Asymptotic properties of third order functional dynamic equations on time scales.

*(English)* Zbl 1218.34086
Summary: 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 $\Bbb{T}$, where $\gamma >0$ is a quotient of odd positive integers, and $p$, $q$, $r$ and $\tau $ are positive right-dense continuous functions defined on $\Bbb{T}$. 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 $C_{i}\ne\emptyset .$ 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.

##### MSC:

34K25 | Asymptotic theory of functional-differential equations |

34K11 | Oscillation theory of functional-differential equations |

34N05 | Dynamic equations on time scales or measure chains |