CLASSIFICATIONS OF POSITIVE SOLUTIONS OF CERTAIN SECOND ORDER FUNCTIONAL DYNAMIC EQUATIONS

TitleCLASSIFICATIONS OF POSITIVE SOLUTIONS OF CERTAIN SECOND ORDER FUNCTIONAL DYNAMIC EQUATIONS
Publication TypeJournal Article
Year of Publication2017
AuthorsXU, ZHITING, YANG, LINLIN
Secondary TitleFunctional Dynamic Equations
Volume21
Issue2
Start Page283
Pagination20
Date Published03/2017
Type of Workscientific: mathematics
ISSN1083-2564
AMS34C10, 34K11, 39A10
Abstract

In this paper, we study the asymptotic behavior of the positive solutions of the following second order functional dynamic equation

\begin{equation*}
\big(r(t)|x^{\Delta}(t)|^{\alpha-1}x^{\Delta}(t)\big)^{\Delta}+f(t,x^{\sigma}(t),x^{\tau}(t))=0,
\end{equation*}

on a time scale $T$, where $\alpha>0$ is a constant, $\sigma(t)$ is the forward jump operator, and $x^{\tau}(t)=x(\tau(t))$. According to the cases, where

$$\int^{\infty}_{t_0}r^{-1/\alpha}(s)\Delta s=\infty\ \text{or}\ \int^{\infty}_{t_0}r^{-1/\alpha}(s)\Delta s<\infty$$

holds, all positive solutions of the above equation are classified into three types by means of their asymptotic behavior, respectively.

Our goal is to establish necessary and sufficient conditions for the existence of certain types of solutions of the dynamic equation. Furthermore, we apply the results obtained this paper to certain second order difference equations and obtain new corresponding results for the difference equations.

Finally, we give an example to illustrate the main results.
URLhttp://www.acadsol.eu/en/articles/21/2/8.pdf
DOI10.12732/caa.v21i2.8
Short TitleFunctional Dynamic Equations
Alternate JournalCAA
Refereed DesignationRefereed
Full Text

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