Asymptotics of eigenvalues of A regular boundary-value problem

Abstract

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: \(\lambda _v = \left( {i2\pi v + c_ \pm + O(|v|^\kappa )} \right)^n \) , 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.