In this paper, an eigenvalue problem for a regular fourth order ordinary differential equation is considered, where one of the boundary conditions linearly depends upon the eigenvalue parameter. The first four terms in the asymptotic expansion of the eigenvalues are derived.
"Asymptotics of the eigenvalues of self-adjoint fourth order differential operators with separated eigenvalue parameter dependent boundary conditions." Rocky Mountain J. Math. 47 (6) 2013 - 2042, 2017. https://doi.org/10.1216/RMJ-2017-47-6-2013