A nonlocal boundary value problem for systems of infinite-order partial differential equations (Q2567815)

The paper deals with solvability conditions, in the framework of suitable classes of Sobolev type, for the boundary value problem \[ \begin{cases} \sum_{| \widehat s| =0}^\infty A_{\widehat s} \partial_t^{s_0}D^s u(t,x)=f(x,t) & \text{in}\;(0,T)\times\Omega,\\ \partial_t^\alpha u| _{t=0}-\mu\partial^\alpha_t u| _{t=T}=0 & \alpha=0,1,\ldots, \end{cases} \] where \(\Omega\) is a \(p\)-dimensional torus, \(\widehat s=(s_0,s),\) \(| \widehat s| =s_0+| s| =s_0+s_1+\ldots+ s_p,\) \(\partial_t=\partial/\partial t,\) \(D^s=D^{s_1}_1\cdots D^{s_p}_p=(-i)^{| s| }\partial^{| s| }/\partial x^{s_1}_1\cdots x^{s_p}_p,\) \(A_{\hat s}\) are square matrices of order \(m\) with complex entries, while \(\mu\) is a nonzero complex number.