Stability of \(p\)-periodic solutions of a degenerating equation (Q2577303)

Let \(H\) be a complex separable Hilbert space and let \({\mathcal P}_1\) the class of all \(p\)-periodic \(H\)-valued processes \(\xi\) with \(\sup_{0 \leq t \leq p}{\mathbf E}\| \xi(t)\| < \infty\). If \(A\) is a bounded selfadjoint operator with zero in the resolvent set and \(\xi \in {\mathcal P}_1\), then the author proves that equation \[ x^\prime(t)=Ax(t) +\xi(t), \quad t \in \mathbb{R}, \] has a unique solution \(x \in {\mathcal P}_1\) and equation \[ \varepsilon x_{\varepsilon}^{\prime \prime}(t) + x_{\varepsilon}^{\prime}(t) = A x_{\varepsilon}(t) + \xi(t), \quad t \in \mathbb{R}, \] has a unique solution \(x_\varepsilon \in {\mathcal P}_1\) for each \(\varepsilon\) less than some positive number \(\varepsilon_1\), and there is \(C>0\) such that \(\sup_{0 \leq t \leq p}{\mathbf E} \| x_\varepsilon(t)-x(t)\| \leq C\varepsilon\).