Asymptotic properties of third order functional dynamic equations on time scales

I. Kubiaczyk¹ ; S. H. Saker²

¹ Faculty of Mathematics and Computer Science Adam Mickiewicz University Umultowska 87 61-614 Poznań, Poland
² Department of Mathematics Skills, PY King Saud University Riyadh 11451, Saudi Arabia and Department of Mathematics Faculty of Science Mansoura University Mansoura, 35516, Egypt

Annales Polonici Mathematici, Tome 100 (2011) no. 3, pp. 203-222

Résumé

The purpose of this paper is to study the asymptotic properties of nonoscillatory solutions of the third order nonlinear functional dynamic equation $$ [ p(t)[ (r(t)x^{\Delta }(t))^{\Delta }] ^{\gamma }] ^{\Delta }+q(t)f(x(\tau (t)))=0,\quad\ t\geq t_{0}, $$ on a time scale $\mathbb{T}$, where $\gamma >0$ is a quotient of odd positive integers, and $p$, $q$, $r$ and $\tau $ are positive right-dense continuous functions defined on $\mathbb{T}$. We classify the nonoscillatory solutions into certain classes $C_{i}$, $i=0,1,2,3$, according to the sign of the $\Delta $-quasi-derivatives and obtain sufficient conditions in order that $C_{i}=\emptyset .$ Also, we establish some sufficient conditions which ensure the property $A$ of the solutions. Our results are new for third order dynamic equations and involve and improve some results previously obtained for differential and difference equations. Some examples are worked out to demonstrate the main results.

Détail

DOI : 10.4064/ap100-3-1

Keywords: purpose paper study asymptotic properties nonoscillatory solutions third order nonlinear functional dynamic equation delta delta gamma delta x tau quad geq time scale mathbb where gamma quotient odd positive integers tau positive right dense continuous functions defined mathbb classify nonoscillatory solutions certain classes according sign delta quasi derivatives obtain sufficient conditions order emptyset establish sufficient conditions which ensure property solutions results third order dynamic equations involve improve results previously obtained differential difference equations examples worked out demonstrate main results