Hysteresis phenomena arise quite often, for example during magnetization,
deformation of plastic materials or during adaption of shares in the market
of mobile phone providers following changes of prices. Here, we have to deal
with processes generating output values that do not depend only on the
current value of the input value but also on former values, such that one
can observe loops in the corresponding input-output diagrams.
The so called hysteresis operators are used for the mathematical modeling of
such effects. The scalar hysteresis operators are defined in the lecture and
some examples (stop, play, Prandtl-Ishlinskii, Preisach) are presented.
The analytic properties of these operators (continuity, piecewise monotonicity)
and their memory properties are investigated.
Finally some evolution equations will be presented wherein hysteresis operators
replace the simple functional dependencies.
It will be shown how one can prove existence and uniqueness
of this equations even if the hysteresis are not differentiable.