Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2017
Volume
Vol. 37
Number
No. 3
Description
Journal Volume
Opuscula Mathematica
Vol. 37 (2017)
Projects
Pages
Articles
Existence of three solutions for impulsive multi-point boundary value problems
(2017) Bohner, Martin; Heidarkhani, Shapour; Salari, Amjad; Caristi, Giuseppe
This paper is devoted to the study of the existence of at least three classical solutions for a second-order multi-point boundary value problem with impulsive effects. We use variational methods for smooth functionals defined on reflexive Banach spaces in order to achieve our results. Also by presenting an example, we ensure the applicability of our results.
On the uniform perfectness of equivariant diffeomorphism groups for principal G manifolds
(2017) Fukui, Kazuhiko
We proved in [K. Abe, K. Fukui, <i>On commutators of equivariant diffeomorphisms</i>, Proc. Japan Acad. 54 (1978), 52–54] that the identity component $\text{Diff}\,^r_{G,c}(M)_0$ of the group of equivariant $C^r$-diffeomorphisms of a principal $G$ bundle $M$ over a manifold $B$ is perfect for a compact connected Lie group $G$ and $1 \leq r \leq \infty$ ($r \neq \dim B + 1$). In this paper, we study the uniform perfectness of the group of equivariant $C^r$-diffeomorphisms for a principal $G$ bundle $M$ over a manifold $B$ by relating it to the uniform perfectness of the group of $C^r$-diffeomorphisms of $B$ and show that under a certain condition, $\text{Diff}\,^r_{G,c}(M)_0$ is uniformly perfect if $B$ belongs to a certain wide class of manifolds. We characterize the uniform perfectness of the group of equivariant $C^r$-diffeomorphisms for principal $G$ bundles over closed manifolds of dimension less than or equal to 3, and in particular we prove the uniform perfectness of the group for the 3-dimensional case and $r\neq 4$.
General solutions of second-order linear difference equations of Euler type
(2017) Hongyo, Akane; Yamaoka, Naoto
The purpose of this paper is to give general solutions of linear difference equations which are related to the Euler-Cauchy differential equation $y^{\prime\prime}+(\lambda/t^2)y=0$ or more general linear differential equations. We also show that the asymptotic behavior of solutions of the linear difference equations are similar to solutions of the linear differential equations.
Sufficient conditions for optimality for a mathematical model of drug treatment with pharmacodynamics
(2017) Leszczyński, Maciej; Ratajczyk, Elżbieta; Ledzewicz, Urszula; Schättler, Heinz M.
We consider an optimal control problem for a general mathematical model of drug treatment with a single agent. The control represents the concentration of the agent and its effect (pharmacodynamics) is modelled by a Hill function (i.e., Michaelis-Menten type kinetics). The aim is to minimize a cost functional consisting of a weighted average related to the state of the system (both at the end and during a fixed therapy horizon) and to the total amount of drugs given. The latter is an indirect measure for the side effects of treatment. It is shown that optimal controls are continuous functions of time that change between full or no dose segments with connecting pieces that take values in the interior of the control set. Sufficient conditions for the strong local optimality of an extremal controlled trajectory in terms of the existence of a solution to a piecewise defined Riccati differential equation are given.
Positive solutions of a singular fractional boundary value problem with a fractional boundary condition
(2017) Lyons, Jeffrey W.; Neugebauer, Jeffrey T.
For $\alpha\in(1,2]$ the singular fractional boundary value problem $D^{\alpha}_{0^+}x+f\left(t,x,D^{\mu}_{0^+}x\right)=0,\quad 0\lt t\lt 1,$ satisfying the boundary conditions $x(0)=D^{\beta}_{0^+}x(1)=0$, where $\beta\in(0,\alpha-1]$, $\mu\in(0,\alpha-1]$, and $D^{\alpha}_{0^+}$, $D^{\beta}_{0^+}$ and $D^{\mu}_{0^+}$ are Riemann-Liouville derivatives of order $\alpha$, $\beta$, and $\mu$ respectively, is considered. Here $f$ satisfies a local Carathéodory condition, and $f(t, x, y)$ may be singular at the value 0 in its space variable $x$. Using regularization and sequential techniques and Krasnosel’skii’s fixed point theorem, it is shown this boundary value problem has a positive solution. An example is given.