Abstract
We consider the boundary value problem generated on the finite closed interval \(x \in [0, 1]\) by the \(2 \times 2\) system of ordinary differential equations \(y' - B(x)y=\lambda A(x)y, \qquad A(x)=\operatorname{diag}\{a_1(x), a_2(x)\}, \quad a_1(x) < 0 < a_2(x),\) and the boundary conditions \(U_0y(0) + U_1y(1)=0,\) where \(y(x)=(y_1(x), y_2(x))^\top\) , \(U_0\) and \(U_1\) are constant \((2 \times 2)\) matrices, and the system coefficients \(a_j\) and \(b_{jk}\) 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 \((L_p[0, 1])^2\) , and in the case of almost regular boundary value problem of order \(m \in \mathbb{N}\) , we prove the Schauder basis property in some subspace of the space \((W_{p}^{m}[0, 1])^2\) with respect to the \((L_p[0, 1])^2\) -norm.
