On Henri Cartan's vectorial mean-value theorem and its applications to Lipschitzian operators and generalized Lebesgue-Bochner-Stieltjes integration theory

H. Cartan in his book on differential calculus proved a theorem generalizing a Cauchy's mean-value theorem to the case of functions taking values in a Banach space. Cartan used this theorem in a masterful way to develop the entire theory of differential calculus and theory of differential equations in finite and infinite dimensional Banach spaces. The author proves a generalization of this theorem to the case when the inequality involving the derivatives holds everywhere with exception of a set of Lebesgue measure zero, and the derivatives are replaced by weaker derivatives. Namely the right-sided Lipschitz derivative and lower right-sided Dini derivative, respectively. He also presents applications of the theorem to the study of Lipschitzian operators in Banach spaces. Lipschitzian operators played pivotal role in the n-body problems of electrodynamics, as also in general n-body problem of Einstein's special theory of relativity. For references see Bogdan arXiv:0909.5240 and arXiv:0910.0538. Using the generalization of Cartan's theorem the author proves a version of the fundamental theorem of calculus in a class of Bochner summable functions. In the process he introduces the reader to the generalized theory of Lebesgue-Bochner-Stieltjes integral and Lebesgue and Bochner spaces of summable functions as developed by Bogdanowicz. \cite{bogdan10}--\cite{bogdan23}.