Stability of a flexible missile and asymptotics of the eigenvalues of fourth order boundary value problems

Fourth order problems with the differential equation \(y^{(4)}-(gy')'=\lambda ^2y\), where \(g\in C^1[0,a]\) and \(a>0\), occur in engineering on stability of elastic rods. They occur as well in aeronautics to describe the stability of a flexible missile. Fourth order Birkhoff regular problems with the differential equation \(y^{(4)}-(gy')'=\lambda ^2y\) and eigenvalue dependent boundary conditions are considered. These problems have quadratic operator representations with non-self-adjoint operators. The first four terms of the asymptotics of the eigenvalues of the problems as well as those of the eigenvalues of the problem describing the stability of a flexible missile are evaluated explicitly.