Asymptotics of eigenvalues of A regular boundary-value problem - Radzievskii, G. V. (Ukrains’kyi Matematychnyi Zhurnal, 1996)
We study a boundary-value problem x (n) + Fx = λx, U h(x) = 0, h = 1,..., n, where functions x are given on the interval [0, 1], a linear continuous operator F acts from a Hölder space H y into a Sobolev space W 1 n+s , U h are linear continuous functional defined in the space \(H^{k_h } \) , and k h ≤ n + s - 1 are nonnegative integers. We introduce a concept of k-regular-boundary conditions U h(x)=0, h = 1, ..., n and deduce the following asymptotic formula for eigenvalues of the boundary-value problem with boundary conditions of the indicated type: \(λ _v = \left( {i2π v + c_ ± + O(|v|^κ )} \right)^n \) , v = ± N, ± N ± 1,..., which is true for upper and lower sets of signs and the constants κ≥0 and c ± depend on boundary conditions.