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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)
##### Keywords:
time scales; Pachpatte inequalities; dynamic inequalities
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