Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2023
Volume
Vol. 43
Number
No. 4
Description
Journal Volume
Opuscula Mathematica
Vol. 43 (2023)
Projects
Pages
Articles
The heat equation on time scales
(Wydawnictwa AGH, 2023) Cuchta, Tom; Ferreira, Rui Alexandre Cardoso
We present the use of a Fourier transform on time scales to solve a dynamic heat IVP. This is done by inverting a certain exponential function via contour integral. We include some specific examples and directions for further study.
Generalized derivations and generalized exponential monomials on hypergroups
(Wydawnictwa AGH, 2023) Fechner, Żywilla; Gselmann, Eszter; Székelyhidi, László
In one of our former papers <i>Endomorphisms of the measure algebra of commutative hypergroups</i> we considered exponential monomials on hypergroups and higher order derivations of the corresponding measure algebra. Continuing with this, we are now looking for the connection between the generalized exponential polynomials of a commutative hypergroup and the higher order derivations of the corresponding measure algebra.
Periodic, nonperiodic, and chaotic solutions for a class of difference equations with negative feedback
(Wydawnictwa AGH, 2023) Kennedy, Benjamin B.
We study the scalar difference equation $x(k+1) = x(k) + \frac{f(x(k-N))}{N},$ where $f$ is nonincreasing with negative feedback. This equation is a discretization of the well-studied differential delay equation $x'(t) = f(x(t-1)).$ We examine explicit families of such equations for which we can find, for infinitely many values of $ and appropriate parameter values, various dynamical behaviors including periodic solutions with large numbers of sign changes per minimal period, solutions that do not converge to periodic solutions, and chaos. We contrast these behaviors with the dynamics of the limiting differential equation. Our primary tool is the analysis of return maps for the difference equations that are conjugate to continuous self-maps of the circle.
On the existence of optimal solutions to the Lagrange problem governed by a nonlinear Goursat-Darboux problem of fractional order
(Wydawnictwa AGH, 2023) Majewski, Marek
In the paper, the Lagrange problem given by a fractional boundary problem with partial derivatives is considered. The main result is the existence of optimal solutions based on the convexity assumption of a certain set. The proof is based on the lower closure theorem and the appropriate implicit measurable function theorem.
The first eigencurve for a Neumann boundary problem involving p-Laplacian with essentially bounded weights
(Wydawnictwa AGH, 2023) Sanhaji, Ahmed; Dakkak, Ahmed; Moussaoui, Mimoun
This article is intended to prove the existence and uniqueness of the first eigencurve, for a homogeneous Neumann problem with singular weights associated with the equation $-\Delta_{p} u=\alpha m_{1}|u|^{p-2}u+\beta m_{2}|u|^{p-2}u$ in a bounded domain $\Omega \subset \mathbb{R}^{N}$. We then establish many properties of this eigencurve, particularly the continuity, variational characterization, asymptotic behavior, concavity and the differentiability.