关于Henri Cartan向量中值定理及其对Lipschitz算子和广义Lebesgue-Bochner-Stieltjes积分理论的应用
On Henri Cartan's vectorial mean-value theorem and its applications to Lipschitzian operators and generalized Lebesgue-Bochner-Stieltjes integration theory
摘要：
H.卡尔丹在他的书中证明了微分定理推广柯西的中值定理的情况下，职能采取的价值在巴拿赫空间。嘉当利用这一定理在一个熟练的方式发展整个理论的微分和理论的微分方程在有限和无限维巴拿赫空间。证明了这个定理的一个推广，即当涉及导数的不等式除一个勒贝格测度为零的集合外处处成立，且导数用较弱的导数代替时.即分别是右侧Lipschitz导数和右下Dini导数。他还提出了应用定理的研究Lipschitz算子在巴拿赫空间。Lipschitz算子在电动力学的n体问题中起着关键作用，在爱因斯坦狭义相对论的一般n体问题中也是如此。参考文献见Bogdan arXiv：0909.5240和arXiv：0910.0538。利用Cartan定理的推广，证明了一类Bochner可和函数的微积分基本定理的一个形式。在这个过程中，他向读者介绍了广义理论的勒贝格，博赫纳-Stieltjes积分和勒贝格和博赫纳空间的可和函数作为开发的波格丹诺维奇。\cite{bogdan10}--\cite{bogdan23}。
英文摘要：
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}.
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