# American Institute of Mathematical Sciences

December  2017, 22(10): 3783-3795. doi: 10.3934/dcdsb.2017190

## Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions

 1 College of Science, Hohai University, Nanjing 210098, China 2 Institute of Mathematics, Czech Academy of Sciences, Žitná 25, CZ-11567 Praha 1, Czech Republic 3 College of Science, Hohai University, Nanjing 210098, China

* Corresponding author: Wei Liu

The hospitality of the Hohai University in Nanjing during the second author's stay in OctoberNovember 2016 is gratefully acknowledged

Received  November 2016 Revised  April 2017 Published  July 2017

Fund Project: This work was supported by the Program of High-end Foreign Experts of the SAFEA (No. GDW20163200216). The work of the second author was partially supported by the GACR GrantČ 15-12227S and RVO: 67985840.

It is well known that the Prandtl-Ishlinskii hysteresis operator is locally Lipschitz continuous in the space of continuous functions provided its primary response curve is convex or concave. This property can easily be extended to any absolutely continuous primary response curve with derivative of locally bounded variation. Under the same condition, the Prandtl-Ishlinskii operator in the Kurzweil integral setting is locally Lipschitz continuous also in the space of regulated functions. This paper shows that the Prandtl-Ishlinskii operator is still continuous if the primary response curve is only monotone and continuous, and that it may not even be locally Hölder continuous for continuously differentiable primary response curves.

Citation: Wei Liu, Pavel Krejčí, Guoju Ye. Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3783-3795. doi: 10.3934/dcdsb.2017190
##### References:

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The hospitality of the Hohai University in Nanjing during the second author's stay in OctoberNovember 2016 is gratefully acknowledged

##### References:
The memory curves $\lambda(r)$ (the bold solid line) and $\hat \lambda(r)$ (the thin solid line)
The memory curves $\lambda(r)$ (the solid line) and $\hat \lambda(r)$ (the {dashed} line)
The primary response curve $\psi_1$ (the bold solid line), its derivative $\psi_1'$ (the bold dashed line), and the piecewise linear regularization $\psi_2'$ of $\psi_1'$ (the thin solid line)
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