Oscillation criteria for half-linear delay dynamic equations on time scales. (English) Zbl 1173.34037

Erbe, Lynn; Hassan, Taher S.; Peterson, Allan; Saker, Samir H.

Nonlinear Dyn. Syst. Theory 9, No. 1, 51-68 (2009).

Summary: This paper is concerned with oscillation of the second-order halflinear delay dynamic equation
\[ (r(t)(x^\Delta)^\gamma)^\Delta+p(t)x^\gamma(\tau(t))=0, \]
on a time scale \(\mathbb T\), where \(\gamma\geq1\) is the quotient of odd positive integers, \(p(t)\), and \(\tau :\mathbb T\to\mathbb T\) are positive rd-continuous functions on \(\mathbb T\), \(r(t)\) is positive and (delta) differentiable, \(\tau(t)\leq t\), and \(\lim_{t\to\infty}r(t) =\infty\). We establish some new sufficient conditions which ensure that every solution oscillates or converges to zero. Our results in the special cases when \(\mathbb T =\mathbb R\) and \(\mathbb T =\mathbb N\) involve and improve some oscillation results for second-order differential and difference equations; and when \(\mathbb T = h\mathbb N\), \(\mathbb T = q^{\mathbb N_0}\) and \(\mathbb T =\mathbb N^2\) our oscillation results are essentially new. Some examples illustrating the importance of our results are also included.

MSC:

34K11	Oscillation theory of functional-differential equations
39A10	Additive difference equations

Keywords:

oscillation; delay half-linear dynamic equations; time scales