Complete WKB asymptotics of high frequency vibrations in a stiff problem

Asymptotic behaviour of eigenvalues and eigenfunctions of a stiff problem is described in the case of the fourth-order ordinary differential operator. Considering the stiffness coefficient that depends on a small parameter epsilon and vanishes as epsilon tends to zero on a subinterval, we prove the existence of low and high frequency resonance vibrations. The low frequency vibrations admit the power series expansions on epsilon but this method is not applicable to the description of high frequency vibrations. However, the nonclassical asymptotics on epsilon of the high frequency vibrations were constructed using the WKB method.