### Hysteresis and Periodic Solutions of Semilinear and Quasilinear Wave Equations.

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The hypothesis that, on the macroscopic level, the accumulated fatigue of an elastoplastic material with kinematic hardening can be identified from the mathematical point of view with the dissipated energy, is used for the construction of a new constitutive elastoplastic fatigue model. Its analytical investigation characterizes conditions for the formation of singularities in a finite time. The corresponding constitutive law is then coupled with the dynamical equation of motion of a one-dimensional...

It is known that the vector stop operator with a convex closed characteristic $Z$ of class ${C}^{1}$ is locally Lipschitz continuous in the space of absolutely continuous functions if the unit outward normal mapping $n$ is Lipschitz continuous on the boundary $\partial Z$ of $Z$. We prove that in the regular case, this condition is also necessary.

This paper deals with the asymptotic behavior as $t\to \infty $ of solutions $u$ to the forced Preisach oscillator equation $\ddot{w}\left(t\right)+u\left(t\right)=\psi \left(t\right)$, $w=u+\mathcal{P}\left[u\right]$, where $\mathcal{P}$ is a Preisach hysteresis operator, $\psi \in {L}^{\infty}(0,\infty )$ is a given function and $t\ge 0$ is the time variable. We establish an explicit asymptotic relation between the Preisach measure and the function $\psi $ (or, in a more physical terminology, a balance condition between the hysteresis dissipation and the external forcing) which guarantees that every solution remains bounded for all times. Examples show...

Energy functionals for the Preisach hysteresis operator are used for proving the existence of weak periodic solutions of the one-dimensional systems of Maxwell equations with hysteresis for not too large right-hand sides. The upper bound for the speed of propagation of waves is independent of the hysteresis operator.

The recent development of mathematical methods of investigation of problems with hysteresis has shown that the structure of the hysteresis memory plays a substantial role. In this paper we characterize the hysteresis operators which exhibit a memory effect of the Preisach type (memory preserving operators). We investigate their properties (continuity, invertibility) and we establish some relations between special classes of such operators (Preisach, Ishlinskii and Nemytskii operators). For a general...

We prove the existence and find necessary and sufficient conditions for the uniqueness of the time-periodic solution to the equations ${u}_{tt}-{\Delta}_{x}u\pm F\left(u\right)=g(x,t)$ for an arbitrary (sufficiently smooth) periodic right-hand side $g$, where ${\Delta}_{x}$ denotes the Laplace operator with respect to $x\in \Omega \subset {R}^{N},N\ge 1$, and $F$ is the Ishlinskii hysteresis operator. For $N=2$ this equation describes e.g. the vibrations of an elastic membrane in an elastico-plastic medium.

The main goal of the paper is to formulate some new properties of the Ishlinskii hysteresis operator $F$, which characterizes e.g. the relation between the deformation and the stress in a non-perfectly elastic (elastico-plastic) material. We introduce two energy functionals and derive the energy inequalities. As an example we investigate the equation ${u}^{\text{'}\text{'}}+F\left(u\right)=0$ describing the motion of a mass point at the extremity of an elastico-plastic spring.

The paper deals with a model for water freezing in a deformable elastoplastic container. The mathematical problem consists of a system of one parabolic equation for temperature, one integrodifferential equation with a hysteresis operator for local volume increment, and one differential inclusion for the water content. The problem is shown to admit a unique global uniformly bounded weak solution.

We propose an extended version of the Kurzweil integral which contains both the Young and the Kurzweil integral as special cases. The construction is based on a reduction of the class of $\delta $-fine partitions by excluding small sets.

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