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Keywords:
Hake-variational McShane integral; variational McShane integral; Banach space; $m$-dimensional Euclidean space
Hake-variational McShane integral; variational McShane integral; Banach space; $m$-dimensional Euclidean space
Summary:
We define the Hake-variational McShane integral of Banach space valued functions defined on an open and bounded subset $G$ of $m$-dimensional Euclidean space $\mathbb {R}^{m}$. It is a "natural" extension of the variational McShane integral (the strong McShane integral) from $m$-dimensional closed non-degenerate intervals to open and bounded subsets of $\mathbb {R}^{m}$. We will show a theorem that characterizes the Hake-variational McShane integral in terms of the variational McShane integral. This theorem reduces the study of our integral to the study of the variational McShane integral. As an application, a full descriptive characterization of the Hake-variational McShane integral is presented in terms of the cubic derivative.
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