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**Bounded variation in time.**
*(English)*
Zbl 0657.28008

Topics in nonsmooth mechanics, 1-74 (1988).

[For the entire collection see Zbl 0646.00014.]

The motivation for this exposition is the study of evolution problems which are not smooth enough to be formulated as differential equations. If the evolution in the system’s state space may be modeled as having a locally bounded variation the differential measure provides a generalization of the notion of derivative, and the system may be described by a measure differential inclusion. Functions of locally bounded variation with values in some Banach space are defined, as well as the associated differential measure. The approach is via duality theory, and the relation to the set-theoretic measure theory is established as well as the connection to the derivative in the sense of distributions. The important Radon-Nikodým theorem together with some decomposition properties conclude the survey.

The motivation for this exposition is the study of evolution problems which are not smooth enough to be formulated as differential equations. If the evolution in the system’s state space may be modeled as having a locally bounded variation the differential measure provides a generalization of the notion of derivative, and the system may be described by a measure differential inclusion. Functions of locally bounded variation with values in some Banach space are defined, as well as the associated differential measure. The approach is via duality theory, and the relation to the set-theoretic measure theory is established as well as the connection to the derivative in the sense of distributions. The important Radon-Nikodým theorem together with some decomposition properties conclude the survey.

Reviewer: H.Matthies

### MSC:

28B05 | Vector-valued set functions, measures and integrals |

46G10 | Vector-valued measures and integration |