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**Differential models of hysteresis.**
*(English)*
Zbl 0820.35004

Applied Mathematical Sciences. 111. Berlin: Springer-Verlag,. xi, 407 p. (1994).

Although hysteresis effects have been studied by physicists and engineers for more than one hundred years, it was not before the sixties of this century that mathematicians started to develop a functional approach to describe hysteresis phenomena. Beginning with a series of papers by Krasnosel’skij and his co-workers and then the fundamental monograph of Krasnosel’skij and Prokrovskij, more and more mathematicians took interest in this field. The author has been concerned with the mathematical investigation of hysteresis effects for more than fifteen years. In this monograph, he gives a thorough insight into the mathematical research in this field.

The book is divided into two parts. The first part deals with a number of mathematical models descrbing hysteresis phenomena. The first two chapters are devoted to presenting some basic models for hysteresis, like the play between two mechanical elements, the Prandtl model of elasto- plasticity and the concept of rheological models as combinations of elementary models for mechanical properties such as elasticity, viscosity and plasticity.

In chaps. III to VI hysteresis operators, defined as rate-independent Volterra operators, are introduced. For the models of Prandtl-Ishlinskij, Preisach and Duhem, and numerous extensions, mathematical properties as continuity in different spaces, piecewise monotonicity and existence of the inverse are investigated in great detail.

In the second part of the book PDEs with nonlinearities of hysteresis- type are investigated. While chap. VII deals with PDE models for elasto- plasticity in terms of variational inequalities, chap. VIII offers a semigroup approach. The following two chapters are concerned with quasi- and semilinear parabolic equations with memory. In chap. XI several PDE models with discontinuous hysteresis, arising for instance in biology, are discussed. In a closing chapter some mathematical tools concerning nonlinear operators in Banach spaces, nonlinear semigroup theory and convergence properties in BV spaces are collected. Each chapter starts with an outline and necessary prerequisites and ends with comments and open problems.

The book can only be warmly recommended to every mathematician who is interested in the state of research in mathematical properties of hysteresis operators, as well as in PDEs having nonlinearities of hysteresis type.

The book is divided into two parts. The first part deals with a number of mathematical models descrbing hysteresis phenomena. The first two chapters are devoted to presenting some basic models for hysteresis, like the play between two mechanical elements, the Prandtl model of elasto- plasticity and the concept of rheological models as combinations of elementary models for mechanical properties such as elasticity, viscosity and plasticity.

In chaps. III to VI hysteresis operators, defined as rate-independent Volterra operators, are introduced. For the models of Prandtl-Ishlinskij, Preisach and Duhem, and numerous extensions, mathematical properties as continuity in different spaces, piecewise monotonicity and existence of the inverse are investigated in great detail.

In the second part of the book PDEs with nonlinearities of hysteresis- type are investigated. While chap. VII deals with PDE models for elasto- plasticity in terms of variational inequalities, chap. VIII offers a semigroup approach. The following two chapters are concerned with quasi- and semilinear parabolic equations with memory. In chap. XI several PDE models with discontinuous hysteresis, arising for instance in biology, are discussed. In a closing chapter some mathematical tools concerning nonlinear operators in Banach spaces, nonlinear semigroup theory and convergence properties in BV spaces are collected. Each chapter starts with an outline and necessary prerequisites and ends with comments and open problems.

The book can only be warmly recommended to every mathematician who is interested in the state of research in mathematical properties of hysteresis operators, as well as in PDEs having nonlinearities of hysteresis type.

Reviewer: J.Sprekels (Berlin)

### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |

47J05 | Equations involving nonlinear operators (general) |

58C07 | Continuity properties of mappings on manifolds |