# American Institute of Mathematical Sciences

June  2015, 35(6): 2591-2614. doi: 10.3934/dcds.2015.35.2591

## On the representation of hysteresis operators acting on vector-valued, left-continuous and piecewise monotaffine and continuous functions

 1 Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, 10117 Berlin

Received  December 2013 Revised  June 2014 Published  December 2014

In Brokate-Sprekels 1996, it is shown that hysteresis operators acting on scalar-valued, continuous, piecewise monotone input functions can be represented by functionals acting on alternating strings. In a number of recent papers, this representation result is extended to hysteresis operators dealing with input functions in a general topological vector space. The input functions have to be continuous and piecewise monotaffine, i.e. being piecewise the composition of two functions such that the output of a monotone increasing function is used as input for an affine function.
In the current paper, a representation result is formulated for hysteresis operators dealing with input functions being left-continuous and piecewise monotaffine and continuous. The operators are generated by functions acting on an admissible subset of the set of all strings of pairs of elements of the vector space.
Citation: Olaf Klein. On the representation of hysteresis operators acting on vector-valued, left-continuous and piecewise monotaffine and continuous functions. Discrete & Continuous Dynamical Systems, 2015, 35 (6) : 2591-2614. doi: 10.3934/dcds.2015.35.2591
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##### References:
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