The paper deals with a scalar diffusion equation c ut = (F[ux])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 data cε and ηε when the spatial period ε tends to zero. The homogenized characteristics c∗ and η∗ are identified and the convergence of the corresponding solutions to the solution of the homogenized equation is proved.
Certain financial market strategies are known to exhibit a hysteretic structure similar to the memory observed in plasticity, ferromagnetism, or magnetostriction. The main difference is that in financial markets, the spontaneous occurrence of discontinuities in the time evolution has to be taken into account. We show that one particular market model considered here admits a representation in terms of Prandtl-Ishlinskii hysteresis operators, which are extended in order to include possible discontinuities both in time and in memory. The main analytical tool is the Kurzweil integral formalism, and the main result proves the well-posedness of the process in the space of right-continuous regulated functions.