Abstract <p>We consider the spectral problem for a fourth-order differential equation, where three boundary conditions contain a spectral parameter. The coefficient of this differential equation is absolutely continuous function. We obtain sharp asymptotics of eigenvalues at the high energy. We also determine this asymptotics for the smooth coefficient. The asymptotics exhibits a non-trivial high-frequency phenomenon generated by the presence of the spectral parameter in the boundary conditions. Moreover, we establish the trace formula for this spectral problem.</p>