Detail publikace

Two classes of asymptotically different positive solutions to advanced differential equations via two different fixed-point principles

Originální název

Two classes of asymptotically different positive solutions to advanced differential equations via two different fixed-point principles

Anglický název

Two classes of asymptotically different positive solutions to advanced differential equations via two different fixed-point principles

Jazyk

en

Originální abstrakt

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.

Anglický abstrakt

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.

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"
}