Basis Property of Eigenfunctions of a Boundary Value Problem for a 2 × 2 System of Ordinary Differential Equations

We consider the boundary value problem generated on the finite closed interval x ∈ [0, 1] by the 2 × 2 system of ordinary differential equations y' - B(x)y=λ A(x)y, A(x)=diag₁(x), a₂(x)\, a₁(x) < 0 < a₂(x), and the boundary conditions U₀y(0) + U₁y(1)=0, where y(x)=(y₁(x), y₂(x))^ , U₀ and U₁ are constant (2 × 2) matrices, and the system coefficients aⱼ and bⱼₖ are assumed to be absolutely continuous. In the regular case, we prove that the system of eigenfunctions and associated functions forms a Schauder basis in the space (Lₚ[0, 1])² , and in the case of almost regular boundary value problem of order m ∈ N , we prove the Schauder basis property in some subspace of the space (Wₚᵐ[0, 1])² with respect to the (Lₚ[0, 1])² -norm.