Nonlocal problem for Laplace equation in Bochner spaces

We study the Laplace equation posed in the unbounded rectangular domain $\Pi = I \times (0,\infty)$ with $I= (0,2\pi)$, and subject to nonlocal boundary conditions on $\partial \Pi$ in the trace sense. The analysis is carried out in the Bochner-Sobolev space $W^2_{p,1}(\Pi;X)$, associated with the Bochner space $L^{p,1}(\Pi;X)$, with $p \in (1,\infty)$ and $X$ is a suitable Banach space. To solve the problem, we employ a generalized spectral method. In particular, we introduce the notion of $\otimes$-basis generated by tensor products and extend the classical scheme known from the scalar case to the present setting. Moreover, we prove that the system of root functions of the corresponding nonlocal spectral problem forms a $\otimes$-basis in $L^p(I;X)$.