On eigenvalues of some boundary value problems for a polynomial pencil of Sturm--Liouville equation (Q1780512)

The asymptotic behavior of eigenvalues is determined for the eigenvalue problem for a polynomial pencil of Sturm-Liouville operators (i) \(Ly(x):= -y^{(n)}(x)+ \sum^{n-1}_{j=0} \lambda^j q_j(x) y(x)= \lambda^{2n} y(x),\quad 0\leq x\leq\pi\), with separable, periodic and antiperiodic boundary conditions, (ii) \(y(0)= y(\pi)= 0\), (iii) \(hy(0)- y'(0)= 0\), \(h_1y(\pi)- y'(\pi)= 0\), (iv) \(y(0)= y'(\pi)= 0\). The results are based on an integral representation of the fundamental solutions \(y(x,\lambda)\) of (i) satisfying \(y_j(0,\lambda)= 1\), \(y_j(0,\lambda)= (-1)^{j+1} i\lambda^n\), \(j= 1,2\).