Pachpatte inequalities on time scales. (English) Zbl 1086.34014

Akin-Bohner, Elvan; Bohner, Martin; Akin, Faysal

JIPAM, J. Inequal. Pure Appl. Math. 6, No. 1, Paper No. 6, 23 p. (2005).

The authors prove a variety of inequalities within the context of the calculus on time scales. Here, the involved functions are assumed to be defined on arbitrary closed subsets \({\mathbb T}\) of the reals and the crucial notion is the so-called delta derivative generalizing the usual derivative (for \({\mathbb T}={\mathbb R}\)) and the forward difference operator (for \({\mathbb T}={\mathbb Z}\)).
More precisely, comparison principles are used to derive time scale versions of certain inequalities, which in the special cases \({\mathbb T}={\mathbb R}\) or \({\mathbb T}={\mathbb Z}\) date back to Gronwall, Gamidov (or in the discrete case, to Pachpatte), Gollwitzer, Norbury and Stuart (Volterra-type inequalities), Green (in the discrete case, to Pachpatte) and Ma. These results are supplemented by several corollaries. Finally, also inequalities involving first- and second-order delta derivatives are addressed (originally due to Pachpatte).

Reviewer: Christian Pötzsche (Minneapolis)

MSC:

34A40	Differential inequalities involving functions of a single real variable
39A10	Additive difference equations
39A13	Difference equations, scaling (\(q\)-differences)

Keywords:

time scales; Pachpatte inequalities; dynamic inequalities