Expansion in eigenfunctions of a differential operator with nonregular boundary conditions (Q1060336)

In this paper, considering the differential equation \[ (1)\quad y^{(n)}+a_ 1(x)y^{(n-1)}+(b(x)\lambda +a_ 2(x))y^{(n- 2)}+\sum^{n}_{j=3}a_ j(x)y^{(n-j)}=0 \] and the so-called nonregular boundary conditions (2) \(y^{(k)}(0)=0\), \(k=0,1,...,n-2\), \(y(1)=0\), we prove that any n times continuously differentiable function f(x), satisfying the boundary conditions (2), can be expanded in a uniformly convergent series with respect to the eigenfunctions of the problem (1), (2). We assume that the coefficients \(a_ j(x)\), \(j=1,2,...,n\), have summable derivatives to the (n-j)-th order in the segment [0,1], and \(b(x)>0\) to the (n-2)-nd order.