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On locally isotone rate independent operators. (English) Zbl 1152.47059
The extension of rate-independent operators (for instance, hysteresis operators) to domains of non continuous functions is an important issue for applications as well as a challenging problem from an analytical point of view.
In this interesting paper, the author, using a previous sufficiency result from his previous work, gives a complete characterization of those rate independent operators which admit a continuous extension to the space of functions of bounded variation. In particular, the result may be expressed as follows. Let us be given of a set of continuous time-dependent functions of bounded variation, containing the set of Lipschitz functions. Then a rate-independent operator on it, which is continuous with respect to the strict topology of \(BV\), and which maps Lipschitz functions to continuous functions, may be continuously extended to \(BV\) if and only if it is locally isotone.
Here, isotone means that the operator locally maintains the monotonicity (but not necessary the sign of it). Also, a representation of such extensions is given.

47J40 Equations with nonlinear hysteresis operators
47J99 Equations and inequalities involving nonlinear operators
26B30 Absolutely continuous real functions of several variables, functions of bounded variation
Full Text: DOI
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