum degree at least $2$. We show that no tree is Class $2$ and characterize Class $1$ trees. We provide an infinite family of Class $2$ bipartite graphs.
Remarks on damped Schrödinger equation of Choquard type
(Wydawnictwa AGH, 2021) Chergui, Lassaad
This paper is devoted to the Schrödinger-Choquard equation with linear damping. Global existence and scattering are proved depending on the size of the damping coefficient.
Spectrum of discrete 2n-th order difference operator with periodic boundary conditions and its applications
(Wydawnictwa AGH, 2021) El Amrouss, Abdelrachid; Hammouti, Omar
Let $n\in\mathbb{N}^{*}$, and $N\geq n$ be an integer. We study the spectrum of discrete linear $2n$-th order eigenvalue problems $\begin{cases}\sum_{k=0}^{n}(-1)^{k}\Delta^{2k}u(t-k) = \lambda u(t) ,\quad & t\in[1, N]_{\mathbb{Z}}, \\ \Delta^{i}u(-(n-1))=\Delta^{i}u(N-(n-1)),\quad & i\in[0, 2n-1]_{\mathbb{Z}},\end{cases}$
where $\lambda$ is a parameter. As an application of this spectrum result, we show the existence of a solution of discrete nonlinear $2n$-th order problems by applying the variational methods and critical point theory.
Asymptotic expansions for the first hitting times of Bessel processes
(Wydawnictwa AGH, 2021) Hamana, Yuji; Kaikura, Ryo; Shinozaki, Kosuke
We study a precise asymptotic behavior of the tail probability of the first hitting time of the Bessel process. We deduce the order of the third term and decide the explicit form of its coefficient.
Reaction-diffusion coupled inclusions with variable exponents and large diffusion
(Wydawnictwa AGH, 2021) Simsen, Jacson; Simsen, Mariza Stefanello; Wittbold, Petra
This work concerns the study of asymptotic behavior of coupled systems of $p(x)$-Laplacian differential inclusions. We obtain that the generalized semiflow generated by the coupled system has a global attractor, we prove continuity of the solutions with respect to initial conditions and a triple of parameters and we prove upper semicontinuity of a family of global attractors for reaction-diffusion systems with spatially variable exponents when the exponents go to constants greater than 2 in the topology of $L^{\infty}(\Omega)$ and the diffusion coefficients go to infinity.