Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2020
Volume
Vol. 40
Number
No. 4
Description
Journal Volume
Opuscula Mathematica
Vol. 40 (2020)
Projects
Pages
Articles
Multiple solutions of boundary value problems on time scales for a φ-Laplacian operator
(Wydawnictwa AGH, 2020) Amster, Pablo; Kuna, Mariel Paula; Santos, Dionicio Pastor
We establish the existence and multiplicity of solutions for some boundary value problems on time scales with a $\varphi$-Laplacian operator. For this purpose, we employ the concept of lower and upper solutions and the Leray-Schauder degree. The results extend and improve known results for analogous problems with discrete $p$-Laplacian as well as those for boundary value problems on time scales.
An inverse backward problem for degenerate two-dimensional parabolic equation
(Wydawnictwa AGH, 2020) Atifi, Khalid; Essoufi, El-Hassan; Khouiti, Bouchra
This paper deals with the determination of an initial condition in the degenerate two-dimensional parabolic equation $\partial_{t}u-\mathrm{div}\left(a(x,y)I_2\nabla u\right)=f,\quad (x,y)\in\Omega,\; t\in(0,T),$ where $\Omega$ is an open, bounded subset of $\mathbb{R}^2$, $a \in C^1(\bar{\Omega})$ with $a\geqslant 0$ everywhere, and $f\in L^{2}(\Omega \times (0,T))$, with initial and boundary conditions $u(x,y,0)=u_0(x,y), \quad u\mid_{\partial\Omega}=0,$ from final observations. This inverse problem is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. To show the convergence of the descent method, we prove the Lipschitz continuity of the gradient of the Tikhonov functional. Also we present some numerical experiments to show the performance and stability of the proposed approach.
Option pricing formulas under a change of numèraire
(Wydawnictwa AGH, 2020) Attalienti, Antonio; Bufalo, Michele
We present some formulations of the Cox-Ross-Rubinstein and Black-Scholes formulas for European options obtained through a suitable change of measure, which corresponds to a change of numèraire for the underlying price process. Among other consequences, a closed formula for the price of an European call option at each node of the multi-period binomial tree is achieved, too. Some of the results contained herein, though comparable with analogous ones appearing elsewhere in the financial literature, provide however a supplementary widening and deepening in view of useful applications in the more challenging framework of incomplete markets. This last issue, having the present paper as a preparatory material, will be treated extensively in a forthcoming paper.
Facial rainbow edge-coloring of simple 3-connected plane graphs
(Wydawnictwa AGH, 2020) Czap, Július
A facial rainbow edge-coloring of a plane graph $G$ is an edge-coloring such that any two edges receive distinct colors if they lie on a common facial path of $G$. The minimum number of colors used in such a coloring is denoted by $\text{erb}(G)$. Trivially, $\text{erb}(G) \geq \text{L}(G)+1$ holds for every plane graph without cut-vertices, where $\text{L}(G)$ denotes the length of a longest facial path in $G$. Jendroľ in 2018 proved that every simple $3$-connected plane graph admits a facial rainbow edge-coloring with at most $\text{L}(G)+2$ colors, moreover, this bound is tight for $\text{L}(G)=3$. He also proved that $\text{erb}(G)=\text{L}(G)+1$ for $\text{L}(G)\not\in\{3,4,5\}$. He posed the following conjecture: There is a simple $3$-connected plane graph $G$ with $\text{L}(G)=4$ and $\text{erb}(G)=\text{L}(G)+2$. In this note we answer the conjecture in the affirmative.
Properties of solutions to some weighted p-Laplacian equation
(Wydawnictwa AGH, 2020) Garain, Prashanta
In this paper, we prove some qualitative properties for the positive solutions to some degenerate elliptic equation given by $-\text{div}\big(w|\nabla u|^{p-2}\nabla u\big)=f(x,u),\quad w\in \mathcal{A}_p,$ on smooth domain and for varying nonlinearity $f$.