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.
",
}