On the distribution of periodic solutions to nonlinear equations (Q2755267)

Let \(B\) be a complex separable Banach space, and let \(L(B)\) be a Banach space of linear bounded operators acting in \(B\). The author considers the equation NEWLINE\[NEWLINEx'(t)=A(t)x(t)+F(t,x(t),\xi(t)), \quad t\in \mathbb{R},NEWLINE\]NEWLINE with \(A\in C(\mathbb{R},L(B))\), \(A(t+T)=A(t)\), \(F:\mathbb{R}\times B\times B\to B\), and \(\xi(t)\) with \(t\in \mathbb{R}\) is a \(T\)-period \(B\)-valued stochastic process. Sufficient conditions for the existence of periodic solutions to the considered equation and the asymptotical periodicity of the solutions to the corresponding Cauchy problem are obtained.