##
**Half-linear dynamic equations.**
*(English)*
Zbl 1056.34049

Agarwal, Ravi P. (ed.) et al., Nonlinear analysis and applications: To V. Lakshmikantham on his 80th birthday. Vol. 1. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1711-1/hbk). 1-57 (2003).

This paper deals with second-order half-linear dynamic equations
\[
(r(t)\Phi(y^\Delta))^\Delta+p(t)\Phi(y^\sigma)=0\tag{\(*\)}
\]
on arbitrary time scales (closed subset of the reals), where \(\Phi(y)=| y| ^{\alpha-1}\text{sgn\,}y\) with \(\alpha>1\). In this generalized framework unifying corresponding results for ODEs and difference equations, \(y^\Delta\) denotes the \(\Delta\)-derivative of \(y\) and \(y^\sigma\) the composition of \(y\) with the forward jump operator \(\sigma\). Moreover, the coefficient functions \(r,p\) are assumed to be rd-continuous and \(r\) is only required to be nonzero (and not necessarily positive).

Beginning with a quite comprehensive introduction, some basic results on derivatives and integrals on time scales are presented, including Gronwall’s inequality, existence theorems for dynamic equations and a second mean value theorem for time scale integrals, before the author addresses the question of global existence and uniqueness for solutions of \((\ast)\). Then a generalized Picone identity is derived and 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 continues with oscillation (Leighton-Wintner, Hinton-Lewis) and various nonoscillation criteria, before further comparison theorems (e.g., of Leighton-type) are established. Some of them are based on Schauder’s fixed-point theorem. Finally, two examples illustrate the results.

For the entire collection see [Zbl 1030.00016].

Beginning with a quite comprehensive introduction, some basic results on derivatives and integrals on time scales are presented, including Gronwall’s inequality, existence theorems for dynamic equations and a second mean value theorem for time scale integrals, before the author addresses the question of global existence and uniqueness for solutions of \((\ast)\). Then a generalized Picone identity is derived and 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 continues with oscillation (Leighton-Wintner, Hinton-Lewis) and various nonoscillation criteria, before further comparison theorems (e.g., of Leighton-type) are established. Some of them are based on Schauder’s fixed-point theorem. Finally, two examples illustrate the results.

For the entire collection see [Zbl 1030.00016].

Reviewer: Christian Poetzsche (Minneapolis)

### MSC:

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

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

34B24 | Sturm-Liouville theory |

39A11 | Stability of difference equations (MSC2000) |