Asymptotic properties of third order functional dynamic equations on time scales

Volume 100 / 2011

I. Kubiaczyk, S. H. Saker Annales Polonici Mathematici 100 (2011), 203-222 MSC: 34K11, 39A10, 39A99. DOI: 10.4064/ap100-3-1


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.


  • I. KubiaczykFaculty of Mathematics and Computer Science
    Adam Mickiewicz University
    Umultowska 87
    61-614 Poznań, Poland
  • S. H. SakerDepartment of Mathematics Skills, PY
    King Saud University
    Riyadh 11451, Saudi Arabia
    Department of Mathematics
    Faculty of Science
    Mansoura University
    Mansoura, 35516, Egypt

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