Solutions periodic in distribution to a boundary value problem for the telegraph equation (Q5951184)

Let \((B,\|\cdot\|)\) be a complex separable Banach space and \({\mathcal L}(B)\) the Banach space of bounded linear operators in \(B\) equipped with the operator norm. Let \(\tau>0\) be a given number, \(A\in C(\mathbb{R}, {\mathcal L}(B))\), \(A(t+\tau) =A(t)\) be a given function. Let \(Q:=\mathbb{R} \times[0,\pi]\), \(C^r_0:= \{g: [0,\pi]\to \mathbb{C}\mid g^{(k)}(0)= g^{(k)}(\pi)=0\), \(0\leq k\leq r-1\}\cap C^r ([0,\pi])\), \((r\in\mathbb{N})\), \(\{u(t,x)|(t,x) \in Q\}\) a \(B\)-valued random function, and \(S\) the class of all \(B\)-valued \(\tau\)-periodic processes \(\xi=\{ \xi(t)\mid t\in\mathbb{R}\}\) continuous with probability 1 and such that \(E(\sup_{0 \leq t\leq\tau} \|\xi(t)\|) <+\infty\). In this paper, for given \(g\in C^1_0\) and \(\xi\in S\), the author considers the following boundary value problem with a periodic perturbation: \[ \begin{cases} u_{tt}'' (t,x)+au_t'(t,x)-u_{xx}'' (t,x)=A(t)u (t,x)+\xi (t)g(x),\quad (t,x)\in Q,\\ u(t,0)= u(t,\pi)= \overline 0,\quad t\in\mathbb{R}. \end{cases}\tag{I} \] He studies a sufficient condition and a necessary condition for a unique solution for the problem (I), respectively. It is a very interesting and nice result.