The Lebesgue Integration. L p-Spaces. Sobolev Spaces

Svetlin G. Georgiev
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Svetlin G. Georgiev: Sofia University St Kliment Ohridski, Faculty of Mathematics & Info

Chapter Chapter 6 in Fuzzy Dynamic Equations, Dynamic Inclusions, and Optimal Control Problems on Time Scales, 2021, pp 361-434 from Springer

Abstract: Abstract In this chapter we define the Lebesgue measure and Lebesgue integral on time scales. We show the difference between the classical Lebesgue integral and the time scale Lebesgue integral. We introduce the absolutely continuous functions and deduct some of their properties. In this chapter we define functions of bounded variation and give some of their properties. We introduce L p spaces, Sobolev spaces, and generalized derivatives. As their applications, we investigate the weak solutions and Euler solutions of dynamic systems. We prove an analogue of the Gronwall type inequality. We define Δ × B $$\varDelta \times \mathcal {B}$$ -measurable set-valued functions and deduct some of their basic properties.

Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-76132-5_6

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DOI: 10.1007/978-3-030-76132-5_6

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