Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2022
Volume
Vol. 42
Number
No. 5
Description
Journal Volume
Opuscula Mathematica
Vol. 42 (2022)
Projects
Pages
Articles
Properties of even order linear functional differential equations with deviating arguments of mixed type
(Wydawnictwa AGH, 2022) Džurina, Jozef
This paper is concerned with oscillatory behavior of linear functional differential equations of the type $y^{(n)}(t)=p(t)y(\tau(t))$ with mixed deviating arguments which means that its both delayed and advanced parts are unbounded subset of $(0,\infty)$. Our attention is oriented to the Euler type of equation, i.e. when $p(t)\sim a/t^n.$
Stability switches in a linear differential equation with two delays
(Wydawnictwa AGH, 2022) Hata, Yuki; Matsunaga, Hideaki
This paper is devoted to the study of the effect of delays on the asymptotic stability of a linear differential equation with two delays $x'(t)=-ax(t)-bx(t-\tau)-cx(t-2\tau),\quad t\geq 0,$ where $a$, $b$, and $c$ are real numbers and $\tau \gt 0$. We establish some explicit conditions for the zero solution of the equation to be asymptotically stable. As a corollary, it is shown that the zero solution becomes unstable eventually after undergoing stability switches finite times when $\tau$ increases only if $c-a\lt 0$ and $\sqrt{-8c(c-a)}\lt |b| \lt a+c$. The explicit stability dependence on the changing $\tau$ is also described.
Nonlinear Choquard equations on hyperbolic space
(Wydawnictwa AGH, 2022) He, Haiyang
In this paper, our purpose is to prove the existence results for the following nonlinear Choquard equation $-\Delta_{\mathbb{B}^{N}}u=\int_{\mathbb{B}^N}\dfrac{|u(y)|^{p}}{|2\sinh\frac{\rho(T_y(x))}{2}|^\mu} dV_y \cdot |u|^{p-2}u +\lambda u$ on the hyperbolic space $\mathbb{B}^{N}$, where $\Delta_{\mathbb{B}^{N}}$ denotes the Laplace-Beltrami operator on $\mathbb{B}^{N}$, $\sinh\frac{\rho(T_y(x))}{2}=\dfrac{|T_y(x)|}{\sqrt{1-|T_y(x)|^2}}=\dfrac{|x-y|}{\sqrt{(1-|x|^2)(1-|y|^2)}},$ $\lambda$ is a real parameter, $0\lt \mu\lt N$, $1\lt p\leq 2_\mu^*$, $N \geq 3$ and $2_\mu^*:=\frac{2N-\mu}{N-2}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.
On the numerical solution of one inverse problem for a linearized two-dimensional system of Navier-Stokes equations
(Wydawnictwa AGH, 2022) Dženaliev, Muvašarhan Tanabievič; Ramazanov, Murat; Ergaliev, Madi Gabidenovič
The paper studies the numerical solution of the inverse problem for a linearized two-dimensional system of Navier-Stokes equations in a circular cylinder with a final overdetermination condition. For a biharmonic operator in a circle, a generalized spectral problem has been posed. For the latter, a system of eigenfunctions and eigenvalues is constructed, which is used in the work for the numerical solution of the inverse problem in a circular cylinder with specific numerical data. Graphs illustrating the results of calculations are presented.
Positive stationary solutions of convection-diffusion equations for superlinear sources
(Wydawnictwa AGH, 2022) Orpel, Aleksandra
We investigate the existence and multiplicity of positive stationary solutions for acertain class of convection-diffusion equations in exterior domains. This problem leads to the following elliptic equation $\Delta u(x)+f(x,u(x))+g(x)x\cdot \nabla u(x)=0,$ for $x\in \Omega_{R}=\{ x \in \mathbb{R}^n, \|x\|\gt R \}$, $n \gt 2$. The goal of this paper is to show that our problem possesses an uncountable number of nondecreasing sequences of minimal solutions with finite energy in a neighborhood of infinity. We also prove that each of these sequences generates another solution of the problem. The case when $f(x,\cdot)$ may be negative at the origin, so-called semipositone problem, is also considered. Our results are based on a certain iteration schema in which we apply the sub and supersolution method developed by Noussair and Swanson. The approach allows us to consider superlinear problems with convection terms containing functional coefficient $g$ without radial symmetry.