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 n+s1 , 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.
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Radzievskii, G.V. Asymptotics of eigenvalues of A regular boundary-value problem. Ukr Math J 48, 537–575 (1996). https://doi.org/10.1007/BF02390614
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DOI: https://doi.org/10.1007/BF02390614