Eigenvalues and eigenfunctions of a differential operator with nonlocal boundary conditions (Q2459692)

The author studies in the space \(E=L_p(0,1)\), \(p\geq 1\), an ordinary differential operator \(L\) which is defined by \[ Ly:=-y''+a(x)y\text{ and }D(L):=\{y\in W^2_p(0,1) : L_iy=0,\;i=1,2\}, \] where \(L_iy:=\int\limits^1_0\varphi_i(x)y(x)dx\), and \(a(x)\) and \(\varphi_i(x)\) are some functions. Asymptotic formulas for the eigenvalues and eigenfunctions have been obtained for the eigenvalue problem \(Ly+\lambda y=0\) under some conditions on \(a(x)\) and \(\varphi_i(x)\), \(i=1,2\).