Thus, before discussing the proposed research in detail, it is worthwhile to describe and summarize the main results achieved in the course of the research work under the above contract.The ongoing research has largely been forcused on the development of mathematical models 6f hysteretic nonlinearities with "nonlocal memories". The distinct feature of these nonlinearities is that their current states depend on past histories of input variations. It turns out that memories of hysteretic nonlinearities are quite selective. Indeed, experimentsshow that only some past input extrema (not the entire input variations) leave their marks upon future states of hysteretic nonlinearities. Thus special mathematical tools are need in order to describe nonlocal selective memories of hysteretic nonlinearities. The origin of such tools can be traced back to the landmark paper of Preisach.Our research has been primarily concerned with Preisach-type models of hysteresis.All these models have a common generic feature; they are constructed as superpositions of simplest hysteretic noiilinearities-rectangular loops. During the past four years, our study has been by and large centered around the folloing topics:Further development of scalar and vector Preisacli-type models of hysteresis.e Experimental testing of Preisach-type models of hysteresis.Development of new models for viscosity (aftereffect) in hysteretic systems. Next, I shall briefly described the main scientific results obtained in the areas outlined A §T above.

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ABSTRACT. For an m-dimensional differential inclusion of the form ie A(t)x(t) + F[t,x(t)],with A and F T-periodic in t, we prove the existence of a nonconstant periodic solution. Our hypotheses require m to be odd, and require F to have different growth behavior for |i| small and |i| large (often the case in applications). The idea is to guarantee that the topological degree associated with the system has different values on two distinct neighborhoods of the origin.We prove the existence of a nonconstant T-periodic solution to a time-dependent differential inclusion of the formWe assume the following: In the following, the maps A(-) and F(-,x) will be considered extended from the interval [0, T] to the real line R by T-periodicity. (L~)m is the Banach space of functions in L1^,T],Rm), extended to R by T-periodicity. In the same way ÍACt)™ and (Gr)m are the Banach spaces of T-periodic functions which are absolutely continuous and continuous endowed with the usual norms ||x||>ic and HxHoo respectively.

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