Half-linear dynamic equations on time scales: IVP and oscillatory properties.

*(English)*Zbl 1037.34002In this long article, it is shown that the basic results of oscillation theory for the classical Sturm-Liouville linear differential equation
\[
(r(t)x')'+p(t)x=0
\]
can be extended to so-called half-linear dynamic equations
\[
(r(t)\Phi(x^\Delta))^\Delta+p(t)\Phi(x^\sigma)=0 \tag \(*\)
\]
on arbitrary time scales, where \(\Phi(x)=| x| ^{\alpha-1}\text{sgn}\,x\) with \(\alpha>0\). Here, a time scale is any closed subset of the reals, \(x^\Delta\) denotes the \(\Delta\)-derivative of \(x\) and \(x^\sigma\) the composition of \(x\) with the forward jump operator \(\sigma\). Additionally, the coefficient functions \(r,p\) are rd-continuous and \(r\) is only required to be nonzero (and not necessarily positive).

After a thorough motivational introduction, some basic results on derivatives and integrals on time scales are presented, including a second mean value theorem for time scale integrals, before the questions of existence and uniqueness for solutions to \((\ast)\) is addressed. The author derives a generalized Picone identity, which is used to prove a version of Reid’s Roundabout theorem characterizing the disconjugacy of \((\ast)\) using Riccati-type dynamic equations or definiteness-properties of quadratic functionals. This Roundabout theorem, in turn, yields comparison and separation theorems of Sturm type for \((\ast)\). Moreover, the Roundabout theorem is utilized to obtain necessary and sufficient conditions for equation \((\ast)\) to be nonoscillatory via a Riccati-technique, as well as a variational principle. The paper ends with oscillation (Leighton-Wintner, Hinton-Lewis) and nonoscillation criteria.

After a thorough motivational introduction, some basic results on derivatives and integrals on time scales are presented, including a second mean value theorem for time scale integrals, before the questions of existence and uniqueness for solutions to \((\ast)\) is addressed. The author derives a generalized Picone identity, which is used to prove a version of Reid’s Roundabout theorem characterizing the disconjugacy of \((\ast)\) using Riccati-type dynamic equations or definiteness-properties of quadratic functionals. This Roundabout theorem, in turn, yields comparison and separation theorems of Sturm type for \((\ast)\). Moreover, the Roundabout theorem is utilized to obtain necessary and sufficient conditions for equation \((\ast)\) to be nonoscillatory via a Riccati-technique, as well as a variational principle. The paper ends with oscillation (Leighton-Wintner, Hinton-Lewis) and nonoscillation criteria.

Reviewer: Christian Pötzsche (Augsburg)

##### MSC:

34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |

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

34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |

39A10 | Additive difference equations |