Asymptotically stable almost-periodic oscillations in systems with hysteresis nonlinearities (Q1578915)

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.