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Pachpatte inequalities on time scales. (English) Zbl 1086.34014
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).

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