On the oscillation of certain class of third-order nonlinear delay differential equations. (English) Zbl 1224.34217

Saker, S. H.; Džurina, J.

Math. Bohem. 135, No. 3, 225-237 (2010).

Summary: We consider the third-order nonlinear delay differential equation \[ ( a(t)\left ( x''(t)\right ) ^{\gamma })' +q(t)x^{\gamma }(\tau (t))=0,\quad t\geq t_{0}, \] where \(a\), \(q\) are positive functions, \(\gamma >0\) is a quotient of odd positive integers and the delay function \(\tau (t)\leq t\) satisfies \(\lim _{t\rightarrow \infty }\tau (t)=\infty \). We establish some sufficient conditions which ensure that the above equation is oscillatory or the solutions converge to zero. Our results in the nondelay case extend and improve some known results and in the delay case the results can be applied to new classes of equations which are not covered by known criteria. Some examples are considered to illustrate the main results.

MSC:

34K11	Oscillation theory of functional-differential equations
34K25	Asymptotic theory of functional-differential equations

Keywords:

third-order differential equation; oscillation; nonoscillation; disconjugacy