On the solvability of a nonlocal boundary value problem with opposite fluxes at a part of the boundary and of the adjoint problem

We consider a nonlocal boundary value problem for the Laplace operator in a circular sector with opposite fluxes on the radii and with zero value of the solution on one of the radii; we also consider the adjoint problem. We prove the unique solvability of these problems and obtain the solution in an explicit form by the spectral method. As a by-product, we study the completeness and the basis property of systems of roots functions for problems of Samarskii-Ionkin type, which may be of interest in itself.