### A hybrid iterative scheme for equilibrium problems, variational inequality problems, and fixed point problems in Banach spaces.

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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.

The asymptotic behaviour for $t\to \infty $ of the solutions to a one-dimensional model for thermo-visco-plastic behaviour is investigated in this paper. The model consists of a coupled system of nonlinear partial differential equations, representing the equation of motion, the balance of the internal energy, and a phase evolution equation, determining the evolution of a phase variable. The phase evolution equation can be used to deal with relaxation processes. Rate-independent hysteresis effects in the strain-stress...

We consider autonomous systems where two scalar differential equations are coupled with the input-output relationship of the Preisach hysteresis operator, which has an infinite-dimensional memory. A prototype system of this type is an LCR electric circuit where the inductive element has a ferromagnetic core with a hysteretic relationship between the magnetic field and the magnetization. Further examples of such systems include lumped hydrological models with two soil layers; they can also appear...

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.

Systems of operator-differential equations with hysteresis operators can have unstable equilibrium points with an open basin of attraction. Such equilibria can have homoclinic orbits attached to them, and these orbits are robust. In this paper a population dynamics model with hysteretic response of the prey to variations of the predator is introduced. In this model the prey moves between two patches, and the derivative of the Preisach operator is used to describe the hysteretic flow between the...

The paper deals with a scalar diffusion equation $c\phantom{\rule{0.166667em}{0ex}}{u}_{t}={\left(F\left[{u}_{x}\right]\right)}_{x}+f,$ where $F$ is a Prandtl-Ishlinskii operator and $c,f$ are given functions. In the diffusion or heat conduction equation the linear constitutive relation is replaced by a scalar Prandtl-Ishlinskii hysteresis spatially dependent operator. We prove existence, uniqueness and regularity of solution to the corresponding initial-boundary value problem. The problem is then homogenized by considering a sequence of equations of the above type with spatially periodic...

This paper is concerned with integral control of systems with hysteresis. Using an input-output approach, it is shown that application of integral control to the series interconnection of either (a) a hysteretic input nonlinearity, an L2-stable, time-invariant linear system and a non-decreasing globally Lipschitz static output nonlinearity, or (b) an L2-stable, time-invariant linear system and a hysteretic output nonlinearity, guarantees, under certain assumptions, tracking of constant reference...

On a closed convex set $Z$ in ${\mathbb{R}}^{N}$ with sufficiently smooth (${\mathcal{W}}^{2,\infty}$) boundary, the stop operator is locally Lipschitz continuous from ${\mathbf{W}}^{1,1}([0,T],{\mathbb{R}}^{N})\times Z$ into ${\mathbf{W}}^{1,1}([0,T],{\mathbb{R}}^{N})$. The smoothness of the boundary is essential: A counterexample shows that ${\mathcal{C}}^{1}$-smoothness is not sufficient.

Modification of behaviour in response to changes in the environment or ambient conditions, based on memory, is typical of the human and, possibly, many animal species.One obvious example of such adaptivity is, for instance, switching to a safer behaviour when in danger, from either a predator or an infectious disease. In human society such switching to safe behaviour is particularly apparent during epidemics. Mathematically, such changes of behaviour...

Hysteresis operators are illustrated, and a weak formulation is studied for an initial- and boundary-value problem associated to the equation $\left({\partial}^{2}/\partial {t}^{2}\right)\left[u+\mathcal{F}\left(u\right)\right]+Au=f$; here $\mathcal{F}$ is a (possibly discontinuous) hysteresis operator, $A$ is a second order elliptic operator, $f$ is a known function. Problems of this sort arise in plasticity, ferromagnetism, ferroelectricity, and so on. In particular an existence result is outlined.

We study systems of reaction-diffusion equations with discontinuous spatially distributed hysteresis on the right-hand side. The input of the hysteresis is given by a vector-valued function of space and time. Such systems describe hysteretic interaction of non-diffusive (bacteria, cells, etc.) and diffusive (nutrient, proteins, etc.) substances leading to formation of spatial patterns. We provide sufficient conditions under which the problem is well posed in spite of the assumed discontinuity of...

An existence and regularity theorem is proved for integral equations of convolution type which contain hysteresis nonlinearities. On the basis of this result, frequency-domain stability criteria are derived for feedback systems with a linear infinite-dimensional system in the forward path and a hysteresis nonlinearity in the feedback path. These stability criteria are reminiscent of the classical circle criterion which applies to static sector-bounded nonlinearities. The class of hysteresis operators...

L'attività di ricerca di chi scrive si è finora indirizzata principalmente verso l'esame dei modelli di transizione di fase, dei modelli di isteresi, e delle relative equazioni non lineari alle derivate parziali. Qui si illustrano brevemente tali problematiche, indicando alcuni degli elementi che le collegano tra di loro. Il lavoro è organizzato come segue. I paragrafi 1, 2, 3 vertono sulle transizioni di fase: si introducono le formulazioni forte e debole del classico modello di Stefan, e si illustrano...