Expansion theorem of eigenfunctions for multipoint and integral conditions (Q1293061)

The authors consider the functional-differential equation \[ y^{(4)}(x)+q(x)y(x)=\lambda y(x)+ay\left(\frac\pi 2\right)+ \int_0^\pi \psi(t)y(t)dt \] together with the boundary conditions \[ y\left(\frac{\sqrt 2\pi}2\right) +\int_0^\pi\beta(t)y(t)dt=0, \quad y'(0)=0, \qquad y''\left(\frac{\sqrt 2\pi}2\right)+y''(\pi)=0, \quad y'''(\pi)=0. \] It is shown that this eigenvalue problem has a uniformly convergent eigenfunction expansion for any function in the domain of the associated operator if all the eigenvalues are simple.