Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2021
Volume
Vol. 41
Number
No. 1
Description
Journal Volume
Opuscula Mathematica
Vol. 41 (2021)
Projects
Pages
Articles
Some existence results for a nonlocal non-isotropic problem
(Wydawnictwa AGH, 2021) Bentifour, Rachid; Miri, Sofiane El-Hadi
In this paper we deal with the following problem $\begin{cases}-\sum\limits_{i=1}^{N}\left[ \left( a+b\int\limits_{\, \Omega }\left\vert \partial _{i}u\right\vert ^{p_{i}}dx\right) \partial _{i}\left( \left\vert \partial _{i}u\right\vert ^{p_{i}-2}\partial _{i}u\right) \right]=\frac{f(x)}{u^{\gamma }}\pm g(x)u^{q-1} & in\ \Omega, \\ u\geq 0 & in\ \Omega, \\ u=0 & on\ \partial \Omega, \end{cases}$ where $\Omega$ is a bounded regular domain in $\mathbb{R}^{N}$. We will assume without loss of generality that $1\leq p_{1}\leq p_{2}\leq \ldots\leq p_{N}$ and that $f$ and $g$ are non-negative functions belonging to a suitable Lebesgue space $L^{m}(\Omega)$, $1\lt q\lt \overline{p}^{\ast}$, $a \gt 0$, $b \gt 0$ and $0\lt\gamma \lt 1.$
Nonlinear parabolic equation having nonstandard growth condition with respect to the gradient and variable exponent
(Wydawnictwa AGH, 2021) Charkaoui, Abderrahim; Fahim, Houda; Alaa, Nour Eddine
We are concerned with the existence of solutions to a class of quasilinear parabolic equations having critical growth nonlinearity with respect to the gradient and variable exponent. Using Schaeffer's fixed point theorem combined with the sub- and supersolution method, we prove the existence results of a weak solutions to the considered problems.
More on linear and metric tree maps
(Wydawnictwa AGH, 2021) Kozerenko, Sergìj Oleksandrovič
We consider linear and metric self-maps on vertex sets of finite combinatorial trees. Linear maps are maps which preserve intervals between pairs of vertices whereas metric maps are maps which do not increase distances between pairs of vertices. We obtain criteria for a given linear or a metric map to be a positive (negative) under some orientation of the edges in a tree, we characterize trees which admit maps with Markov graphs being paths and prove that the converse of any partial functional digraph is isomorphic to a Markov graph for some suitable map on a tree.
Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, I
(Wydawnictwa AGH, 2021) Naitō, Manabu
We consider the half-linear differential equation of the form $(p(t)|x'|^{\alpha}\mathrm{sgn} x')' + q(t)|x|^{\alpha}\mathrm{sgn} x = 0, \quad t\geq t_{0},$ under the assumption $\int_{t_{0}}^{\infty}p(s)^{-1/\alpha}ds =\infty$. It is shown that if a certain condition is satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as $t \to \infty$.
On the crossing numbers of join products of W4+Pn and W4+Cn
(Wydawnictwa AGH, 2021) Staš, Michal; Valiska, Juraj
The crossing number $cr(G)$ of a graph $G$ is the minimum number of edge crossings over all drawings of $G$ in the plane. The main aim of the paper is to give the crossing number of the join product $W_{4}+P_{n}$ and $W_{4}+C_{n}$ for the wheel $W_4$ on five vertices, where $P_n$ and $C_n$ are the path and the cycle on $n$ vertices, respectively. Yue et al. conjectured that the crossing number of $W_{m}+C_{n}$ is equal to $Z(m+1)Z(n)+(Z(m)-1) \big \lfloor \frac{n}{2} \big \rfloor + n+ \big\lceil\frac{m}{2}\big\rceil +2$, for all $m,n \geq 3$, and where the Zarankiewicz's number $Z(n)=\big \lfloor \frac{n}{2} \big \rfloor \big \lfloor \frac{n-1}{2} \big \rfloor$ is defined for $n \geq 1$. Recently, this conjecture was proved for $W_{3}+C_{n}$ by Klešč. We establish the validity of this conjecture for $W_{4}+C_{n}$ and we also offer a new conjecture for the crossing number of the join product $W_{m}+P_{n}$ for $m \geq 3$ and $n \geq 2$.