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DCDS

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,

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.

*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.

DCDS-B

Suggested by conversations in 1991 (Mark Krasnosel'skiĭ and Aleksei Pokrovskiĭ with TIS), this paper generalizes earlier work [7] of theirs
by defining a setting of hybrid systems with isotone switching rules
for a partially ordered set of modes and then obtaining a periodicity
result in that context. An application is given to a partial
differential equation modeling calcium release and diffusion in
cardiac cells.

keywords:
fixed point
,
isotone
,
Periodic
,
discontinuous
,
hybrid system
,
calcium waves.
,
hysteresis

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