# 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:

show all references

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)
 [1] Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021020 [2] Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247 [3] Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2739-2776. doi: 10.3934/dcds.2020384 [4] Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3109-3140. doi: 10.3934/dcds.2020400 [5] Sara Munday. On the derivative of the $\alpha$-Farey-Minkowski function. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 709-732. doi: 10.3934/dcds.2014.34.709 [6] Joel Fotso Tachago, Giuliano Gargiulo, Hubert Nnang, Elvira Zappale. Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting. Evolution Equations & Control Theory, 2021, 10 (2) : 297-320. doi: 10.3934/eect.2020067 [7] Oleksandr Boichuk, Victor Feruk. Boundary-value problems for weakly singular integral equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021094 [8] Yumi Yahagi. Construction of unique mild solution and continuity of solution for the small initial data to 1-D Keller-Segel system. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021099 [9] Craig Cowan. Supercritical elliptic problems involving a Cordes like operator. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021037 [10] John Villavert. On problems with weighted elliptic operator and general growth nonlinearities. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021023 [11] Ralf Hielscher, Michael Quellmalz. Reconstructing a function on the sphere from its means along vertical slices. Inverse Problems & Imaging, 2016, 10 (3) : 711-739. doi: 10.3934/ipi.2016018 [12] Raimund Bürger, Christophe Chalons, Rafael Ordoñez, Luis Miguel Villada. A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function. Networks & Heterogeneous Media, 2021, 16 (2) : 187-219. doi: 10.3934/nhm.2021004 [13] Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137 [14] Elena K. Kostousova. External polyhedral estimates of reachable sets of discrete-time systems with integral bounds on additive terms. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021015 [15] Sergey E. Mikhailov, Carlos F. Portillo. Boundary-Domain Integral Equations equivalent to an exterior mixed BVP for the variable-viscosity compressible Stokes PDEs. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1103-1133. doi: 10.3934/cpaa.2021009 [16] Claudia Lederman, Noemi Wolanski. An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2907-2946. doi: 10.3934/dcds.2020391 [17] Markus Harju, Jaakko Kultima, Valery Serov, Teemu Tyni. Two-dimensional inverse scattering for quasi-linear biharmonic operator. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021026 [18] Zhaoqiang Ge. Controllability and observability of stochastic implicit systems and stochastic GE-evolution operator. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021009 [19] Zhang Chao, Minghua Yang. BMO type space associated with Neumann operator and application to a class of parabolic equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021104 [20] Joel Coacalle, Andrew Raich. Compactness of the complex Green operator on non-pseudoconvex CR manifolds. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021061

2019 Impact Factor: 1.27