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')' = ?(2)y, where g ? 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')' = ?(2)y 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.