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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.

##### MSC:
 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
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##### References:
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