## Asymptotically stable almost-periodic oscillations in systems with hysteresis nonlinearities.(English)Zbl 0972.47064

This article deals with forced almost periodic oscillations in nonlinear systems of type $x'= f(t,x)+ \varepsilon g(x,z(t)),\quad z(t)= (\Gamma(t_0, z(t_0))Lx)(t);\tag{1}$ here $$L: \mathbb{R}^d\to\mathbb{R}^m$$ is a linear mapping and $$\Gamma(t_0,z_0)$$ an operator with initial memory, $$z_0\in Z$$, which transforms functions $$u:[t_0,\infty)\to \mathbb{R}^m$$ to functions $$z: [t_0,\infty)\to Z$$ ($$Z$$ is a Banach space or its subset); it is assumed that $$\Gamma(t_0, z_0)$$ satisfies the Volterra property, the semigroup property and is autonomous. The main result presents the description of conditions for $$f(t,x)$$ and $$\Gamma(t_0, z_0)$$ under which (1) has a unique almost periodic solution $$x_\varepsilon$$ closed to a fixed almost periodic solution $$x_0$$ of $$x'= f(t,x)$$.
As applications, systems with one-dimensional nonlinearities $$f(t,x)$$ and with smooth nonlinearities $$f(t,x)$$ are considered; in the second case one can study the stability of almost periodic solutions discussed above.

### MSC:

 47J40 Equations with nonlinear hysteresis operators 34D05 Asymptotic properties of solutions to ordinary differential equations
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### References:

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