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

journal article in Web of Science

English

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.

Advanced differential equation, monotone iterative method, Schauder-Tychonoff theorem, positive solution, asymptotic behavior of solutions, nonlinear system.

DIBLÍK, J.; KÚDELČÍKOVÁ, M.

6. 3. 2017

John Wiley & Sons

1099-1476

Mathematical Methods in the Applied Sciences

40

3

United Kingdom of Great Britain and Northern Ireland

1422

1437

16

http://onlinelibrary.wiley.com/doi/10.1002/mma.4064/full