On strong solvability of one boundary value problem for X -valued Laplace equation on half disc

A boundary value problem is considered for an X-valued Laplace equation, where X is a Hilbert space. The considered problem has specific features: the solution is sought in an X-valued Sobolev space; the boundary conditions are nonlocal in nature; the boundary conditions of the corresponding spectral problem are regular, but not strongly regular in the sense of Birkhoff. To solve this problem, we use the concept of a t-basis generated by the tensor product Lp(0,π)⊗X. The spectral problem possesses two series of asymptotically close eigenfunctions, which therefore do not form a basis in Lp(0,1). We construct a linear combination of eigenfunctions and establish the t-basisness of the resulting system for Lp([0,π];X). Using this fact, we prove the unique solvability of the BVP in Sobelev space Wp2(D;X), where D=(0,1)×(0,π). We also emphasize that the results obtained are new even in the scalar-valued case.