Vitali Lemma approach to differentiation on a time scale

Chuan Jen Chyan ¹   ; Andrzej Fryszkowski ²

¹ Department of Mathematics Tamkang University Taipei 251, Taiwan
² Faculty of Mathematics and Information Science Warsaw University of Technology Plac Politechniki 1 00-661 Warszawa, Poland

Studia Mathematica, Tome 162 (2004) no. 2, pp. 161-173

Résumé

A new approach to differentiation on a time scale ${{\mathbb T}}$ is presented. We give a suitable generalization of the Vitali Lemma and apply it to prove that every increasing function $f:{{\mathbb T}}\rightarrow {\mathbb R}$ has a right derivative $f_{+}^{\prime } ( x) $ for $\mu _{\Delta } $-almost all $x\in {{\mathbb T}}$. Moreover, $\int _{[ a,b) }f_{+}^{\prime } ( x) \kern .16667em d\mu _{\Delta }\leq f ( b) -f ( a) .$

Détail

DOI : 10.4064/sm162-2-4

Keywords: approach differentiation time scale mathbb presented suitable generalization vitali lemma apply prove every increasing function mathbb rightarrow mathbb has right derivative prime delta almost mathbb moreover int prime kern delta leq f