Publication detail
Two classes of asymptotically different positive solutions to advanced differential equations via two different fixed-point principles
DIBLÍK, J. KÚDELČÍKOVÁ, M.
Original Title
Two classes of asymptotically different positive solutions to advanced differential equations via two different fixed-point principles
English Title
Two classes of asymptotically different positive solutions to advanced differential equations via two different fixed-point principles
Type
journal article in Web of Science
Language
en
Original Abstract
The paper considers a system of advanced-type functional differential equations $$ \dot{x}(t) = F(t,x^t) $$ where $F$ is a given functional, $x^t \in C([0,r],{\mathbb R}^n)$, $r>0$ and $x^t(\theta)=x(t+\theta)$, $\theta \in [0,r]$. Two different results on the existence of solutions, with coordinates bounded above and below by the coordinates of the given vector functions if $t\to\infty$, are proved using two different fixed-point principles. It is illustrated by examples that, applying both results simultaneously to the same equation yields two positive solutions asymptotically different for $t\to\infty$. The equation $$ \dot{x}(t) = \left(a+{b}/{t}\right)\,x(t+\tau) $$ where $a, \tau \in (0,\infty)$, $a<1/(\tau\e)$, $b \in {\mathbb R}$ are constants can serve as a linear example. The existence of a pair of positive solutions asymptotically different for $t\to\infty$ is proved and their asymptotic behavior is investigated. The results are also illustrated by a nonlinear equation.
English abstract
The paper considers a system of advanced-type functional differential equations $$ \dot{x}(t) = F(t,x^t) $$ where $F$ is a given functional, $x^t \in C([0,r],{\mathbb R}^n)$, $r>0$ and $x^t(\theta)=x(t+\theta)$, $\theta \in [0,r]$. Two different results on the existence of solutions, with coordinates bounded above and below by the coordinates of the given vector functions if $t\to\infty$, are proved using two different fixed-point principles. It is illustrated by examples that, applying both results simultaneously to the same equation yields two positive solutions asymptotically different for $t\to\infty$. The equation $$ \dot{x}(t) = \left(a+{b}/{t}\right)\,x(t+\tau) $$ where $a, \tau \in (0,\infty)$, $a<1/(\tau\e)$, $b \in {\mathbb R}$ are constants can serve as a linear example. The existence of a pair of positive solutions asymptotically different for $t\to\infty$ is proved and their asymptotic behavior is investigated. The results are also illustrated by a nonlinear equation.
Keywords
Advanced differential equation, monotone iterative method, Schauder-Tychonoff theorem, positive solution, asymptotic behavior of solutions, nonlinear system.
Released
06.03.2017
Publisher
John Wiley & Sons
ISBN
1099-1476
Periodical
Mathematical Methods in the Applied Sciences
Year of study
40
Number
3
State
GB
Pages from
1422
Pages to
1437
Pages count
16
URL
Documents
BibTex
@article{BUT137193,
author="Josef {Diblík} and Mária {Kúdelčíková}",
title="Two classes of asymptotically different positive solutions to advanced differential equations via two different fixed-point principles",
annote="The paper considers a system of advanced-type functional differential equations
$$
\dot{x}(t) = F(t,x^t)
$$
where $F$ is a given functional, $x^t \in C([0,r],{\mathbb R}^n)$, $r>0$
and $x^t(\theta)=x(t+\theta)$, $\theta \in [0,r]$.
Two different results on the existence of solutions, with coordinates
bounded above and below by the coordinates of the given vector functions
if $t\to\infty$,
are proved using two different fixed-point principles.
It is illustrated by examples that, applying both results simultaneously to the same equation
yields
two positive solutions asymptotically different for $t\to\infty$.
The equation
$$
\dot{x}(t) = \left(a+{b}/{t}\right)\,x(t+\tau)
$$
where $a, \tau \in (0,\infty)$, $a<1/(\tau\e)$, $b \in {\mathbb R}$ are constants
can serve as a linear example.
The existence of a pair of positive solutions
asymptotically different for $t\to\infty$ is proved
and their asymptotic behavior is investigated.
The results are also illustrated by a nonlinear equation.
",
address="John Wiley & Sons",
chapter="137193",
doi="10.1002/mma.4064",
howpublished="print",
institution="John Wiley & Sons",
number="3",
volume="40",
year="2017",
month="march",
pages="1422--1437",
publisher="John Wiley & Sons",
type="journal article in Web of Science"
}