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A hybrid model is proposed for characterization of the hysteresis nonlinearity of the well-known play operator. The proposed model holds the hysteresis nonlinearity and the memory effects of the play operator. Simulation results are also presented to show the capability of the hybrid model to present the hysteresis nonlinearity with memory effects.

We extend the well-known inversion result for Prandtl–Ishlinskii operators to the case of time dependent thresholds. The exact inversion formula is shown to hold under the condition that the distances between the thresholds do not decrease in time. Examples show that this threshold dilation condition cannot be completely omitted.

The convexity inequality for small amplitude loops is used, together with the strict local monotonicity of large amplitude branches in the Preisach model, to derive a global second-order energy bound. Such an estimate is then used for solving a singular Preisach oscillator problem.

In some recent papers, the concept of outward pointing operators has been used as a tool within the mathematical investigation of equations involving hysteresis operators, and has been considered for the stop operator, the play operator, and Prandtl–Ishlinskii operators. Now, for Preisach operators, conditions will be discussed which ensure that this condition is satisfied.

The paper deals with the long-time behaviour of evolution systems involving hysteresis operators. Introducing the concept of outwards pointing hysteresis operators allows to generalize methods that have been used to derive uniform a priori bounds for solutions to equations involving the superposition with nonlinear functions. This is done to derive a large-time asymptotic result for a coupled system of PDEs and ODEs modelling nonlinear thermo-visco-plastic developments. Moreover, a class of generalized Prandtl–Ishlinskii operator will be introduced and results concerning the thermodynamic consistency and the outwards pointing property of these operators will be presented.

The explicit inversion formula for rate dependent Prandtl–Ishlinskii operators is extended to cases without the threshold dilation condition. This solves a problem in hysteresis and creep modeling of magnetostrictive behavior.

In this paper, a class of multi-dimensional hysteresis operators of generalized Prandtl–Ishlinskii form is introduced which can be viewed as a possibility to extend the famous one-dimensional Preisach model of hysteresis to higher space dimensions. We study the continuity properties of these hysteresis operators and of the hysteresis potential operators associated with them. It is also indicated, how this class of hysteresis operators can be applied to a class of phase-field systems occurring in the theory of phase transitions with non-conserved order parameters.